Auxiliary-lag Dependent Gaussian Process Model for Forecasting Rainfall Data Using Proposed Kernels and Multi-start Optimization Method

Auxiliary-lag Dependent Gaussian Process Model for Forecasting Rainfall Data Using Proposed Kernels and Multi-start Optimization Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Auxiliary-lag Dependent Gaussian Process Model for Forecasting Rainfall Data Using Proposed Kernels and Multi-start Optimization Method Muhammad Ahmed Shehzad, Haris Khurram, Aamna Khan, Muhammad Mutahir Iqbal This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2486388/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Pakistan is currently facing the biggest flood of history due to monsoon rains. The rainfall forecasting is very important for policy making. In this paper, we have presented an auxiliary-lag dependent Gaussian process, a Bayesian non-parametric machine learning model, for modeling the rainfall data using auxiliary lags. We have also introduced some new multifeatured kernel functions that are versatile in dealing with seasonal data. A simplex cluster-based multi-start technique using the Nelder-Mead optimizer has also been proposed for optimizing the hyperparameters of the kernel functions, which can be used for any available or proposed kernel function(s). For comparison of the proposed model, we have used the autoregressive random forest model, autoregressive artificial neural network model, seasonal autoregressive moving average models, and exponential smoothing models. Results confirmed the superiority of the proposed model over conventional models. The proposed methodology will be helpful for other researchers and local experts in making more reliable forecasting which will be helpful in policymaking relevant to agriculture systems, water management systems, climate change, and natural disasters such as droughts and floods. Gaussian Process Machine learning models Auxiliary Lags Kernel functions Forecasting Rainfall Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction Rainfall in a country is an important variable that impacts agriculture systems, especially on productions, water management systems, climate change, and natural disasters such as droughts and floods. Pakistan (currently facing hard days due to flood caused by high rainfall) is agricultural land, and the economy is highly dependent on agriculture products and productions. Simultaneously, crop production is highly dependent on the amount of rainfall and has highly affected by high or low rainfall in the area. Similarly, water storage in Pakistan highly depends on the amount of rainfall. Pakistan is among the top ten countries where climate change has a very adverse effect and has critical water resources (Khan et al., 2020 ). To keep all these facts in view, rainfall foresting is very important and helpful in policymaking. There are various research studies related to the forecasting of rainfall in a different area of the world. Most of them are related to seasonal ARIMA (SARIMA) modeling, and few are based on a machine learning approach. Afrifa-Yamoah et al. ( 2016 ) use the SARIMA model to forecast the rainfall data in a region of Ghana. Kaur and Rakshit ( 2020 ) also used SARIMA models to forecast Punjab's rainfall data, India. Murthy et al. ( 2018 ) use SARIMA models to forecast the rainfall data in North-East India. Ortiz-Garcia et al. (2014) forecast the rainfall in Spin using support vector classifiers. Ni et al. ( 2020 ) used short term memory-based models to forecast rainfall and streamflow. Yu et al. ( 2017 ) used random forest and support vector models for rainfall forecasting. Yasmeen and Hameed ( 2018 ) used sliced functional time series to forecast the rainfall in Pakistan. Adnan et al. ( 2020 ) predict the monsoon rainfall in summer using multiple linear regression and principal component regression models in Pakistan. Xiang et al. ( 2018 ) use support vector model and artificial neural network model jointly for forecasting rainfall. Dwivedi, Kelajya and Sharma (2019) used ANN model for forecasting monthly rainfall in India. Unnikrishnan, Poornima and Jothiprakash use hybrid models by combining machine learning and statistical approaches for forecasting daily rainfall data. Improved forecasting using advanced models is always a significant contribution in literature. In this article to perform the improved forecasting of the rainfall, we suggest an auxiliary-lag dependent Gaussian process (ALD-GP), a Bayesian non-parametric machine learning model. We have also suggested some multifeatured kernels to be used with the proposed model. 2. Proposed Methodology The Gaussian process is known to be a stochastic process over infinite-dimension latent function indexed by some known variables. Any finite and known subset of these functions follows a multivariate normal distribution (Shi and Choi, 2011 ; Gramacy, 2020 ). Formally we can define a Gaussian process as the distribution over the function \(f\left(X\right)\) where $$f\left(X\right)~GP\left(m\left(X\right),g(X,X{\prime })\right).$$ 1 such that \(f\left(X\right):\mathbb{X}\to {\mathbb{R}}^{\infty }\text{and }X\in \mathbb{X}\) with specified mean function \(m\left(X\right)=E\left[f\right(X\left)\right]\) and covariance function \(g\left(X,{X}^{{\prime }}\right)\) . Auxiliary-Lags Dependent Gaussian Process Now, considered a variable \({y}_{t}\) is indexed over some fixed time interval t , known to be a time series variable. The primary aim is to model the \({y}_{t}\) for forecasting purposes. Let the l- th auxiliary-lags of \({y}_{t}\) are denoted as \({L}^{l}y\) where \({L}^{l}y\) of order \(n\times l.\) These are the selective lags of a series that are highly correlated with the series. Thus, this lag dependent model's purpose is to use selective lags, which are considered auxiliary lags of the series of interest. Then the functional form of the model of \({y}_{t}\) is $${y}_{t}=f\left({L}^{l}y\right)+{e}_{t}$$ 2 where \({e}_{t}\) is supposed to be a stationary disturbance term such that \({e}_{t}~N(0,{\sigma }^{2})\) . Then, the distribution of \({y}_{t}\) is auxiliary-lag dependent Gaussian process (ALD-GP) as $${y}_{t}~ALD-GP\left(\mu \left({L}^{l}y\right),\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)+{\sigma }^{2}{\delta }_{{y}_{tt}}\right)$$ 3 where \(\mu \left(\cdot \right)\) is the process mean function, \(\gamma \left(\cdot \right)\) is the autocovariance function of the process, and \({\delta }_{{y}_{tt}}\) is the Kronecker delta. Consider the h- step ahead forecasting of series \({y}_{t}\) that is \({y}_{t+h}\) . This h- step ahead forecasting is based on l auxiliary-lags \({L}^{l}{y}_{h}\) . For h- step ahead forecasting the conditional distribution of \({y}_{t+h}\) given \({y}_{t}\) will be distributed as auxiliary-lag dependent Gaussian process. Theorem 1 Considered the time series distributed as ALD-GP, with zero mean prior, the Posterior distribution of the h- step ahead forecast is also distributed as ALD-GP as following $$p\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right)~ALD-GP\left(\mu \left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right),\gamma \left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right)\right)$$ 4 where the posterior mean is $$\mu \left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right)=\gamma \left({L}^{l}{y}_{h},{L}^{l}y\right){\left[\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)+{\sigma }^{2}{\delta }_{yy}\right]}^{-1}{y}_{t}$$ 5 and the posterior autocovariance function is $$\begin{gathered} \gamma \left( {{y_{t+h}}|{L^l}{y_h},{L^l}y,{y_t}} \right)=\gamma \left( {{L^l}{y_h},{L^l}{y_h}^{\prime }} \right) - \gamma \left( {{L^l}{y_h},{L^l}y} \right) \times \\ {\left[ {\gamma \left( {{L^l}y,{L^l}y^{\prime}} \right)+{\sigma ^2}{\delta _{yy}}} \right]^{ - 1}}\gamma \left( {{L^l}y,{L^l}{y_h}} \right) \\ \end{gathered}$$ 6 Proof By using (O Hagan, 1978), we can prove it on similar lines as the distribution of the observation t + 1 is similar to the distribution of the t observation. Thus, the \(\mu \left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right)\) is the posterior mean function, which is considered as the forecast at t+h time. Similarly, \(\gamma \left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\right)\) is the posterior autocovariance function of the forecast and used to draw a 95% confidence band of the forecast. b. Autocovariance Function Covariance function of ALD-GP is based on lag values of the main series, so we considered it as autocovariance. Here, autocovariance function is defined as the covariance between two matrices of the different time-dependent lags. The elementary properties of the autocovariance function \(\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)\) for the ALD-GP model are as follows. The autocovariance functions should be symmetric as $$\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)=\gamma \left({L}^{l}{y}^{{\prime }},{L}^{l}y\right)$$ 7 The autocovariance function for a vector \(a={a}_{1},{a}_{2},...,{a}_{n}\in {\mathbb{R}}^{n}\) should be at least positive semidefinite. $${\sum }_{i=1}^{n}{\sum }_{j=1}^{n}{a}_{i}\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right){a}_{j}\ge 0$$ 8 For adding the non-parametric flavor, autocovariance functions will be defined using kernel functions. A kernel function that is symmetric and at least positive semidefinite will be considered as the autocovariance function. The stationarity of the autocovariance function is based on the translation-invariant property. So, if the covariance function can be written as $$\gamma \left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)=\gamma \left(‖{L}^{l}{y}^{{\prime }}-{L}^{l}y‖\right)$$ 9 and the positive semidefinite property as $${\sum }_{i=1}^{n}{\sum }_{j=1}^{n}{a}_{i}\gamma \left({L}^{l}{y}_{i}-{L}^{l}{y}_{j}\right){a}_{j}\ge 0$$ 10 then the autocovariance function will be stationary. Common kernel functions for lagged series Gaussian Process is defined over kernel functions. Some pre-defined kernel functions used in Gaussian Process modelling are reconsidered here for auxiliary lag series. Squared exponential (SE) is an essential and commonly used kernel function in kernel-based modeling. It is also known as redial based function (RBF) kernel. SE kernel function for lag series is $${g}_{SE}\left(Ly,L{y}^{{\prime }}\right)={\sigma }^{2}{exp}\left(-\frac{‖Ly-L{y}^{{\prime }}‖}{2{l}^{2}}\right)$$ 11 where σ is the scale, and l is the length-scale parameters. This kernel function is an isotropic stationary kernel function. Additionally, the prominent feature of this kernel function is that it can be infinitely differentiable. The rational quadratic kernel (RQ) function is the scale mixture, or in other words, the infinite sum of the SE kernel. RQ kernel function for some lag series $${{g}}_{{R}{Q}}\left({L}{y},{L}{{y}}^{{{\prime }}}\right)={\left(1+\frac{‖{L}{y}-{L}{{y}}^{{{\prime }}}‖}{2{\alpha }{{l}}^{2}}\right)}^{-{\alpha }}$$ 12 where α is the scale mixture parameter and l is the length-scale parameter. Both parameters \(\alpha ,l>0\) and for \(\alpha \to \infty\) the RQ kernel function will be converted into the SE kernel function. This is also isotropic stationary kernel function. The periodic (P) kernel was firstly used by MacKay ( 1998 ) to access the periodic repetition in data. For lag variables, the periodic kernel is defined as $${g}_{P}\left(Ly,L{y}^{{\prime }}\right)={exp}\left[\frac{-2}{{l}^{2}}{\left[{sin}\left(\frac{\pi }{w}‖Ly-L{y}^{{\prime }}‖\right)\right]}^{2}\right]$$ 13 where w is the periodic parameter, l is the length-scale parameters and \(w,l>0\) . This also an isotropic stationary kernel function. The Matérn (M P ) class of kernel was proposed by Matérn ( 1960 ) is the generalization of the RBF kernel function. For a lag series, the general Matérn class of kernel function in the form of modified Bessel function is defined as $${g}_{M}(Ly,L{y}^{{\prime }})=\frac{1}{{2}^{1-v}\varGamma \left(v\right)}{\left(\frac{\sqrt{2v}‖Ly-L{y}^{{\prime }}‖}{l}\right)}^{v}{K}_{v}\left(\frac{\sqrt{2v}‖Ly-L{y}^{{\prime }}‖}{l}\right)$$ 14 where v is the order parameter, and l is the length-scale parameter with \(v,l>0\) . \({K}_{v}\) is the modified Bessel function and \(\varGamma \left(.\right)\) is a gamma function. A simplified form of Matérn class of kernel can be extracted by assigning \(v=P+\frac{1}{2}\) . For \(P=2\) we have the commonly used form of Matérn class of kernel that is $${g}_{{M}_{2}}(Ly,L{y}^{{\prime }})=\left(1+\frac{\sqrt{5}‖Ly-L{y}^{{\prime }}‖}{l}+\frac{5}{3}{\left(\frac{‖Ly-L{y}^{{\prime }}‖}{l}\right)}^{2}\right){exp}\left(-\frac{\sqrt{5}‖Ly-L{y}^{{\prime }}‖}{l}\right)$$ 15 A wavelet (W) kernel function based on some mother wavelet function is considered by Zhang et al. ( 2004 ). The translation-invariant W kernel function is defined as $${g}_{W}\left(Ly,L{y}^{{\prime }}\right)={\prod }_{i=1}^{p}h\left(\frac{L{y}_{i}-L{y}_{i}^{{\prime }}}{a}\right)$$ 16 and for mother wavelet, it can be defined as $${g}_{W}\left(Ly,L{y}^{{\prime }}\right)={\prod }_{i=1}^{p}\left[{cos}\left(\frac{1.75\left(L{y}_{i}-L{y}_{i}^{{\prime }}\right)}{a}\right){exp}\left(\frac{-‖L{y}_{i}-L{y}_{i}^{{\prime }}‖}{2{a}^{2}}\right)\right]$$ 17 where a is the parameter and x is the realization in one dimension (Antoniadis, 2006). New kernel functions for seasonal series Squared Exponential Periodic Kernel (SEP) is the squared exponential times Periodic Kernel. This is the stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is $${\gamma }_{SE\times P}\left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)={{\sigma }_{f}}^{2}{exp}\left(-\frac{‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{2{l}^{2}}-\frac{2}{{l}_{1}^{2}}{\left[{sin}\left(\frac{\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖\right)\right]}^{2}\right)$$ 18 where σ, l, l 1 , and w are the hyperparameters which will be tune using an optimization algorithm. Squared Exponential with Wavelet Kernel (SE + W) is the special kernel which the sum of the squared exponential and wavelet kernel where the mother wavelet is the same as defined by Zhang et al. ( 2004 ). It is the locally stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is $${\gamma }_{SE+W}\left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)={{\sigma }_{f}}^{2}{exp}\left(-\frac{‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{2{l}^{2}}\right)+{\prod }_{i=1}^{p}\left[{cos}\left(\frac{1.75\left({L}^{l}y-{L}^{l}{y}^{{\prime }}\right)}{a}\right){exp}\left(\frac{-‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{2{a}_{1}^{2}}\right)\right]$$ 19 where σ, l, a , and a 1 , are the hyperparameters that will be tune using an optimization algorithm. Rational Quadratic with Wavelet Kernel (RQ + W) is the sum of the Rational Quadratic and wavelet kernel function. It is also a locally stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is as $${{\gamma }}_{{R}{Q}+{W}}\left({{L}}^{{l}}{y},{{L}}^{{l}}{{y}}^{{{\prime }}}\right)={\left(1+\frac{‖{{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\prime }}}‖}{2{\alpha }{{l}}^{2}}\right)}^{-{\alpha }}+{\prod }_{{i}=1}^{{p}}\left[{cos}\left(\frac{1.75\left({{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\prime }}}\right)}{{a}}\right){exp}\left(\frac{-‖{{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\prime }}}‖}{2{{a}}_{1}^{2}}\right)\right]$$ 20 where 𝛼, l, a , and a 1 , are the hyperparameters. Squared Exponential Periodic with Matérn Class Kernel (SEP + M 2 ) is the sum of the Squared Exponential times periodic and Matern kernel function. It is a stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is as $$\begin{gathered} {\gamma _{SE \times P+M2}}\left( {{L^l}y,{L^l}y^{\prime}} \right)=\sigma _{f}^{2}\exp \left( { - \frac{{\left\| {{L^l}y - {L^l}y^{\prime}} \right\|}}{{2{l^2}}} - \frac{2}{{l_{1}^{2}}}{{\left[ {\sin \left( {\frac{\pi }{w}\left\| {{L^l}y - {L^l}y^{\prime}} \right\|} \right)} \right]}^2}} \right)+ \\ \left( {1+\frac{{\sqrt 5 \left\| {{L^l}y - {L^l}y^{\prime}} \right\|}}{{{l_2}}}+\frac{5}{3}{{\left( {\frac{{\left\| {{L^l}y - {L^l}y^{\prime}} \right\|}}{{{l_2}}}} \right)}^2}} \right)\exp \left( { - \frac{{\sqrt 5 \left\| {{L^l}y - {L^l}y^{\prime}} \right\|}}{{{l_3}}}} \right) \\ \end{gathered}$$ 21 where 𝜎, l, l 1 , w, l 2 , and l 3 , are the hyperparameters. Squared Squared Exponential Periodic Kernel (SSEP) is the squared of Squared Exponential kernel, times periodic kernel function. It is a stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is as $${\gamma }_{S{E}^{2}\times P}\left({L}^{l}y-{L}^{l}{y}^{{\prime }}\right)={\sigma }_{f}^{2}{exp}\left(-\frac{‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{{l}^{2}}-\frac{2}{{l}_{1}^{2}}{\left[{sin}\left(\frac{\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖\right)\right]}^{2}\right)$$ 22 where 𝜎, l, l 1 , and w , are the hyperparameters. Squared Exponential Squared Periodic (SESP) kernel is the Squared Exponential kernel times squared of the periodic kernel function. It is also a stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is as $${\gamma }_{SE\times {P}^{2}}\left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)={\sigma }_{f}^{2}{exp}\left(-\frac{‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{2{l}^{2}}-\frac{4}{{l}_{1}^{2}}{\left[{sin}\left(\frac{\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖\right)\right]}^{2}\right)$$ 23 where 𝜎, l, l 1 , and w , are the hyperparameters. Squared Exponential with Squared Exponential Periodic (2SEP) is the sum of the Squared Exponential and squared exponential times periodic kernel function. It is also a stationary kernel function. For the matrix of auxiliary lag series, \({L}^{l}y\) the functional form of the kernel is as $${\gamma }_{2SE\times P}\left({L}^{l}y,{L}^{l}{y}^{{\prime }}\right)=2{\sigma }_{f}^{2}{exp}\left(-\frac{‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖}{2{l}^{2}}\right){exp}\left(-\frac{2}{{l}_{1}^{2}}{\left[{sin}\left(\frac{\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\prime }}‖\right)\right]}^{2}\right)$$ 24 where 𝜎, l, l 1 , and w are the hyperparameters. Proposition 1 The proposed kernels defined in Eq. 18 – 24 are all at least a positive semidefinite kernel. As the product and sum of all at least positive semidefinite kernels will also atleast positive semidefinite kernels, so this proposition is true for all the defined kernel functions. All these suggested kernel functions and their detailed properties were discussed in Khurram and Iqbal ( 2021 ). Marginal Likelihood Using Bayesian Model Selection The marginal likelihood of ALD-GP is obtained by using the approach of Bayesian model selection. Consider a set of models \(\mathcal{M}\) for some data y with a set of parameters θ such that the probability function of y is \(p\left(y|\theta ,\mathcal{M}\right)\) . By using the Bayes rule the posterior probability model is $$p\left(\mathcal{M}{}_{i} |y\right)=\frac{p\left(y|\mathcal{M}{}_{i}\right)p\left(\mathcal{M}{}_{i}\right)}{{\sum }_{\varOmega }p\left(y|\mathcal{M}\mathcal{}\right)p\left(\mathcal{M}\mathcal{}\right)}$$ 25 the \(p\left({\mathcal{M}}_{i}\right)\) is the prior distribution over models. The posterior distribution of parameter over data and model is $$p\left({\theta }_{i}|y,{\mathcal{M}}_{i}\right)=\frac{p\left(y|{\theta }_{i},{\mathcal{M}}_{i}\right)p\left({\theta }_{i}|{\mathcal{M}}_{i}\right)}{p\left(y|{\mathcal{M}}_{i}\right)}$$ 26 $$p\left({\theta }_{i}|y,{\mathcal{M}}_{i}\right)=\frac{p\left(y|{\theta }_{i},{\mathcal{M}}_{i}\right)p\left({\theta }_{i}|{\mathcal{M}}_{i}\right)}{\int p\left(y|{\theta }_{i},{\mathcal{M}}_{i}\right)p\left({\theta }_{i}|{\mathcal{M}}_{i}\right)d{\theta }_{i}}$$ 27 So, the best model and their hyperparameters can be accessed using the Bayesian model selection procedure. Using the Bayesian model selection approach, the marginal likelihood function of ALD-GP is $${log}p\left({y}_{t}|{L}^{l}y,\beta \right)=-\frac{1}{2}{y}_{t}^{{\prime }}{\left(\gamma ({L}^{l}y,{L}^{l}{y}^{{\prime }})+{\sigma }^{2}{\delta }_{{y}_{tt}}\right)}^{-1}{y}_{t}-\frac{1}{2}{log}\left|\gamma ({L}^{l}y,{L}^{l}{y}^{{\prime }})+{\sigma }^{2}{\delta }_{{y}_{tt}}\right|-\frac{n}{2}{log}2\pi$$ 28 where β is the vector of hyperparameters that need to be carefully optimized. Proposed Optimization Method A computationally stable optimization method was introduced by Butler et al. ( 2014 ) and Macdonald et al. ( 2015 ), which suggest choosing multiple input starter for avoiding the problem of misleading maxima. These authors used the SE and Matern class of kernels, and their methods used the deviance function of the correlation matrix for specified kernel settings. This approach also needs a starting point that needs information related to kernel before the start, which is subjective based. This method can be used for some available class of kernels but not for a newly proposed kernel. Now, we will present our proposed method, which is the modified version of the cluster-based multi-start technique. The proposed simplex cluster-based multi-start technique is based on the optimization of negative marginal loglikelihood. The feature of this proposed method is that it is computationally stable and able to work for any newly proposed kernel. It does not require any initial values and range to be used. The optimization method is based on the standard Nelder-Mead approach proposed by Nelder and Mead ( 1965 ). The following are the steps of the proposed method. Consider that d is the total number of hyperparameters β to be optimized for a kernel and negative marginal likelihood. Then the proposed method is as follows Use maximin criteria to draw 200d Latin hypercube samples over the interval \({\left[\text{0,1}\right]}^{d}\) for each hyperparameter. Select 80d points having minimum \(-{log}p\left({y}_{t}|{L}^{l}y,\beta \right)\) from the 200d points. Apply k-mean clustering methods on these selected 80d points. Use five different starters of k-mean clustering and select 3d group. Use these the 3d point in running the NM algorithm and selected the best starting point. The selected points using a simplex cluster-based multi-starter technique based on NM optimizer are considered for the values of the respective hyperparameters. To avoid the problem of singularity, we have also suggested the lower bound of a nugget value by following Ranjan et al. ( 2011 ). The lower bound of the nugget value \(\kappa\) is defined as $$\kappa =\left|{min}\left({\lambda }^{\left[1\right]},0\right)\right|$$ 29 where \({\lambda }^{\left[1\right]}\) is the lowest eigenvalue of the covariance or autocovariance matrix. Thus, instead of using a simple autocorrelation matrix if this nugget value will be added in the matrix's diagonal, \({\gamma }_{B}\left(\cdot \right)=\gamma \left(\cdot \right)+\kappa I\) then this jittering will remove the problem of singularity. 3. Other Models Seasonal Autoregressive Moving Average Models Autoregressive moving average (ARMA) model was firstly considered by Box and Jenkins (1976) as the mixture of autoregressive (AR) and moving average (MA) models. For a series of seasonal nature, there exist seasonal autoregressive moving average (SARMA) models. The parameters of SARIMA models are p, d, q, P, D, Q and m respectively. Where p is the order of autoregressive terms, d is the difference, q is the order of moving average terms, P is the order of seasonal autoregressive terms, D is the seasonal difference, Q is the order of moving average terms and m is the order of seasonal difference which is 12 for monthly time series data. Thus, a SARIMA \(\left({p},{d},{q}\right)\times {\left({P},{D},{Q}\right)}_{{m}}\) model will be $${\phi }_{p}\left(B\right)\cdot \varPhi \left({B}^{m}\right)\cdot {\nabla }^{d}\cdot {\nabla }_{m}^{D}\cdot {Y}_{t}={\theta }_{q}\left(B\right)\cdot {\varTheta }_{Q}\left({B}^{m}\right)\cdot {\epsilon }_{t}$$ 30 where, B is the back-shift operator and \(\phi ,\varPhi ,\nabla ,\theta \text{, and }{\Theta }\) are the parameters of, non-seasonal AR, seasonal AR, differencing, non-seasonal moving average and seasonal moving average terms. \({\epsilon }_{t}\) is the disturbance term assumed to be a normally distributed with constant mean \({\mu }\) and variance \({{\sigma }}^{2}\) . Exponential Smoothing Models Exponential Smoothing (ES) models are very known models for forecasting time series. The robustness and versatility of this model inspire researchers to be used for forecasting. A simple ES model is based on simple mean, linear trend and quadratic trend methods. Quadratic ES can be used for seasonal data but there should be trend in the data. There is one parameter of quadratic ES method. Holt-Winter’s ES models are mainly used when series is seasonal. The model parameters are associated with level, trend and seasonal terms. These three parameters optimized for forecasting. Artificial Neural Network Model Artificial Neural network (ANN) model is comprised of multiple nodes which are also known as neurons and hidden layers. This hidden layer is the connect input layer with output layer. Weights of hidden layer connect different nodes and layers. Mostly the weights were optimize using backpropagation optimization methods. ANN is capable to capture any kind of nonlinear relationship between input and output series. For an input series, the ANN model can be: $$Y{}_{t}=\beta {}_{0}+{\sum }_{j=1}^{k}{\alpha }_{j}f\left({\gamma }_{0,j}+{\sum }_{i=1}^{p}{\alpha }_{ij}{Y}_{t-i}\right)+{\epsilon }_{t}$$ 31 where \({a}_{ij};\left(i=\text{1,2},\dots ,p ;j=\text{1,2},\dots ,k\right)\) are the weights that connect the layers, k are the hidden nodes, \({\beta }_{j}\) is the bias of the j -th unit and \(f\left(.\right)\) is the activation function. The commonly used transform function is the logistic function. Random Forest Model Random Forest (RF) model is a modified form of decision trees proposed by Breiman (2001). The model is based on two parameters which are number of trees \({{n}}_{{t}{r}{e}{e}}\) and number of candidate variables randomly sampled at each split \({{n}}_{{t}{r}{y}}\) . The model makes tree of the series and forecast using it. 4. Data Description And Analysis Methodology The average monthly rainfall (mm) is taken from Regional Meteorology Center (RMC) Lahore. This average monthly rainfall data is from Jan-2000 to Dec-2019. This historically recorded data is of Lahore city at latitude 31 o 33' and longitude 74 o 20'. For training purposes, we have taken the data from Jan-2000 to Aug 2017 as train data and from Sep-2017 to Dec-2019 as a test data set. Figure 1 shows the line plot of the rainfall series, which shows that the series is seasonal. We have used conventional models, including SARIMA and Exponential Smoothing (ES) models on train data, and select the best model among them. After that, we have used autoregressive artificial Neural network (ANN) and random forest (RF) models by taking two autoregressive lags of the series as explanatory variables. The R packages randomForest and neuralnet were used for training the models. Then a one-step ahead forecast using these models were calculated. Both of these models show best results when using two autoregressive lags. For ANN we have used three layers with ten nodes because these provides optimal results. For applying the ALD-GP using proposed methods, we have to select the auxiliary lags for the model. For that purpose, we have present autocorrelation plot at the different lag periods in Fig. 2 , and from this graph, we observe that the series is highly correlated with the 12th, 24th, and 36th lag term. We simultaneously use all the selected lags in the model. Thus, there are three models used for comparison purpose as $${y}_{t}=f\left({y}_{t-12}\right)+{e}_{t}$$ 32 $${y}_{t}=f\left({y}_{t-12},{y}_{t-24}\right)+{e}_{t}$$ 33 $${y}_{t}=f\left({y}_{t-12},{y}_{t-24},{y}_{t-36}\right)+{e}_{t}$$ 34 The first model is the one lag model, the second is the two lags model, and the third is the three lags model. To measure the performance of different purpose kernel functions in this demonstration, we have used all the proposed kernel functions in the forecasting process. Performance Evaluation Measures We have used the following performance evaluation measures based on the actual and forecasted observations for measuring the forecasting performance of proposed and conventional models. Root mean square error (RMSE) of the forecasted value F t of a series y t at h- step ahead forecast is defined as: $$RMSE=\sqrt{\frac{{\sum }_{t=1}^{h}{\left({y}_{t}-{F}_{t}\right)}^{2}}{h}}$$ 35 Mean Absolute Percentage Error measure is a valid measure for measuring the forecasting performance, but If any of the actual value is zero, this measure did not converge. For such an issue, Makridakis ( 1993 ) suggest using Symmetric Mean Absolute Percentage Error (sMAPE) which is a bounded measure and more resistant to the outlier. For h- step ahead forecast, the sMAPE is defined as: $$sMAPE=\frac{{\sum }_{i=1}^{h}\frac{2\times \left|{y}_{t}-{F}_{t}\right|}{\left|{y}_{t}\right|+\left|{F}_{t}\right|}}{h}\times 100$$ 36 Table 1 Model comparison for Selecting best ARIMA and ES model Model RMSE AIC Quadratic ES with α = 0.0304 66.325 8.502 Holt-Winter's ES with α = 0.208, β = 0.2026, γ = 0.2171 240.932 10.997 ARIMA(1,0,1)x(1,1,2) 12 54.404 8.040 ARIMA(0,0,0)x(1,0,2) 12 55.256 8.052 ARIMA(0,0,1)x(1,0,2) 12 55.069 8.054 ARIMA(1,0,0)x(1,0,2) 12 55.090 8.055 5. Results Table 1 shows the results of the comparison of the conventional models. The Quadratic ES and Holt-Winter ES was used to model the seasonal data. Similarly, some best-fitted SARIMA models using conventional Box-Jenkins’s methodology are also presented. After a comparison of the selected model, it is noticed that the SARIMA(1,0,1)×(1,1,2) 12 model has the lowest AIC and RMSE. Thus, it is considered the best selected conventional model. Table 2 Optimized Values of Hyperparameters of Kernels using Proposed Optimization Method Lags One Two Three One Two Three One Two Three One Two Three SE SEP SSEP SE + W σ 5.475 5.298 5.203 4.739 5.277 6.033 4.736 5.583 6.995 5.403 5.534 6.824 σ f 7.321 13.184 7.542 6.297 9.724 9.882 7.375 9.388 5.929 8.974 7.955 4.463 l -9.214 -11.797 -5.730 -5.825 -8.192 -10.057 -4.766 -14.953 18.533 11.070 -8.302 -7.108 w - - - -0.006 -3.262 9.921 -0.957 -8.873 47.790 - - - l 1 - - - 0.669 36.192 -14.580 -2.145 -8.597 -72.170 - - - a - - - - - - - - - -3.836 0.855 1.564 l 2 - - - - - - - - - - - - l 3 - - - - - - - - - - - - a 1 - - - - - - - - - -6.880 -6.209 -11.836 𝛼 - - - - - - - - - - - - PSE + M2 SESP 2SEP RQ + W σ 5.035 3.525 6.054 4.647 6.203 6.813 5.364 5.777 6.781 5.347 0.741 4.476 σ f 8.474 14.950 7.898 6.797 6.522 4.464 6.393 5.260 4.346 6.500 7.897 11.048 l -8.667 6.828 -8.105 -4.444 15.827 -6.569 5.959 -5.874 -5.828 -6.031 0.978 -5.266 w -2.828 -3.062 -1.690 -2.221 10.177 2.785 -4.420 1.459 3.220 - - - l 1 1.966 -11.995 6.491 -1.452 -21.536 -3.798 -3.244 -1.014 -5.494 - - - a - - - - - - 1.765 2.188 -3.241 l 2 1.385 -0.465 -4.877 - - - - - - l 3 1.302 1.043 1.972 - - - - - - a 1 - - - - - - -3.781 -1.424 -6.362 𝛼 - - - - - - 0.563 0.458 0.466 Figure 3 present the ANN plot with three layers and ten nodes. Table 2 presents the tune values of hyperparameters for purposed kernel functions at different sets of auxiliary lags using the proposed optimization scheme. The evaluation of the proposed models' forecast performance in terms of RMSE's and sMAPE's is presented in Table 3 , where the table contains the forecast performance results at selected lags. After comparing all these proposed models with SARIMA(1,0,1)×(1,1,2) 12 , RF and ANN models we can conclude that the ALD-GP model is more efficient than the conventional models. Table 3 Performance comparison of different models based on test data Models Kernel (1,0,1)x(1,1,2) One lag Two lags Three lags RMSE sMAPE RMSE sMAPE RMSE sMAPE RMSE sMAPE SARIMA - 51.965 92.117 - - - - - - ANN - 46.47 108.56 RF - 47.26 102.91 ALD-GP SE - - 44.495 93.805 46.483 89.141 43.196 89.397 ALD-GP SEP - - 41.975 89.812 46.554 89.200 46.956 91.485 ALD-GP SE + W - - 43.085 90.624 46.601 88.246 42.531 86.329 ALD-GP RQ + W - - 43.604 90.606 44.858 88.942 46.709 92.904 ALD-GP PSE + M 2 - - 46.838 99.575 46.923 89.178 44.599 85.258 ALD-GP SSEP - - 46.578 88.964 47.367 87.220 46.185 92.863 ALD-GP SESP - - 43.490 88.347 47.081 90.243 41.877 84.665 ALD-GP 2SEP - - 44.544 92.673 45.611 88.291 45.077 93.085 The least efficient ALD-GP model is at least 7% more efficient than the SARIMA, RF and ANN models and the best selected ALD-GP model is about 25% more efficient than the SARIMA, RF and ANN models. Thus, the lowest performance of the proposed model is far better than the best selected conventional model. The compassion of the ALD-GP models suggests that the SESP kernel with three lags is the best-selected model with RMSE is 41.8, and sMAPE is 84%. For this model, we have minimum RMSE and sMAPE. At second, ALD-GP with SE + W kernel at three lags is selected as the best model with minimum average RMSE 42.5 and sMAPE 86%. At third, ALD-GP with PSE + M 2 also at three lags is selected as the best model with average RMSE 44.5 and sMAPE 85%. While the SEP kernel at one lag, SESP kernel at one lag, RQ + W kernel at two lags, and SE at three lags also perform very good. Additionally, for the one lag situation, SESP and SEP are the best kernels for forecasting. For the two lags situation, RQ + W is the best-selected kernel function. Figure 4 presents the graphical comparison of the forecast of each ALD-GP model with the actual data set. Also, we have drawn the 95% confidence band of the forecast which is the shaded region along with the forecasted points. The covariance metric of Prior and posterior ALD-GP using different kernel function at different lag was presented by using heat plot. The Figure S1-S24 of the supplementary materials shows these heat plots. 6. Conclusion Overall results show that the ALD-GP is better than the SARIMA, RF and ANN models for forecasting the rainfall data. The comparison of different lags and proposed kernel functions suggested that the performance of ALD-GP using three auxiliary lags is far better compared to others. Additionally, SESP is the best-selected kernel function for rainfall series. So, the efficient forecasting can be made by using ALD-GP model with suggested kernel functions. The proposed optimization technique is also valid for modelling using the Gaussian process regression model. Similarly, the value of the suggested nugget can handle ill-conditioning while using Gaussian process regression. Declarations Acknowledgments. We are very thankful to the Regional Meteorology Department officials, Lahore, Pakistan, for sharing their recorded data. No funds, grants, or other support was received. The authors have no relevant financial or non-financial interests to disclose. Funding No funds, grants, or other support was received Conflict of Interest The authors have no relevant financial or non-financial interests to disclose. Ethics Approval This is an observational study. The Board of Advanced Studies and Research of Bahauddin Zakariya University, Multan, Pakistan has confirmed that no ethical approval is required. Consent to Participate Not Applicable Consent to Publication Not Applicable Availability of data The data that support the findings of this study are available from Regional Meteorology Department Lahore, but restrictions apply to the availability of these data, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of Regional Meteorology Department Lahore. Code availability The code in R-language used in the current study are available from the corresponding author on reasonable request. Authors Contribution Haris Khurram and Muhammad Mutahir Iqbal contributed equally to the conceptualization, framework and analysis of this paper. Both authors read and approved the final manuscript. References Adnan, M., Khan, F., Rehman, N., Ali, S., Hassan, S. S., Dogar, M. M., ... & Hasson, S. (2020). Variability and Predictability of Summer Monsoon Rainfall over Pakistan. Asia-Pacific Journal of Atmospheric Sciences , 1-9. Afrifa-Yamoah, E., Saeed, B. I., & Karim, A. (2016). Sarima modelling and forecasting of monthly rainfall in the Brong Ahafo Region of Ghana. World Environment , 6(1), 1-9. Antoniadis, A., Paparoditis, E., & Sapatinas, T. (2006). A functional wavelet–kernel approach for time series prediction. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 68(5), 837-857. Butler, A., Haynes, R. D., Humphries, T. D., & Ranjan, P. (2014). Efficient optimization of the likelihood function in Gaussian process modelling. Computational Statistics & Data Analysis , 73, 40-52. Dwivedi, D. K., Kelaiya, J. H., & Sharma, G. R. (2019). Forecasting monthly rainfall using autoregressive integrated moving average model (ARIMA) and artificial neural network (ANN) model: A case study of Junagadh, Gujarat, India. Journal of Applied and Natural Science, 11(1), 35-41. Gramacy, R. B. (2020). Surrogates: Gaussian Process Modeling, Design, and Optimization for the Applied Sciences. CRC Press . Kaur, S., & Rakshit, M. (2020). Seasonal and Periodic Autoregressive Time Series Models Used for Forecasting Analysis of Rainfall Data. International Journal of Advanced Research in Engineering and Technology , 10(1), 2019. Khan, S., Guan, Y., Khan, F., & Khan, Z. (2020). A Comprehensive Index for Measuring Water Security in an Urbanizing World: The Case of Pakistan’s Capital. Water , 12(1), 166. Khurram, H. and Iqbal, M. M. (2021). Kernel Functions for Seasonal And Non-Seasonal Time Series. Proc. 18th International Conference on Statistical Sciences Lahore, Pakistan – February 18-20, 2021, Vol. 35, pp. 119-13. MacDonald, B., Ranjan, P., & Chipman, H. (2015). GPfit: An R package for fitting a Gaussian process model to deterministic simulator outputs. Journal of Statistical Software, 64(i12). MacKay, D. J. (1998). Introduction to Gaussian processes. NATO ASI Series F Computer and Systems Sciences , 168, 133-166. Makridakis S. (1993). Accuracy measures: theoretical and practical concerns. International Journal of Forecasting . 9(4):527–529. Matérn, B. (1960). Spatial variation, meddelanden fran statens skogsforskningsinstitut . Lecture Notes in Statistics, 36, 21. Murthy, K. N., Saravana, R., & Kumar, K. V. (2018). Modeling and forecasting rainfall patterns of southwest monsoons in North–East India as a SARIMA process. Meteorology and Atmospheric Physics , 130(1), 99-106. Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The computer journal, 7(4), 308-313. Ni, L., Wang, D., Singh, V. P., Wu, J., Wang, Y., Tao, Y., & Zhang, J. (2020). Streamflow and rainfall forecasting by two long short-term memory-based models. Journal of Hydrology , 583, 124296. O'Hagan, A. (1978). Curve fitting and optimal design for prediction. Journal of the Royal Statistical Society: Series B (Methodological) , 40(1), 1-24. Ortiz-García, E. G., Salcedo-Sanz, S., & Casanova-Mateo, C. (2014). Accurate precipitation prediction with support vector classifiers: A study including novel predictive variables and observational data. Atmospheric research , 139, 128-136. Ranjan, P., Haynes, R., & Karsten, R. (2011). A computationally stable approach to Gaussian process interpolation of deterministic computer simulation data. Technometrics , 53(4), 366-378. Shi, J. Q., & Choi, T. (2011). Gaussian process regression analysis for functional data. CRC Press . Unnikrishnan, P., & Jothiprakash, V. (2020). Hybrid SSA-ARIMA-ANN model for forecasting daily rainfall. Water Resources Management , 34 (11), 3609-3623. Xiang, Y., Gou, L., He, L., Xia, S., & Wang, W. (2018). A SVR–ANN combined model based on ensemble EMD for rainfall prediction. Applied Soft Computing , 73 , 874-883. Yasmeen, F., & Hameed, S. (2018). Forecasting of Rainfall in Pakistan via Sliced Functional Times Series (SFTS). World Environment 2018, 8(1), 1-14. Yu, P. S., Yang, T. C., Chen, S. Y., Kuo, C. M., & Tseng, H. W. (2017). Comparison of random forests and support vector machine for real-time radar-derived rainfall forecasting. Journal of Hydrology , 552, 92-104. Zhang, L., Zhou, W., & Jiao, L. (2004). Wavelet support vector machine. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) , 34(1), 34-39. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2486388","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":172115345,"identity":"a9f31daa-10c8-44e5-aaaf-99501ee3785a","order_by":0,"name":"Muhammad Ahmed Shehzad","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVRIiWNgGAWjYDACCSDmYWBIgDAqDsAFDYjUcoZkLbxtRGjhn92d+OENQ10ev3T7ww9v592xNzjAfPA2D8MdY5yW3Dm7WXIOw+FiyTkHkiXnbnvGbHCALdmah+GZGU6H3cjdIM3DcCBxw42EA9K82w6zGRzgMQOKHLbBpUP+Ru7m3zwMdYn7byQ2/+adc5jH4AD/N7xaDG7kbgMqYE7cIJHMJs3bcFgCaAsbSAtOhxkCtVjOMTicOONGGpvlnGPPDCQPsxkDRZ7h9L4c0GE33lTUJfbPSH98403NHXu+480PgSJ3DBtw+h/sPGQOM1jkAF4NWAEZWkbBKBgFo2C4AgCJG1vBY36r1AAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-4440-6809","institution":"Bahauddin Zakariya University Faculty of Science","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"Ahmed","lastName":"Shehzad","suffix":""},{"id":172115346,"identity":"e8177c50-538b-49cf-88eb-f59222e3e2cc","order_by":1,"name":"Haris Khurram","email":"","orcid":"","institution":"National University of Computer and Emerging Sciences","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Haris","middleName":"","lastName":"Khurram","suffix":""},{"id":172115347,"identity":"5db0329c-3545-49ce-b2f6-bffce96b3027","order_by":2,"name":"Aamna Khan","email":"","orcid":"","institution":"Bahauddin Zakariya University Faculty of Science","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Aamna","middleName":"","lastName":"Khan","suffix":""},{"id":172115348,"identity":"159559be-034a-4d4d-8d94-9dbed6525245","order_by":3,"name":"Muhammad Mutahir Iqbal","email":"","orcid":"","institution":"Bahauddin Zakariya University Faculty of Science","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"Mutahir","lastName":"Iqbal","suffix":""}],"badges":[],"createdAt":"2023-01-17 07:03:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2486388/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2486388/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":32390572,"identity":"133b3d5c-14c8-42ca-9531-4d238780553b","added_by":"auto","created_at":"2023-02-02 16:14:34","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":64116,"visible":true,"origin":"","legend":"\u003cp\u003eLine Plot of Monthly Rainfall in Lahore, Pakistan\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-2486388/v1/43c4173b414d602ba91985db.jpg"},{"id":32390556,"identity":"74c5062f-a91c-4bc3-9999-bbda21d6abbf","added_by":"auto","created_at":"2023-02-02 16:14:34","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":60793,"visible":true,"origin":"","legend":"\u003cp\u003eAutocorrelation at different lags\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-2486388/v1/b4043d40042638790cb46f6f.jpg"},{"id":32390590,"identity":"ac33dbce-09c7-44cc-8a4c-f97952813625","added_by":"auto","created_at":"2023-02-02 16:14:35","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":152411,"visible":true,"origin":"","legend":"\u003cp\u003eArtificial Neural Network Plot\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-2486388/v1/23f82107030e0c497b1ee9e5.jpg"},{"id":32390585,"identity":"2c3ef654-c4e6-4b53-9949-79c329fc4ce3","added_by":"auto","created_at":"2023-02-02 16:14:35","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":113351,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the actual and forecast data of ALD-GP model using different kernel with a 95% confidence band at different lags. (i) SE kernel at one lag. (ii) SE kernel at two lags. (iii) SE kernel at three lags. (iv) SEP kernel at one lag. (v) SEP kernel at two lags. (vi) SEP kernel at three lags. (vii) SE+W kernel at one lag. (viii) SE+W kernel at two lags. (ix) SE+W kernel at three lags. (x) PSE+M2 kernel at one lag. (xi) PSE+M2 kernel at two lags. (xii) PSE+M2 kernel at three lags. (xiii) SSEP kernel at one lag. (xiv) SSEP kernel at two lags. (xv) SSEP kernel at three lags. (xvi) SESP kernel at one lag. (xvii) SESP kernel at two lags. (xviii) SESP kernel at three lags. (xix) 2SEP kernel at one lag. (xx) 2SEP kernel at two lags. (xxi) 2SEP kernel at three lags. (xxii) 2SEP kernel at one lag. (xxiii) 2SEP kernel at two lags. (xxiv) 2SEP kernel at three lags.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-2486388/v1/460bffe5e2c3cf90eeb323be.jpg"},{"id":33863318,"identity":"68ceb39c-f50e-40a7-b27d-9da01d5376f1","added_by":"auto","created_at":"2023-03-06 23:15:56","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":705577,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2486388/v1/5690a160-20ed-48cc-bd02-4659d50f9136.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eAuxiliary-lag Dependent Gaussian Process Model for Forecasting Rainfall Data Using Proposed Kernels and Multi-start Optimization Method\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eRainfall in a country is an important variable that impacts agriculture systems, especially on productions, water management systems, climate change, and natural disasters such as droughts and floods. Pakistan (currently facing hard days due to flood caused by high rainfall) is agricultural land, and the economy is highly dependent on agriculture products and productions. Simultaneously, crop production is highly dependent on the amount of rainfall and has highly affected by high or low rainfall in the area. Similarly, water storage in Pakistan highly depends on the amount of rainfall. Pakistan is among the top ten countries where climate change has a very adverse effect and has critical water resources (Khan et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). To keep all these facts in view, rainfall foresting is very important and helpful in policymaking.\u003c/p\u003e \u003cp\u003eThere are various research studies related to the forecasting of rainfall in a different area of the world. Most of them are related to seasonal ARIMA (SARIMA) modeling, and few are based on a machine learning approach. Afrifa-Yamoah et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) use the SARIMA model to forecast the rainfall data in a region of Ghana. Kaur and Rakshit (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) also used SARIMA models to forecast Punjab's rainfall data, India. Murthy et al. (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) use SARIMA models to forecast the rainfall data in North-East India. Ortiz-Garcia et al. (2014) forecast the rainfall in Spin using support vector classifiers. Ni et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) used short term memory-based models to forecast rainfall and streamflow. Yu et al. (\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) used random forest and support vector models for rainfall forecasting. Yasmeen and Hameed (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) used sliced functional time series to forecast the rainfall in Pakistan. Adnan et al. (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) predict the monsoon rainfall in summer using multiple linear regression and principal component regression models in Pakistan. Xiang et al. (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) use support vector model and artificial neural network model jointly for forecasting rainfall. Dwivedi, Kelajya and Sharma (2019) used ANN model for forecasting monthly rainfall in India. Unnikrishnan, Poornima and Jothiprakash use hybrid models by combining machine learning and statistical approaches for forecasting daily rainfall data. Improved forecasting using advanced models is always a significant contribution in literature.\u003c/p\u003e \u003cp\u003eIn this article to perform the improved forecasting of the rainfall, we suggest an auxiliary-lag dependent Gaussian process (ALD-GP), a Bayesian non-parametric machine learning model. We have also suggested some multifeatured kernels to be used with the proposed model.\u003c/p\u003e"},{"header":"2. Proposed Methodology","content":"\u003cp\u003eThe Gaussian process is known to be a stochastic process over infinite-dimension latent function indexed by some known variables. Any finite and known subset of these functions follows a multivariate normal distribution (Shi and Choi, \u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e; Gramacy, \u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e). Formally we can define a Gaussian process as the distribution over the function\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(f\\left(X\\right)\\)\u003c/span\u003e\u003c/span\u003e where\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ1\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$f\\left(X\\right)~GP\\left(m\\left(X\\right),g(X,X{\\prime })\\right).$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003esuch that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(f\\left(X\\right):\\mathbb{X}\\to {\\mathbb{R}}^{\\infty }\\text{and }X\\in \\mathbb{X}\\)\u003c/span\u003e\u003c/span\u003e with specified mean function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\left(X\\right)=E\\left[f\\right(X\\left)\\right]\\)\u003c/span\u003e\u003c/span\u003e and covariance function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(g\\left(X,{X}^{{\\prime }}\\right)\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eAuxiliary-Lags Dependent Gaussian Process\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eNow, considered a variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e is indexed over some fixed time interval \u003cem\u003et\u003c/em\u003e, known to be a time series variable. The primary aim is to model the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e for forecasting purposes. Let the \u003cem\u003el-\u003c/em\u003eth auxiliary-lags of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e are denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003eof order \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\times l.\\)\u003c/span\u003e\u003c/span\u003e These are the selective lags of a series that are highly correlated with the series. Thus, this lag dependent model\u0026apos;s purpose is to use selective lags, which are considered auxiliary lags of the series of interest. Then the functional form of the model of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ2\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$${y}_{t}=f\\left({L}^{l}y\\right)+{e}_{t}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}_{t}\\)\u003c/span\u003e\u003c/span\u003eis supposed to be a stationary disturbance term such that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}_{t}~N(0,{\\sigma }^{2})\\)\u003c/span\u003e\u003c/span\u003e. Then, the distribution of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e is auxiliary-lag dependent Gaussian process (ALD-GP) as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ3\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$${y}_{t}~ALD-GP\\left(\\mu \\left({L}^{l}y\\right),\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)+{\\sigma }^{2}{\\delta }_{{y}_{tt}}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu \\left(\\cdot \\right)\\)\u003c/span\u003e\u003c/span\u003e is the process mean function, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma \\left(\\cdot \\right)\\)\u003c/span\u003e\u003c/span\u003e is the autocovariance function of the process, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta }_{{y}_{tt}}\\)\u003c/span\u003e\u003c/span\u003eis the Kronecker delta. Consider the \u003cem\u003eh-\u003c/em\u003estep ahead forecasting of series\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003ethat is\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t+h}\\)\u003c/span\u003e\u003c/span\u003e. This \u003cem\u003eh-\u003c/em\u003estep ahead forecasting is based on \u003cem\u003el\u003c/em\u003e auxiliary-lags \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}{y}_{h}\\)\u003c/span\u003e\u003c/span\u003e. For \u003cem\u003eh-\u003c/em\u003estep ahead forecasting the conditional distribution of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t+h}\\)\u003c/span\u003e\u003c/span\u003e given \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{t}\\)\u003c/span\u003e\u003c/span\u003e will be distributed as auxiliary-lag dependent Gaussian process.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 1\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConsidered the time series distributed as ALD-GP, with zero mean prior, the Posterior distribution of the \u003cem\u003eh-\u003c/em\u003estep ahead forecast is also distributed as ALD-GP as following\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ4\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$$p\\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right)~ALD-GP\\left(\\mu \\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right),\\gamma \\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right)\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere the posterior mean is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ5\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$$\\mu \\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right)=\\gamma \\left({L}^{l}{y}_{h},{L}^{l}y\\right){\\left[\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)+{\\sigma }^{2}{\\delta }_{yy}\\right]}^{-1}{y}_{t}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eand the posterior autocovariance function is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ6\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e$$\\begin{gathered} \\gamma \\left( {{y_{t+h}}|{L^l}{y_h},{L^l}y,{y_t}} \\right)=\\gamma \\left( {{L^l}{y_h},{L^l}{y_h}^{\\prime }} \\right) - \\gamma \\left( {{L^l}{y_h},{L^l}y} \\right) \\times \\\\ {\\left[ {\\gamma \\left( {{L^l}y,{L^l}y^{\\prime}} \\right)+{\\sigma ^2}{\\delta _{yy}}} \\right]^{ - 1}}\\gamma \\left( {{L^l}y,{L^l}{y_h}} \\right) \\\\ \\end{gathered}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eBy using (O Hagan, 1978), we can prove it on similar lines as the distribution of the observation \u003cem\u003et\u0026thinsp;+\u0026thinsp;1\u003c/em\u003e is similar to the distribution of the \u003cem\u003et\u003c/em\u003e observation.\u003c/p\u003e\n\u003cp\u003eThus, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu \\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right)\\)\u003c/span\u003e\u003c/span\u003e is the posterior mean function, which is considered as the forecast at \u003cem\u003et+h\u003c/em\u003e time. Similarly, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma \\left({y}_{t+h}|{L}^{l}{y}_{h},{L}^{l}y,{y}_{t}\\right)\\)\u003c/span\u003e\u003c/span\u003eis the posterior autocovariance function of the forecast and used to draw a 95% confidence band of the forecast.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eb. Autocovariance Function\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eCovariance function of ALD-GP is based on lag values of the main series, so we considered it as autocovariance. Here, autocovariance function is defined as the covariance between two matrices of the different time-dependent lags. The elementary properties of the autocovariance function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)\\)\u003c/span\u003e\u003c/span\u003e for the ALD-GP model are as follows. The autocovariance functions should be symmetric as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ7\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e$$\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)=\\gamma \\left({L}^{l}{y}^{{\\prime }},{L}^{l}y\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe autocovariance function for a vector \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(a={a}_{1},{a}_{2},...,{a}_{n}\\in {\\mathbb{R}}^{n}\\)\u003c/span\u003e\u003c/span\u003e should be at least positive semidefinite.\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ8\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e$${\\sum }_{i=1}^{n}{\\sum }_{j=1}^{n}{a}_{i}\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right){a}_{j}\\ge 0$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eFor adding the non-parametric flavor, autocovariance functions will be defined using kernel functions. A kernel function that is symmetric and at least positive semidefinite will be considered as the autocovariance function.\u003c/p\u003e\n\u003cp\u003eThe stationarity of the autocovariance function is based on the translation-invariant property. So, if the covariance function can be written as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ9\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e$$\\gamma \\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)=\\gamma \\left(‖{L}^{l}{y}^{{\\prime }}-{L}^{l}y‖\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eand the positive semidefinite property as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ10\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e$${\\sum }_{i=1}^{n}{\\sum }_{j=1}^{n}{a}_{i}\\gamma \\left({L}^{l}{y}_{i}-{L}^{l}{y}_{j}\\right){a}_{j}\\ge 0$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ethen the autocovariance function will be stationary.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eCommon kernel functions for lagged series\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eGaussian Process is defined over kernel functions. Some pre-defined kernel functions used in Gaussian Process modelling are reconsidered here for auxiliary lag series.\u003c/p\u003e\n\u003cp\u003eSquared exponential (SE) is an essential and commonly used kernel function in kernel-based modeling. It is also known as redial based function (RBF) kernel. SE kernel function for lag series is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ11\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e$${g}_{SE}\\left(Ly,L{y}^{{\\prime }}\\right)={\\sigma }^{2}{exp}\\left(-\\frac{‖Ly-L{y}^{{\\prime }}‖}{2{l}^{2}}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026sigma;\u003c/em\u003e is the scale, and \u003cem\u003el\u003c/em\u003e is the length-scale parameters. This kernel function is an isotropic stationary kernel function. Additionally, the prominent feature of this kernel function is that it can be infinitely differentiable.\u003c/p\u003e\n\u003cp\u003eThe rational quadratic kernel (RQ) function is the scale mixture, or in other words, the infinite sum of the SE kernel. RQ kernel function for some lag series\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ12\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e$${{g}}_{{R}{Q}}\\left({L}{y},{L}{{y}}^{{{\\prime }}}\\right)={\\left(1+\\frac{‖{L}{y}-{L}{{y}}^{{{\\prime }}}‖}{2{\\alpha }{{l}}^{2}}\\right)}^{-{\\alpha }}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026alpha;\u003c/em\u003e is the scale mixture parameter and \u003cem\u003el\u003c/em\u003e is the length-scale parameter. Both parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha ,l\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e and for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha \\to \\infty\\)\u003c/span\u003e\u003c/span\u003e the RQ kernel function will be converted into the SE kernel function. This is also isotropic stationary kernel function.\u003c/p\u003e\n\u003cp\u003eThe periodic (P) kernel was firstly used by MacKay (\u003cspan class=\"CitationRef\"\u003e1998\u003c/span\u003e) to access the periodic repetition in data. For lag variables, the periodic kernel is defined as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ13\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e$${g}_{P}\\left(Ly,L{y}^{{\\prime }}\\right)={exp}\\left[\\frac{-2}{{l}^{2}}{\\left[{sin}\\left(\\frac{\\pi }{w}‖Ly-L{y}^{{\\prime }}‖\\right)\\right]}^{2}\\right]$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003ew\u003c/em\u003e is the periodic parameter, \u003cem\u003el\u003c/em\u003e is the length-scale parameters and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(w,l\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e. This also an isotropic stationary kernel function.\u003c/p\u003e\n\u003cp\u003eThe Mat\u0026eacute;rn (M\u003csub\u003eP\u003c/sub\u003e) class of kernel was proposed by Mat\u0026eacute;rn (\u003cspan class=\"CitationRef\"\u003e1960\u003c/span\u003e) is the generalization of the RBF kernel function. For a lag series, the general Mat\u0026eacute;rn class of kernel function in the form of modified Bessel function is defined as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ14\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e$${g}_{M}(Ly,L{y}^{{\\prime }})=\\frac{1}{{2}^{1-v}\\varGamma \\left(v\\right)}{\\left(\\frac{\\sqrt{2v}‖Ly-L{y}^{{\\prime }}‖}{l}\\right)}^{v}{K}_{v}\\left(\\frac{\\sqrt{2v}‖Ly-L{y}^{{\\prime }}‖}{l}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003ev\u003c/em\u003e is the order parameter, and \u003cem\u003el\u003c/em\u003e is the length-scale parameter with\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v,l\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{v}\\)\u003c/span\u003e\u003c/span\u003e is the modified Bessel function and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varGamma \\left(.\\right)\\)\u003c/span\u003e\u003c/span\u003e is a gamma function. A simplified form of Mat\u0026eacute;rn class of kernel can be extracted by assigning \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v=P+\\frac{1}{2}\\)\u003c/span\u003e\u003c/span\u003e. For \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P=2\\)\u003c/span\u003e\u003c/span\u003e we have the commonly used form of Mat\u0026eacute;rn class of kernel that is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ15\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e$${g}_{{M}_{2}}(Ly,L{y}^{{\\prime }})=\\left(1+\\frac{\\sqrt{5}‖Ly-L{y}^{{\\prime }}‖}{l}+\\frac{5}{3}{\\left(\\frac{‖Ly-L{y}^{{\\prime }}‖}{l}\\right)}^{2}\\right){exp}\\left(-\\frac{\\sqrt{5}‖Ly-L{y}^{{\\prime }}‖}{l}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eA wavelet (W) kernel function based on some mother wavelet function is considered by Zhang et al. (\u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e). The translation-invariant W kernel function is defined as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ16\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e$${g}_{W}\\left(Ly,L{y}^{{\\prime }}\\right)={\\prod }_{i=1}^{p}h\\left(\\frac{L{y}_{i}-L{y}_{i}^{{\\prime }}}{a}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eand for mother wavelet, it can be defined as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ17\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e$${g}_{W}\\left(Ly,L{y}^{{\\prime }}\\right)={\\prod }_{i=1}^{p}\\left[{cos}\\left(\\frac{1.75\\left(L{y}_{i}-L{y}_{i}^{{\\prime }}\\right)}{a}\\right){exp}\\left(\\frac{-‖L{y}_{i}-L{y}_{i}^{{\\prime }}‖}{2{a}^{2}}\\right)\\right]$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere a is the parameter and \u003cem\u003ex\u003c/em\u003e is the realization in one dimension (Antoniadis, 2006).\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eNew kernel functions for seasonal series\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eSquared Exponential Periodic Kernel (SEP) is the squared exponential times Periodic Kernel. This is the stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ18\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e$${\\gamma }_{SE\\times P}\\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)={{\\sigma }_{f}}^{2}{exp}\\left(-\\frac{‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{2{l}^{2}}-\\frac{2}{{l}_{1}^{2}}{\\left[{sin}\\left(\\frac{\\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖\\right)\\right]}^{2}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026sigma;, l, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003ew\u003c/em\u003e are the hyperparameters which will be tune using an optimization algorithm.\u003c/p\u003e\n\u003cp\u003eSquared Exponential with Wavelet Kernel (SE\u0026thinsp;+\u0026thinsp;W) is the special kernel which the sum of the squared exponential and wavelet kernel where the mother wavelet is the same as defined by Zhang et al. (\u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e). It is the locally stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ19\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e$${\\gamma }_{SE+W}\\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)={{\\sigma }_{f}}^{2}{exp}\\left(-\\frac{‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{2{l}^{2}}\\right)+{\\prod }_{i=1}^{p}\\left[{cos}\\left(\\frac{1.75\\left({L}^{l}y-{L}^{l}{y}^{{\\prime }}\\right)}{a}\\right){exp}\\left(\\frac{-‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{2{a}_{1}^{2}}\\right)\\right]$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026sigma;, l, a\u003c/em\u003e, and \u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, are the hyperparameters that will be tune using an optimization algorithm.\u003c/p\u003e\n\u003cp\u003eRational Quadratic with Wavelet Kernel (RQ\u0026thinsp;+\u0026thinsp;W) is the sum of the Rational Quadratic and wavelet kernel function. It is also a locally stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ20\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e$${{\\gamma }}_{{R}{Q}+{W}}\\left({{L}}^{{l}}{y},{{L}}^{{l}}{{y}}^{{{\\prime }}}\\right)={\\left(1+\\frac{‖{{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\\prime }}}‖}{2{\\alpha }{{l}}^{2}}\\right)}^{-{\\alpha }}+{\\prod }_{{i}=1}^{{p}}\\left[{cos}\\left(\\frac{1.75\\left({{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\\prime }}}\\right)}{{a}}\\right){exp}\\left(\\frac{-‖{{L}}^{{l}}{y}-{{L}}^{{l}}{{y}}^{{{\\prime }}}‖}{2{{a}}_{1}^{2}}\\right)\\right]$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e𝛼, l, a\u003c/em\u003e, and \u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, are the hyperparameters.\u003c/p\u003e\n\u003cp\u003eSquared Exponential Periodic with Mat\u0026eacute;rn Class Kernel (SEP\u0026thinsp;+\u0026thinsp;M\u003csub\u003e2\u003c/sub\u003e) is the sum of the Squared Exponential times periodic and Matern kernel function. It is a stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ21\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e$$\\begin{gathered} {\\gamma _{SE \\times P+M2}}\\left( {{L^l}y,{L^l}y^{\\prime}} \\right)=\\sigma _{f}^{2}\\exp \\left( { - \\frac{{\\left\\| {{L^l}y - {L^l}y^{\\prime}} \\right\\|}}{{2{l^2}}} - \\frac{2}{{l_{1}^{2}}}{{\\left[ {\\sin \\left( {\\frac{\\pi }{w}\\left\\| {{L^l}y - {L^l}y^{\\prime}} \\right\\|} \\right)} \\right]}^2}} \\right)+ \\\\ \\left( {1+\\frac{{\\sqrt 5 \\left\\| {{L^l}y - {L^l}y^{\\prime}} \\right\\|}}{{{l_2}}}+\\frac{5}{3}{{\\left( {\\frac{{\\left\\| {{L^l}y - {L^l}y^{\\prime}} \\right\\|}}{{{l_2}}}} \\right)}^2}} \\right)\\exp \\left( { - \\frac{{\\sqrt 5 \\left\\| {{L^l}y - {L^l}y^{\\prime}} \\right\\|}}{{{l_3}}}} \\right) \\\\ \\end{gathered}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e𝜎, l, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003ew, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e, are the hyperparameters.\u003c/p\u003e\n\u003cp\u003eSquared Squared Exponential Periodic Kernel (SSEP) is the squared of Squared Exponential kernel, times periodic kernel function. It is a stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ22\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e$${\\gamma }_{S{E}^{2}\\times P}\\left({L}^{l}y-{L}^{l}{y}^{{\\prime }}\\right)={\\sigma }_{f}^{2}{exp}\\left(-\\frac{‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{{l}^{2}}-\\frac{2}{{l}_{1}^{2}}{\\left[{sin}\\left(\\frac{\\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖\\right)\\right]}^{2}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e𝜎, l, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003ew\u003c/em\u003e, are the hyperparameters.\u003c/p\u003e\n\u003cp\u003eSquared Exponential Squared Periodic (SESP) kernel is the Squared Exponential kernel times squared of the periodic kernel function. It is also a stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ23\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e$${\\gamma }_{SE\\times {P}^{2}}\\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)={\\sigma }_{f}^{2}{exp}\\left(-\\frac{‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{2{l}^{2}}-\\frac{4}{{l}_{1}^{2}}{\\left[{sin}\\left(\\frac{\\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖\\right)\\right]}^{2}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e𝜎, l, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003ew\u003c/em\u003e, are the hyperparameters.\u003c/p\u003e\n\u003cp\u003eSquared Exponential with Squared Exponential Periodic (2SEP) is the sum of the Squared Exponential and squared exponential times periodic kernel function. It is also a stationary kernel function. For the matrix of auxiliary lag series, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}^{l}y\\)\u003c/span\u003e\u003c/span\u003e the functional form of the kernel is as\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ24\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e$${\\gamma }_{2SE\\times P}\\left({L}^{l}y,{L}^{l}{y}^{{\\prime }}\\right)=2{\\sigma }_{f}^{2}{exp}\\left(-\\frac{‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖}{2{l}^{2}}\\right){exp}\\left(-\\frac{2}{{l}_{1}^{2}}{\\left[{sin}\\left(\\frac{\\pi }{w}‖{L}^{l}y-{L}^{l}{y}^{{\\prime }}‖\\right)\\right]}^{2}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e𝜎, l, l\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003ew\u003c/em\u003e are the hyperparameters.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProposition 1\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe proposed kernels defined in Eq. \u003cspan class=\"InternalRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan class=\"InternalRef\"\u003e24\u003c/span\u003e are all at least a positive semidefinite kernel.\u003c/p\u003e\n\u003cp\u003eAs the product and sum of all at least positive semidefinite kernels will also atleast positive semidefinite kernels, so this proposition is true for all the defined kernel functions.\u003c/p\u003e\n\u003cp\u003eAll these suggested kernel functions and their detailed properties were discussed in Khurram and Iqbal (\u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eMarginal Likelihood Using Bayesian Model Selection\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eThe marginal likelihood of ALD-GP is obtained by using the approach of Bayesian model selection. Consider a set of models\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mathcal{M}\\)\u003c/span\u003e\u003c/span\u003efor some data \u003cem\u003ey\u003c/em\u003e with a set of parameters \u003cem\u003e\u0026theta;\u003c/em\u003e such that the probability function of \u003cem\u003ey\u003c/em\u003e is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p\\left(y|\\theta ,\\mathcal{M}\\right)\\)\u003c/span\u003e\u003c/span\u003e. By using the Bayes rule the posterior probability model is\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equ25\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e$$p\\left(\\mathcal{M}{}_{i} |y\\right)=\\frac{p\\left(y|\\mathcal{M}{}_{i}\\right)p\\left(\\mathcal{M}{}_{i}\\right)}{{\\sum }_{\\varOmega }p\\left(y|\\mathcal{M}\\mathcal{}\\right)p\\left(\\mathcal{M}\\mathcal{}\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003cp\u003ethe \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p\\left({\\mathcal{M}}_{i}\\right)\\)\u003c/span\u003e\u003c/span\u003eis the prior distribution over models. The posterior distribution of parameter over data and model is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ26\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e$$p\\left({\\theta }_{i}|y,{\\mathcal{M}}_{i}\\right)=\\frac{p\\left(y|{\\theta }_{i},{\\mathcal{M}}_{i}\\right)p\\left({\\theta }_{i}|{\\mathcal{M}}_{i}\\right)}{p\\left(y|{\\mathcal{M}}_{i}\\right)}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Equation\" id=\"Equ27\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e$$p\\left({\\theta }_{i}|y,{\\mathcal{M}}_{i}\\right)=\\frac{p\\left(y|{\\theta }_{i},{\\mathcal{M}}_{i}\\right)p\\left({\\theta }_{i}|{\\mathcal{M}}_{i}\\right)}{\\int p\\left(y|{\\theta }_{i},{\\mathcal{M}}_{i}\\right)p\\left({\\theta }_{i}|{\\mathcal{M}}_{i}\\right)d{\\theta }_{i}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eSo, the best model and their hyperparameters can be accessed using the Bayesian model selection procedure. Using the Bayesian model selection approach, the marginal likelihood function of ALD-GP is\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ28\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ28\" name=\"EquationSource\"\u003e$${log}p\\left({y}_{t}|{L}^{l}y,\\beta \\right)=-\\frac{1}{2}{y}_{t}^{{\\prime }}{\\left(\\gamma ({L}^{l}y,{L}^{l}{y}^{{\\prime }})+{\\sigma }^{2}{\\delta }_{{y}_{tt}}\\right)}^{-1}{y}_{t}-\\frac{1}{2}{log}\\left|\\gamma ({L}^{l}y,{L}^{l}{y}^{{\\prime }})+{\\sigma }^{2}{\\delta }_{{y}_{tt}}\\right|-\\frac{n}{2}{log}2\\pi$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cem\u003e\u0026beta;\u003c/em\u003e is the vector of hyperparameters that need to be carefully optimized.\u003c/p\u003e\n \u003cp\u003e\u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003eProposed Optimization Method\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eA computationally stable optimization method was introduced by Butler et al. (\u003cspan class=\"CitationRef\"\u003e2014\u003c/span\u003e) and Macdonald et al. (\u003cspan class=\"CitationRef\"\u003e2015\u003c/span\u003e), which suggest choosing multiple input starter for avoiding the problem of misleading maxima. These authors used the SE and Matern class of kernels, and their methods used the deviance function of the correlation matrix for specified kernel settings. This approach also needs a starting point that needs information related to kernel before the start, which is subjective based. This method can be used for some available class of kernels but not for a newly proposed kernel. Now, we will present our proposed method, which is the modified version of the cluster-based multi-start technique.\u003c/p\u003e\n \u003cp\u003eThe proposed simplex cluster-based multi-start technique is based on the optimization of negative marginal loglikelihood. The feature of this proposed method is that it is computationally stable and able to work for any newly proposed kernel. It does not require any initial values and range to be used. The optimization method is based on the standard Nelder-Mead approach proposed by Nelder and Mead (\u003cspan class=\"CitationRef\"\u003e1965\u003c/span\u003e). The following are the steps of the proposed method.\u003c/p\u003e\n \u003cp\u003eConsider that d is the total number of hyperparameters \u0026beta; to be optimized for a kernel and negative marginal likelihood. Then the proposed method is as follows\u003c/p\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003eUse maximin criteria to draw 200d Latin hypercube samples over the interval \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left[\\text{0,1}\\right]}^{d}\\)\u003c/span\u003e\u003c/span\u003e for each hyperparameter.\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003c/span\u003e \u003cspan\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003eSelect 80d points having minimum \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(-{log}p\\left({y}_{t}|{L}^{l}y,\\beta \\right)\\)\u003c/span\u003e\u003c/span\u003efrom the 200d points.\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003c/span\u003e \u003cspan\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003eApply k-mean clustering methods on these selected 80d points. Use five different starters of k-mean clustering and select 3d group.\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003c/span\u003e \u003cspan\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003eUse these the 3d point in running the NM algorithm and selected the best starting point.\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003cp\u003eThe selected points using a simplex cluster-based multi-starter technique based on NM optimizer are considered for the values of the respective hyperparameters.\u003c/p\u003e\n \u003cp\u003eTo avoid the problem of singularity, we have also suggested the lower bound of a nugget value by following Ranjan et al. (\u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e). The lower bound of the nugget value \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\kappa\\)\u003c/span\u003e\u003c/span\u003e is defined as\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ29\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ29\" name=\"EquationSource\"\u003e$$\\kappa =\\left|{min}\\left({\\lambda }^{\\left[1\\right]},0\\right)\\right|$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\lambda }^{\\left[1\\right]}\\)\u003c/span\u003e\u003c/span\u003e is the lowest eigenvalue of the covariance or autocovariance matrix. Thus, instead of using a simple autocorrelation matrix if this nugget value will be added in the matrix\u0026apos;s diagonal, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\gamma }_{B}\\left(\\cdot \\right)=\\gamma \\left(\\cdot \\right)+\\kappa I\\)\u003c/span\u003e\u003c/span\u003e then this jittering will remove the problem of singularity.\u003c/p\u003e"},{"header":"3. Other Models","content":"\u003cp\u003e \u003cspan type=\"BoldItalic\" class=\"BoldItalic\" name=\"Emphasis\"\u003eSeasonal Autoregressive Moving Average Models\u003c/span\u003e \u003c/p\u003e \u003cp\u003eAutoregressive moving average (ARMA) model was firstly considered by Box and Jenkins (1976) as the mixture of autoregressive (AR) and moving average (MA) models. For a series of seasonal nature, there exist seasonal autoregressive moving average (SARMA) models. The parameters of SARIMA models are p, d, q, P, D, Q and m respectively. Where p is the order of autoregressive terms, d is the difference, q is the order of moving average terms, P is the order of seasonal autoregressive terms, D is the seasonal difference, Q is the order of moving average terms and m is the order of seasonal difference which is 12 for monthly time series data. Thus, a SARIMA \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({p},{d},{q}\\right)\\times {\\left({P},{D},{Q}\\right)}_{{m}}\\)\u003c/span\u003e\u003c/span\u003e model will be\u003cdiv id=\"Equ30\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ30\" name=\"EquationSource\"\u003e\n$${\\phi }_{p}\\left(B\\right)\\cdot \\varPhi \\left({B}^{m}\\right)\\cdot {\\nabla }^{d}\\cdot {\\nabla }_{m}^{D}\\cdot {Y}_{t}={\\theta }_{q}\\left(B\\right)\\cdot {\\varTheta }_{Q}\\left({B}^{m}\\right)\\cdot {\\epsilon }_{t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e30\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cem\u003eB\u003c/em\u003e is the back-shift operator and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi ,\\varPhi ,\\nabla ,\\theta \\text{, and }{\\Theta }\\)\u003c/span\u003e\u003c/span\u003eare the parameters of, non-seasonal AR, seasonal AR, differencing, non-seasonal moving average and seasonal moving average terms. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\epsilon }_{t}\\)\u003c/span\u003e\u003c/span\u003e is the disturbance term assumed to be a normally distributed with constant mean \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }\\)\u003c/span\u003e\u003c/span\u003e and variance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\sigma }}^{2}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"BoldItalic\" class=\"BoldItalic\" name=\"Emphasis\"\u003eExponential Smoothing Models\u003c/span\u003e \u003c/p\u003e \u003cp\u003eExponential Smoothing (ES) models are very known models for forecasting time series. The robustness and versatility of this model inspire researchers to be used for forecasting. A simple ES model is based on simple mean, linear trend and quadratic trend methods.\u003c/p\u003e \u003cp\u003eQuadratic ES can be used for seasonal data but there should be trend in the data. There is one parameter of quadratic ES method. Holt-Winter\u0026rsquo;s ES models are mainly used when series is seasonal. The model parameters are associated with level, trend and seasonal terms. These three parameters optimized for forecasting.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"BoldItalic\" class=\"BoldItalic\" name=\"Emphasis\"\u003eArtificial Neural Network Model\u003c/span\u003e \u003c/p\u003e \u003cp\u003eArtificial Neural network (ANN) model is comprised of multiple nodes which are also known as neurons and hidden layers. This hidden layer is the connect input layer with output layer. Weights of hidden layer connect different nodes and layers. Mostly the weights were optimize using backpropagation optimization methods. ANN is capable to capture any kind of nonlinear relationship between input and output series. For an input series, the ANN model can be:\u003cdiv id=\"Equ31\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ31\" name=\"EquationSource\"\u003e\n$$Y{}_{t}=\\beta {}_{0}+{\\sum }_{j=1}^{k}{\\alpha }_{j}f\\left({\\gamma }_{0,j}+{\\sum }_{i=1}^{p}{\\alpha }_{ij}{Y}_{t-i}\\right)+{\\epsilon }_{t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e31\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}_{ij};\\left(i=\\text{1,2},\\dots ,p ;j=\\text{1,2},\\dots ,k\\right)\\)\u003c/span\u003e\u003c/span\u003e are the weights that connect the layers, \u003cem\u003ek\u003c/em\u003e are the hidden nodes, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{j}\\)\u003c/span\u003e\u003c/span\u003e is the bias of the \u003cem\u003ej\u003c/em\u003e-th unit and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(f\\left(.\\right)\\)\u003c/span\u003e\u003c/span\u003e is the activation function. The commonly used transform function is the logistic function.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"BoldItalic\" class=\"BoldItalic\" name=\"Emphasis\"\u003eRandom Forest Model\u003c/span\u003e \u003c/p\u003e \u003cp\u003eRandom Forest (RF) model is a modified form of decision trees proposed by Breiman (2001). The model is based on two parameters which are number of trees \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{n}}_{{t}{r}{e}{e}}\\)\u003c/span\u003e\u003c/span\u003e and number of candidate variables randomly sampled at each split \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{n}}_{{t}{r}{y}}\\)\u003c/span\u003e\u003c/span\u003e. The model makes tree of the series and forecast using it.\u003c/p\u003e\n"},{"header":"4. Data Description And Analysis Methodology","content":"\u003cp\u003eThe average monthly rainfall (mm) is taken from Regional Meteorology Center (RMC) Lahore. This average monthly rainfall data is from Jan-2000 to Dec-2019. This historically recorded data is of Lahore city at latitude 31\u003csup\u003eo\u003c/sup\u003e 33' and longitude 74\u003csup\u003eo\u003c/sup\u003e 20'. For training purposes, we have taken the data from Jan-2000 to Aug 2017 as train data and from Sep-2017 to Dec-2019 as a test data set.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the line plot of the rainfall series, which shows that the series is seasonal. We have used conventional models, including SARIMA and Exponential Smoothing (ES) models on train data, and select the best model among them. After that, we have used autoregressive artificial Neural network (ANN) and random forest (RF) models by taking two autoregressive lags of the series as explanatory variables. The R packages \u003cem\u003erandomForest\u003c/em\u003e and \u003cem\u003eneuralnet\u003c/em\u003e were used for training the models. Then a one-step ahead forecast using these models were calculated. Both of these models show best results when using two autoregressive lags. For ANN we have used three layers with ten nodes because these provides optimal results.\u003c/p\u003e \u003cp\u003eFor applying the ALD-GP using proposed methods, we have to select the auxiliary lags for the model. For that purpose, we have present autocorrelation plot at the different lag periods in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, and from this graph, we observe that the series is highly correlated with the 12th, 24th, and 36th lag term. We simultaneously use all the selected lags in the model. Thus, there are three models used for comparison purpose as\u003cdiv id=\"Equ32\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ32\" name=\"EquationSource\"\u003e\n$${y}_{t}=f\\left({y}_{t-12}\\right)+{e}_{t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e32\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ33\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ33\" name=\"EquationSource\"\u003e\n$${y}_{t}=f\\left({y}_{t-12},{y}_{t-24}\\right)+{e}_{t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e33\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ34\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ34\" name=\"EquationSource\"\u003e\n$${y}_{t}=f\\left({y}_{t-12},{y}_{t-24},{y}_{t-36}\\right)+{e}_{t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e34\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe first model is the one lag model, the second is the two lags model, and the third is the three lags model. To measure the performance of different purpose kernel functions in this demonstration, we have used all the proposed kernel functions in the forecasting process.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"BoldItalic\" class=\"BoldItalic\" name=\"Emphasis\"\u003ePerformance Evaluation Measures\u003c/span\u003e \u003c/p\u003e \u003cp\u003eWe have used the following performance evaluation measures based on the actual and forecasted observations for measuring the forecasting performance of proposed and conventional models. Root mean square error (RMSE) of the forecasted value \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of a series \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e at \u003cem\u003eh-\u003c/em\u003estep ahead forecast is defined as:\u003cdiv id=\"Equ35\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ35\" name=\"EquationSource\"\u003e\n$$RMSE=\\sqrt{\\frac{{\\sum }_{t=1}^{h}{\\left({y}_{t}-{F}_{t}\\right)}^{2}}{h}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e35\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMean Absolute Percentage Error measure is a valid measure for measuring the forecasting performance, but If any of the actual value is zero, this measure did not converge. For such an issue, Makridakis (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1993\u003c/span\u003e) suggest using Symmetric Mean Absolute Percentage Error (sMAPE) which is a bounded measure and more resistant to the outlier. For \u003cem\u003eh-\u003c/em\u003estep ahead forecast, the sMAPE is defined as:\u003cdiv id=\"Equ36\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ36\" name=\"EquationSource\"\u003e\n$$sMAPE=\\frac{{\\sum }_{i=1}^{h}\\frac{2\\times \\left|{y}_{t}-{F}_{t}\\right|}{\\left|{y}_{t}\\right|+\\left|{F}_{t}\\right|}}{h}\\times 100$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e36\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eModel comparison for Selecting best ARIMA and ES model\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAIC\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuadratic ES with α\u0026thinsp;=\u0026thinsp;0.0304\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e66.325\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.502\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHolt-Winter's ES with α\u0026thinsp;=\u0026thinsp;0.208, β\u0026thinsp;=\u0026thinsp;0.2026, γ\u0026thinsp;=\u0026thinsp;0.2171\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e240.932\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eARIMA(1,0,1)x(1,1,2)\u003c/b\u003e\u003csub\u003e\u003cb\u003e12\u003c/b\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e54.404\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e8.040\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eARIMA(0,0,0)x(1,0,2)\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e55.256\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.052\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eARIMA(0,0,1)x(1,0,2)\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e55.069\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.054\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eARIMA(1,0,0)x(1,0,2)\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e55.090\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.055\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"5. Results","content":"\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the results of the comparison of the conventional models. The Quadratic ES and Holt-Winter ES was used to model the seasonal data. Similarly, some best-fitted SARIMA models using conventional Box-Jenkins\u0026rsquo;s methodology are also presented. After a comparison of the selected model, it is noticed that the SARIMA(1,0,1)\u0026times;(1,1,2)\u003csub\u003e12\u003c/sub\u003e model has the lowest AIC and RMSE. Thus, it is considered the best selected conventional model.\u003c/p\u003e\u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOptimized Values of Hyperparameters of Kernels using Proposed Optimization Method\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"13\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c13\" colnum=\"13\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLags\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOne\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTwo\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eThree\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eOne\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eTwo\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eThree\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eOne\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eTwo\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003eThree\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c11\"\u003e \u003cp\u003eOne\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c12\"\u003e \u003cp\u003eTwo\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c13\"\u003e \u003cp\u003eThree\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c7\" namest=\"c5\"\u003e \u003cp\u003eSEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c10\" namest=\"c8\"\u003e \u003cp\u003eSSEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c13\" namest=\"c11\"\u003e \u003cp\u003eSE\u0026thinsp;+\u0026thinsp;W\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eσ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.475\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.203\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.739\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6.033\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e4.736\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e5.583\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e6.995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e5.403\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e5.534\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e6.824\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eσ\u003csub\u003ef\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7.321\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e13.184\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.542\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.297\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e9.724\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e9.882\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e7.375\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e9.388\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e5.929\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e8.974\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e7.955\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e4.463\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-9.214\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-11.797\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-5.730\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-5.825\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-8.192\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-10.057\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-4.766\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-14.953\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e18.533\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e11.070\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-8.302\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-7.108\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ew\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.006\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-3.262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e9.921\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-0.957\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-8.873\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e47.790\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.669\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e36.192\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-14.580\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-2.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-8.597\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-72.170\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-3.836\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.855\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e1.564\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-6.880\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-6.209\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-11.836\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003e\u0026#120572;\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003ePSE\u0026thinsp;+\u0026thinsp;M2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c7\" namest=\"c5\"\u003e \u003cp\u003eSESP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c10\" namest=\"c8\"\u003e \u003cp\u003e2SEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c13\" namest=\"c11\"\u003e \u003cp\u003eRQ\u0026thinsp;+\u0026thinsp;W\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eσ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.035\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.525\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e6.054\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.647\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e6.203\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6.813\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e5.364\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e5.777\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e6.781\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e5.347\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.741\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e4.476\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eσ\u003csub\u003ef\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8.474\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e14.950\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.898\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.797\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e6.522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e4.464\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e6.393\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e5.260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e4.346\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e6.500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e7.897\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e11.048\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-8.667\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.828\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-8.105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-4.444\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e15.827\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-6.569\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e5.959\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-5.874\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-5.828\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-6.031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.978\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-5.266\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ew\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.828\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-3.062\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-1.690\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-2.221\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e10.177\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.785\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-4.420\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e1.459\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e3.220\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.966\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-11.995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e6.491\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-1.452\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-21.536\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-3.798\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-3.244\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-1.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-5.494\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e1.765\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e2.188\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-3.241\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.465\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-4.877\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.302\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.043\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e-3.781\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e-1.424\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e-6.362\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003e\u0026#120572;\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.458\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.466\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e present the ANN plot with three layers and ten nodes. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the tune values of hyperparameters for purposed kernel functions at different sets of auxiliary lags using the proposed optimization scheme. The evaluation of the proposed models' forecast performance in terms of RMSE's and sMAPE's is presented in Table \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, where the table contains the forecast performance results at selected lags. After comparing all these proposed models with SARIMA(1,0,1)\u0026times;(1,1,2)\u003csub\u003e12\u003c/sub\u003e, RF and ANN models we can conclude that the ALD-GP model is more efficient than the conventional models.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePerformance comparison of different models based on test data\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eModels\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eKernel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e(1,0,1)x(1,1,2)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003eOne lag\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c8\" namest=\"c7\"\u003e \u003cp\u003eTwo lags\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003eThree lags\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003esMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003esMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003esMAPE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003esMAPE\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSARIMA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e51.965\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e92.117\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eANN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e46.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e108.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e47.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e102.91\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e44.495\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e93.805\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e46.483\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e89.141\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e43.196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e89.397\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e41.975\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e89.812\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e46.554\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e89.200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e46.956\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e91.485\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSE\u0026thinsp;+\u0026thinsp;W\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e43.085\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e90.624\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e46.601\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e88.246\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e42.531\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e86.329\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRQ\u0026thinsp;+\u0026thinsp;W\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e43.604\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e90.606\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e44.858\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e88.942\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e46.709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e92.904\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePSE\u0026thinsp;+\u0026thinsp;M\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e46.838\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e99.575\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e46.923\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e89.178\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e44.599\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e85.258\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSSEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e46.578\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e88.964\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e47.367\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e87.220\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e46.185\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e92.863\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSESP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e43.490\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e88.347\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e47.081\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e90.243\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e41.877\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e84.665\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eALD-GP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2SEP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e44.544\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e92.673\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e45.611\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e88.291\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e45.077\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e93.085\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe least efficient ALD-GP model is at least 7% more efficient than the SARIMA, RF and ANN models and the best selected ALD-GP model is about 25% more efficient than the SARIMA, RF and ANN models. Thus, the lowest performance of the proposed model is far better than the best selected conventional model. The compassion of the ALD-GP models suggests that the SESP kernel with three lags is the best-selected model with RMSE is 41.8, and sMAPE is 84%. For this model, we have minimum RMSE and sMAPE. At second, ALD-GP with SE\u0026thinsp;+\u0026thinsp;W kernel at three lags is selected as the best model with minimum average RMSE 42.5 and sMAPE 86%. At third, ALD-GP with PSE\u0026thinsp;+\u0026thinsp;M\u003csub\u003e2\u003c/sub\u003e also at three lags is selected as the best model with average RMSE 44.5 and sMAPE 85%. While the SEP kernel at one lag, SESP kernel at one lag, RQ\u0026thinsp;+\u0026thinsp;W kernel at two lags, and SE at three lags also perform very good. Additionally, for the one lag situation, SESP and SEP are the best kernels for forecasting. For the two lags situation, RQ\u0026thinsp;+\u0026thinsp;W is the best-selected kernel function. Figure\u0026nbsp;4 presents the graphical comparison of the forecast of each ALD-GP model with the actual data set. Also, we have drawn the 95% confidence band of the forecast which is the shaded region along with the forecasted points.\u003c/p\u003e \u003cp\u003eThe covariance metric of Prior and posterior ALD-GP using different kernel function at different lag was presented by using heat plot. The Figure S1-S24 of the supplementary materials shows these heat plots.\u003c/p\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eOverall results show that the ALD-GP is better than the SARIMA, RF and ANN models for forecasting the rainfall data. The comparison of different lags and proposed kernel functions suggested that the performance of ALD-GP using three auxiliary lags is far better compared to others. Additionally, SESP is the best-selected kernel function for rainfall series. So, the efficient forecasting can be made by using ALD-GP model with suggested kernel functions. The proposed optimization technique is also valid for modelling using the Gaussian process regression model. Similarly, the value of the suggested nugget can handle ill-conditioning while using Gaussian process regression.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe are very thankful to the Regional Meteorology Department officials, Lahore, Pakistan, for sharing their recorded data.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003eNo funds, grants, or other support was received. The authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo funds, grants, or other support was received\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics Approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis is an observational study. The Board of Advanced Studies and Research of Bahauddin Zakariya University, Multan, Pakistan has confirmed that no ethical approval is required.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot Applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot Applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data that support the findings of this study are available from\u0026nbsp;Regional Meteorology Department Lahore,\u0026nbsp;but restrictions apply to the availability of these data, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of\u0026nbsp;Regional Meteorology Department Lahore.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCode availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe code in R-language used in the current study are available from the corresponding author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors Contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eHaris Khurram and Muhammad Mutahir Iqbal contributed equally to the conceptualization, framework and analysis of this paper. Both authors read and approved the final manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAdnan, M., Khan, F., Rehman, N., Ali, S., Hassan, S. S., Dogar, M. M., ... \u0026amp; Hasson, S. (2020). 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Wavelet support vector machine. \u003cem\u003eIEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics)\u003c/em\u003e, 34(1), 34-39.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Gaussian Process, Machine learning models, Auxiliary Lags, Kernel functions, Forecasting, Rainfall","lastPublishedDoi":"10.21203/rs.3.rs-2486388/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2486388/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePakistan is currently facing the biggest flood of history due to monsoon rains. The rainfall forecasting is very important for policy making. In this paper, we have presented an auxiliary-lag dependent Gaussian process, a Bayesian non-parametric machine learning model, for modeling the rainfall data using auxiliary lags. We have also introduced some new multifeatured kernel functions that are versatile in dealing with seasonal data. A simplex cluster-based multi-start technique using the Nelder-Mead optimizer has also been proposed for optimizing the hyperparameters of the kernel functions, which can be used for any available or proposed kernel function(s). For comparison of the proposed model, we have used the autoregressive random forest model, autoregressive artificial neural network model, seasonal autoregressive moving average models, and exponential smoothing models. Results confirmed the superiority of the proposed model over conventional models. The proposed methodology will be helpful for other researchers and local experts in making more reliable forecasting which will be helpful in policymaking relevant to agriculture systems, water management systems, climate change, and natural disasters such as droughts and floods.\u003c/p\u003e","manuscriptTitle":"Auxiliary-lag Dependent Gaussian Process Model for Forecasting Rainfall Data Using Proposed Kernels and Multi-start Optimization Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-02-02 16:14:19","doi":"10.21203/rs.3.rs-2486388/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a151073a-76d6-46f5-9306-c7b498e01f3f","owner":[],"postedDate":"February 2nd, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-07-21T07:27:51+00:00","versionOfRecord":[],"versionCreatedAt":"2023-02-02 16:14:19","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-2486388","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2486388","identity":"rs-2486388","version":["v1"]},"buildId":"FbvkV6FR0MCFSLy54lSbu","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}