simulateLM
simulateLM simulates linear regression data with prespecified values of statistical indexes.
Description
simulateLM simulates linear regression data. It is possible to specify: - 1) the requested value of R2 (or equivaletly its SNR);
- 2) the values of the beta coefficients (possibly sparse);
- 3) the correlation (covariance) matrix among the explanatory variables.
- 4) the value of the intercept term.
- 5) the distribution to use to generate the Xs;
- 6) the distribution to use to generate the ys.
- 7) the MSOM contamination in Xs and ys.
- 8) the VIOM contamination in ys.
Simulate with prefixed value of R2.out
=simulateLM(n,
Name, Value)
Examples
Use all defaul options.
Use all defaul options.- Simulate 100 observations y and X (uncorrelated with y) using standard normal distribution.
out=simulateLM(100,'plots',true);
Simulate with prefixed value of R2.
Simulate with prefixed value of R2.- Set value of R2;
R2=0.82; -n=10000; -out=simulateLM(n,'R2',R2); -outLM=fitlm(out.X,out.y);
Related Examples
Use prefixed correlation matrix for cov(X).
Use prefixed correlation matrix for cov(X).- Set value of R2;
R2=0.26;
-n=10000;
-A = gallery('moler',5,0.2);
-out=simulateLM(n,'R2',R2,'SigmaX',A);
+
simulateLM
simulateLM
simulateLM simulates linear regression data with pre-specified values of statistical indexes.
Description
simulateLM simulates linear regression data. It is possible to specify:
+ 1) the requested value of R2 (or equivalently its SNR);
+ 2) the values of the beta coefficients (possibly sparse);
+ 3) the correlation (covariance) matrix among the explanatory variables.
+ 4) the value of the intercept term.
+ 5) the distribution to use to generate the Xs;
+ 6) the distribution to use to generate the ys.
+ 7) the MSOM contamination in Xs and ys.
+ 8) the VIOM contamination in ys.
out
=simulateLM(n,
Name, Value)
Simulate with prefixed value of R2.
Examples
Use all default options.
+ Simulate 100 observations y and X (uncorrelated with y) using standard normal distribution.
out=simulateLM(100,'plots',true);
Simulate with prefixed value of R2.
+ Set value of R2;
R2=0.82;
+n=10000;
+out=simulateLM(n,'R2',R2);
+outLM=fitlm(out.X,out.y);
Related Examples
Use prefixed correlation matrix for cov(X).
+ Set value of R2;
R2=0.26;
+n=10000;
+A = gallery('moler',5,0.2);
+out=simulateLM(n,'R2',R2,'SigmaX',A);
outLM=fitlm(out.X,out.y)
outLM =
@@ -29,139 +24,147 @@
y ~ 1 + x1 + x2 + x3 + x4 + x5
Estimated Coefficients:
- Estimate SE tStat pValue
- _________ ________ _______ __________
+ Estimate SE tStat pValue
+ ________ ________ ______ __________
- (Intercept) -0.076653 0.053898 -1.4222 0.15501
- x1 1.0515 0.056414 18.638 3.0647e-76
- x2 0.92447 0.056001 16.508 2.0145e-60
- x3 1.0394 0.055297 18.797 1.72e-77
- x4 0.92012 0.055248 16.654 1.8903e-61
- x5 1.029 0.054653 18.828 9.8022e-78
+ (Intercept) 0.075868 0.053908 1.4073 0.15936
+ x1 1.1242 0.056285 19.974 4.6082e-87
+ x2 1.0032 0.055788 17.982 3.5569e-71
+ x3 0.95948 0.055435 17.308 3.7476e-66
+ x4 0.97913 0.055195 17.739 2.3933e-69
+ x5 0.98381 0.054149 18.169 1.3412e-72
Number of observations: 10000, Error degrees of freedom: 9994
Root Mean Squared Error: 5.39
-R-squared: 0.253, Adjusted R-Squared: 0.253
-F-statistic vs. constant model: 679, p-value = 0
-
Use prefixed values of R2, beta and intercept.
- Set value of R2.
R2=0.92;
-beta=[3; 4; 5; 2; 7];
-intercept=true;
-n=100000;
-out=simulateLM(n,'R2',R2,'beta',beta);
-outLM=fitlm(out.X,out.y);
Use SNR and include MSOM (on active features) and VIOM contamination
- SNR=3;
- beta=[2, 2, 0, 0];
- intercept=true;
- n=100;
- out=simulateLM(n,'SNR',SNR,'beta',beta, 'pMSOM', 0.
%% Use SNR and include MSOM (on active features) and VIOM contamination
-SNR=3;
-beta=[2, 2, 0, 0];
-intercept=true;
-n=100;
-out=simulateLM(n,'SNR',SNR,'beta',beta, 'pMSOM', 0.1, 'pVIOM', 0.2, 'plots', 1);
-X = out.X;
-y = out.y;
-outLM=fitlm(X,y);
-Xc = out.Xc;
-yc = out.yc;
-outLM2=fitlm(Xc,yc);
Input Arguments
Output Arguments
References
Insolia, L., F. Chiaromonte, and M. Riani (2020a).
- "A Robust Estimation Approach for Mean-Shift and Variance-Inflation Outliers".
+R-squared: 0.26, Adjusted R-Squared: 0.259
+F-statistic vs. constant model: 701, p-value = 0
+
Use prefixed values of R2, beta and intercept.
+ Set value of R2.
R2=0.92;
+beta=[3; 4; 5; 2; 7];
+intercept=true;
+n=100000;
+out=simulateLM(n,'R2',R2,'beta',beta);
+outLM=fitlm(out.X,out.y);
Use SNR and include MSOM (on active features) and VIOM contamination.
%% Use SNR and include MSOM (on active features) and VIOM contamination.
+SNR=3;
+beta=[2, 2, 0, 0];
+intercept=true;
+n=100;
+out=simulateLM(n,'SNR',SNR,'beta',beta, 'pMSOM', 0.1, 'pVIOM', 0.2, 'plots', 1);
+X = out.X;
+y = out.y;
+outLM=fitlm(X,y);
+Xc = out.Xc;
+yc = out.yc;
+outLM2=fitlm(Xc,yc);
Input Arguments
Output Arguments
References
Insolia, L., F. Chiaromonte, and M. Riani (2020a).
+ "A Robust Estimation Approach for Mean-Shift and Variance-Inflation Outliers".
Festschrift in Honor of R. Dennis Cook pp 17–41.
See Also
This page has been automatically generated by our routine publishFS
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diff --git a/toolbox/regression/simulateLM.m b/toolbox/regression/simulateLM.m
index 1f88f279f..f0dd769ac 100644
--- a/toolbox/regression/simulateLM.m
+++ b/toolbox/regression/simulateLM.m
@@ -12,7 +12,7 @@
% 6) the distribution to use to generate the ys.
% 7) the MSOM contamination in Xs and ys.
% 8) the VIOM contamination in ys.
-%
+%
% Required input arguments:
%
% n : sample size. Scalar. n is a positive integer
@@ -22,9 +22,15 @@
% Optional input arguments:
%
% R2 : Squared multiple correlation coefficient (R2). Scalar. The
-% requested value of R2. A number in the
-% interval [0 1] that specifies the requested value of R2.
-% The default is to simulate regression data with R2=0;
+% requested value of R2. A number in the interval [0 1] that
+% specifies the asymptotic requested value of R2. The default is
+% to simulate regression data with R2=0; Note that the value of
+% R2 is the one in the population not in the sample in the sense
+% that if, for example 'R2',00 the sample data (expecially if n
+% is very small) can have a value which is slightly different
+% from the prefixed one. If the exact value of R2 is required
+% then the user has to use option exactR2. See below for further
+% details.
% Example - 'R2',0.90
% Data Types - double
%
@@ -59,7 +65,7 @@
% Data Types - double
% distriby : distribution to use to simulate the response. Character.
% Character that specifies the distribution to use to
-% simulate the values of the explanatory variables. The
+% simulate the values of the response. The
% default is to use the Standard normal distribution.
% Example - 'distriby', 'Lognormal'
% Data Types - double
@@ -71,6 +77,15 @@
% with parameters mu and sigma respectively equal to 2 and 10.
% Example - 'distribypars', '[2 10]'
% Data Types - double
+%
+% exactR2 : exact value of R2. Boolean.
+% If exactR2 is the sample data have the requested value of
+% R2. The default is exactR2 equal to false, that is just
+% asymptotically, the sample data have a value of R2 equal
+% to the one which is specified in option R2.
+% Example - 'exactR2', true
+% Data Types - logical
+%
% nexpl : number of explanatory variables. If vector beta is
% supplied, nexpl is equal to length(beta). Similarly if
% SigmaX is supplied nexpl is set equal to size(SigmaX,1).
@@ -91,33 +106,33 @@
% Example - 'plots',false
% Data Types - single | double
%
-% pMSOM : Proportion of MSOM outliers. The default is 10% MSOM
+% pMSOM : Proportion of MSOM outliers. The default is 10% MSOM
% contamination.
% Example - 'pMSOM',0.25
% Data Types - double
-%
-% pVIOM : Proportion of VIOM outliers (non-overlapping with MSOM).
+%
+% pVIOM : Proportion of VIOM outliers (non-overlapping with MSOM).
% The default is 10% VIOM contamination.
% Example - 'pVIOM',0.25
% Data Types - double
-%
+%
% shiftMSOMe : Mean-shift on the error terms for MSOM outliers.
% Default value shiftMSOMe==10.
% Example - 'shiftMSOMe',-3
% Data Types - double
-%
-% predxMSOM : Predictors subject to a mean shift by MSOM. It is a
+%
+% predxMSOM : Predictors subject to a mean shift by MSOM. It is a
% p-dimensional vector indexing design matrix columns.
% Default value is to contaminate only the non-zero
% entries of beta_true (excluding the intercept).
% Example - 'predxMSOM',true(2,1)
% Data Types - boolean
-%
+%
% shiftMSOMx : Mean-shift on the predictor terms for MSOM outliers.
% Default value shiftMSOMx==10.
% Example - 'shiftMSOMx',3
% Data Types - double
-%
+%
% inflVIOMe : Variance-inflation for the errors subject to a VIOM.
% Default value is inflVIOMe==10.
% Example - 'inflVIOMe',5
@@ -147,8 +162,8 @@
%
% References:
%
-% Insolia, L., F. Chiaromonte, and M. Riani (2020a).
-% "A Robust Estimation Approach for Mean-Shift and Variance-Inflation Outliers".
+% Insolia, L., F. Chiaromonte, and M. Riani (2020a).
+% "A Robust Estimation Approach for Mean-Shift and Variance-Inflation Outliers".
% Festschrift in Honor of R. Dennis Cook pp 17–41.
%
%
@@ -262,7 +277,7 @@
predxMSOM = '';
shiftMSOMx = 10;
inflVIOMe = 10;
-
+exactR2=false;
options=struct('R2',R2,...
'beta',beta,'SigmaX',SigmaX,...
@@ -271,7 +286,7 @@
'nexpl',nexpl,'intercept',intercept,'plots',plots, ...
'SNR', SNR, 'pMSOM', pMSOM, 'pVIOM', pVIOM, ...
'shiftMSOMe', shiftMSOMe, 'predxMSOM', predxMSOM, ...
- 'shiftMSOMx', shiftMSOMx, 'inflVIOMe', inflVIOMe);
+ 'shiftMSOMx', shiftMSOMx, 'inflVIOMe', inflVIOMe,'exactR2',exactR2);
%% User options
@@ -280,12 +295,12 @@
[varargin{:}] = convertStringsToChars(varargin{:});
UserOptions=varargin(1:2:length(varargin));
if ~isempty(UserOptions)
-
+
% Check if number of supplied options is valid
if length(varargin) ~= 2*length(UserOptions)
error('FSDA:simulateLM:WrongInputOpt','Number of supplied options is invalid. Probably values for some parameters are missing.');
end
-
+
% Check if all the specified optional arguments were present in
% structure options. Remark: the nocheck option has already been dealt
% by routine chkinputR.
@@ -295,17 +310,17 @@
disp(strcat('Non existent user option found->', char(WrongOptions{:})))
error('FSDA:simulateLM:NonExistInputOpt','In total %d non-existent user options found.', length(WrongOptions));
end
-
+
% Check the presence of input options beta, SigmaX and nexpl.
betaboo=max(strcmp(UserOptions,'beta'))==1;
SigmaXboo=max(strcmp(UserOptions,'SigmaX'))==1;
nexplboo=max(strcmp(UserOptions,'nexpl'))==1;
-
+
% Write in structure 'options' the options chosen by the user.
for i=1:2:length(varargin)
options.(varargin{i})=varargin{i+1};
end
-
+
R2=options.R2;
nexpl=options.nexpl;
beta=options.beta;
@@ -317,6 +332,7 @@
distribypars = options.distribypars;
plots=options.plots;
intercept=options.intercept;
+ exactR2=options.exactR2;
SNR = options.SNR;
pMSOM = options.pMSOM;
pVIOM = options.pVIOM;
@@ -335,21 +351,21 @@
if SNR<=0
error('FSDA:simulateLM:WrongOpt','SNR must be greater than zero');
end
-
+
% Preliminary checks both beta, nexpl and SigmaX have been supplied.
if betaboo==true && SigmaXboo==true && nexplboo==true
-
+
if nexpl~=size(SigmaX,1)
error('FSDA:simulateLM:WrongOpt',['Length of supplied vector beta ' ...
'must be equal to number of rows (columns) of matrix SigmaX']);
end
-
+
if nexpl~=length(beta)
error('FSDA:simulateLM:WrongOpt',['Length of supplied vector beta ' ...
'must be equal to number of rows (columns) of matrix SigmaX']);
end
end
-
+
% Preliminary checks just beta and SigmaX have been supplied.
if betaboo==true && SigmaXboo==true && nexplboo==false
nexpl=length(betaboo);
@@ -358,7 +374,7 @@
'must be equal to number of rows (columns) of matrix SigmaX']);
end
end
-
+
% Preliminary checks just beta and nexpl have been supplied.
if betaboo==true && SigmaXboo==false && nexpl==true
nexpl=length(beta);
@@ -368,7 +384,7 @@
end
SigmaX=eye(nexpl);
end
-
+
% Preliminary checks just SigmaX and nexpl have been supplied.
if betaboo==false && SigmaXboo==true && nexplboo==true
nexplchk=size(SigmaXboo,1);
@@ -378,60 +394,60 @@
end
beta=ones(nexpl,1);
end
-
+
% Preliminary checks just beta has been supplied.
if betaboo==true && SigmaXboo == false && nexplboo==false
nexpl=length(beta);
SigmaX=eye(nexpl);
end
-
+
% Preliminary checks just SigmaX has been supplied.
if betaboo==false && SigmaXboo == true && nexplboo==false
nexpl=size(SigmaX,1);
beta=ones(nexpl,1);
end
-
+
% Preliminary checks just nexpl has been supplied.
if betaboo==false && SigmaXboo == false && nexplboo==true
beta=ones(nexpl,1);
SigmaX=eye(nexpl);
end
-
+
[T,err] = cholcov(SigmaX);
if err ~= 0
error('FSDA:mvnrnd:BadCovariance2DSymPos','WrongSigma');
end
lXpars=length(distribXpars);
-
+
%% checks on the explanatory variables.
if ischar(distribX)
- if lXpars==1
- X = random(distribX,distribXpars,n,nexpl);
- elseif lXpars==2
- X = random(distribX,distribXpars(1),distribXpars(2),n,nexpl);
- elseif lXpars==3
- X = random(distribX,distribXpars(1),distribXpars(2),distribXpars(3),n,nexpl);
- else
- X = random(distribX,distribXpars(1),distribXpars(2),distribXpars(3),distribXpars(4),n,nexpl);
- end
- % Generate the X in such a way their corr is SigmaX.
- X=X*T;
+ if lXpars==1
+ X = random(distribX,distribXpars,n,nexpl);
+ elseif lXpars==2
+ X = random(distribX,distribXpars(1),distribXpars(2),n,nexpl);
+ elseif lXpars==3
+ X = random(distribX,distribXpars(1),distribXpars(2),distribXpars(3),n,nexpl);
+ else
+ X = random(distribX,distribXpars(1),distribXpars(2),distribXpars(3),distribXpars(4),n,nexpl);
+ end
+ % Generate the X in such a way their corr is SigmaX.
+ X=X*T;
else
% In this case, the user has directly supplied matrix X.
% Make sure that the size of X is n-by-nexpl.
X=distribX;
[nchk,nexplchk]=size(X);
- if nchk~=n
+ if nchk~=n
error('FSDA:simulateLM:WrongOpt',['supplied matrix X must have ' ...
num2str(n) ' rows']);
- end
- if nexpl~=nexplchk
+ end
+ if nexpl~=nexplchk
error('FSDA:simulateLM:WrongOpt',['supplied matrix X must have ' ...
num2str(nexpl) ' columns']);
- end
+ end
end
-
-
+
+
lypars=length(distribypars);
if lypars==1
err = random(distriby,distribypars,n,1);
@@ -442,35 +458,57 @@
else
err = random(distriby,distribypars(1),distribypars(2),distribypars(3),distribypars(4),n,1);
end
-
+
p=nexpl+intercept;
-
+
% Divide by std and multiply by a small sample correction factor.
- err=sqrt((n)/(n-p))*err/std(err,1);
+ err=sqrt((n)/(n-p))*err/std(err,1);
% err=err/std(err,1);
-
+
if R2>0 && isempty(SNR)
% Find var(\epsilon) which produces a value of R2 centered around
% the one which has been requested.
vareps=(intercept+beta'*SigmaX*beta)*((1 - R2)/R2);
y=intercept+X*beta(:)+err*sqrt(vareps);
+ if exactR2==true
+ step=100;
+ outTMP=fitlm(X,y);
+ while abs(outTMP.Rsquared.Ordinary-R2)>0.01
+ if outTMP.Rsquared.Ordinary >R2
+ while outTMP.Rsquared.Ordinary>R2
+ vareps=vareps+step;
+ y=intercept+X*beta(:)+err*sqrt(vareps);
+ outTMP=fitlm(X,y);
+ end
+ step=step/2;
+ elseif outTMP.Rsquared.Ordinary 0
-
+
eu = err*sqrt(vareps);
Xu = X;
@@ -480,7 +518,7 @@
indMSOM = randperm(n, nMSOM);
% MSOM (also on X).
Xc = Xu;
- Xc(indMSOM, predxMSOM) = Xu(indMSOM, predxMSOM) + shiftMSOMx;
+ Xc(indMSOM, predxMSOM) = Xu(indMSOM, predxMSOM) + shiftMSOMx;
ec = eu;
ec(indMSOM) = eu(indMSOM) + shiftMSOMe;
@@ -497,7 +535,7 @@
indcont = unique([indMSOM, indVIOM]);
% clean obs indexes
indkeep = setdiff(1:n, indcont);
-
+
if plots==true
indunit = zeros(n, 1);
indunit(indMSOM) = 1;
@@ -513,7 +551,7 @@
out.ind_MSOM = indMSOM;
out.ind_VIOM = indVIOM;
out.vareps = vareps;
-
+
end
out.X = X;