Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application

Employing Fixed-Point Theory for Fuzzy Regression... | F1000Research "use strict";function _typeof(t){return(_typeof="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}!function(){var t=function(){var t,e,o=[],n=window,r=n;for(;r;){try{if(r.frames.__tcfapiLocator){t=r;break}}catch(t){}if(r===n.top)break;r=r.parent}t||(!function t(){var e=n.document,o=!!n.frames.__tcfapiLocator;if(!o)if(e.body){var r=e.createElement("iframe");r.style.cssText="display:none",r.name="__tcfapiLocator",e.body.appendChild(r)}else setTimeout(t,5);return!o}(),n.__tcfapi=function(){for(var t=arguments.length,n=new Array(t),r=0;r 3&&2===parseInt(n[1],10)&&"boolean"==typeof n[3]&&(e=n[3],"function"==typeof n[2]&&n[2]("set",!0)):"ping"===n[0]?"function"==typeof n[2]&&n[2]({gdprApplies:e,cmpLoaded:!1,cmpStatus:"stub"}):o.push(n)},n.addEventListener("message",(function(t){var e="string"==typeof t.data,o={};if(e)try{o=JSON.parse(t.data)}catch(t){}else o=t.data;var n="object"===_typeof(o)&&null!==o?o.__tcfapiCall:null;n&&window.__tcfapi(n.command,n.version,(function(o,r){var a={__tcfapiReturn:{returnValue:o,success:r,callId:n.callId}};t&&t.source&&t.source.postMessage&&t.source.postMessage(e?JSON.stringify(a):a,"*")}),n.parameter)}),!1))};"undefined"!=typeof module?module.exports=t:t()}(); dataLayer = dataLayer || []; // Standard GTM initialization - Google Consent Mode handles consent automatically (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+ '>m_auth=hzk0Vc3qFsQYhCrIoHz68A>m_preview=env-1>m_cookies_win=x';f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-MWFK8L5J'); ;window.NREUM||(NREUM={});NREUM.init={distributed_tracing:{enabled:true},privacy:{cookies_enabled:true},ajax:{deny_list:["bam.nr-data.net"]}}; ;NREUM.loader_config={accountID:"438030",trustKey:"438030",agentID:"772317073",licenseKey:"97f8f67f26",applicationID:"772317073"} ;NREUM.info={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net",licenseKey:"97f8f67f26",applicationID:"772317073",sa:1} ;/*! For license information please see nr-loader-spa-1.236.0.min.js.LICENSE.txt */ (()=>{"use strict";var e,t,r={5763:(e,t,r)=>{r.d(t,{P_:()=>l,Mt:()=>g,C5:()=>s,DL:()=>v,OP:()=>T,lF:()=>D,Yu:()=>y,Dg:()=>h,CX:()=>c,GE:()=>b,sU:()=>_});var n=r(8632),i=r(9567);const o={beacon:n.ce.beacon,errorBeacon:n.ce.errorBeacon,licenseKey:void 0,applicationID:void 0,sa:void 0,queueTime:void 0,applicationTime:void 0,ttGuid:void 0,user:void 0,account:void 0,product:void 0,extra:void 0,jsAttributes:{},userAttributes:void 0,atts:void 0,transactionName:void 0,tNamePlain:void 0},a={};function s(e){if(!e)throw new Error("All info objects require an agent identifier!");if(!a[e])throw new Error("Info for ".concat(e," was never set"));return a[e]}function c(e,t){if(!e)throw new Error("All info objects require an agent identifier!");a[e]=(0,i.D)(t,o),(0,n.Qy)(e,a[e],"info")}var u=r(7056);const d=()=>{const e={blockSelector:"[data-nr-block]",maskInputOptions:{password:!0}};return{allow_bfcache:!0,privacy:{cookies_enabled:!0},ajax:{deny_list:void 0,enabled:!0,harvestTimeSeconds:10},distributed_tracing:{enabled:void 0,exclude_newrelic_header:void 0,cors_use_newrelic_header:void 0,cors_use_tracecontext_headers:void 0,allowed_origins:void 0},session:{domain:void 0,expiresMs:u.oD,inactiveMs:u.Hb},ssl:void 0,obfuscate:void 0,jserrors:{enabled:!0,harvestTimeSeconds:10},metrics:{enabled:!0},page_action:{enabled:!0,harvestTimeSeconds:30},page_view_event:{enabled:!0},page_view_timing:{enabled:!0,harvestTimeSeconds:30,long_task:!1},session_trace:{enabled:!0,harvestTimeSeconds:10},harvest:{tooManyRequestsDelay:60},session_replay:{enabled:!1,harvestTimeSeconds:60,sampleRate:.1,errorSampleRate:.1,maskTextSelector:"*",maskAllInputs:!0,get blockClass(){return"nr-block"},get ignoreClass(){return"nr-ignore"},get maskTextClass(){return"nr-mask"},get blockSelector(){return e.blockSelector},set blockSelector(t){e.blockSelector+=",".concat(t)},get maskInputOptions(){return e.maskInputOptions},set maskInputOptions(t){e.maskInputOptions={...t,password:!0}}},spa:{enabled:!0,harvestTimeSeconds:10}}},f={};function l(e){if(!e)throw new Error("All configuration objects require an agent identifier!");if(!f[e])throw new Error("Configuration for ".concat(e," was never set"));return f[e]}function h(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");f[e]=(0,i.D)(t,d()),(0,n.Qy)(e,f[e],"config")}function g(e,t){if(!e)throw new Error("All configuration objects require an agent identifier!");var r=l(e);if(r){for(var n=t.split("."),i=0;i {r.d(t,{D:()=>i});var n=r(50);function i(e,t){try{if(!e||"object"!=typeof e)return(0,n.Z)("Setting a Configurable requires an object as input");if(!t||"object"!=typeof t)return(0,n.Z)("Setting a Configurable requires a model to set its initial properties");const r=Object.create(Object.getPrototypeOf(t),Object.getOwnPropertyDescriptors(t)),o=0===Object.keys(r).length?e:r;for(let a in o)if(void 0!==e[a])try{"object"==typeof e[a]&&"object"==typeof t[a]?r[a]=i(e[a],t[a]):r[a]=e[a]}catch(e){(0,n.Z)("An error occurred while setting a property of a Configurable",e)}return r}catch(e){(0,n.Z)("An error occured while setting a Configurable",e)}}},6818:(e,t,r)=>{r.d(t,{Re:()=>i,gF:()=>o,q4:()=>n});const n="1.236.0",i="PROD",o="CDN"},385:(e,t,r)=>{r.d(t,{FN:()=>a,IF:()=>u,Nk:()=>f,Tt:()=>s,_A:()=>o,il:()=>n,pL:()=>c,v6:()=>i,w1:()=>d});const n="undefined"!=typeof window&&!!window.document,i="undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self.navigator instanceof WorkerNavigator||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis.navigator instanceof WorkerNavigator),o=n?window:"undefined"!=typeof WorkerGlobalScope&&("undefined"!=typeof self&&self instanceof WorkerGlobalScope&&self||"undefined"!=typeof globalThis&&globalThis instanceof WorkerGlobalScope&&globalThis),a=""+o?.location,s=/iPad|iPhone|iPod/.test(navigator.userAgent),c=s&&"undefined"==typeof SharedWorker,u=(()=>{const e=navigator.userAgent.match(/Firefox[/\s](\d+\.\d+)/);return Array.isArray(e)&&e.length>=2?+e[1]:0})(),d=Boolean(n&&window.document.documentMode),f=!!navigator.sendBeacon},1117:(e,t,r)=>{r.d(t,{w:()=>o});var n=r(50);const i={agentIdentifier:"",ee:void 0};class o{constructor(e){try{if("object"!=typeof e)return(0,n.Z)("shared context requires an object as input");this.sharedContext={},Object.assign(this.sharedContext,i),Object.entries(e).forEach((e=>{let[t,r]=e;Object.keys(i).includes(t)&&(this.sharedContext[t]=r)}))}catch(e){(0,n.Z)("An error occured while setting SharedContext",e)}}}},8e3:(e,t,r)=>{r.d(t,{L:()=>d,R:()=>c});var n=r(2177),i=r(1284),o=r(4322),a=r(3325);const s={};function c(e,t){const r={staged:!1,priority:a.p[t]||0};u(e),s[e].get(t)||s[e].set(t,r)}function u(e){e&&(s[e]||(s[e]=new Map))}function d(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:"",t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:"feature";if(u(e),!e||!s[e].get(t))return a(t);s[e].get(t).staged=!0;const r=[...s[e]];function a(t){const r=e?n.ee.get(e):n.ee,a=o.X.handlers;if(r.backlog&&a){var s=r.backlog[t],c=a[t];if(c){for(var u=0;s&&u {let[t,r]=e;return r.staged}))&&(r.sort(((e,t)=>e[1].priority-t[1].priority)),r.forEach((e=>{let[t]=e;a(t)})))}function f(e,t){var r=e[1];(0,i.D)(t[r],(function(t,r){var n=e[0];if(r[0]===n){var i=r[1],o=e[3],a=e[2];i.apply(o,a)}}))}},2177:(e,t,r)=>{r.d(t,{c:()=>f,ee:()=>u});var n=r(8632),i=r(2210),o=r(1284),a=r(5763),s="nr@context";let c=(0,n.fP)();var u;function d(){}function f(e){return(0,i.X)(e,s,l)}function l(){return new d}function h(){u.aborted=!0,u.backlog={}}c.ee?u=c.ee:(u=function e(t,r){var n={},c={},f={},g=!1;try{g=16===r.length&&(0,a.OP)(r).isolatedBacklog}catch(e){}var p={on:b,addEventListener:b,removeEventListener:y,emit:v,get:x,listeners:w,context:m,buffer:A,abort:h,aborted:!1,isBuffering:E,debugId:r,backlog:g?{}:t&&"object"==typeof t.backlog?t.backlog:{}};return p;function m(e){return e&&e instanceof d?e:e?(0,i.X)(e,s,l):l()}function v(e,r,n,i,o){if(!1!==o&&(o=!0),!u.aborted||i){t&&o&&t.emit(e,r,n);for(var a=m(n),s=w(e),d=s.length,f=0;fn,p:()=>i});var n=r(2177).ee.get("handle");function i(e,t,r,i,o){o?(o.buffer([e],i),o.emit(e,t,r)):(n.buffer([e],i),n.emit(e,t,r))}},4322:(e,t,r)=>{r.d(t,{X:()=>o});var n=r(5546);o.on=a;var i=o.handlers={};function o(e,t,r,o){a(o||n.E,i,e,t,r)}function a(e,t,r,i,o){o||(o="feature"),e||(e=n.E);var a=t[o]=t[o]||{};(a[r]=a[r]||[]).push([e,i])}},3239:(e,t,r)=>{r.d(t,{bP:()=>s,iz:()=>c,m$:()=>a});var n=r(385);let i=!1,o=!1;try{const e={get passive(){return i=!0,!1},get signal(){return o=!0,!1}};n._A.addEventListener("test",null,e),n._A.removeEventListener("test",null,e)}catch(e){}function a(e,t){return i||o?{capture:!!e,passive:i,signal:t}:!!e}function s(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;window.addEventListener(e,t,a(r,n))}function c(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2],n=arguments.length>3?arguments[3]:void 0;document.addEventListener(e,t,a(r,n))}},4402:(e,t,r)=>{r.d(t,{Ht:()=>u,M:()=>c,Rl:()=>a,ky:()=>s});var n=r(385);const i="xxxxxxxx-xxxx-4xxx-yxxx-xxxxxxxxxxxx";function o(e,t){return e?15&e[t]:16*Math.random()|0}function a(){const e=n._A?.crypto||n._A?.msCrypto;let t,r=0;return e&&e.getRandomValues&&(t=e.getRandomValues(new Uint8Array(31))),i.split("").map((e=>"x"===e?o(t,++r).toString(16):"y"===e?(3&o()|8).toString(16):e)).join("")}function s(e){const t=n._A?.crypto||n._A?.msCrypto;let r,i=0;t&&t.getRandomValues&&(r=t.getRandomValues(new Uint8Array(31)));const a=[];for(var s=0;s {r.d(t,{Bq:()=>n,Hb:()=>o,oD:()=>i});const n="NRBA",i=144e5,o=18e5},7894:(e,t,r)=>{function n(){return Math.round(performance.now())}r.d(t,{z:()=>n})},7243:(e,t,r)=>{r.d(t,{e:()=>o});var n=r(385),i={};function o(e){if(e in i)return i[e];if(0===(e||"").indexOf("data:"))return{protocol:"data"};let t;var r=n._A?.location,o={};if(n.il)t=document.createElement("a"),t.href=e;else try{t=new URL(e,r.href)}catch(e){return o}o.port=t.port;var a=t.href.split("://");!o.port&&a[1]&&(o.port=a[1].split("/")[0].split("@").pop().split(":")[1]),o.port&&"0"!==o.port||(o.port="https"===a[0]?"443":"80"),o.hostname=t.hostname||r.hostname,o.pathname=t.pathname,o.protocol=a[0],"/"!==o.pathname.charAt(0)&&(o.pathname="/"+o.pathname);var s=!t.protocol||":"===t.protocol||t.protocol===r.protocol,c=t.hostname===r.hostname&&t.port===r.port;return o.sameOrigin=s&&(!t.hostname||c),"/"===o.pathname&&(i[e]=o),o}},50:(e,t,r)=>{function n(e,t){"function"==typeof console.warn&&(console.warn("New Relic: ".concat(e)),t&&console.warn(t))}r.d(t,{Z:()=>n})},2587:(e,t,r)=>{r.d(t,{N:()=>c,T:()=>u});var n=r(2177),i=r(5546),o=r(8e3),a=r(3325);const s={stn:[a.D.sessionTrace],err:[a.D.jserrors,a.D.metrics],ins:[a.D.pageAction],spa:[a.D.spa],sr:[a.D.sessionReplay,a.D.sessionTrace]};function c(e,t){const r=n.ee.get(t);e&&"object"==typeof e&&(Object.entries(e).forEach((e=>{let[t,n]=e;void 0===u[t]&&(s[t]?s[t].forEach((e=>{n?(0,i.p)("feat-"+t,[],void 0,e,r):(0,i.p)("block-"+t,[],void 0,e,r),(0,i.p)("rumresp-"+t,[Boolean(n)],void 0,e,r)})):n&&(0,i.p)("feat-"+t,[],void 0,void 0,r),u[t]=Boolean(n))})),Object.keys(s).forEach((e=>{void 0===u[e]&&(s[e]?.forEach((t=>(0,i.p)("rumresp-"+e,[!1],void 0,t,r))),u[e]=!1)})),(0,o.L)(t,a.D.pageViewEvent))}const u={}},2210:(e,t,r)=>{r.d(t,{X:()=>i});var n=Object.prototype.hasOwnProperty;function i(e,t,r){if(n.call(e,t))return e[t];var i=r();if(Object.defineProperty&&Object.keys)try{return Object.defineProperty(e,t,{value:i,writable:!0,enumerable:!1}),i}catch(e){}return e[t]=i,i}},1284:(e,t,r)=>{r.d(t,{D:()=>n});const n=(e,t)=>Object.entries(e||{}).map((e=>{let[r,n]=e;return t(r,n)}))},4351:(e,t,r)=>{r.d(t,{P:()=>o});var n=r(2177);const i=()=>{const e=new WeakSet;return(t,r)=>{if("object"==typeof r&&null!==r){if(e.has(r))return;e.add(r)}return r}};function o(e){try{return JSON.stringify(e,i())}catch(e){try{n.ee.emit("internal-error",[e])}catch(e){}}}},3960:(e,t,r)=>{r.d(t,{K:()=>a,b:()=>o});var n=r(3239);function i(){return"undefined"==typeof document||"complete"===document.readyState}function o(e,t){if(i())return e();(0,n.bP)("load",e,t)}function a(e){if(i())return e();(0,n.iz)("DOMContentLoaded",e)}},8632:(e,t,r)=>{r.d(t,{EZ:()=>u,Qy:()=>c,ce:()=>o,fP:()=>a,gG:()=>d,mF:()=>s});var n=r(7894),i=r(385);const o={beacon:"bam.nr-data.net",errorBeacon:"bam.nr-data.net"};function a(){return i._A.NREUM||(i._A.NREUM={}),void 0===i._A.newrelic&&(i._A.newrelic=i._A.NREUM),i._A.NREUM}function s(){let e=a();return e.o||(e.o={ST:i._A.setTimeout,SI:i._A.setImmediate,CT:i._A.clearTimeout,XHR:i._A.XMLHttpRequest,REQ:i._A.Request,EV:i._A.Event,PR:i._A.Promise,MO:i._A.MutationObserver,FETCH:i._A.fetch}),e}function c(e,t,r){let i=a();const o=i.initializedAgents||{},s=o[e]||{};return Object.keys(s).length||(s.initializedAt={ms:(0,n.z)(),date:new Date}),i.initializedAgents={...o,[e]:{...s,[r]:t}},i}function u(e,t){a()[e]=t}function d(){return function(){let e=a();const t=e.info||{};e.info={beacon:o.beacon,errorBeacon:o.errorBeacon,...t}}(),function(){let e=a();const t=e.init||{};e.init={...t}}(),s(),function(){let e=a();const t=e.loader_config||{};e.loader_config={...t}}(),a()}},7956:(e,t,r)=>{r.d(t,{N:()=>i});var n=r(3239);function i(e){let t=arguments.length>1&&void 0!==arguments[1]&&arguments[1],r=arguments.length>2?arguments[2]:void 0,i=arguments.length>3?arguments[3]:void 0;return void(0,n.iz)("visibilitychange",(function(){if(t)return void("hidden"==document.visibilityState&&e());e(document.visibilityState)}),r,i)}},1214:(e,t,r)=>{r.d(t,{em:()=>v,u5:()=>N,QU:()=>S,_L:()=>I,Gm:()=>L,Lg:()=>M,gy:()=>U,BV:()=>Q,Kf:()=>ee});var n=r(2177);const i="nr@original";var o=Object.prototype.hasOwnProperty,a=!1;function s(e,t){return e||(e=n.ee),r.inPlace=function(e,t,n,i,o){n||(n="");var a,s,c,u="-"===n.charAt(0);for(c=0;c 2?n-2:0),o=2;o {r(A[T],e,w),r(E[T],e,w)})),r(l._A,"fetch",y),t.on(y+"end",(function(e,r){var n=this;if(r){var i=r.headers.get("content-length");null!==i&&(n.rxSize=i),t.emit(y+"done",[null,r],n)}else t.emit(y+"done",[e],n)})),t}const O={},j=["pushState","replaceState"];function S(e){const t=function(e){return(e||n.ee).get("history")}(e);return!l.il||O[t.debugId]++||(O[t.debugId]=1,s(t).inPlace(window.history,j,"-")),t}var P=r(3239);const C={},R=["appendChild","insertBefore","replaceChild"];function I(e){const t=function(e){return(e||n.ee).get("jsonp")}(e);if(!l.il||C[t.debugId])return t;C[t.debugId]=!0;var r=s(t),i=/[?&](?:callback|cb)=([^&#]+)/,o=/(.*)\.([^.]+)/,a=/^(\w+)(\.|$)(.*)$/;function c(e,t){var r=e.match(a),n=r[1],i=r[3];return i?c(i,t[n]):t[n]}return r.inPlace(Node.prototype,R,"dom-"),t.on("dom-start",(function(e){!function(e){if(!e||"string"!=typeof e.nodeName||"script"!==e.nodeName.toLowerCase())return;if("function"!=typeof e.addEventListener)return;var n=(a=e.src,s=a.match(i),s?s[1]:null);var a,s;if(!n)return;var u=function(e){var t=e.match(o);if(t&&t.length>=3)return{key:t[2],parent:c(t[1],window)};return{key:e,parent:window}}(n);if("function"!=typeof u.parent[u.key])return;var d={};function f(){t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}function l(){t.emit("jsonp-error",[],d),t.emit("jsonp-end",[],d),e.removeEventListener("load",f,(0,P.m$)(!1)),e.removeEventListener("error",l,(0,P.m$)(!1))}r.inPlace(u.parent,[u.key],"cb-",d),e.addEventListener("load",f,(0,P.m$)(!1)),e.addEventListener("error",l,(0,P.m$)(!1)),t.emit("new-jsonp",[e.src],d)}(e[0])})),t}var k=r(5763);const H={};function L(e){const t=function(e){return(e||n.ee).get("mutation")}(e);if(!l.il||H[t.debugId])return t;H[t.debugId]=!0;var r=s(t),i=k.Yu.MO;return i&&(window.MutationObserver=function(e){return this instanceof i?new i(r(e,"fn-")):i.apply(this,arguments)},MutationObserver.prototype=i.prototype),t}const z={};function M(e){const t=function(e){return(e||n.ee).get("promise")}(e);if(z[t.debugId])return t;z[t.debugId]=!0;var r=n.c,o=s(t),a=k.Yu.PR;return a&&function(){function e(r){var n=t.context(),i=o(r,"executor-",n,null,!1);const s=Reflect.construct(a,[i],e);return t.context(s).getCtx=function(){return n},s}l._A.Promise=e,Object.defineProperty(e,"name",{value:"Promise"}),e.toString=function(){return a.toString()},Object.setPrototypeOf(e,a),["all","race"].forEach((function(r){const n=a[r];e[r]=function(e){let i=!1;[...e||[]].forEach((e=>{this.resolve(e).then(a("all"===r),a(!1))}));const o=n.apply(this,arguments);return o;function a(e){return function(){t.emit("propagate",[null,!i],o,!1,!1),i=i||!e}}}})),["resolve","reject"].forEach((function(r){const n=a[r];e[r]=function(e){const r=n.apply(this,arguments);return e!==r&&t.emit("propagate",[e,!0],r,!1,!1),r}})),e.prototype=a.prototype;const n=a.prototype.then;a.prototype.then=function(){var e=this,i=r(e);i.promise=e;for(var a=arguments.length,s=new Array(a),c=0;c e())),t};function m(e,t){i.inPlace(t,["onreadystatechange"],"fn-",E)}function b(){var e=this,t=r.context(e);e.readyState>3&&!t.resolved&&(t.resolved=!0,r.emit("xhr-resolved",[],e)),i.inPlace(e,f,"fn-",E)}if(function(e,t){for(var r in e)t[r]=e[r]}(o,p),p.prototype=o.prototype,i.inPlace(p.prototype,J,"-xhr-",E),r.on("send-xhr-start",(function(e,t){m(e,t),function(e){h.push(e),a&&(y?y.then(A):u?u(A):(w=-w,x.data=w))}(t)})),r.on("open-xhr-start",m),a){var y=c&&c.resolve();if(!u&&!c){var w=1,x=document.createTextNode(w);new a(A).observe(x,{characterData:!0})}}else t.on("fn-end",(function(e){e[0]&&e[0].type===d||A()}));function A(){for(var e=0;e {r.d(t,{t:()=>n});const n=r(3325).D.ajax},6660:(e,t,r)=>{r.d(t,{A:()=>i,t:()=>n});const n=r(3325).D.jserrors,i="nr@seenError"},3081:(e,t,r)=>{r.d(t,{gF:()=>o,mY:()=>i,t9:()=>n,vz:()=>s,xS:()=>a});const n=r(3325).D.metrics,i="sm",o="cm",a="storeSupportabilityMetrics",s="storeEventMetrics"},4649:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageAction},7633:(e,t,r)=>{r.d(t,{Dz:()=>i,OJ:()=>a,qw:()=>o,t9:()=>n});const n=r(3325).D.pageViewEvent,i="firstbyte",o="domcontent",a="windowload"},9251:(e,t,r)=>{r.d(t,{t:()=>n});const n=r(3325).D.pageViewTiming},3614:(e,t,r)=>{r.d(t,{BST_RESOURCE:()=>i,END:()=>s,FEATURE_NAME:()=>n,FN_END:()=>u,FN_START:()=>c,PUSH_STATE:()=>d,RESOURCE:()=>o,START:()=>a});const n=r(3325).D.sessionTrace,i="bstResource",o="resource",a="-start",s="-end",c="fn"+a,u="fn"+s,d="pushState"},7836:(e,t,r)=>{r.d(t,{BODY:()=>A,CB_END:()=>E,CB_START:()=>u,END:()=>x,FEATURE_NAME:()=>i,FETCH:()=>_,FETCH_BODY:()=>v,FETCH_DONE:()=>m,FETCH_START:()=>p,FN_END:()=>c,FN_START:()=>s,INTERACTION:()=>l,INTERACTION_API:()=>d,INTERACTION_EVENTS:()=>o,JSONP_END:()=>b,JSONP_NODE:()=>g,JS_TIME:()=>T,MAX_TIMER_BUDGET:()=>a,REMAINING:()=>f,SPA_NODE:()=>h,START:()=>w,originalSetTimeout:()=>y});var n=r(5763);const i=r(3325).D.spa,o=["click","submit","keypress","keydown","keyup","change"],a=999,s="fn-start",c="fn-end",u="cb-start",d="api-ixn-",f="remaining",l="interaction",h="spaNode",g="jsonpNode",p="fetch-start",m="fetch-done",v="fetch-body-",b="jsonp-end",y=n.Yu.ST,w="-start",x="-end",A="-body",E="cb"+x,T="jsTime",_="fetch"},5938:(e,t,r)=>{r.d(t,{W:()=>o});var n=r(5763),i=r(2177);class o{constructor(e,t,r){this.agentIdentifier=e,this.aggregator=t,this.ee=i.ee.get(e,(0,n.OP)(this.agentIdentifier).isolatedBacklog),this.featureName=r,this.blocked=!1}}},9144:(e,t,r)=>{r.d(t,{j:()=>m});var n=r(3325),i=r(5763),o=r(5546),a=r(2177),s=r(7894),c=r(8e3),u=r(3960),d=r(385),f=r(50),l=r(3081),h=r(8632);function g(){const e=(0,h.gG)();["setErrorHandler","finished","addToTrace","inlineHit","addRelease","addPageAction","setCurrentRouteName","setPageViewName","setCustomAttribute","interaction","noticeError","setUserId"].forEach((t=>{e[t]=function(){for(var r=arguments.length,n=new Array(r),i=0;i 1?r-1:0),i=1;i {e.exposed&&e.api[t]&&o.push(e.api[t](...n))})),o.length>1?o:o[0]}(t,...n)}}))}var p=r(2587);function m(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:{},m=arguments.length>2?arguments[2]:void 0,v=arguments.length>3?arguments[3]:void 0,{init:b,info:y,loader_config:w,runtime:x={loaderType:m},exposed:A=!0}=t;const E=(0,h.gG)();y||(b=E.init,y=E.info,w=E.loader_config),(0,i.Dg)(e,b||{}),(0,i.GE)(e,w||{}),(0,i.sU)(e,x),y.jsAttributes??={},d.v6&&(y.jsAttributes.isWorker=!0),(0,i.CX)(e,y),g();const T=function(e,t){t||(0,c.R)(e,"api");const h={};var g=a.ee.get(e),p=g.get("tracer"),m="api-",v=m+"ixn-";function b(t,r,n,o){const a=(0,i.C5)(e);return null===r?delete a.jsAttributes[t]:(0,i.CX)(e,{...a,jsAttributes:{...a.jsAttributes,[t]:r}}),x(m,n,!0,o||null===r?"session":void 0)(t,r)}function y(){}["setErrorHandler","finished","addToTrace","inlineHit","addRelease"].forEach((e=>h[e]=x(m,e,!0,"api"))),h.addPageAction=x(m,"addPageAction",!0,n.D.pageAction),h.setCurrentRouteName=x(m,"routeName",!0,n.D.spa),h.setPageViewName=function(t,r){if("string"==typeof t)return"/"!==t.charAt(0)&&(t="/"+t),(0,i.OP)(e).customTransaction=(r||"http://custom.transaction")+t,x(m,"setPageViewName",!0)()},h.setCustomAttribute=function(e,t){let r=arguments.length>2&&void 0!==arguments[2]&&arguments[2];if("string"==typeof e){if(["string","number"].includes(typeof t)||null===t)return b(e,t,"setCustomAttribute",r);(0,f.Z)("Failed to execute setCustomAttribute.\nNon-null value must be a string or number type, but a type of was provided."))}else(0,f.Z)("Failed to execute setCustomAttribute.\nName must be a string type, but a type of was provided."))},h.setUserId=function(e){if("string"==typeof e||null===e)return b("enduser.id",e,"setUserId",!0);(0,f.Z)("Failed to execute setUserId.\nNon-null value must be a string type, but a type of was provided."))},h.interaction=function(){return(new y).get()};var w=y.prototype={createTracer:function(e,t){var r={},i=this,a="function"==typeof t;return(0,o.p)(v+"tracer",[(0,s.z)(),e,r],i,n.D.spa,g),function(){if(p.emit((a?"":"no-")+"fn-start",[(0,s.z)(),i,a],r),a)try{return t.apply(this,arguments)}catch(e){throw p.emit("fn-err",[arguments,this,"string"==typeof e?new Error(e):e],r),e}finally{p.emit("fn-end",[(0,s.z)()],r)}}}};function x(e,t,r,i){return function(){return(0,o.p)(l.xS,["API/"+t+"/called"],void 0,n.D.metrics,g),i&&(0,o.p)(e+t,[(0,s.z)(),...arguments],r?null:this,i,g),r?void 0:this}}function A(){r.e(439).then(r.bind(r,7438)).then((t=>{let{setAPI:r}=t;r(e),(0,c.L)(e,"api")})).catch((()=>(0,f.Z)("Downloading runtime APIs failed...")))}return["actionText","setName","setAttribute","save","ignore","onEnd","getContext","end","get"].forEach((e=>{w[e]=x(v,e,void 0,n.D.spa)})),h.noticeError=function(e,t){"string"==typeof e&&(e=new Error(e)),(0,o.p)(l.xS,["API/noticeError/called"],void 0,n.D.metrics,g),(0,o.p)("err",[e,(0,s.z)(),!1,t],void 0,n.D.jserrors,g)},d.il?(0,u.b)((()=>A()),!0):A(),h}(e,v);return(0,h.Qy)(e,T,"api"),(0,h.Qy)(e,A,"exposed"),(0,h.EZ)("activatedFeatures",p.T),T}},3325:(e,t,r)=>{r.d(t,{D:()=>n,p:()=>i});const n={ajax:"ajax",jserrors:"jserrors",metrics:"metrics",pageAction:"page_action",pageViewEvent:"page_view_event",pageViewTiming:"page_view_timing",sessionReplay:"session_replay",sessionTrace:"session_trace",spa:"spa"},i={[n.pageViewEvent]:1,[n.pageViewTiming]:2,[n.metrics]:3,[n.jserrors]:4,[n.ajax]:5,[n.sessionTrace]:6,[n.pageAction]:7,[n.spa]:8,[n.sessionReplay]:9}}},n={};function i(e){var t=n[e];if(void 0!==t)return t.exports;var o=n[e]={exports:{}};return r[e](o,o.exports,i),o.exports}i.m=r,i.d=(e,t)=>{for(var r in t)i.o(t,r)&&!i.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},i.f={},i.e=e=>Promise.all(Object.keys(i.f).reduce(((t,r)=>(i.f[r](e,t),t)),[])),i.u=e=>(({78:"page_action-aggregate",147:"metrics-aggregate",242:"session-manager",317:"jserrors-aggregate",348:"page_view_timing-aggregate",412:"lazy-feature-loader",439:"async-api",538:"recorder",590:"session_replay-aggregate",675:"compressor",733:"session_trace-aggregate",786:"page_view_event-aggregate",873:"spa-aggregate",898:"ajax-aggregate"}[e]||e)+"."+{78:"ac76d497",147:"3dc53903",148:"1a20d5fe",242:"2a64278a",317:"49e41428",348:"bd6de33a",412:"2f55ce66",439:"30bd804e",538:"1b18459f",590:"cf0efb30",675:"ae9f91a8",733:"83105561",786:"06482edd",860:"03a8b7a5",873:"e6b09d52",898:"998ef92b"}[e]+"-1.236.0.min.js"),i.o=(e,t)=>Object.prototype.hasOwnProperty.call(e,t),e={},t="NRBA:",i.l=(r,n,o,a)=>{if(e[r])e[r].push(n);else{var s,c;if(void 0!==o)for(var u=document.getElementsByTagName("script"),d=0;d {s.onerror=s.onload=null,clearTimeout(h);var i=e[r];if(delete e[r],s.parentNode&&s.parentNode.removeChild(s),i&&i.forEach((e=>e(n))),t)return t(n)},h=setTimeout(l.bind(null,void 0,{type:"timeout",target:s}),12e4);s.onerror=l.bind(null,s.onerror),s.onload=l.bind(null,s.onload),c&&document.head.appendChild(s)}},i.r=e=>{"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},i.j=364,i.p="https://js-agent.newrelic.com/",(()=>{var e={364:0,953:0};i.f.j=(t,r)=>{var n=i.o(e,t)?e[t]:void 0;if(0!==n)if(n)r.push(n[2]);else{var o=new Promise(((r,i)=>n=e[t]=[r,i]));r.push(n[2]=o);var a=i.p+i.u(t),s=new Error;i.l(a,(r=>{if(i.o(e,t)&&(0!==(n=e[t])&&(e[t]=void 0),n)){var o=r&&("load"===r.type?"missing":r.type),a=r&&r.target&&r.target.src;s.message="Loading chunk "+t+" failed.\n("+o+": "+a+")",s.name="ChunkLoadError",s.type=o,s.request=a,n[1](s)}}),"chunk-"+t,t)}};var t=(t,r)=>{var n,o,[a,s,c]=r,u=0;if(a.some((t=>0!==e[t]))){for(n in s)i.o(s,n)&&(i.m[n]=s[n]);if(c)c(i)}for(t&&t(r);u {i.r(o);var e=i(3325),t=i(5763);const r=Object.values(e.D);function n(e){const n={};return r.forEach((r=>{n[r]=function(e,r){return!1!==(0,t.Mt)(r,"".concat(e,".enabled"))}(r,e)})),n}var a=i(9144);var s=i(5546),c=i(385),u=i(8e3),d=i(5938),f=i(3960),l=i(50);class h extends d.W{constructor(e,t,r){let n=!(arguments.length>3&&void 0!==arguments[3])||arguments[3];super(e,t,r),this.auto=n,this.abortHandler,this.featAggregate,this.onAggregateImported,n&&(0,u.R)(e,r)}importAggregator(){let e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:{};if(this.featAggregate||!this.auto)return;const r=c.il&&!0===(0,t.Mt)(this.agentIdentifier,"privacy.cookies_enabled");let n;this.onAggregateImported=new Promise((e=>{n=e}));const o=async()=>{let t;try{if(r){const{setupAgentSession:e}=await Promise.all([i.e(860),i.e(242)]).then(i.bind(i,3228));t=e(this.agentIdentifier)}}catch(e){(0,l.Z)("A problem occurred when starting up session manager. This page will not start or extend any session.",e)}try{if(!this.shouldImportAgg(this.featureName,t))return void(0,u.L)(this.agentIdentifier,this.featureName);const{lazyFeatureLoader:r}=await i.e(412).then(i.bind(i,8582)),{Aggregate:o}=await r(this.featureName,"aggregate");this.featAggregate=new o(this.agentIdentifier,this.aggregator,e),n(!0)}catch(e){(0,l.Z)("Downloading and initializing ".concat(this.featureName," failed..."),e),this.abortHandler?.(),n(!1)}};c.il?(0,f.b)((()=>o()),!0):o()}shouldImportAgg(r,n){return r!==e.D.sessionReplay||!1!==(0,t.Mt)(this.agentIdentifier,"session_trace.enabled")&&(!!n?.isNew||!!n?.state.sessionReplay)}}var g=i(7633),p=i(7894);class m extends h{static featureName=g.t9;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];if(super(r,n,g.t9,i),("undefined"==typeof PerformanceNavigationTiming||c.Tt)&&"undefined"!=typeof PerformanceTiming){const n=(0,t.OP)(r);n[g.Dz]=Math.max(Date.now()-n.offset,0),(0,f.K)((()=>n[g.qw]=Math.max((0,p.z)()-n[g.Dz],0))),(0,f.b)((()=>{const t=(0,p.z)();n[g.OJ]=Math.max(t-n[g.Dz],0),(0,s.p)("timing",["load",t],void 0,e.D.pageViewTiming,this.ee)}))}this.importAggregator()}}var v=i(1117),b=i(1284);class y extends v.w{constructor(e){super(e),this.aggregatedData={}}store(e,t,r,n,i){var o=this.getBucket(e,t,r,i);return o.metrics=function(e,t){t||(t={count:0});return t.count+=1,(0,b.D)(e,(function(e,r){t[e]=w(r,t[e])})),t}(n,o.metrics),o}merge(e,t,r,n,i){var o=this.getBucket(e,t,n,i);if(o.metrics){var a=o.metrics;a.count+=r.count,(0,b.D)(r,(function(e,t){if("count"!==e){var n=a[e],i=r[e];i&&!i.c?a[e]=w(i.t,n):a[e]=function(e,t){if(!t)return e;t.c||(t=x(t.t));return t.min=Math.min(e.min,t.min),t.max=Math.max(e.max,t.max),t.t+=e.t,t.sos+=e.sos,t.c+=e.c,t}(i,a[e])}}))}else o.metrics=r}storeMetric(e,t,r,n){var i=this.getBucket(e,t,r);return i.stats=w(n,i.stats),i}getBucket(e,t,r,n){this.aggregatedData[e]||(this.aggregatedData[e]={});var i=this.aggregatedData[e][t];return i||(i=this.aggregatedData[e][t]={params:r||{}},n&&(i.custom=n)),i}get(e,t){return t?this.aggregatedData[e]&&this.aggregatedData[e][t]:this.aggregatedData[e]}take(e){for(var t={},r="",n=!1,i=0;i t.max&&(t.max=e),e 2&&void 0!==arguments[2])||arguments[2];super(e,r,j.t,n),c.il&&((0,t.OP)(e).initHidden=Boolean("hidden"===document.visibilityState),(0,N.N)((()=>(0,s.p)("docHidden",[(0,p.z)()],void 0,j.t,this.ee)),!0),(0,O.bP)("pagehide",(()=>(0,s.p)("winPagehide",[(0,p.z)()],void 0,j.t,this.ee))),this.importAggregator())}}var P=i(3081);class C extends h{static featureName=P.t9;constructor(e,t){let r=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(e,t,P.t9,r),this.importAggregator()}}var R,I=i(2210),k=i(1214),H=i(2177),L={};try{R=localStorage.getItem("__nr_flags").split(","),console&&"function"==typeof console.log&&(L.console=!0,-1!==R.indexOf("dev")&&(L.dev=!0),-1!==R.indexOf("nr_dev")&&(L.nrDev=!0))}catch(e){}function z(e){try{L.console&&z(e)}catch(e){}}L.nrDev&&H.ee.on("internal-error",(function(e){z(e.stack)})),L.dev&&H.ee.on("fn-err",(function(e,t,r){z(r.stack)})),L.dev&&(z("NR AGENT IN DEVELOPMENT MODE"),z("flags: "+(0,b.D)(L,(function(e,t){return e})).join(", ")));var M=i(6660);class B extends h{static featureName=M.t;constructor(r,n){let i=!(arguments.length>2&&void 0!==arguments[2])||arguments[2];super(r,n,M.t,i),this.skipNext=0;try{this.removeOnAbort=new AbortController}catch(e){}const o=this;o.ee.on("fn-start",(function(e,t,r){o.abortHandler&&(o.skipNext+=1)})),o.ee.on("fn-err",(function(t,r,n){o.abortHandler&&!n[M.A]&&((0,I.X)(n,M.A,(function(){return!0})),this.thrown=!0,(0,s.p)("err",[n,(0,p.z)()],void 0,e.D.jserrors,o.ee))})),o.ee.on("fn-end",(function(){o.abortHandler&&!this.thrown&&o.skipNext>0&&(o.skipNext-=1)})),o.ee.on("internal-error",(function(t){(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,o.ee)})),this.origOnerror=c._A.onerror,c._A.onerror=this.onerrorHandler.bind(this),c._A.addEventListener("unhandledrejection",(t=>{const r=function(e){let t="Unhandled Promise Rejection: ";if(e instanceof Error)try{return e.message=t+e.message,e}catch(t){return e}if(void 0===e)return new Error(t);try{return new Error(t+(0,D.P)(e))}catch(e){return new Error(t)}}(t.reason);(0,s.p)("err",[r,(0,p.z)(),!1,{unhandledPromiseRejection:1}],void 0,e.D.jserrors,this.ee)}),(0,O.m$)(!1,this.removeOnAbort?.signal)),(0,k.gy)(this.ee),(0,k.BV)(this.ee),(0,k.em)(this.ee),(0,t.OP)(r).xhrWrappable&&(0,k.Kf)(this.ee),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}onerrorHandler(t,r,n,i,o){"function"==typeof this.origOnerror&&this.origOnerror(...arguments);try{this.skipNext?this.skipNext-=1:(0,s.p)("err",[o||new F(t,r,n),(0,p.z)()],void 0,e.D.jserrors,this.ee)}catch(t){try{(0,s.p)("ierr",[t,(0,p.z)(),!0],void 0,e.D.jserrors,this.ee)}catch(e){}}return!1}}function F(e,t,r){this.message=e||"Uncaught error with no additional information",this.sourceURL=t,this.line=r}let U=1;const q="nr@id";function G(e){const t=typeof e;return!e||"object"!==t&&"function"!==t?-1:e===c._A?0:(0,I.X)(e,q,(function(){return U++}))}function V(e){if("string"==typeof e&&e.length)return e.length;if("object"==typeof e){if("undefined"!=typeof ArrayBuffer&&e instanceof ArrayBuffer&&e.byteLength)return e.byteLength;if("undefined"!=typeof Blob&&e instanceof Blob&&e.size)return e.size;if(!("undefined"!=typeof FormData&&e instanceof FormData))try{return(0,D.P)(e).length}catch(e){return}}}var X=i(7243);class W{constructor(e){this.agentIdentifier=e,this.generateTracePayload=this.generateTracePayload.bind(this),this.shouldGenerateTrace=this.shouldGenerateTrace.bind(this)}generateTracePayload(e){if(!this.shouldGenerateTrace(e))return null;var r=(0,t.DL)(this.agentIdentifier);if(!r)return null;var n=(r.accountID||"").toString()||null,i=(r.agentID||"").toString()||null,o=(r.trustKey||"").toString()||null;if(!n||!i)return null;var a=(0,_.M)(),s=(0,_.Ht)(),c=Date.now(),u={spanId:a,traceId:s,timestamp:c};return(e.sameOrigin||this.isAllowedOrigin(e)&&this.useTraceContextHeadersForCors())&&(u.traceContextParentHeader=this.generateTraceContextParentHeader(a,s),u.traceContextStateHeader=this.generateTraceContextStateHeader(a,c,n,i,o)),(e.sameOrigin&&!this.excludeNewrelicHeader()||!e.sameOrigin&&this.isAllowedOrigin(e)&&this.useNewrelicHeaderForCors())&&(u.newrelicHeader=this.generateTraceHeader(a,s,c,n,i,o)),u}generateTraceContextParentHeader(e,t){return"00-"+t+"-"+e+"-01"}generateTraceContextStateHeader(e,t,r,n,i){return i+"@nr=0-1-"+r+"-"+n+"-"+e+"----"+t}generateTraceHeader(e,t,r,n,i,o){if(!("function"==typeof c._A?.btoa))return null;var a={v:[0,1],d:{ty:"Browser",ac:n,ap:i,id:e,tr:t,ti:r}};return o&&n!==o&&(a.d.tk=o),btoa((0,D.P)(a))}shouldGenerateTrace(e){return this.isDtEnabled()&&this.isAllowedOrigin(e)}isAllowedOrigin(e){var r=!1,n={};if((0,t.Mt)(this.agentIdentifier,"distributed_tracing")&&(n=(0,t.P_)(this.agentIdentifier).distributed_tracing),e.sameOrigin)r=!0;else if(n.allowed_origins instanceof Array)for(var i=0;i 2&&void 0!==arguments[2])||arguments[2];super(r,n,Z.t,i),(0,t.OP)(r).xhrWrappable&&(this.dt=new W(r),this.handler=(e,t,r,n)=>(0,s.p)(e,t,r,n,this.ee),(0,k.u5)(this.ee),(0,k.Kf)(this.ee),function(r,n,i,o){function a(e){var t=this;t.totalCbs=0,t.called=0,t.cbTime=0,t.end=E,t.ended=!1,t.xhrGuids={},t.lastSize=null,t.loadCaptureCalled=!1,t.params=this.params||{},t.metrics=this.metrics||{},e.addEventListener("load",(function(r){_(t,e)}),(0,O.m$)(!1)),c.IF||e.addEventListener("progress",(function(e){t.lastSize=e.loaded}),(0,O.m$)(!1))}function s(e){this.params={method:e[0]},T(this,e[1]),this.metrics={}}function u(e,n){var i=(0,t.DL)(r);i.xpid&&this.sameOrigin&&n.setRequestHeader("X-NewRelic-ID",i.xpid);var a=o.generateTracePayload(this.parsedOrigin);if(a){var s=!1;a.newrelicHeader&&(n.setRequestHeader("newrelic",a.newrelicHeader),s=!0),a.traceContextParentHeader&&(n.setRequestHeader("traceparent",a.traceContextParentHeader),a.traceContextStateHeader&&n.setRequestHeader("tracestate",a.traceContextStateHeader),s=!0),s&&(this.dt=a)}}function d(e,t){var r=this.metrics,i=e[0],o=this;if(r&&i){var a=V(i);a&&(r.txSize=a)}this.startTime=(0,p.z)(),this.listener=function(e){try{"abort"!==e.type||o.loadCaptureCalled||(o.params.aborted=!0),("load"!==e.type||o.called===o.totalCbs&&(o.onloadCalled||"function"!=typeof t.onload)&&"function"==typeof o.end)&&o.end(t)}catch(e){try{n.emit("internal-error",[e])}catch(e){}}};for(var s=0;s 1?e[1]=i:e.push(i)}else e[0]&&e[0].headers&&s(e[0].headers,n)&&(this.dt=n);function s(e,t){var r=!1;return t.newrelicHeader&&(e.set("newrelic",t.newrelicHeader),r=!0),t.traceContextParentHeader&&(e.set("traceparent",t.traceContextParentHeader),t.traceContextStateHeader&&e.set("tracestate",t.traceContextStateHeader),r=!0),r}}function x(e,t){this.params={},this.metrics={},this.startTime=(0,p.z)(),this.dt=t,e.length>=1&&(this.target=e[0]),e.length>=2&&(this.opts=e[1]);var r,n=this.opts||{},i=this.target;"string"==typeof i?r=i:"object"==typeof i&&i instanceof Y?r=i.url:c._A?.URL&&"object"==typeof i&&i instanceof URL&&(r=i.href),T(this,r);var o=(""+(i&&i instanceof Y&&i.method||n.method||"GET")).toUpperCase();this.params.method=o,this.txSize=V(n.body)||0}function A(t,r){var n;this.endTime=(0,p.z)(),this.params||(this.params={}),this.params.status=r?r.status:0,"string"==typeof this.rxSize&&this.rxSize.length>0&&(n=+this.rxSize);var o={txSize:this.txSize,rxSize:n,duration:(0,p.z)()-this.startTime};i("xhr",[this.params,o,this.startTime,this.endTime,"fetch"],this,e.D.ajax)}function E(t){var r=this.params,n=this.metrics;if(!this.ended){this.ended=!0;for(var o=0;o 2&&void 0!==arguments[2])||arguments[2];super(e,t,we.t,r),this.importAggregator()}}new class{constructor(e){let t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:(0,_.ky)(16);c._A?(this.agentIdentifier=t,this.sharedAggregator=new y({agentIdentifier:this.agentIdentifier}),this.features={},this.desiredFeatures=new Set(e.features||[]),this.desiredFeatures.add(m),Object.assign(this,(0,a.j)(this.agentIdentifier,e,e.loaderType||"agent")),this.start()):(0,l.Z)("Failed to initial the agent. Could not determine the runtime environment.")}get config(){return{info:(0,t.C5)(this.agentIdentifier),init:(0,t.P_)(this.agentIdentifier),loader_config:(0,t.DL)(this.agentIdentifier),runtime:(0,t.OP)(this.agentIdentifier)}}start(){const t="features";try{const r=n(this.agentIdentifier),i=[...this.desiredFeatures];i.sort(((t,r)=>e.p[t.featureName]-e.p[r.featureName])),i.forEach((t=>{if(r[t.featureName]||t.featureName===e.D.pageViewEvent){const n=function(t){switch(t){case e.D.ajax:return[e.D.jserrors];case e.D.sessionTrace:return[e.D.ajax,e.D.pageViewEvent];case e.D.sessionReplay:return[e.D.sessionTrace];case e.D.pageViewTiming:return[e.D.pageViewEvent];default:return[]}}(t.featureName);n.every((e=>r[e]))||(0,l.Z)("".concat(t.featureName," is enabled but one or more dependent features has been disabled (").concat((0,D.P)(n),"). This may cause unintended consequences or missing data...")),this.features[t.featureName]=new t(this.agentIdentifier,this.sharedAggregator)}})),(0,T.Qy)(this.agentIdentifier,this.features,t)}catch(e){(0,l.Z)("Failed to initialize all enabled instrument classes (agent aborted) -",e);for(const e in this.features)this.features[e].abortHandler?.();const r=(0,T.fP)();return delete r.initializedAgents[this.agentIdentifier]?.api,delete r.initializedAgents[this.agentIdentifier]?.[t],delete this.sharedAggregator,r.ee?.abort(),delete r.ee?.get(this.agentIdentifier),!1}}}({features:[J,m,S,class extends h{static featureName=oe;constructor(t,r){if(super(t,r,oe,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;const n=this.ee;let i;(0,k.QU)(n),this.eventsEE=(0,k.em)(n),this.eventsEE.on(se,(function(e,t){this.bstStart=(0,p.z)()})),this.eventsEE.on(ae,(function(t,r){(0,s.p)("bst",[t[0],r,this.bstStart,(0,p.z)()],void 0,e.D.sessionTrace,n)})),n.on(ce+ne,(function(e){this.time=(0,p.z)(),this.startPath=location.pathname+location.hash})),n.on(ce+ie,(function(t){(0,s.p)("bstHist",[location.pathname+location.hash,this.startPath,this.time],void 0,e.D.sessionTrace,n)}));try{i=new PerformanceObserver((t=>{const r=t.getEntries();(0,s.p)(te,[r],void 0,e.D.sessionTrace,n)})),i.observe({type:re,buffered:!0})}catch(e){}this.importAggregator({resourceObserver:i})}},C,xe,B,class extends h{static featureName=de;constructor(e,r){if(super(e,r,de,!(arguments.length>2&&void 0!==arguments[2])||arguments[2]),!c.il)return;if(!(0,t.OP)(e).xhrWrappable)return;try{this.removeOnAbort=new AbortController}catch(e){}let n,i=0;const o=this.ee.get("tracer"),a=(0,k._L)(this.ee),s=(0,k.Lg)(this.ee),u=(0,k.BV)(this.ee),d=(0,k.Kf)(this.ee),f=this.ee.get("events"),l=(0,k.u5)(this.ee),h=(0,k.QU)(this.ee),g=(0,k.Gm)(this.ee);function m(e,t){h.emit("newURL",[""+window.location,t])}function v(){i++,n=window.location.hash,this[ve]=(0,p.z)()}function b(){i--,window.location.hash!==n&&m(0,!0);var e=(0,p.z)();this[pe]=~~this[pe]+e-this[ve],this[ye]=e}function y(e,t){e.on(t,(function(){this[t]=(0,p.z)()}))}this.ee.on(ve,v),s.on(be,v),a.on(be,v),this.ee.on(ye,b),s.on(ge,b),a.on(ge,b),this.ee.buffer([ve,ye,"xhr-resolved"],this.featureName),f.buffer([ve],this.featureName),u.buffer(["setTimeout"+le,"clearTimeout"+fe,ve],this.featureName),d.buffer([ve,"new-xhr","send-xhr"+fe],this.featureName),l.buffer([me+fe,me+"-done",me+he+fe,me+he+le],this.featureName),h.buffer(["newURL"],this.featureName),g.buffer([ve],this.featureName),s.buffer(["propagate",be,ge,"executor-err","resolve"+fe],this.featureName),o.buffer([ve,"no-"+ve],this.featureName),a.buffer(["new-jsonp","cb-start","jsonp-error","jsonp-end"],this.featureName),y(l,me+fe),y(l,me+"-done"),y(a,"new-jsonp"),y(a,"jsonp-end"),y(a,"cb-start"),h.on("pushState-end",m),h.on("replaceState-end",m),window.addEventListener("hashchange",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("load",m,(0,O.m$)(!0,this.removeOnAbort?.signal)),window.addEventListener("popstate",(function(){m(0,i>1)}),(0,O.m$)(!0,this.removeOnAbort?.signal)),this.abortHandler=this.#e,this.importAggregator()}#e(){this.removeOnAbort?.abort(),this.abortHandler=void 0}}],loaderType:"spa"})})(),window.NRBA=o})(); window.jQuery || document.write(' ') CKEDITOR_BASEPATH='https://f1000research.com/js/vendor/ckeditor/' window.reactTheme = 'research'; window.MathJax = { CommonHTML: { linebreaks: { automatic: true } }, 'HTML-CSS': { linebreaks: { automatic: true } }, SVG: { linebreaks: { automatic: true } }, AuthorInit: function() { MathJax.Hub.Register.MessageHook('End Process', function () { let timeout = false; // holder for timeout id const delay = 250; // delay after event is "complete" to run callback const reflowMath = function() { const dispFormulas = document.querySelectorAll('.disp-formula.panel'); if (!dispFormulas) { return; } for (const dispFormula of dispFormulas) { const child = dispFormula.querySelector('.MathJax_Preview').nextSibling.firstChild; const isMultiline = MathJax.Hub.getAllJax(dispFormula)[0].root.isMultiline; if (dispFormula.offsetWidth < child.offsetWidth || isMultiline) { MathJax.Hub.Queue(['Rerender', MathJax.Hub, dispFormula]); } } }; window.addEventListener('resize', function() { clearTimeout(timeout); // clear the timeout timeout = setTimeout(reflowMath, delay); // start timing for event "completion" }); }); }, }; if (window.location.hash == '#_=_'){ window.location = window.location.href.split('#')[0] } !function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function() {n.callMethod? n.callMethod.apply(n,arguments):n.queue.push(arguments)} ;if(!f._fbq)f._fbq=n; n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0; t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.insertBefore(t,s)}(window, document,'script','https://connect.facebook.net/en_US/fbevents.js'); fbq('init', '1641728616063202'); fbq('track', "PixelInitialized", {}); (function(h,o,t,j,a,r){ h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)}; h._hjSettings={hjid:2318163,hjsv:6}; a=o.getElementsByTagName('head')[0]; r=o.createElement('script');r.async=1; r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv; a.appendChild(r); })(window,document,'https://static.hotjar.com/c/hotjar-','.js?sv='); search file_upload Submit your research search menu close search Browse Gateways & Collections How to Publish Submit your Research My Submissions Article Guidelines Article Guidelines (New Versions) Open Data, Software and Code Guidelines Open Data and Accessible Source Materials Guidelines (HSS) Open Data, Software and Code Guidelines (PSE) Prepublication Checks Production Process Posters and Slides Guidelines Document Guidelines Article Processing Charges Peer Review Finding Article Reviewers About How it Works For Reviewers Our Advisors Policies Glossary FAQs For Developers Newsroom Contact My Research Submissions Content and Tracking Alerts My Details Sign In file_upload Submit your research { "@context": "https://schema.org", "@type": "ScholarlyArticle", "mainEntityOfPage": { "@type": "WebPage", "@id": "https://f1000research.com/articles/14-1435" }, "headline": "Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application", "datePublished": "2025-12-23T17:44:01", "dateModified": "2025-12-23T17:44:01", "author": [ { "@type": "Person", "name": "Naeem Malik Jasim" }, { "@type": "Person", "name": "Mushtaq K. Abdalrahem" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": " Background Traditional fuzzy regression approaches, such as Tanaka’s fuzzy minimum method and fuzzy least squares, often lack theoretical guarantees of existence, uniqueness, and numerical stability. These limitations are of paramount importance in engineering applications involving uncertainty, such as predicting the compressive strength of concrete. This study addresses these issues by presenting a mathematically rigorous fuzzy regression model, based on fixed-point theory, formulated within the full metric space of trapezoidal fuzzy numbers using the metric scale d ∞ . Methodology We define a shrinkage coefficient on the trapezoidal fuzzy coefficient vector space and prove, using Banach’s fixed-point theory, the existence and uniqueness of the regression solution. An iterative algorithm is constructed to estimate the coefficients, using alpha computation and Lipschitz continuity to ensure convergence. The University of California, Irvine concrete compressive strength dataset was amplified using ASTM and ACI-based uncertainty coefficients, and the proposed fixed-point model was evaluated against the Tanaka method and least-squares fuzzy regression. Performance was assessed by mean squared error (MSE), coefficient ambiguity, convergence behavior, and toughness under ±5% noise. Results The proposed method demonstrated consistent geometric convergence with an average of 12.3 iterations and zero divergence across all experiments. It also reduced the overall mean squared error by an average of 12.5%, and by up to 25.1% at best, compared to the comparator methods. Coefficient ambiguity—measured by ambiguity width—was reduced by 18.3% compared to the Tanaka method and by 13.3% compared to ambiguity squares. Under noise perturbation, the model exhibits a significantly smaller increase in the mean error (+6.2%) compared to the Tanaka approach (+24.7%) and the LS-based approach (+18.3%), indicating a substantial improvement in robustness. Conclusions Incorporating fuzzy regression within a fixed-point theoretical framework helps resolve the stability, existence, and singularity challenges that have long plagued classical fuzzy regression models. The proposed approach offers a mathematically consistent, computationally stable, uncertainty-aware regression tool suitable for engineering applications involving imprecise measurements. Future work includes extending the model to nonlinear fuzzy structures, Gaussian/LR fuzzy representations, and broader applications in data-driven prediction under uncertainty. " } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/14-1435", "name": "Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology..." } } ] } Home Browse Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Malik Jasim N and K. Abdalrahem M. Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.12688/f1000research.173734.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] Naeem Malik Jasim 1 , Mushtaq K. Abdalrahem https://orcid.org/0000-0002-9719-2197 1,2 Naeem Malik Jasim 1 , Mushtaq K. Abdalrahem https://orcid.org/0000-0002-9719-2197 1,2 PUBLISHED 23 Dec 2025 Author details Author details 1 University of Kerbala, Karbala, Karbala Governorate, Iraq 2 University of Al-Ameed, Karbala City, Karbala, 56001, Iraq Naeem Malik Jasim Roles: Data Curation, Funding Acquisition, Writing – Original Draft Preparation Mushtaq K. Abdalrahem Roles: Project Administration, Supervision, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Background Traditional fuzzy regression approaches, such as Tanaka’s fuzzy minimum method and fuzzy least squares, often lack theoretical guarantees of existence, uniqueness, and numerical stability. These limitations are of paramount importance in engineering applications involving uncertainty, such as predicting the compressive strength of concrete. This study addresses these issues by presenting a mathematically rigorous fuzzy regression model, based on fixed-point theory, formulated within the full metric space of trapezoidal fuzzy numbers using the metric scale d ∞ . Methodology We define a shrinkage coefficient on the trapezoidal fuzzy coefficient vector space and prove, using Banach’s fixed-point theory, the existence and uniqueness of the regression solution. An iterative algorithm is constructed to estimate the coefficients, using alpha computation and Lipschitz continuity to ensure convergence. The University of California, Irvine concrete compressive strength dataset was amplified using ASTM and ACI-based uncertainty coefficients, and the proposed fixed-point model was evaluated against the Tanaka method and least-squares fuzzy regression. Performance was assessed by mean squared error (MSE), coefficient ambiguity, convergence behavior, and toughness under ±5% noise. Results The proposed method demonstrated consistent geometric convergence with an average of 12.3 iterations and zero divergence across all experiments. It also reduced the overall mean squared error by an average of 12.5%, and by up to 25.1% at best, compared to the comparator methods. Coefficient ambiguity—measured by ambiguity width—was reduced by 18.3% compared to the Tanaka method and by 13.3% compared to ambiguity squares. Under noise perturbation, the model exhibits a significantly smaller increase in the mean error (+6.2%) compared to the Tanaka approach (+24.7%) and the LS-based approach (+18.3%), indicating a substantial improvement in robustness. Conclusions Incorporating fuzzy regression within a fixed-point theoretical framework helps resolve the stability, existence, and singularity challenges that have long plagued classical fuzzy regression models. The proposed approach offers a mathematically consistent, computationally stable, uncertainty-aware regression tool suitable for engineering applications involving imprecise measurements. Future work includes extending the model to nonlinear fuzzy structures, Gaussian/LR fuzzy representations, and broader applications in data-driven prediction under uncertainty. READ ALL READ LESS Keywords Fixed-Point Theory, Fuzzy Regression Analysis, Trapezoidal Fuzzy Numbers, d∞-Metric, Concrete Compressive Strength. Corresponding Author(s) Mushtaq K. Abdalrahem ( [email protected] ) Close Corresponding author: Mushtaq K. Abdalrahem Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Malik Jasim N and K. Abdalrahem M. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Malik Jasim N and K. Abdalrahem M. Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.12688/f1000research.173734.1 ) First published: 23 Dec 2025, 14 :1435 ( https://doi.org/10.12688/f1000research.173734.1 ) Latest published: 23 Dec 2025, 14 :1435 ( https://doi.org/10.12688/f1000research.173734.1 ) Introduction Regression analysis is a fundamental form of quantitative modeling across fields of science; it provides a way of distilling actionable relations from real data. However, all forms of regression analysis have a deterministic basis that is increasingly incongruent with modern data, where epistemic uncertainty, or reducible error based on measurement constraints, environmental changeability, and incomplete knowledge of the system itself, defines the boundaries of the analysis. This uncertainty is most acutely present in the common engineering fields still reliant on forecasting a dependent variable, for example, in predicting the compressive strength of concrete, where the conjunction of measured laboratory results with real-world variability of composition, retentional effects of age, and variability inherent in the curing conditions will have a very real impact on the preciseness of prediction required in a managed environment. It is important to recognize that while traditional least-squares regression is statistically robust also in the presence of random errors, it is not inherently considered as a formal method for quantification of structured imprecision, to the extent that the regression coefficients produced can have a false-sense of preciseness that misrepresents the potential complexity of the system. Fuzzy regression was originally introduced by Tanaka et al., 4 for exactly this reason; fuzzy sets were introduced to be used to model the parameters and the observations. The primary goal was to model the variables with membership functions, as opposed to single point values. This allows one to explicitly encode uncertainty bounds within the framework of analysis. Many fuzzy regression studies were published under various disciplines (e.g., economics, engineering design) but the conceptual and methodological limitations imposed by the established framework remain, 2 principle of minimum fuzziness, while it provides an intuitive way to select parameters, still can provide multiple solutions depending on how the constraints are defined, as highlighted by, 7 , 12 fuzzy least squares approaches, while innovative, maintain high degree of numerical instability due to ill-posed linear systems; this issue was also confirmed empirically by Riali, 14 particularly in structural engineering, 14 demonstrated that the ambiguity of the solutions produced by traditional fuzzy regression can exceed 25% in cases with high variability in the data, showing relatively small predictive validity. The endurance of such concerns illustrates an important research gap: There has not been a theoretical counting/estimation framework that ensures existence, uniqueness, and algorithmic stability for fuzzy parameters. Existing approaches are all based on heuristic optimization or purely algebraic approaches that lack any functional-analytic grounding. As 15 correspondingly noted, this separation is problematic and inhibits fuzzy regression from accessing useful mathematical tools, such as fixed-point theory – even though such methods are effective and have proven to be successful at stabilizing ill-conditioned problems in numerical analysis and differential equations. 6 The potential opportunity cost to latent fuzzy regression is significant: without formal convergence guarantees, industries employing fuzzy regression theory (such as safety in structural assessments) will appropriately remain hesitant to utilize fuzzy regression, no matter how ideal the concept is for the topic. In order to address this gap, we introduce a new fixed-point-theoretic framework for fuzzy linear regression in the complete metric space of trapezoidal fuzzy numbers within the d∞-metric. Our primary contribution is the first mathematically verified conditions for reliable parameter estimation: 1. We defined a contraction operator T that defines the iterative parameter update and can prove via Banach’s Fixed-Point Theorem that there is a unique solution that converges exponentially fast based on measurable Lipschitz conditions. This resolves the existence/uniqueness limitations of methods like Tanaka’s. 2. We implement a stabilized iterative algorithm with precision-controlled termination (∥ θ (k+1) − θ (k)∥ < 10−6), achieving computational reliability absent in LS-based formulations. 3. We validate the framework through rigorous empirical analysis using the UCI Concrete Compressive Strength Dataset, demonstrating statistically significant improvements in prediction accuracy (12.5% MSE reduction) and solution clarity (18.3% ambiguity decrease) over state-of-the-art alternatives under noise perturbation. The subsequent sections are structured as follows: Section 2 critically reviews fuzzy regression methodologies and fixed-point applications. Section 3 formalizes our mathematical framework and proofs. Section 4 details experimental results, and Section 5 discusses engineering implications and limitations. Section 6 concludes with future research trajectories. Literature review Fuzzy regression methodologies have evolved significantly since Tanaka’s seminal minimum fuzziness principle, 2 which framed parameter estimation as a linear programming problem minimizing total spread. Although, as a methodology, it ultimately addressed the input-output vagueness problem, Diamond (1988) showed a tendency toward estimating overwide intervals, which increased solution vagueness. Later least-squares (LS) versions by Škrjanc 16 and again 18 sought to improve accuracy, by minimizing the quadratic distance between observed and predicted fuzzy sets. However, the work of Eren and Baets 9 rigorously demonstrated that these formulations most often violate the primary existence-unique duality: and that their algebraic solutions routinely collapse when rank deficient data matrices or lorsque les functions d’appartenance sont asymétriques, producing just temporary resolutions. This theoretical fragility translates into practical context as a practical computational instability where 1 reported solution divergence rates of greater than 30% for high-dimensional concrete strength models via Tanaka’s method. These restrictions prompted alternative strategies, including distance-based approaches that utilize metrics between fuzzy sets. The d∞-metric proposed by Hussain et al. 11 and axiomatized by 22 acquired particular significance because of the important trapezoidal shape remained under arithmetic operations; this is a vital criterion for establishing interpretability in engineering applications. 13 Unlike metrics of probabilities, the triangle inequality and completeness were satisfied in the space of trapezoidal fuzzy numbers (TrFNs). 15 This metric has defined relationships mathematically which allowed critical error estimation; however, it remained theoretically untapped for including estimation problems. At the same time, fixed-point theory (FPT) advanced a stabilization technique for poorly posed numeric problems. Banach’s contraction principle, and 17 extension to metric spaces, provides verifiable conditions for the existence and uniqueness of solutions, and the convergence of algorithmically derived solutions—exactly the guarantees you lack in fuzzy regression. 6 successfully applied FPT to stabilize fuzzy differential equations, while 24 demonstrated its efficacy in fuzzy optimization. Remarkably, despite this proven utility, FPT saw minimal integration into fuzzy regression. Recent surveys by Allahviranloo et al. 2 and Gopal and Moreno 10 confirm only two nascent attempts 8 : applied fixed-point iterations to simple fuzzy equations without regression context, and Turab 20 explored stochastic variants without addressing parameter uniqueness. This gap persists despite the mathematical compatibility between contraction mappings and the d ∞-metric’s completeness, as noted in Sing ’s 2024 call for “functional-analytic foundations in fuzzy inference.” The choice of fuzzy representation further influences methodological robustness. While triangular fuzzy numbers (TFNs) simplify computation, 5 their asymmetric real-world uncertainty modeling limitations drive adoption of trapezoidal representations. TrFNs provide superior flexibility in capturing measurement imprecision through distinct core and support intervals, as validated in 14 concrete strength uncertainty analysis. Recent empirical work by Zhang et al. 23 Further confirms that TrFNs reduce prediction bandwidth by 18–22% compared to TFNs in material science applications. This advantage, however, remains constrained by estimation instability in conventional methods. Synthesizing these strands reveals a critical research void: no existing framework unites the d ∞-metric’s completeness, TrFNs’ representational flexibility, and FPT’s convergence guarantees to resolve the existence-uniqueness-stability trilemma in fuzzy regression. Our study addresses this by constructing the first Banach-space formulation of fuzzy linear regression within the TrFN- d ∞ metric space, establishing mathematically verifiable solution properties absent in all reviewed methodologies. Methodology Mathematical foundations The use of fixed-point theory in solving mathematical modeling problems has gained increasing attention in recent literature, 21 further validating its suitability for stabilizing fuzzy regression models under uncertainty. The methodological framework is anchored in three rigorously defined mathematical constructs that establish the foundation for robust fuzzy regression analysis. A trapezoidal fuzzy number (TrFN) A ~ is formally characterized by a quadruple ( a , b , c , d ) ∈ ℝ 4 where a ≤ b ≤ c ≤ d , with the membership function: (1) μ A ~ ( x ) = { x − a b − a a ≤ x < b 1 b ≤ x ≤ c d − x d − c c < x ≤ d } This representation distinguishes the core [ b , c ] (values with full membership) from the support [ a , d ] (values with non-zero membership), enabling nuanced encoding of epistemic uncertainty. The d ∞ -metric between two TrFNs A ~ 1 = ( a 1 , b 1 , c 1 , d 1 ) and A ~ 2 = ( a 2 , b 2 , c 2 , d 2 ) is defined as: (2) d ∞ ( A ~ 1 , A ~ 2 ) = max ( | a 1 − a 2 | , | b 1 − b 2 | , | c 1 − c 2 | , | d 1 − d 2 | ) Theorem 1. 22 The space ( F ( ℝ ) , d ∞ ) of TrFNs equipped with the d ∞ -metric forms a complete metric space . Proof sketch: Pointwise convergence of quadruples ( a n , b n , c n , d n ) implies d ∞ -convergence. The triangle inequality follows from supremum norm properties. Remark 1. Closure of TrFNs under Arithmetic Operations Let each trapezoidal fuzzy number (TrFN) be denoted by A ~ = ( a 1 , a 2 , a 3 , a 4 ) , B ~ = ( b 1 , b 2 , b 3 , b 4 ) , where a 1 ≤ a 2 ≤ a 3 ≤ a 4 and the membership function is μ A ~ ( x ) = { 0 , x a 4 . The α-cut of a TrFN is the closed interval [ A ~ ] α = [ a 1 + ( a 2 − a 1 ) α , | a 4 − ( a 4 − a 3 ) α ] , α ∈ [ 0 , 1 ] . Using the extension principle , basic operations are defined as: [ A ~ + B ~ ] α = [ a 1 + b 1 + ( a 2 − a 1 + b 2 − b 1 ) α , | a 4 + b 4 − ( a 4 − a 3 + b 4 − b 3 ) α ] , which is again trapezoidal with parameters A ~ + B ~ = ( a 1 + b 1 , a 2 + b 2 , a 3 + b 3 , a 4 + b 4 ) . Similarly, for any real scalar c > 0 , [ c A ~ ] α = [ c a 1 + c ( a 2 − a 1 ) α , | c a 4 − c ( a 4 − a 3 ) α ] , yielding c A ~ = ( c a 1 , c a 2 , c a 3 , c a 4 ) , which is also trapezoidal. The α-cut based fuzzy arithmetic operations used in this study are summarized in Table 1 . Therefore, the set of TrFNs is closed under addition and positive scalar multiplication. This property ensures that all intermediate computations and the final regression outputs remain trapezoidal fuzzy numbers, preserving model consistency under the fixed-point framework. The fuzzy linear regression model for p predictors is expressed as: (3) Y ~ = A ~ 0 ⊕ ( A ~ 1 ⊗ X ~ 1 ) ⊕ ⋯ ⊕ ( A ~ p ⊗ X ~ p ) where Y ~ , A ~ j , and X ~ j are TrFNs, with ⊕ and ⊗ denoting α-cut-based arithmetic: Arithmetic operations preserve the trapezoidal form. For multiplication, the interval hull of vertex products ensures computational tractability. Table 1. Fuzzy arithmetic via α-cuts ( α ∈ [ 0 , 1 ] ). Operation α-cut Interval A ~ ⊕ B ~ [ A ~ ] α + [ B ~ ] α A ~ ⊗ B ~ hull ( { a α L b α L , a α L b α U , … } ) k ⊗ A ~ k ⋅ [ A ~ ] α Fixed-point theoretical framework The parameter space Θ = F ( ℝ ) p + 1 of TrFN vectors θ = ( A ~ 0 , … , A ~ p ) is equipped with the extended d ∞ -metric: (4) ‖ θ 1 − θ 2 ‖ ∞ = max 0 ≤ j ≤ p d ∞ ( A ~ j ( 1 ) , A ~ j ( 2 ) ) Lemma 1. ( Θ , ∥ ⋅ ∥ ∞ ) is a Banach space. Proof. Completeness follows from finite-dimensional extension of ( F ( ℝ ) , d ∞ ) . The contraction operator T : Θ → Θ is constructed through gradient-based optimization: (5) θ ( k + 1 ) = T ( θ ( k ) ) = θ ( k ) − η ∇ J ( θ ( k ) ) , J ( θ ) = ∑ i = 1 n d ∞ 2 ( Y ~ i , ⨁ j A ~ j ⊗ X ~ ij ) Differentiability and continuity The cost function J ( θ ) = ∑ i d ∞ 2 ( y ~ i , y ~ i est ( θ ) ) is defined within the complete metric space ( T , d ∞ ) of trapezoidal fuzzy numbers. Because d ∞ represents a metric rather than an inner product, J ( θ ) is not differentiable in the classical sense . Instead, it possesses Fréchet continuity with respect to the metric topology, ensuring smooth variation of J ( θ ) under infinitesimal perturbations of fuzzy parameters. The iterative operator T ( θ ) associated with this cost function satisfies the contraction property: ∥ T ( θ 1 ) − T ( θ 2 ) ∥ d ∞ ≤ L ∥ θ 1 − θ 2 ∥ d ∞ , L < 1 , guaranteeing convergence toward a unique fixed point that minimizes J ( θ ) . Therefore, the estimation procedure avoids explicit gradient computation and instead relies on Banach’s Fixed-Point Theorem as a theoretically sound alternative to gradient-based optimization in fuzzy regression analysis. The empirical Lipschitz constants computed for the key predictors are presented in Table 2 . Theorem 2. (Contraction). Under the Lipschitz condition ∥ ∇ J ( θ 1 ) − ∇ J ( θ 2 ) ∥ ∞ ≤ K ∥ θ 1 − θ 2 ∥ ∞ with K ⋅ max ∥ X ~ ∥ < 1 , T satisfies: (6) ‖ T ( θ 1 ) − T ( θ 2 ) ‖ ∞ ≤ L ‖ θ 1 − θ 2 ‖ ∞ , L = 1 − η + ηK max ‖ X ~ ‖ < 1 Derived via mean value theorem and bounded gradient variation. Empirical verification: K quantifies gradient sensitivity. The max ∥ X ~ j ∥ values derive from the UCI dataset. All L < 1 confirm contractivity. Theorem 3. (Banach Fixed-Point Theorem). For contraction T on complete Θ , ∃ ! θ ∗ ∈ Θ such that: • θ ∗ = T ( θ ∗ ) (Existence) • lim k → ∞ θ ( k ) = θ ∗ (Convergence) • ∥ θ ( k ) − θ ∗ ∥ ∞ ≤ L k 1 − L ∥ θ ( 1 ) − θ ( 0 ) ∥ ∞ (Geometric rate) Table 2. Empirical Lipschitz constants (UCI Dataset). Predictor max ∥ X ~ j ∥ K L Cement 540.0 0.0014 0.756 Age 365.0 0.0021 0.767 Water 247.0 0.0032 0.790 Aggregate - - 0.819 Iterative algorithm The parameter estimation procedure is formalized as: Initialization : θ ( 0 ) ← Tanaka ’ s solution Iteration : For k = 0 , 1 , … a. Compute predictions: Y ~ i ( k ) = ⨁ j A ~ j ( k ) ⊗ X ~ ij b. Evaluate loss: J ( θ ( k ) ) = ∑ i d ∞ 2 ( Y ~ i , Y ~ i ( k ) ) c. Update parameters: A ~ j ( k + 1 ) = A ~ j ( k ) − η ∂ J ∂ A ~ j Termination : ∥ θ ( k + 1 ) − θ ( k ) ∥ ∞ < 10 − 6 The per-iteration computational requirements of the proposed method are listed in Table 3 . Table 3. Computational complexity ( n = 1 , 030 , p = 8 ). Component Operations Time (ms) Prediction O ( np ) 8.2 Loss O ( n ) 3.7 Gradient O ( np ) 9.6 Total/Iteration O ( np ) 21.5 Computational Implementation : • TrFNs represented as 4D vectors ( a , b , c , d ) • α-cut arithmetic at α = { 0 , 0.5 , 1 } • Parallelized gradient computation Linear complexity enables scalability. Benchmarks performed on Intel i7-12700H. Data fuzzification For the UCI Concrete Dataset, crisp values x i are transformed to TrFNs X ~ i via: (7) X ~ i = { ( x i ( 1 − δ 2 ) , x i ( 1 − δ 1 ) , x i ( 1 + δ 1 ) , x i ( 1 + δ 2 ) ) ( symmetric ) ( x i − δ 2 − , x i − δ 1 − , x i + δ 1 + , x i + δ 2 + ) ( asymmetric ) } Core ( δ 1 ) and support ( δ 2 ) widths derive from material uncertainty sources: Asymmetric fuzzification applied where physical constraints exist (e.g., strength ≥0). Robustness validation Noise sensitivity is quantified via perturbations: (8) X ~ i ϵ = ( x i ( 1 − ϵ ) , x i ( 1 − ϵ 2 ) , x i ( 1 + ϵ 2 ) , x i ( 1 + ϵ ) ) , ϵ ∼ U ( 0,0.05 ) The fuzzification parameters used to model input uncertainty are shown in Table 4 . While a deeper justification of the fuzzification intervals is provided in Table 5 . Table 4. Fuzzification parameters. Variable Core ( δ 1 ) Support ( δ 2 ) Source Cement ±1.5% ±4.0% ASTM C150 Age ±0.5 days ±3.0 days ACI 214R Strength ±1.8% ±6.5% ASTM C39 Table 5. Fuzzification parameters and rationale. Variable Core (±δ 1 ) Support (±δ 2 ) Uncertainty source Cement ±1.5% ±4.0% Batch mixing variability (ASTM C150) Age ±0.5 days ±3.0 days Curing time documentation errors Fly Ash ±2.0% ±5.0% Material heterogeneity Strength ±1.8% ±6.5% Testing machine calibration (ASTM C39) Synthesis This methodology establishes a theoretically rigorous fusion of fixed-point theory and fuzzy regression within a Banach space. The contraction operator ensures existence, uniqueness, and convergence—resolving foundational limitations in prior approaches. The fuzzification protocol embeds real-world uncertainty, while computational design ensures practical feasibility. Results and analysis Dataset description and fuzzification The empirical validation employs the UCI Concrete Compressive Strength Dataset, 19 comprising 1,030 observations of high-performance concrete formulations. Key variables include: • Predictors : ○ Cement (kg/m³), Blast Furnace Slag (kg/m³), Fly Ash (kg/m³), Water (kg/m³) ○ Superplasticizer (kg/m³), Coarse Aggregate (kg/m³), Fine Aggregate (kg/m³), Age (days) • Response : Compressive strength (MPa) measured at 28 days. Fuzzification transformed crisp values into trapezoidal fuzzy numbers (TrFNs) using industry-derived uncertainty parameters: For example, a 35 MPa strength measurement becomes: Strength ~ = ( 35 × 0.935 , 35 × 0.982 , 35 × 1.018 , 35 × 1.065 ) = ( 32.7 , 34.4 , 35.6 , 37.3 ) Algorithm performance Convergence analysis The fixed-point algorithm demonstrated consistent convergence across 100 trials with randomized initial parameters. The convergence behavior across repeated runs is summarized in Table 6 : • Geometric decay of error ∥θ (k+1) − θ (k) ∥∞ ( Figure 1 ) aligns with Theorem 3 predictions • No divergence observed across 100 trials, confirming numerical stability • Initialization with Tanaka’s solution reduced iterations by 22% vs. random starts Table 6. Convergence statistics. Metric Mean Std Dev Range Iterations 12.3 2.7 9–18 Runtime 0.26 s 0.07 s 0.19–0.42 s Figure 1. Iteration error decay curve. This figure illustrates the geometric convergence of the fixed-point algorithm, showing the decrease in ∥θ (k+1) − θ (k) ∥∞ across iterations for the UCI Concrete dataset. Figure 2. Comparison of uncertainty width of coefficients. This Figure 2 shows the ambiguity (fuzzy width) of the estimated trapezoidal coefficients for the proposed method compared to the Tanaka and LS methods. Computational stability The comparative ambiguity widths of model coefficients are given in Table 7 . • Zero overflow/underflow occurrences in 1.03 × 10 5 arithmetic operations • Ill-conditioning score (κ = ∥J′J∥ · ∥(J′J) −1 ∥) remained below 10 3 (well-conditioned) • Memory footprint : 42 MB for n = 1,030, p = 8 (efficient for engineering workstations) Table 7. Ambiguity analysis. Method Avg. width Reduction p-value Proposed (Fixed-Point) 0.85 MPa – – Tanaka’s Min-Fuzziness 1.04 MPa 18.3% <0.001 LS-Based 0.98 MPa 13.3% <0.01 Predictive performance comparison Accuracy (MSE) Mean Squared Error evaluated at α = 0 cut (support boundaries) and α = 1 (core boundaries): Improvement MSE Reduction = 7.68 − 5.75 7.68 × 100 % = 25.1 % ( support : 12.5 % ) Solution ambiguity Average width of fuzzy parameters (measure of model uncertainty): Critical Insight : Tighter parameter distributions indicate enhanced estimation precision, reducing epistemic uncertainty in strength predictions. Robustness testing Noise perturbation protocol The robustness of all methods under ±5% noise is reported in Table 8 . • Training data contaminated with ±5% uniform noise: X ~ noisy = ( x ( 1 − ϵ ) , x ( 1 − ϵ / 2 ) , x ( 1 + ϵ / 2 ) , x ( 1 + ϵ ) ) , ϵ ∼ U ( 0 , 0.05 ) • Tested on original (unperturbed) test set (n = 309) Table 8. Noise robustness comparison (±5% Training Noise). Method ΔMSE ΔAmbiguity Failure rate Proposed (Fixed-Point) +6.2% +7.8% 0% Tanaka’s Min-Fuzziness +24.7% +31.5% 12% LS-Based +18.3% +22.1% 8% ΔMSE = Percentage increase in test MSE after noise exposure Failure Rate : Instances where ∥θ∥ → ∞ or MSE > 50 MPa² Stability visualization Fixed-point method maintains prediction coherence ( Figure 3 ) due to contractive properties 18.3% lower ambiguity persists under noise (validating Theorem 2 ) Figure 3. Stability of prediction periods under noise. This figure 3 shows the behavior of the prediction periods under ±5% noise, highlighting the strength of the fixed-point model. Failure cases in benchmark methods linked to unbounded error growth Synthesis of key results The fixed-point approach demonstrates triple superiority : Accuracy : 25.1% lower MSE than Tanaka’s method (p < 0.001) Precision : 18.3% narrower solution ambiguity bands Robustness : 4× lower MSE degradation under noise vs. LS-based methods These empirical outcomes directly validate the theoretical framework: contractive operators suppress error propagation, while the d ∞ -metric’s completeness ensures solution stability. The convergence profile ( Figure 1 ) further confirms the geometric decay rate predicted by Banach’s theorem. Discussion The empirical and theoretical outcomes of this study collectively affirm that fixed-point theory provides a mathematically rigorous foundation for fuzzy regression analysis, fundamentally resolving longstanding stability and reliability issues in parameter estimation. As anticipated, the convergence (12.3 ± 2.7 iterations) and stability of the algorithm conforms exactly to Banach’s fixed-point theorem. This theorem guarantees that contraction mappings in a complete metric space will decay exponentally in error. The geometric convergence illustrated in Figure 1 is an immediate result of the Lipschitz continuity of our operator T (L ≤ 0.82) which halted the propagation of error over the updated iterates. The ambiguity of the solution was reduced to a remarkable extent (18.3% narrower from the widths fo the parameters) than Tanaka’s method resulting from the uniqueness of the solution due to Banach’s constraints. While heuristic methods are free to explore multiple minimized fuzziness sets of parameters, the operator T, as a contraction mapping through Banach, compresses the solution space into a single, clearly defined attractor in parameter space. This quantifiable reduction in epistemic uncertainty in predicting was enabled by Banach’s assertions. The reduction in prediction MSE of 25.1% was further evidence of the effectiveness of this approach. We formulated the regression problem within the d_∞-metric space of trapezoidal fuzzy numbers (TrFNs), while still retaining the structural relationships between variables via arithmetic operations ( Table 5 ). We did not introduce the distortions incurred by the use of linear programming forms by Tanaka or least squares forms by Diamond. For example, in the case of Tanaka, parameter widths will be inflated in order to incorporate outliers. LS-adopted forms often suffer from matrix ill-conditioning with improperly defined asymmetric uncertainties. In contrast, our gradient-based fixed-point iterations ( Equation 7 ), which iteratively tune the parameters, retain their geometric definitions, allowing for tighter and more accurate predictions intervals. Comparative analysis of prior work The existential fragility of traditional fuzzy regression techniques is evident when systematically considered. Tanaka’s minimum fuzziness principle fails to guarantee solutions on multicollinear correlative cases, 12 noted; while in traditional LS-based approaches propagation of noise also results in unbounded error, because algebraic inversions do not consider topological constraints. Our approach avoids these issues by replacing deterministic optimization with a contractive mapping that converges uniquely (as proven) under certain conditions ( Theorem 2 ). The mathematics here underscores the statistically greater 4 × robustness to ±5 % input noise that we witness ( Table 8 ): where Tanaka’s and LS approaches are subject to error propagation, and unhelpful to degrees of freedom given that there is mathematical inherent failure, when the Lipschitz contraction condition L < 1 is met, perturbation to convergence is limited by default. Practical implications for engineering These advancements have significant ramifications for uncertainty-aware modeling in material science and structural engineering. In high-stakes applications like concrete strength prediction, our method not only yields useful point estimates, but also estimate quantifiable uncertainty bounds based on the widths of the TrFNs (e.g., 32.7-37.3 MPa for a given 35 MPa specimen). This enables engineers to propagate imprecision through design calculations, supporting reliability-based decision-making. For instance, the 18.3% ambiguity reduction directly translates to narrower safety margins in load-bearing calculations, potentially reducing material overdesign by 12–15% while maintaining safety standards (ACI 318-19). Limitations and methodological considerations Three limitations warrant emphasis. First, the contraction property hinges on the Lipschitz condition K ⋅ max ∥ X ~ ∥ < 1 , which may require data scaling for high-magnitude predictors (e.g., aggregate content exceeding 1,000 kg/m³). Second, while TrFNs efficiently model symmetric uncertainty, they may require extensions to Gaussian or LR-type fuzzy numbers for skewed distributions. Third, computational complexity scales linearly with predictors ( O ( np ) ), but for p > 50 , the iterative gradient updates ( Equation 7 ) may benefit from quasi-Newton acceleration. Critically, performance depends on appropriate fuzzification parameters ( Table 5 ). Overly narrow supports (e.g., δ 2 < 2 % ) artificially suppress uncertainty, while excessive widths inflate ambiguity. We recommend deriving δ 1 , δ 2 from domain-specific standards (e.g., ASTM tolerances) or bootstrap resampling. Future work should explore automated fuzzification via uncertainty quantification techniques like Monte Carlo dropout. Synthesis This study demonstrates that fixed-point theory transcends theoretical elegance to deliver tangible improvements in fuzzy regression’s practicality. By embedding estimation within a complete metric space and leveraging Banach’s contractive principles, we resolve the existence-uniqueness-stability trilemma that has hindered the field since Tanaka’s pioneering work. The resulting framework bridges mathematical rigor with engineering utility—a critical step toward trustworthy uncertainty quantification in data-driven design. Conclusion This study introduces a mathematically rigorous framework for fuzzy regression by employing fixed-point theory within the complete metric space of trapezoidal fuzzy numbers using the d∞-metric. Unlike classical approaches such as Tanaka’s minimum fuzziness or least-squares-based methods, which often lack guarantees of stability and uniqueness, our model ensures existence, uniqueness, and convergence through Banach’s Fixed-Point Theorem under verifiable Lipschitz conditions. We developed a reliable iterative algorithm with a strict convergence criterion, implemented efficiently in Python. Empirical validation using the UCI Concrete Compressive Strength Dataset showed significant improvements in predictive accuracy (MSE reduction of 12.5%) and reduced uncertainty (18.3% decrease in parameter ambiguity) compared to benchmark methods. The model also demonstrated strong robustness under noise perturbation, making it practical for real-world engineering scenarios. This work lays a solid mathematical foundation for fuzzy regression under uncertainty and supports its practical application in data-driven domains. Future research will explore extensions to non-trapezoidal fuzzy numbers (e.g., Gaussian, LR-shaped), nonlinear model structures, machine-learning-based fuzzification, and applications in economics and biomedical fields. Data availability All data supporting the findings of this study are openly available in Zenodo under a CC-BY 4.0 license . The dataset includes fuzzified trapezoidal data, numerical values underlying all tables and figures, simulation outputs, and extended materials. The complete dataset and extended data package can be accessed at: https://doi.org/10.5281/zenodo.17772389 (Abdalrahem, M. (2025).) 3 References 1. Adulaimi AAA, Pradhan B, Chakraborty S, et al. : Traffic noise modelling using land use regression models based on machine learning, statistical regression and GIS. Energies. 2021; 14 (16): 5095. Publisher Full Text 2. Allahviranloo T, Pedrycz W, Esfandiari A: Advances in numerical analysis emphasizing interval data. CRC Press; 2022. 3. Abdalrahem M: Extended Data Package. Zenodo. 2025. Publisher Full Text 4. American Society for Testing and Materials. Committee C09 on Concrete and Concrete Aggregates: Standard test method for compressive strength of cylindrical concrete specimens. ASTM International; 2021. 5. Asai HTSUK, Tanaka S, Uegima K: Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybern. 1982; 12 : 903–907. Publisher Full Text 6. Chachi J: A multi-attribute assessment of fuzzy regression models. Iranian Journal of Fuzzy Systems. 2021. 7. Debnath P, Konwar N, Radenović S: Metric fixed point theory. Singapore: Springer; 2021. 8. Diamond P: Fuzzy least squares. Inf. Sci. 1988; 46 (3): 141–157. Publisher Full Text 9. Dzitac S, Oros H, Deac D, et al. : Fixed point theory in fuzzy normed linear spaces: a general view. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL. 2021; 16 (6). Publisher Full Text 10. Eren M, Baets BD: Forecasting Turkey’s Primary Energy Demand Based on Fuzzy Auto-regressive Distributed Lag Models with Symmetric and Non-symmetric Triangular Coefficients. International Journal of Fuzzy Systems. 2025; 27 (1): 237–249. Publisher Full Text 11. Gopal D, Moreno JM: Recent advances and applications of fuzzy metric fixed point theory. Chapman and Hall/CRC; 2023. 12. Hussain A, Ishtiaq U, Ahmed K, et al. : On pentagonal controlled fuzzy metric spaces with an application to dynamic market equilibrium. Journal of function spaces. 2022; 2022 (1): 1–8. Publisher Full Text 13. Kumar R, Dhiman G, Rakhra M: Disseminate Reduce Flexible Fuzzy linear regression model to the analysis of an IoT-based Intelligent Transportation System.2024. 14. Peddi PBR, Abdullah L: Nearest Triangular Fuzzy Quantity Approximation of Trapezoidal Fuzzy Number with Vagueness in Lower Decision Level for Ranking Fuzzy Numbers. Malaysian Journal of Mathematical Sciences. 2025; 19 (2): 727–748. Publisher Full Text 15. Riali I, Fareh M, Bobillo F: ProbFuzzOnto: A Fuzzy Ontology-Driven Uncertainty Approach Using Fuzzy Bayesian Networks. International Journal of Fuzzy Systems. 2024; 1–21. 16. Sing P, Rahaman M, Sankar SPM: Solution of fuzzy system of linear equation under different fuzzy difference ideology. Spectrum of operational research. 2024; 1 (1): 64–74. Publisher Full Text 17. Škrjanc I: An evolving concept in the identification of an interval fuzzy model of Wiener-Hammerstein nonlinear dynamic systems. Inf. Sci. 2021; 581 : 73–87. Publisher Full Text 18. Smart DR: Fixed Point Theorems, Cambridge Tracts in Mathematics, no. 6.1974. 19. Tanveer M, Ganaie MA, Bhattacharjee A, et al. : Intuitionistic fuzzy weighted least squares twin SVMs. IEEE Transactions on Cybernetics. 2022; 53 (7): 4400–4409. Publisher Full Text 20. Tipu RK, Panchal VR, Pandya KS: An ensemble approach to improve BPNN model precision for predicting compressive strength of high-performance concrete. Structures. Elsevier; 2022, November; Vol. 45 . : pp. 500–508. 21. Turab A, Ali W, Park C: A unified fixed-point approach to study the existence and uniqueness of solutions to the generalized stochastic functional equation emerging in the psychological theory of learning. AIMS Mathematics. 2022; 7 (4): 5291–5304. Publisher Full Text 22. Abood ZH, Abd Al-Rahem MK, Kadhim DA: Fixed point theorem over orthogonal vector metric space and application for solve integral equation. Journal of Interdisciplinary Mathematics. 2025; 28 (3-B): 1021–1030. Publisher Full Text 23. Zadeh LA: Fuzzy logic. Granular, fuzzy, and soft computing. New York, NY: Springer US; 2023; pp. 19–49. 24. Zhang H, Chen W, Iyer A, et al. : Uncertainty-aware mixed-variable machine learning for materials design. Sci. Rep. 2022; 12 (1): 19760. PubMed Abstract | Publisher Full Text | Free Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 23 Dec 2025 ADD YOUR COMMENT Comment Author details Author details 1 University of Kerbala, Karbala, Karbala Governorate, Iraq 2 University of Al-Ameed, Karbala City, Karbala, 56001, Iraq Naeem Malik Jasim Roles: Data Curation, Funding Acquisition, Writing – Original Draft Preparation Mushtaq K. Abdalrahem Roles: Project Administration, Supervision, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 23 Dec 2025, 14:1435 https://doi.org/10.12688/f1000research.173734.1 Copyright © 2025 Malik Jasim N and K. Abdalrahem M. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Malik Jasim N and K. Abdalrahem M. Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.12688/f1000research.173734.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 23 Dec 2025 Views 0 Cite How to cite this report: Hameed AT. Reviewer Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r450666 ) The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-450666 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 28 Jan 2026 Areej Tawfeeq Hameed , University of Baghdad, Baghdad, Baghdad Governorate, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.191573.r450666 Hello, ​I have reviewed the submitted manuscript and find that the paper is highly suitable for publication. The authors have presented several appropriate and mathematically sound results that contribute significantly to the current field of study. Furthermore, the ... Continue reading READ ALL Hello, ​I have reviewed the submitted manuscript and find that the paper is highly suitable for publication. The authors have presented several appropriate and mathematically sound results that contribute significantly to the current field of study. Furthermore, the practical implications of this research produce useful applications for society. I recommend its acceptance as it meets the necessary academic standards for this journal. ​Prof. Dr. Areej Tawfeeq Hameed" Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: math. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Hameed AT. Reviewer Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r450666 ) The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-450666 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Shafaie V. Reviewer Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r448782 ) The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-448782 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 28 Jan 2026 Vahid Shafaie , Széchenyi István University, Győr, Hungary Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.191573.r448782 Overall Evaluation The manuscript proposes a mathematically grounded fuzzy linear regression framework that embeds trapezoidal fuzzy-number (TrFN) regression in the complete metric space induced by the d ∞ metric and estimates fuzzy coefficients via a Banach fixed-point (contraction mapping) formulation. The ... Continue reading READ ALL Overall Evaluation The manuscript proposes a mathematically grounded fuzzy linear regression framework that embeds trapezoidal fuzzy-number (TrFN) regression in the complete metric space induced by the d ∞ metric and estimates fuzzy coefficients via a Banach fixed-point (contraction mapping) formulation. The central contribution is the introduction of a provably unique and convergent estimation scheme for fuzzy regression coefficients—an important advance over classical Tanaka-type linear programming and fuzzy least-squares variants that may suffer from ambiguity and instability—followed by an empirical demonstration on the UCI Concrete Compressive Strength dataset with reported improvements in prediction error, coefficient ambiguity, and robustness under ±5% noise. The work is timely and potentially impactful for uncertainty-aware engineering prediction (notably concrete strength), and the manuscript is generally well-structured with relevant theoretical and empirical components. However, several technical, interpretive, and presentation aspects are insufficiently clear or underdeveloped and should be addressed as outlined below. Specific comments: What is the exact definition of the contraction operator T and the objective J(θ) in fully explicit mathematical form (including the precise predicted output expression and how d ∞ enters the loss)? Please provide the full equations as implemented, not only verbal descriptions. The text states the method “avoids explicit gradient computation,” yet the algorithm and complexity table indicate gradient computation and an update step using ∂ J /∂ A j . Which is correct? If gradients are computed, specify how derivatives/subgradients are defined under d ∞ (a non-smooth metric) and how this is reconciled with the fixed-point formulation. The contraction condition is pivotal. How were predictors scaled/normalized to ensure L<1 in practice, and is this condition guaranteed across datasets or only for the current scaling? Please report the exact preprocessing, the step-size/relaxation parameter (if any), and a formal statement of when the method may fail (e.g., L≥1 ). How is MSE computed for fuzzy outputs? The manuscript mentions evaluation at α=0 and α=1 . Do you compute MSE on (i) the support endpoints, (ii) the core endpoints, (iii) the d ∞ distance, or (iv) another scalarization? Provide the precise formula(s) and justify why this is the appropriate accuracy metric for fuzzy predictions. How exactly were Tanaka’s minimum-fuzziness and the LS-based fuzzy regression implemented (software/package, constraint formulation, solver settings, handling of infeasibility/divergence)? The manuscript should briefly position its fixed-point TrFN fuzzy regression (with existence/uniqueness/convergence guarantees) against the mainstream applied fuzzy-logic practice in concrete materials research, where rule-based fuzzy inference systems are integrated with experimental bond-test programs. In particular, the recent studies “Slant Shear Tests and Fuzzy Logic Integration for Evaluating Shear Bond Strength in SCC and FRSCC Repair Applications (2025)” and “Integrating push-out test validation and fuzzy logic for bond strength study of fiber-reinforced self-compacting concrete (2024)” exemplify how fuzzy logic is used for SCC/FRSCC bond-strength prediction and test-method validation; the authors should add a short paragraph clarifying what their approach adds relative to such FIS-based studies (and what it does not), and whether the framework is readily extendable beyond compressive strength to bond-related targets. Precisely define “ambiguity” (e.g., average coefficient trapezoid width? output interval width? aggregated across coefficients?) and give the computation formula. Clarify whether Table 7 width values refer to coefficients or predicted outputs, and whether widths are measured at α=0 , α=1 , or across all α . Tables show fuzzification parameters for only some variables in one table while another includes additional variables. Provide a complete fuzzification table covering all predictors and the response, with sources/justifications for each δ 1 δ 2 . Explicitly state whether the response is fuzzified during training and how. The manuscript contains systematic reference-number mismatches (e.g., key foundational citations appear misaligned with the bibliography). This must be fully corrected so every in-text citation points to the correct source; also ensure standards referenced in text (ACI documents, ASTM standards) are properly and consistently cited. Decision Minor Revision The manuscript’s core idea—casting fuzzy regression estimation as a contraction mapping in a complete TrFN d ∞ -metric space with a unique fixed point—is novel and technically promising, and the reported empirical outcomes are consistent with the stated objectives. Nevertheless, several issues must be corrected before the work is suitable for indexing. Addressing the numbered items above should be sufficient to make the manuscript verifiable, technically unambiguous, and editorially sound. ===== · Is the work clearly and accurately presented and does it cite the current literature? Partly (presentation is generally structured, but citation-numbering integrity and some clarity issues must be corrected; literature coverage is mostly current but referencing is unreliable until fixed). · Is the study design appropriate and is the work technically sound? Yes (the theoretical framework and empirical benchmarking design are appropriate; the core approach appears technically sound). · Are sufficient details of methods and analysis provided to allow replication by others? Partly (key elements—explicit definition of T , exact loss/MSE computation for fuzzy outputs, preprocessing/scaling choices, baseline implementation details, and statistical testing protocol—need clearer specification). · If applicable, is the statistical analysis and its interpretation appropriate? Partly (reporting p-values is positive, but the protocol generating them—repetitions/splits/tests—must be explicitly documented to validate interpretation). · Are all the source data underlying the results available to ensure full reproducibility? Partly (a data package is indicated, but full reproducibility requires confirmation that code, preprocessing/fuzzification scripts, and baseline solver settings are available; otherwise it remains partial). · Are the conclusions drawn adequately supported by the results? Yes (within the scope of the presented dataset and tests, the conclusions align with the reported results; claims should be carefully scoped to avoid overgeneralization). Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Concrete, Numerical modeling, AI (Machine learning and Deep Learning), Fuzzy system, I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Shafaie V. Reviewer Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r448782 ) The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-448782 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 23 Dec 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 1 23 Dec 25 read read Vahid Shafaie , Széchenyi István University, Győr, Hungary Areej Tawfeeq Hameed , University of Baghdad, Baghdad, Iraq Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Hameed A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 28 Jan 2026 | for Version 1 Areej Tawfeeq Hameed , University of Baghdad, Baghdad, Baghdad Governorate, Iraq 0 Views copyright © 2026 Hameed A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Hello, ​I have reviewed the submitted manuscript and find that the paper is highly suitable for publication. The authors have presented several appropriate and mathematically sound results that contribute significantly to the current field of study. Furthermore, the practical implications of this research produce useful applications for society. I recommend its acceptance as it meets the necessary academic standards for this journal. ​Prof. Dr. Areej Tawfeeq Hameed" Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise math. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Hameed AT. Peer Review Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r450666) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-450666 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Shafaie V. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 28 Jan 2026 | for Version 1 Vahid Shafaie , Széchenyi István University, Győr, Hungary 0 Views copyright © 2026 Shafaie V. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Overall Evaluation The manuscript proposes a mathematically grounded fuzzy linear regression framework that embeds trapezoidal fuzzy-number (TrFN) regression in the complete metric space induced by the d ∞ metric and estimates fuzzy coefficients via a Banach fixed-point (contraction mapping) formulation. The central contribution is the introduction of a provably unique and convergent estimation scheme for fuzzy regression coefficients—an important advance over classical Tanaka-type linear programming and fuzzy least-squares variants that may suffer from ambiguity and instability—followed by an empirical demonstration on the UCI Concrete Compressive Strength dataset with reported improvements in prediction error, coefficient ambiguity, and robustness under ±5% noise. The work is timely and potentially impactful for uncertainty-aware engineering prediction (notably concrete strength), and the manuscript is generally well-structured with relevant theoretical and empirical components. However, several technical, interpretive, and presentation aspects are insufficiently clear or underdeveloped and should be addressed as outlined below. Specific comments: What is the exact definition of the contraction operator T and the objective J(θ) in fully explicit mathematical form (including the precise predicted output expression and how d ∞ enters the loss)? Please provide the full equations as implemented, not only verbal descriptions. The text states the method “avoids explicit gradient computation,” yet the algorithm and complexity table indicate gradient computation and an update step using ∂ J /∂ A j . Which is correct? If gradients are computed, specify how derivatives/subgradients are defined under d ∞ (a non-smooth metric) and how this is reconciled with the fixed-point formulation. The contraction condition is pivotal. How were predictors scaled/normalized to ensure L<1 in practice, and is this condition guaranteed across datasets or only for the current scaling? Please report the exact preprocessing, the step-size/relaxation parameter (if any), and a formal statement of when the method may fail (e.g., L≥1 ). How is MSE computed for fuzzy outputs? The manuscript mentions evaluation at α=0 and α=1 . Do you compute MSE on (i) the support endpoints, (ii) the core endpoints, (iii) the d ∞ distance, or (iv) another scalarization? Provide the precise formula(s) and justify why this is the appropriate accuracy metric for fuzzy predictions. How exactly were Tanaka’s minimum-fuzziness and the LS-based fuzzy regression implemented (software/package, constraint formulation, solver settings, handling of infeasibility/divergence)? The manuscript should briefly position its fixed-point TrFN fuzzy regression (with existence/uniqueness/convergence guarantees) against the mainstream applied fuzzy-logic practice in concrete materials research, where rule-based fuzzy inference systems are integrated with experimental bond-test programs. In particular, the recent studies “Slant Shear Tests and Fuzzy Logic Integration for Evaluating Shear Bond Strength in SCC and FRSCC Repair Applications (2025)” and “Integrating push-out test validation and fuzzy logic for bond strength study of fiber-reinforced self-compacting concrete (2024)” exemplify how fuzzy logic is used for SCC/FRSCC bond-strength prediction and test-method validation; the authors should add a short paragraph clarifying what their approach adds relative to such FIS-based studies (and what it does not), and whether the framework is readily extendable beyond compressive strength to bond-related targets. Precisely define “ambiguity” (e.g., average coefficient trapezoid width? output interval width? aggregated across coefficients?) and give the computation formula. Clarify whether Table 7 width values refer to coefficients or predicted outputs, and whether widths are measured at α=0 , α=1 , or across all α . Tables show fuzzification parameters for only some variables in one table while another includes additional variables. Provide a complete fuzzification table covering all predictors and the response, with sources/justifications for each δ 1 δ 2 . Explicitly state whether the response is fuzzified during training and how. The manuscript contains systematic reference-number mismatches (e.g., key foundational citations appear misaligned with the bibliography). This must be fully corrected so every in-text citation points to the correct source; also ensure standards referenced in text (ACI documents, ASTM standards) are properly and consistently cited. Decision Minor Revision The manuscript’s core idea—casting fuzzy regression estimation as a contraction mapping in a complete TrFN d ∞ -metric space with a unique fixed point—is novel and technically promising, and the reported empirical outcomes are consistent with the stated objectives. Nevertheless, several issues must be corrected before the work is suitable for indexing. Addressing the numbered items above should be sufficient to make the manuscript verifiable, technically unambiguous, and editorially sound. ===== · Is the work clearly and accurately presented and does it cite the current literature? Partly (presentation is generally structured, but citation-numbering integrity and some clarity issues must be corrected; literature coverage is mostly current but referencing is unreliable until fixed). · Is the study design appropriate and is the work technically sound? Yes (the theoretical framework and empirical benchmarking design are appropriate; the core approach appears technically sound). · Are sufficient details of methods and analysis provided to allow replication by others? Partly (key elements—explicit definition of T , exact loss/MSE computation for fuzzy outputs, preprocessing/scaling choices, baseline implementation details, and statistical testing protocol—need clearer specification). · If applicable, is the statistical analysis and its interpretation appropriate? Partly (reporting p-values is positive, but the protocol generating them—repetitions/splits/tests—must be explicitly documented to validate interpretation). · Are all the source data underlying the results available to ensure full reproducibility? Partly (a data package is indicated, but full reproducibility requires confirmation that code, preprocessing/fuzzification scripts, and baseline solver settings are available; otherwise it remains partial). · Are the conclusions drawn adequately supported by the results? Yes (within the scope of the presented dataset and tests, the conclusions align with the reported results; claims should be carefully scoped to avoid overgeneralization). Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Concrete, Numerical modeling, AI (Machine learning and Deep Learning), Fuzzy system, I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Shafaie V. Peer Review Report For: Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application [version 1; peer review: 1 approved, 1 approved with reservations] . F1000Research 2025, 14 :1435 ( https://doi.org/10.5256/f1000research.191573.r448782) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1435/v1#referee-response-448782 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Stay Updated Sign up for content alerts and receive a weekly or monthly email with all newly published articles Register with F1000Research Already registered? Sign in Not now, thanks close PLEASE NOTE If you are an AUTHOR of this article, please check that you signed in with the account associated with this article otherwise we cannot automatically identify your role as an author and your comment will be labelled as a “User Comment”. If you are a REVIEWER of this article, please check that you have signed in with the account associated with this article and then go to your account to submit your report, please do not post your review here. If you do not have access to your original account, please contact us . All commenters must hold a formal affiliation as per our Policies . The information that you give us will be displayed next to your comment. User comments must be in English, comprehensible and relevant to the article under discussion. We reserve the right to remove any comments that we consider to be inappropriate, offensive or otherwise in breach of the User Comment Terms and Conditions . Commenters must not use a comment for personal attacks. When criticisms of the article are based on unpublished data, the data should be made available. I accept the User Comment Terms and Conditions Please confirm that you accept the User Comment Terms and Conditions. Affiliation ✕ refresh Please enter your institution. Note: To add your institution or organisation, start typing the name and then select the correct name from the list. Where applicable, the name will appear in both the original language and in English. Do not paste in the name. If the name does not appear in the drop-down list, we will display the information you have entered. ✕ refresh Country/Region * USA UK Canada China France Germany Afghanistan Aland Islands Albania Algeria American Samoa Andorra Angola Anguilla Antarctica Antigua and Barbuda Argentina Armenia Aruba Australia Austria Azerbaijan Bahamas Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia and Herzegovina Botswana Bouvet Island Brazil British Indian Ocean Territory British Virgin Islands Brunei Bulgaria Burkina Faso Burundi Cambodia Cameroon Canada Cape Verde Cayman Islands Central African Republic Chad Chile China Christmas Island Cocos (Keeling) Islands Colombia Comoros Congo Cook Islands Costa Rica Cote d'Ivoire Croatia Cuba Cyprus Czech Republic Democratic Republic of the Congo Denmark Djibouti Dominica Dominican Republic Ecuador Egypt El Salvador Equatorial Guinea Eritrea Estonia Ethiopia Falkland Islands Faroe Islands Federated States of Micronesia Fiji Finland France French Guiana French Polynesia French Southern Territories Gabon Georgia Germany Ghana Gibraltar Greece Greenland Grenada Guadeloupe Guam Guatemala Guernsey Guinea Guinea-Bissau Guyana Haiti Heard Island and Mcdonald Islands Holy See (Vatican City State) Honduras Hong Kong Hungary Iceland India Indonesia Iran Iraq Ireland Israel Italy Jamaica Japan Jersey Jordan Kazakhstan Kenya Kiribati Kosovo (Serbia and Montenegro) Kuwait Kyrgyzstan Lao People's Democratic Republic Latvia Lebanon Lesotho Liberia Libya Liechtenstein Lithuania Luxembourg Macao Madagascar Malawi Malaysia Maldives Mali Malta Marshall Islands Martinique Mauritania Mauritius Mayotte Mexico Minor Outlying Islands of the United States Moldova Monaco Mongolia Montenegro Montserrat Morocco Mozambique Myanmar Namibia Nauru Nepal Netherlands Antilles New Caledonia New Zealand Nicaragua Niger Nigeria Niue Norfolk Island North Korea North Macedonia Northern Mariana Islands Norway Oman Pakistan Palau Palestinian Territory Panama Papua New Guinea Paraguay Peru Philippines Pitcairn Poland Portugal Puerto Rico Qatar Reunion Romania Russian Federation Rwanda Saint Helena Saint Kitts and Nevis Saint Lucia Saint Pierre and Miquelon Saint Vincent and the Grenadines Samoa San Marino Sao Tome and Principe Saudi Arabia Senegal Serbia Seychelles Sierra Leone Singapore Slovakia Slovenia Solomon Islands Somalia South Africa South Georgia and the South Sandwich Is South Korea South Sudan Spain Sri Lanka Sudan Suriname Svalbard and Jan Mayen Swaziland Sweden Switzerland Syria Taiwan Tajikistan Tanzania Thailand The Gambia The Netherlands Timor-Leste Togo Tokelau Tonga Trinidad and Tobago Tunisia Turkey Turkmenistan Turks and Caicos Islands Tuvalu UK USA Uganda Ukraine United Arab Emirates United States Virgin Islands Uruguay Uzbekistan Vanuatu Venezuela Vietnam Wallis and Futuna West Bank and Gaza Strip Western Sahara Yemen Zambia Zimbabwe Please select your country/region. You must enter a comment. Competing Interests Please disclose any competing interests that might be construed to influence your judgment of the article's or peer review report's validity or importance. Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. You hope/expect to benefit (e.g. favour or employment) as a result of your submission. You are an Editor for the journal in which the article is published. Examples of 'Financial Competing Interests' You expect to receive, or in the past 4 years have received, any of the following from any commercial organisation that may gain financially from your submission: a salary, fees, funding, reimbursements. You expect to receive, or in the past 4 years have received, shared grant support or other funding with any of the authors. You hold, or are currently applying for, any patents or significant stocks/shares relating to the subject matter of the paper you are commenting on. Please state your competing interests The comment has been saved. An error has occurred. Please try again. Cancel Post var lTitle = "Employing Fixed-Point Theory for Fuzzy Regression...".replace("'", ''); var linkedInUrl = "http://www.linkedin.com/shareArticle?url=https://f1000research.com/articles/14-1435/v1" + "&title=" + encodeURIComponent(lTitle) + "&summary=" + encodeURIComponent('Read the article by '); var deliciousUrl = "https://del.icio.us/post?url=https://f1000research.com/articles/14-1435/v1&title=" + encodeURIComponent(lTitle); var redditUrl = "http://reddit.com/submit?url=https://f1000research.com/articles/14-1435/v1" + "&title=" + encodeURIComponent(lTitle); linkedInUrl += encodeURIComponent('Malik Jasim N and K. Abdalrahem M'); var offsetTop = /chrome/i.test( navigator.userAgent ) ? 4 : -10; var addthis_config = { ui_offset_top: offsetTop, services_compact : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_expanded : "facebook,twitter,www.linkedin.com,www.mendeley.com,reddit.com", services_custom : [ { name: "LinkedIn", url: linkedInUrl, icon:"/img/icon/at_linkedin.svg" }, { name: "Mendeley", url: "http://www.mendeley.com/import/?url=https://f1000research.com/articles/14-1435/v1/mendeley", icon:"/img/icon/at_mendeley.svg" }, { name: "Reddit", url: redditUrl, icon:"/img/icon/at_reddit.svg" }, ] }; var addthis_share = { url: "https://f1000research.com/articles/14-1435", templates : { twitter : "Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology.... Malik Jasim N and K. Abdalrahem M, published by " + "@F1000Research" + ", https://f1000research.com/articles/14-1435/v1" } }; if (typeof(addthis) != "undefined"){ addthis.addEventListener('addthis.ready', checkCount); addthis.addEventListener('addthis.menu.share', checkCount); } $(".f1r-shares-twitter").attr("href", "https://twitter.com/intent/tweet?text=" + addthis_share.templates.twitter); $(".f1r-shares-facebook").attr("href", "https://www.facebook.com/sharer/sharer.php?u=" + addthis_share.url); $(".f1r-shares-linkedin").attr("href", addthis_config.services_custom[0].url); $(".f1r-shares-reddit").attr("href", addthis_config.services_custom[2].url); $(".f1r-shares-mendelay").attr("href", addthis_config.services_custom[1].url); function checkCount(){ setTimeout(function(){ $(".addthis_button_expanded").each(function(){ var count = $(this).text(); if (count !== "" && count != "0") $(this).removeClass("is-hidden"); else $(this).addClass("is-hidden"); }); }, 1000); } close How to cite this report {{reportCitation}} Cancel Copy Citation Details $(function(){R.ui.buttonDropdowns('.dropdown-for-downloads');}); $(function(){R.ui.toolbarDropdowns('.toolbar-dropdown-for-downloads');}); $.get("/articles/acj/173734/191573") new F1000.Clipboard(); new F1000.ThesaurusTermsDisplay("articles", "article", "191573"); $(document).ready(function() { $( "#frame1" ).on('load', function() { var mydiv = $(this).contents().find("div"); var h = mydiv.height(); console.log(h) }); var tooltipLivingFigure = jQuery(".interactive-living-figure-label .icon-more-info"), titleLivingFigure = tooltipLivingFigure.attr("title"); tooltipLivingFigure.simpletip({ fixed: true, position: ["-115", "30"], baseClass: 'small-tooltip', content:titleLivingFigure + " " }); tooltipLivingFigure.removeAttr("title"); $("body").on("click", ".cite-living-figure", function(e) { e.preventDefault(); var ref = $(this).attr("data-ref"); $(this).closest(".living-figure-list-container").find("#" + ref).fadeIn(200); }); $("body").on("click", ".close-cite-living-figure", function(e) { e.preventDefault(); $(this).closest(".popup-window-wrapper").fadeOut(200); }); $(document).on("mouseup", function(e) { var metricsContainer = $(".article-metrics-popover-wrapper"); if (!metricsContainer.is(e.target) && metricsContainer.has(e.target).length === 0) { $(".article-metrics-close-button").click(); } }); var articleId = $('#articleId').val(); if($("#main-article-count-box").attachArticleMetrics) { $("#main-article-count-box").attachArticleMetrics(articleId, { articleMetricsView: true }); } }); var figshareWidget = $(".new_figshare_widget"); if (figshareWidget.length > 0) { window.figshare.load("f1000", function(Widget) { // Select a tag/tags defined in your page. In this tag we will place the widget. _.map(figshareWidget, function(el){ var widget = new Widget({ articleId: $(el).attr("figshare_articleId") //height:300 // this is the height of the viewer part. [Default: 550] }); widget.initialize(); // initialize the widget widget.mount(el); // mount it in a tag that's on your page // this will save the widget on the global scope for later use from // your JS scripts. This line is optional. //window.widget = widget; }); }); } close Error Close Add Reset F1000.MICROSERVICES.AFFILIATION = ''; $(document).ready(function () { $('.js-affiliations-form').each((index, form) => { new AffiliationForm({ formId: form.id, institutionErrorSelector: '.comment-enter-institution', departmentErrorSelector: '.comment-enter-department', placeSelector: '.js-add-comment-place', stateSelector: '.js-add-comment-state', zipCodeSelector: '.js-add-comment-zipcode', countrySelector: '.js-add-comment-country', countryErrorSelector: '.comment-enter-country', }); }); }); $(document).ready(function () { var reportIds = { "448782": 5, "448783": 0, "448780": 0, "448781": 0, "448778": 0, "448779": 0, "448786": 0, "448787": 0, "448784": 0, "448785": 0, "444766": 0, "446750": 0, "444767": 0, "446751": 0, "444764": 0, "444765": 0, "446749": 0, "444762": 0, "444763": 0, "444760": 0, "444761": 0, "446758": 0, "450662": 0, "450663": 0, "446756": 0, "450660": 0, "446757": 0, "450661": 0, "446754": 0, "450658": 0, "446755": 0, "450659": 0, "444768": 0, "446752": 0, "444769": 0, "446753": 0, "450666": 4, "450667": 0, "450664": 0, "450665": 0, }; $(".referee-response-container,.js-referee-report").each(function(index, el) { var reportId = $(el).attr("data-reportid"), reportCount = reportIds[reportId] || 0; $(el).find(".comments-count-container,.js-referee-report-views").html(reportCount); }); var uuidInput = $("#article_uuid"), oldUUId = uuidInput.val(), newUUId = "231f2b51-6e3d-4567-a982-d35a9b6c0b37"; uuidInput.val(newUUId); $("a[href*='article_uuid=']").each(function(index, el) { var newHref = $(el).attr("href").replace(oldUUId, newUUId); $(el).attr("href", newHref); }); }); An innovative open access publishing platform offering rapid publication and open peer review, whilst supporting data deposition and sharing. Browse Gateways Collections How it Works Contact For Developers Cookie Notice Privacy Notice RSS Submit Your Research Follow us © 2012-2026 F1000 Research Ltd. ISSN 2046-1402 | Legal | Partner of Research4Life • CrossRef • ORCID • FAIRSharing R.templateTests.simpleTemplate = R.template(' $text $text $text $text $text '); R.templateTests.runTests(); var F1000platform = new F1000.Platform({ name: "f1000research", displayName: "F1000Research", hostName: "f1000research.com", id: "1", editorialEmail: "[email protected]", infoEmail: "[email protected]", usePmcStats: true }); $(function(){R.ui.dropdowns('.dropdown-for-authors, .dropdown-for-about, .dropdown-for-myresearch');}); // $(function(){R.ui.dropdowns('.dropdown-for-referees');}); $(document).ready(function () { if ($(".cookie-warning").is(":visible")) { $(".sticky").css("margin-bottom", "35px"); $(".devices").addClass("devices-and-cookie-warning"); } $(".cookie-warning .close-button").click(function (e) { $(".devices").removeClass("devices-and-cookie-warning"); $(".sticky").css("margin-bottom", "0"); }); $("#tweeter-feed .tweet-message").each(function (i, message) { var self = $(message); self.html(linkify(self.html())); }); $(".partner").on("mouseenter mouseleave", function() { $(this).find(".gray-scale, .colour").toggleClass("is-hidden"); }); }); Sign In Remember me Forgotten your password? Sign In Cancel Email or password not correct. Please try again Please wait... $(function(){ // Note: All the setup needs to run against a name attribute and *not* the id due the clonish // nature of facebox... $("a[id=googleSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("GOOGLE"); $("form[id=oAuthForm]").submit(); }); $("a[id=facebookSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("FACEBOOK"); $("form[id=oAuthForm]").submit(); }); $("a[id=orcidSignInButton]").click(function(event){ event.preventDefault(); $("input[id=oAuthSystem]").val("ORCID"); $("form[id=oAuthForm]").submit(); }); }); If you've forgotten your password, please enter your email address below and we'll send you instructions on how to reset your password. The email address should be the one you originally registered with F1000. Email address not valid, please try again You registered with F1000 via Google, so we cannot reset your password. To sign in, please click here . If you still need help with your Google account password, please click here . You registered with F1000 via Facebook, so we cannot reset your password. To sign in, please click here . If you still need help with your Facebook account password, please click here . Code not correct, please try again Reset password Cancel Email us for further assistance. Server error, please try again. If your email address is registered with us, we will email you instructions to reset your password. If you think you should have received this email but it has not arrived, please check your spam filters and/or contact for further assistance. Please wait... Register $(document).ready(function () { signIn.createSignInAsRow($("#sign-in-form-gfb-popup")); $(".target-field").each(function () { var uris = $(this).val().split("/"); if (uris.pop() === "login") { $(this).val(uris.toString().replace(",","/")); } }); });