infoeqv 6.0.4

Information Equivalent Designs (Two-Point Reduction)

<img src="https://latex.codecogs.com/svg.image?$$---&hash;&hash;🧠Theory:Information&space;Equivalent&space;Designs&space;in&space;the&space;Cubic&space;Regression&space;Model&space;and&space;the&space;De&space;La&space;Garza&space;Phenomenon&hash;&hash;&hash;🔷Introduction&space;In&space;the&space;theory&space;of&space;optimal&space;experimental&space;designs,the**De&space;La&space;Garza&space;Phenomenon**refers&space;to&space;the&space;striking&space;result&space;that,for&space;polynomial&space;regression&space;models&space;of&space;degree$$p$$,any&space;optimal&space;design(with&space;respect&space;to&space;the&space;Loewner&space;ordering&space;of&space;information&space;matrices)can&space;be&space;replaced&space;by&space;a&space;design&space;with&space;at&space;most$$p&plus;1$$support&space;points.This&space;significantly&space;simplifies&space;the&space;search&space;for&space;optimal&space;designs.For&space;a**cubic&space;regression&space;model**,this&space;means&space;that&space;although&space;one&space;may&space;start&space;with&space;a&space;design&space;involving&space;many&space;design&space;points,an&space;equivalent&space;design&space;with&space;just**four&space;support&space;points**(or&space;even&space;fewer&space;under&space;special&space;symmetry&space;conditions)can&space;exist&space;and&space;be**Loewner&space;superior&space;or&space;equivalent**in&space;terms&space;of&space;information&space;content.&hash;&hash;&hash;🔷Cubic&space;Regression&space;Model&space;Consider&space;the&space;cubic&space;polynomial&space;regression&space;model&space;on&space;the&space;interval$$[-1,1]$$:$$y=\beta_0&plus;\beta_1&space;x&plus;\beta_2&space;x^2&plus;\beta_3&space;x^3&plus;\varepsilon$$where$$\varepsilon\sim&space;N(0,\sigma^2)$$,and$$x\in[-1,1]$$is&space;the&space;design&space;variable.For&space;each&space;design&space;point$$x_i$$,the&space;information&space;matrix&space;contribution&space;is:$$f(x_i)f(x_i)^T\quad\text{where}\quad&space;f(x_i)=[1,x_i,x_i^2,x_i^3]^T$$The&space;overall&space;information&space;matrix&space;of&space;a&space;design$$D_n=\{x_1,\dots,x_n;f_1,\dots,f_n\}$$with&space;frequencies$$f_i$$is:$$M(D_n)=\sum_{i=1}^n&space;f_i\cdot&space;f(x_i)f(x_i)^T$$&hash;&hash;&hash;🔷De&space;La&space;Garza&space;Phenomenon&space;for&space;Cubic&space;Models&space;According&space;to&space;the**De&space;La&space;Garza&space;Phenomenon**,for&space;a&space;cubic&space;model(degree&space;3),a&space;design&space;supported&space;on**more&space;than&space;four&space;points**can&space;be**dominated&space;in&space;the&space;Loewner&space;order**by&space;a**four-point&space;design**,possibly&space;symmetric,with&space;equivalent&space;or&space;better&space;information.The**Loewner&space;order**$$A\succeq&space;B$$means&space;that$$A-B$$is**positive&space;semi-definite**,implying$$A$$contains&space;more&space;or&space;equal&space;information&space;than$$B$$.&hash;&hash;&hash;🔷Information&space;Equivalent&space;Design&space;An**Information&space;Equivalent&space;Design**is&space;a&space;reduced&space;design(often&space;with&space;fewer&space;support&space;points)that&space;matches&space;the&space;original&space;design&space;in&space;key&space;information&space;metrics&space;typically&space;the**moments**of&space;the&space;design.For&space;symmetric&space;designs&space;on$$[-1,1]$$,the**odd&space;central&space;moments&space;vanish**,simplifying&space;the&space;comparison.The&space;relevant**even-order&space;central&space;moments**are:*$$\mu_2=\frac{1}{n}\sum&space;f_i&space;x_i^2$$*$$\mu_4=\frac{1}{n}\sum&space;f_i&space;x_i^4$$*$$\mu_6=\frac{1}{n}\sum&space;f_i&space;x_i^6$$These&space;moments&space;define&space;the&space;key&space;entries&space;of&space;the**Fisher&space;Information&space;Matrix**for&space;the&space;cubic&space;model.&hash;&hash;&hash;🔷Theorem:Two-Point&space;Information&space;Equivalent&space;Design**Given**:*A&space;symmetric&space;design$$D_n$$with$$n=2k$$total&space;frequency.*Known&space;second-order&space;moment$$\mu_2'$$.*You&space;seek&space;a&space;2-point&space;symmetric&space;design$$D_4^*$$of&space;the&space;form:<img&space;src="https://latex.codecogs.com/svg.image?$$D_4^*=\left\{(\pm&space;a^*,\frac{n-2f}{2}),(\pm&space;b^*,f)\right\},\quad-1\leq&space;a^*title="$$D_4^*=\left\{(\pm&space;a^*,\frac{n-2f}{2}),(\pm&space;b^*,f)\right\},\quad-1\leq&space;a^*/>**Then**,under&space;the&space;following&space;conditions:1.$$k(1-\mu_2')<f<k$$2.$$f&space;b^{2}<k\mu_2'$$and$$\mu_2'<b^{2}<1$$3.$$f\mu_2'^3\geq&space;k\mu_6'$$thereexistssuch&space;a&space;design$$D_4^$$that&space;isinformation&space;equivalentto$$D_n$$and&space;satisfies:$$\text{Moment&space;2(quadratic):}\quad(n-2f)a^2+2f&space;b^2=n\mu_2'$$$$\text{Moment&space;4(quartic):}\quad(n-2f)a^4+2f&space;b^4=n\mu_4'$$$$\text{Moment&space;6(sextic):}\quad(n-2f)a^6+2f&space;b^6=n\mu_6'$$Thus,thistwo-point&space;design&space;dominatesthe&space;original&space;multivariate&space;design&space;in&space;theLoewner&space;orderof&space;information&space;matrices.&hash;&hash;&hash;🔷SignificanceSimplifies&space;implementation&space;in&space;real-world&space;designs:only&space;two&space;support&space;points&space;needed.Shows&space;that&space;information&space;redundancy&space;exists&space;in&space;symmetric&space;high-point&space;designs.Key&space;incomputational&space;efficiencyand&space;in&space;theoretical&space;understanding&space;ofoptimal&space;design&space;spaces.&hash;&hash;&hash;🔷ApplicationsEngineering&space;experimentsinvolving&space;polynomial&space;approximations.Cricket&space;pitch&space;analysis(predicting&space;ball&space;trajectory).Educational&space;psychology,e.g.,modeling&space;learning&space;curves.Agriculture,especially&space;in&space;response&space;surface&space;methodology&space;for&space;input-output&space;modeling.&hash;&hash;🧠About&space;the&space;PackageinfoeqvThe&space;Python&space;packageinfoeqvimplements&space;the&space;construction&space;ofInformation&space;Equivalent&space;Designsfor&space;thecubic&space;regression&space;model*,based&space;on&space;a&space;theoretical&space;result&space;that&space;simplifiesn-point&space;symmetric&space;designs&space;into&space;2-or&space;4-point&space;designs,often&space;achieving&space;equal&space;or&space;better&space;statistical&space;efficiency.This&space;is&space;aligned&space;with&space;the&space;classicalDe&space;La&space;Garza&space;phenomenon,which&space;states&space;that&space;under&space;certain&space;regression&space;models,optimal&space;designs&space;use&space;fewer&space;support&space;points.The&space;core&space;utility&space;of&space;the&space;package&space;is&space;to&space;automate&space;the&space;process&space;of:Validating&space;a&space;given&space;symmetric&space;design(user-definedxandf)Computing&space;information&space;equivalenttwo-point&space;designsChecking&space;if&space;these&space;designs&space;satisfy&space;moment-matching&space;conditions&space;andLoewner&space;dominationDisplaying&space;results&space;graphically&space;and&space;numerically&space;This&space;tool&space;can&space;be&space;useful&space;forstatisticians,engineers,data&space;scientists,and&space;researchers&space;working&space;inoptimal&space;experimental&space;designorregression&space;modeling.---&hash;&hash;🔧Core&space;Function:info_eqv_design(x,f)This&space;is&space;themain&space;functionexported&space;by&space;theinfoeqvpackage.It&space;accepts&space;two&space;arguments:x:A&space;1D&space;list&space;or&space;array&space;of&space;symmetric&space;design&space;points&space;in&space;the&space;interval[-1,1]f:A&space;1D&space;list&space;or&space;array&space;of&space;their&space;corresponding&space;frequencies(must&space;be&space;even&space;in&space;total)python&space;from&space;infoeqv&space;import&space;info_eqv_design&space;x=[-0.8,-0.4,0.4,0.8]f=[4,3,3,4]info_eqv_design(x,f)When&space;executed,this&space;function:1.Validates&space;inputto&space;ensure&space;symmetry&space;and&space;proper&space;frequency&space;format.2.Standardizesdesign&space;for&space;analysis&space;within&space;the&space;symmetric&space;interval.3.Computes&space;moments,,.4.Iteratively&space;searches&space;for&space;validtwo-point&space;information&space;equivalent&space;designs(a*,b*)using&space;the&space;theoretical&space;formulas.5.For&space;each&space;design,checks:Condition(i):Second&space;moment&space;feasibility&space;and&space;bounds&space;onfCondition(ii):Bounds&space;onbCondition(iii):Sixth&space;moment&space;inequality(dominance)6.Displays&space;output:A&space;table&space;summarizing&space;valid/invalid&space;design&space;pointsA&space;graph&space;showing&space;the&space;conditions&space;met&space;by&space;each&space;candidate---&hash;&hash;🧪How&space;the&space;Theory&space;Was&space;Translated&space;into&space;Code&space;The&space;theoretical&space;foundations&space;involve&space;solving&space;three&space;nonlinear&space;moment&space;equations(second,fourth,and&space;sixth)under&space;symmetry,and&space;checking&space;if&space;the&space;reduced&space;design&space;dominates&space;the&space;original&space;in&space;theLoewner&space;sense.The&space;following&space;components&space;were&space;used:&hash;&hash;&hash;🧩Algorithmic&space;Implementation(Step-by-step):Moment&space;EquationsUsing&space;frequency-weighted&space;formulas,the&space;code&space;computes:$$=E[X],=E[X],=E[X]$$Candidate&space;SearchIterate&space;over&space;possiblefvalues(usingceil,floor)that&space;satisfy:$$k(1-\mu_2')<f<k$$$$fb^{2}<k\mu_2'$$$$\mu_2'<b^{2}<1$$Design&space;ConstructionUsing:$$a^=\pm\sqrt{\frac{k\mu_2'-f&space;b^{2}}{k-f}}$$where$$a<b\in[-1,1]$$VerificationEach&space;candidate&space;is&space;checked&space;against:(i)Lower&space;and&space;upper&space;bounds&space;forf(ii)Condition&space;onbrange(iii)$$f\mu_2'^3\geq&space;k\mu_6'$$PlottingMatplotlib&space;is&space;used&space;to&space;visually&space;indicate&space;which&space;candidates&space;satisfy&space;each&space;subset&space;of&space;the&space;theorem's&space;conditions.---&hash;&hash;📦Installation&space;Instructions&space;You&space;can&space;install&space;theinfoeqvpackage&space;via&space;pip&space;once&space;it&space;is&space;published&space;to&space;PyPI(or&space;install&space;directly&space;from&space;a&space;local&space;folder&space;for&space;testing):&hash;&hash;&hash;🔹Option&space;1:From&space;PyPI(after&space;publishing)bash&space;pip&space;install&space;infoeqv&hash;&hash;&hash;🔹Option&space;2:From&space;a&space;Local&space;Folder&space;If&space;you're&space;still&space;testing&space;the&space;package&space;locally:bash&space;pip&space;install.&hash;&hash;🛠Usage&space;Example&space;Once&space;installed,you&space;can&space;use&space;the&space;functioninfo_eqv_design()to&space;analyze&space;a&space;symmetric&space;design:python&hash;Example:symmetric&space;cubic&space;regression&space;design&space;with&space;even&space;frequency&space;import&space;numpy&space;as&space;np&space;from&space;infoeqv&space;import&space;info_eqv_design&space;x=[-1.5,-1,-0.5,0,0.5,1,1.5]f=[2,3,5,6,5,3,2]info_eqv_design(x,f)``````python&space;Original&space;x&space;values:[-1.5-1.-0.5&space;0.0.5&space;1.1.5]Frequency&space;f&space;values:[2.3.5.6.5.3.2.]Total&space;number&space;of&space;observations(N):26.0&space;Mean&space;of&space;x(x):0.0000&space;Standardized&space;values(d):[-1.-0.6667-0.3333&space;0.0.3333&space;0.6667&space;1.]Weighted&space;mean():0.0000&space;Weighted&space;second&space;moment():0.2991&space;Central&space;moment():0.2991:0.1746&space;Bounds:L=5.9868,U=20.0132&space;Ceiling&space;of&space;L(S):6,Floor&space;of&space;U(T):20&space;Designs&space;satisfying(i)&(ii),but&space;failing(iii):n1&space;n2&space;d1&space;d2&space;Status&space;14&space;12-0.5064&space;0.5908(iii)15&space;11-0.4684&space;0.6387(iii)16&space;10-0.4324&space;0.6918(iii)This&space;will&space;output:The&space;values&space;of&space;moments,,A&space;table&space;of&space;possible(a*,b*)values&space;and&space;correspondingn1,n2allocationsConditions&space;satisfied&space;for&space;each&space;caseA&space;graph&space;showing&space;which&space;points&space;satisfy&space;conditions(i),(ii),and(iii)---&hash;&hash;🧪Dependencies&space;Make&space;sure&space;the&space;following&space;libraries&space;are&space;available(they&space;ll&space;be&space;installed&space;automatically&space;if&space;usingpip&space;install):numpymatplotlibpandas---&hash;&hash;👤AuthorRohit&space;Kumar&space;Behera📍Odisha,India📧\[e-mail]🧪Statistical&space;Python&space;Developer&space;This&space;package&space;and&space;algorithm&space;were&space;created&space;based&space;on&space;original&space;theoretical&space;research&space;onInformation&space;Equivalent&space;Designsfor&space;cubic&space;regression&space;models&space;and&space;implemented&space;as&space;a&space;Python&space;packageinfoeqv.The&space;package&space;can&space;be&space;used&space;as&space;a&space;research&space;tool,teaching&space;aid,or&space;practical&space;software&space;for&space;statisticians&space;and&space;data&space;scientists.---" title="$$---##🧠Theory:Information Equivalent Designs in the Cubic Regression Model and the De La Garza Phenomenon###🔷Introduction In the theory of optimal experimental designs,theDe La Garza Phenomenonrefers to the striking result that,for polynomial regression models of degree$$p$$,any optimal design(with respect to the Loewner ordering of information matrices)can be replaced by a design with at most$$p+1$$support points.This significantly simplifies the search for optimal designs.For acubic regression model,this means that although one may start with a design involving many design points,an equivalent design with justfour support points(or even fewer under special symmetry conditions)can exist and beLoewner superior or equivalentin terms of information content.###🔷Cubic Regression Model Consider the cubic polynomial regression model on the interval$$[-1,1]$$:$$y=\beta_0+\beta_1 x+\beta_2 x^2+\beta_3 x^3+\varepsilon$$where$$\varepsilon\sim N(0,\sigma^2)$$,and$$x\in[-1,1]$$is the design variable.For each design point$$x_i$$,the information matrix contribution is:$$f(x_i)f(x_i)^T\quad\text{where}\quad f(x_i)=[1,x_i,x_i^2,x_i^3]^T$$The overall information matrix of a design$$D_n={x_1,\dots,x_n;f_1,\dots,f_n}$$with frequencies$$f_i$$is:$$M(D_n)=\sum_{i=1}^n f_i\cdot f(x_i)f(x_i)^T$$###🔷De La Garza Phenomenon for Cubic Models According to theDe La Garza Phenomenon,for a cubic model(degree 3),a design supported onmore than four pointscan bedominated in the Loewner orderby afour-point design,possibly symmetric,with equivalent or better information.TheLoewner order$$A\succeq B$$means that$$A-B$$ispositive semi-definite,implying$$A$$contains more or equal information than$$B$$.###🔷Information Equivalent Design AnInformation Equivalent Designis a reduced design(often with fewer support points)that matches the original design in key information metrics typically themomentsof the design.For symmetric designs on$$[-1,1]$$,theodd central moments vanish,simplifying the comparison.The relevanteven-order central momentsare:$$\mu_2=\frac{1}{n}\sum f_i x_i^2$$$$\mu_4=\frac{1}{n}\sum f_i x_i^4$$$$\mu_6=\frac{1}{n}\sum f_i x_i^6$$These moments define the key entries of theFisher Information Matrixfor the cubic model.###🔷Theorem:Two-Point Information Equivalent DesignGiven:A symmetric design$$D_n$$with$$n=2k$$total frequency.Known second-order moment$$\mu_2'$$.You seek a 2-point symmetric design$$D_4^$$of the form:<img src="https://latex.codecogs.com/svg.image?$$D_4^*=\left\{(\pm&space;a^*,\frac{n-2f}{2}),(\pm&space;b^*,f)\right\},\quad-1\leq&space;a^*title="$$D_4^*=\left\{(\pm a^,\frac{n-2f}{2}),(\pm b^,f)\right},\quad-1\leq a^/>Then,under the following conditions:1.$$k(1-\mu_2')<f<k$$2.$$f b^{2}<k\mu_2'$$and$$\mu_2'<b^{2}<1$$3.$$f\mu_2'^3\geq k\mu_6'$$thereexistssuch a design$$D_4^$$that isinformation equivalentto$$D_n$$and satisfies:$$\text{Moment 2(quadratic):}\quad(n-2f)a^2+2f b^2=n\mu_2'$$$$\text{Moment 4(quartic):}\quad(n-2f)a^4+2f b^4=n\mu_4'$$$$\text{Moment 6(sextic):}\quad(n-2f)a^6+2f b^6=n\mu_6'$$Thus,thistwo-point design dominatesthe original multivariate design in theLoewner orderof information matrices.###🔷SignificanceSimplifies implementation in real-world designs:only two support points needed.Shows that information redundancy exists in symmetric high-point designs.Key incomputational efficiencyand in theoretical understanding ofoptimal design spaces.###🔷ApplicationsEngineering experimentsinvolving polynomial approximations.Cricket pitch analysis(predicting ball trajectory).Educational psychology,e.g.,modeling learning curves.Agriculture,especially in response surface methodology for input-output modeling.##🧠About the PackageinfoeqvThe Python packageinfoeqvimplements the construction ofInformation Equivalent Designsfor thecubic regression model,based on a theoretical result that simplifiesn-point symmetric designs into 2-or 4-point designs,often achieving equal or better statistical efficiency.This is aligned with the classicalDe La Garza phenomenon,which states that under certain regression models,optimal designs use fewer support points.The core utility of the package is to automate the process of:Validating a given symmetric design(user-definedxandf)Computing information equivalenttwo-point designsChecking if these designs satisfy moment-matching conditions andLoewner dominationDisplaying results graphically and numerically This tool can be useful forstatisticians,engineers,data scientists,and researchers working inoptimal experimental designorregression modeling.---##🔧Core Function:info_eqv_design(x,f)This is themain functionexported by theinfoeqvpackage.It accepts two arguments:x:A 1D list or array of symmetric design points in the interval[-1,1]f:A 1D list or array of their corresponding frequencies(must be even in total)python from infoeqv import info_eqv_design x=[-0.8,-0.4,0.4,0.8]f=[4,3,3,4]info_eqv_design(x,f)When executed,this function:1.Validates inputto ensure symmetry and proper frequency format.2.Standardizesdesign for analysis within the symmetric interval.3.Computes moments,,.4.Iteratively searches for validtwo-point information equivalent designs(a*,b*)using the theoretical formulas.5.For each design,checks:*Condition(i):Second moment feasibility and bounds onf*Condition(ii):Bounds onbCondition(iii):Sixth moment inequality(dominance)6.Displays output:A table summarizing valid/invalid design pointsA graph showing the conditions met by each candidate---##🧪How the Theory Was Translated into Code The theoretical foundations involve solving three nonlinear moment equations(second,fourth,and sixth)under symmetry,and checking if the reduced design dominates the original in theLoewner sense.The following components were used:###🧩Algorithmic Implementation(Step-by-step):Moment EquationsUsing frequency-weighted formulas,the code computes:$$=E[X],=E[X],=E[X]$$Candidate SearchIterate over possiblefvalues(usingceil,floor)that satisfy:$$k(1-\mu_2')<f<k$$$$fb^{2}<k\mu_2'$$$$\mu_2'<b^{*2}<1$$Design ConstructionUsing:$$a^=\pm\sqrt{\frac{k\mu_2'-f b^{2}}{k-f}}$$where$$a<b\in[-1,1]$$VerificationEach candidate is checked against:(i)Lower and upper bounds forf(ii)Condition onbrange(iii)$$f\mu_2'^3\geq k\mu_6'$$PlottingMatplotlib is used to visually indicate which candidates satisfy each subset of the theorem's conditions.---##📦Installation Instructions You can install theinfoeqvpackage via pip once it is published to PyPI(or install directly from a local folder for testing):###🔹Option 1:From PyPI(after publishing)bash pip install infoeqv###🔹Option 2:From a Local Folder If you're still testing the package locally:bash pip install.##🛠Usage Example Once installed,you can use the functioninfo_eqv_design()to analyze a symmetric design:python#Example:symmetric cubic regression design with even frequency import numpy as np from infoeqv import info_eqv_design x=[-1.5,-1,-0.5,0,0.5,1,1.5]f=[2,3,5,6,5,3,2]info_eqv_design(x,f)``````python Original x values:[-1.5-1.-0.5 0.0.5 1.1.5]Frequency f values:[2.3.5.6.5.3.2.]Total number of observations(N):26.0 Mean of x(x):0.0000 Standardized values(d):[-1.-0.6667-0.3333 0.0.3333 0.6667 1.]Weighted mean():0.0000 Weighted second moment():0.2991 Central moment():0.2991:0.1746 Bounds:L=5.9868,U=20.0132 Ceiling of L(S):6,Floor of U(T):20 Designs satisfying(i)&(ii),but failing(iii):n1 n2 d1 d2 Status 14 12-0.5064 0.5908(iii)15 11-0.4684 0.6387(iii)16 10-0.4324 0.6918(iii)This will output:The values of moments,,A table of possible(a*,b*)values and correspondingn1,n2allocationsConditions satisfied for each caseA graph showing which points satisfy conditions(i),(ii),and(iii)---##🧪Dependencies Make sure the following libraries are available(they ll be installed automatically if usingpip install):numpymatplotlibpandas---##👤AuthorRohit Kumar Behera📍Odisha,India📧\[e-mail]🧪Statistical Python Developer This package and algorithm were created based on original theoretical research onInformation Equivalent Designsfor cubic regression models and implemented as a Python packageinfoeqv.The package can be used as a research tool,teaching aid,or practical software for statisticians and data scientists.---" />