FSRHeda enables to monitor several quantities in each step of the forward search

Common part to all examples: load tradeH dataset (used in the paper ART).

XX=load('tradeH.txt'); y=XX(:,2); X=XX(:,1); X=X./max(X); Z=log(X); [out]=LXS(y,X,'nsamp',1000); out1=FSRHeda(y,X,Z,out.bs);

Total estimated time to complete LMS: 0.06 seconds Warning: interchange greater than 10 when m=26 Warning: interchange greater than 10 when m=27 Warning: interchange greater than 10 when m=67 Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: interchange greater than 10 when m=68 Warning: interchange greater than 10 when m=69 Warning: interchange greater than 10 when m=70 m=100 Warning: interchange greater than 10 when m=165 Warning: interchange greater than 10 when m=166 Warning: interchange greater than 10 when m=167 Warning: interchange greater than 10 when m=168 Warning: interchange greater than 10 when m=169 Warning: interchange greater than 10 when m=170 Warning: interchange greater than 10 when m=171 Warning: interchange greater than 10 when m=172 Warning: interchange greater than 10 when m=180 Warning: interchange greater than 10 when m=181 Warning: interchange greater than 10 when m=182 Warning: interchange greater than 10 when m=183 m=200 Warning: interchange greater than 10 when m=225 m=300 m=400 Warning: interchange greater than 10 when m=482 Warning: interchange greater than 10 when m=483 Warning: interchange greater than 10 when m=484 Warning: interchange greater than 10 when m=485 Warning: interchange greater than 10 when m=486 m=500 m=600 m=700 m=800 m=900 m=1000 m=1100

Example of use of function FSRHeda using a random start and traditional t-stat monitoring.

XX=load('tradeH.txt'); y=XX(:,2); X=XX(:,1); X=X./max(X); Z=log(X); [out]=LXS(y,X,'nsamp',1000); out=FSRHeda(y,X,Z,0,'tstat','trad','init',800); subplot(2,2,1) plot(out.Tgls(:,1),out.Tgls(:,2)) title('t stat for Intercept (traditional)') subplot(2,2,2) plot(out.Tgls(:,1),out.Tgls(:,3)) title('t stat for slope (traditional)') out1=FSRHeda(y,X,Z,0,'tstat','scal','init',800); subplot(2,2,3) plot(out1.Tgls(:,1),out1.Tgls(:,2)) title('t stat for Intercept (using unbiased estimate of sigma)') subplot(2,2,4) plot(out1.Tgls(:,1),out1.Tgls(:,3)) title('t stat for slope (using unbiased estimate of sigma)')

% In this example, figure 3 of ART (see References) is created. XX=load('tradeH.txt'); y=XX(:,2); X=XX(:,1); X=X./max(X); Z=log(X); % Call procedure FSRH to automatically find the outliers outtmp=FSRH(y,X,Z,'plots',0,'msg',0); [out]=LXS(y,X,'nsamp',1000); out=FSRHeda(y,X,Z,out.bs,'init',round(length(y)/2)); out.ListOut=outtmp.ListOut; figure subplot(2,2,1) n=length(y); seq=1:n; sel=setdiff(seq,out.ListOut); hold('on') plot(X(sel),y(sel),'o') plot(X(out.ListOut),y(out.ListOut),'rx','MarkerSize',12,'LineWidth',2) fs=12; ylabel('Value','FontSize',fs) xlabel('Quantity','FontSize',fs) set(gca,'FontSize',fs) subplot(2,2,2) plot(out.Hetero(:,1),out.Hetero(:,3)) xlabel('Subset size m') kk=20; xlim([out.Hetero(1,1) out.Hetero(end,1)+kk]) ylim([1.7 2.7]) title('\alpha') subplot(2,2,3) plot(out.Hetero(:,1),out.Hetero(:,2)) title('log(\theta)') xlim([out.Hetero(1,1) out.Hetero(end,1)+kk]) %ylim([5 7.5]) xlabel('Subset size m') subplot(2,2,4) plot(out.S2(:,1),out.S2(:,2)) xlim([out.Hetero(1,1) out.Hetero(end,1)+kk]) ylim([0 300000]) title('\sigma^2') xlabel('Subset size m')

xx=load('wool.txt'); X=xx(:,1:3); y=log(xx(:,4)); [out]=LXS(y,X,'nsamp',0); [out]=FSRHeda(y,X,X,out.bs,'tstat','scal');

n=100; p=8; state=1; randn('state', state); X=randn(n,p); y=randn(n,1); y(1:10)=y(1:10)+5; % Run the forward search with Exploratory Data Analysis purposes % LMS using 10000 subsamples [outLXS]=LXS(y,X,'nsamp',10000); % Forward Search [out]=FSRHeda(y,X,log(X),outLXS.bs); %The monitoring residuals plot shows a set of positive residuals which %starting from the central part of the search tend to have a residual much %larger than that of the other units. resfwdplot(out); %The minimum deletion residual from m=90 starts going above the 99% threshold. mdrplot(out); %The curve which monitors the normality test shows a sudden big increase with the outliers are included figure; lwdenv=2; xlimx=[10 100]; subplot(2,2,1); plot(out.nor(:,1),out.nor(:,2)); title('Asimmetry test'); xlim(xlimx); quant=chi2inv(0.99,1); v=axis; line([v(1),v(2)],[quant,quant],'color','r','LineWidth',lwdenv); subplot(2,2,2) plot(out.nor(:,1),out.nor(:,3)) title('Kurtosis test'); xlim(xlimx); v=axis; line([v(1),v(2)],[quant,quant],'color','r','LineWidth',lwdenv); subplot(2,2,3:4) plot(out.nor(:,1),out.nor(:,4)); xlim(xlimx); quant=chi2inv(0.99,2); v=axis; line([v(1),v(2)],[quant,quant],'color','r','LineWidth',lwdenv); title('Normality test'); xlabel('Subset size m');

Warning: Using 'state' to set RANDN's internal state causes RAND, RANDI, and RANDN to use legacy random number generators. This syntax is not recommended. See <a href="matlab:helpview([docroot '\techdoc\math\math.map'],'update_random_number_generator')">Replace Discouraged Syntaxes of rand and randn</a> to use RNG to replace the old syntax. Total estimated time to complete LMS: 0.04 seconds Warning: Rank deficient, rank = 7, tol = 8.759214e-04. Warning: Rank deficient, rank = 7, tol = 1.140309e-03. Warning: Rank deficient, rank = 3, tol = 1.480703e-02. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.832917e-16. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.338014e-20. Warning: Rank deficient, rank = 2, tol = 1.082400e+03. Warning: Rank deficient, rank = 2, tol = 3.007266e+03. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.564858e-17. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.647638e-19. Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: Rank deficient, rank = 8, tol = 3.376048e+00. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.275251e-21. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 5.189491e-20. Warning: Rank deficient, rank = 1, tol = 1.282522e+03. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.246314e-20. Warning: Rank deficient, rank = 2, tol = 1.237249e+03. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.913104e-20. Warning: Rank deficient, rank = 1, tol = 1.344331e+03. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.246314e-20. Warning: Rank deficient, rank = 2, tol = 2.495266e+03. Warning: Rank deficient, rank = 2, tol = 6.170946e+03. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.720547e-16. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.146527e-20. Warning: Imaginary parts of complex X and/or Y arguments ignored.

% House price data load hprice.txt; n=size(hprice,1); y=hprice(:,1); X=hprice(:,2:5); % init = point to start monitoring diagnostics along the FS init=450; [outLXS]=LXS(y,X,'nsamp',10000); outEDA=FSRHeda(y,X,log(X),outLXS.bs,'conflev',[0.95 0.99],'init',init,'typeH','har'); p=size(X,2)+1; % Set font size, line width and line style figure; lwd=2.5; FontSize=14; linst={'-','--',':','-.','--',':'}; nr=3; nc=2; xlimL=init; % lower value fo xlim xlimU=n+1; % upper value of xlim close all for j=1:p subplot(nr,nc,j); hold('on') % plot 95% and 99% HPD trajectories plot(outEDA.Bgls(:,1),outEDA.betaINT(:,1:2,j),'LineStyle',linst{4},'LineWidth',lwd,'Color','b') plot(outEDA.Bgls(:,1),outEDA.betaINT(:,3:4,j),'LineStyle',linst{4},'LineWidth',lwd,'Color','r') % plot estimate of beta1_j plot(outEDA.Bgls(:,1),outEDA.Bgls(:,j+1)','LineStyle',linst{1},'LineWidth',lwd,'Color','k') % Set ylim ylimU=max(outEDA.betaINT(:,4,j)); ylimL=min(outEDA.betaINT(:,3,j)); ylim([ylimL ylimU]) % Set xlim xlim([xlimL xlimU]); ylabel(['$\hat{\beta_' num2str(j-1) '}$'],'Interpreter','LaTeX','FontSize',20,'rot',-360); set(gca,'FontSize',FontSize); if j>(nr-1)*nc xlabel('Subset size m','FontSize',FontSize); end end % Subplot associated with the monitoring of sigma^2 subplot(nr,nc,6); hold('on') % 99% plot(outEDA.sigma2INT(:,1),outEDA.sigma2INT(:,4:5),'LineStyle',linst{4},'LineWidth',lwd,'Color','r') % 95% plot(outEDA.sigma2INT(:,1),outEDA.sigma2INT(:,2:3),'LineStyle',linst{2},'LineWidth',lwd,'Color','b') % Plot rescaled S2 plot(outEDA.S2(:,1),outEDA.S2(:,4),'LineWidth',lwd,'Color','k') ylabel('$\hat{\sigma}^2$','Interpreter','LaTeX','FontSize',20,'rot',-360); set(gca,'FontSize',FontSize); % Set ylim ylimU=max(outEDA.sigma2INT(:,5)); ylimL=min(outEDA.sigma2INT(:,4)); ylim([ylimL ylimU]) % Set xlim xlim([xlimL xlimU]); xlabel('Subset size m','FontSize',FontSize); % Add multiple title suplabel(['Housing data; confidence intervals of the parameters monitored in the interval ['... num2str(xlimL) ',' num2str(xlimU) ... ']'],'t');

Total estimated time to complete LMS: 0.17 seconds ------------------------------ Warning: Number of subsets without full rank equal to 37.5% Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: Rank problem in step 6: Beta coefficients are used from the most recent correctly computed step Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. Warning: Maximum number of iteration has been reached without convergence Warning: Maximum number of iteration has been reached without convergence Warning: Maximum number of iteration has been reached without convergence Warning: Maximum number of iteration has been reached without convergence Warning: Maximum number of iteration has been reached without convergence Warning: Maximum number of iteration has been reached without convergence Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.229039e-24. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.531461e-22. Warning: Rank problem in step 16: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 17: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 18: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 19: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 20: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 21: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 22: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 23: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 24: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 25: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 26: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 27: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 28: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 29: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 30: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 31: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 32: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 33: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 34: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 35: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 36: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 37: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 38: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 39: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 40: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 41: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 42: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 43: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 44: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 45: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 46: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 47: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 48: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 49: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 50: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 51: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 52: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 53: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 54: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 55: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 56: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 57: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 58: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 59: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 60: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 61: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 62: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 63: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 64: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 65: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 66: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 67: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 68: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 69: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 70: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 71: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 72: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 73: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 74: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 75: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 76: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 77: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 78: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 79: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 80: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 81: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 82: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 83: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 84: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 85: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 86: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 87: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 88: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 89: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 90: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 91: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 92: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 93: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 94: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 95: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 96: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 97: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 98: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 99: Beta coefficients are used from the most recent correctly computed step m=100 Warning: Rank problem in step 100: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 101: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 102: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 103: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 104: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 105: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 106: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 107: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 108: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 109: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 110: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 111: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 112: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 113: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 114: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 115: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 116: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 117: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 118: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 119: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 120: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 121: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 122: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 123: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 124: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 125: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 126: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 127: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 128: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 129: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 130: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 131: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 132: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 133: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 134: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 135: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 136: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 137: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 138: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 139: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 140: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 141: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 142: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 143: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 144: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 145: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 146: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 147: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 148: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 149: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 150: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 151: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 152: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 153: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 154: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 155: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 156: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 157: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 158: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 159: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 160: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 161: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 162: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 163: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 164: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 165: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 166: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 167: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 168: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 169: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 170: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 171: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 172: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 173: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 174: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 175: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 176: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 177: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 178: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 179: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 180: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 181: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 182: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 183: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 184: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 185: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 186: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 187: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 188: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 189: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 190: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 191: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 192: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 193: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 194: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 195: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 196: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 197: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 198: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 199: Beta coefficients are used from the most recent correctly computed step m=200 Warning: Rank problem in step 200: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 201: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 202: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 203: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 204: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 205: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 206: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 207: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 208: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 209: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 210: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 211: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 212: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 213: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 214: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 215: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 216: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 217: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 218: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 219: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 220: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 221: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 222: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 223: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 224: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 225: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 226: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 227: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 228: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 229: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 230: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 231: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 232: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 233: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 234: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 235: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 236: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 237: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 238: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 239: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 240: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 241: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 242: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 243: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 244: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 245: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 246: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 247: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 248: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 249: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 250: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 251: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 252: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 253: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 254: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 255: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 256: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 257: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 258: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 259: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 260: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 261: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 262: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 263: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 264: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 265: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 266: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 267: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 268: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 269: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 270: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 271: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 272: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 273: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 274: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 275: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 276: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 277: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 278: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 279: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 280: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 281: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 282: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 283: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 284: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 285: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 286: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 287: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 288: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 289: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 290: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 291: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 292: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 293: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 294: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 295: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 296: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 297: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 298: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 299: Beta coefficients are used from the most recent correctly computed step m=300 Warning: Rank problem in step 300: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 301: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 302: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 303: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 304: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 305: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 306: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 307: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 308: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 309: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 310: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 311: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 312: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 313: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 314: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 315: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 316: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 317: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 318: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 319: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 320: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 321: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 322: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 323: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 324: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 325: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 326: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 327: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 328: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 329: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 330: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 331: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 332: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 333: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 334: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 335: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 336: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 337: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 338: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 339: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 340: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 341: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 342: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 343: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 344: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 345: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 346: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 347: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 348: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 349: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 350: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 351: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 352: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 353: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 354: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 355: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 356: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 357: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 358: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 359: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 360: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 361: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 362: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 363: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 364: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 365: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 366: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 367: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 368: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 369: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 370: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 371: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 372: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 373: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 374: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 375: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 376: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 377: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 378: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 379: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 380: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 381: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 382: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 383: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 384: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 385: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 386: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 387: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 388: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 389: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 390: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 391: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 392: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 393: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 394: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 395: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 396: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 397: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 398: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 399: Beta coefficients are used from the most recent correctly computed step m=400 Warning: Rank problem in step 400: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 401: Beta coefficients are used from the most recent correctly computed step Warning: Rank problem in step 402: Beta coefficients are used from the most recent correctly computed step Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.601941e-24. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.837818e-20. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.069719e-18. m=500

`y`

— A vector with n elements that contains the response variables.
Missing values (NaN's) and infinite values (Inf's) are
allowed, since observations (rows) with missing or infinite
values will automatically be excluded from the
computations.**
Data Types: **`single| double`

`X`

— Data matrix of explanatory variables (also called 'regressors')
of dimension (n x p-1).
Rows of X represent observations, and
columns represent variables.Missing values (NaN's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

**
Data Types: **`single| double`

`Z`

— Predictor variables in the scedastic equation.
n x r matrix or index vector of length r.If Z is a n x r matrix it contains the r variables which form the scedastic function as follows (if input option art==1) \[ \omega_i = 1 + exp(\gamma_0 + \gamma_1 Z(i,1) + ...+ \gamma_{r} Z(i,r)) \] If Z is a vector of length r it contains the indexes of the columns of matrix X which form the scedastic function as follows \[ \omega_i = 1 + exp(\gamma_0 + \gamma_1 X(i,Z(1)) + ...+ \gamma_{r} X(i,Z(r))) \] Therefore, if for example the explanatory variables responsible for heteroscedasticity are columns 3 and 5 of matrix X, it is possible to use both the sintax: FSRHeda(y,X,X(:,[3 5])) or the sintax: FSRHeda(y,X,[3 5])

**
Data Types: **`single| double`

`bsb`

— list of units forming the initial
subset.
Vector or scalar.If bsb=0 (default) then the procedure starts with p units randomly chosen else if bsb is not 0 the search will start with m0=length(bsb).

**
Data Types: **`single| double`

Specify optional comma-separated pairs of `Name,Value`

arguments.
`Name`

is the argument name and `Value`

is the corresponding value. `Name`

must appear
inside single quotes (`' '`

).
You can specify several name and value pair arguments in any order as ```
Name1,Value1,...,NameN,ValueN
```

.

```
'intercept',false
```

,```
'init',100 starts monitoring from step m=100
```

,```
'tstat','trad'
```

,```
'nocheck',true
```

,```
'conflev',[0.90 0.93]
```

,```
'gridsearch',0
```

,```
'typeH','har'
```

,```
'constr',[1 6 3]
```

`intercept`

—Indicator for constant term.true (default) | false.Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

**Example: **```
'intercept',false
```

**Data Types: **`boolean`

`init`

—Search initialization.scalar.It specifies the point where to initialize the search and start monitoring required diagnostics. if init is not specified it will be set equal to : p+1, if the sample size is smaller than 40;

min(3*p+1,floor(0.5*(n+p+1))), otherwise.

**Example: **```
'init',100 starts monitoring from step m=100
```

**Data Types: **`double`

`tstat`

—the kind of t-statistics which have to be monitored.character.tstat = 'trad' implies monitoring of traditional t statistics (out.Tgls). In this case the estimate of \sigma^2 at step m is based on s^2_m (notice that s^2_m<<\sigma^2 when m/n is small) tstat = 'scal' (default) implies monitoring of rescaled t statistics In this scale the estimate of \sigma^2 at step m is based on s^_m / var_truncnorm(m/n) where var_truncnorm(m/n) is the variance of the truncated normal distribution.

**Example: **```
'tstat','trad'
```

**Data Types: **`char`

`nocheck`

—Check input arguments.boolean.If nocheck is equal to true no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=false. The controls on h, alpha and nsamp still remain

**Example: **```
'nocheck',true
```

**Data Types: **`boolean`

`conflev`

—confidence levels to be used to compute confidence interval
for the elements of $\beta$ and for $\sigma^2$.vector.The default value of conflev is [0.95 0.99] that is 95% and 99% confidence intervals are computed.

**Example: **```
'conflev',[0.90 0.93]
```

**Data Types: **`double`

`gridsearch`

—Algorithm to be used.scalar.If gridsearch ==1 grid search will be used else the scoring algorith will be used.

REMARK: the grid search has only been implemented when there is just one explantory variable which controls heteroskedasticity Example - 'gridsearch',1 Data Types - double

**Example: **```
'gridsearch',0
```

**Data Types: **`double`

`typeH`

—Parametric function to be used in the skedastic equation.string.If typeH is 'art' (default) than the skedastic function is modelled as follows \[ \sigma^2_i = \sigma^2 (1 + \exp(\gamma_0 + \gamma_1 Z(i,1) + \cdots + \gamma_{r} Z(i,r))) \] on the other hand, if typeH is 'har' then traditional formulation due to Harvey is used as follows \[ \sigma^2_i = \exp(\gamma_0 + \gamma_1 Z(i,1) + \cdots + \gamma_{r} Z(i,r)) =\sigma^2 \exp(\gamma_1 Z(i,1) + \cdots + \gamma_{r} Z(i,r)) \]

**Example: **```
'typeH','har'
```

**Data Types: **`string`

`constr`

—units which are forced to join the search in the last r steps.vector.r x 1 vector. The default is constr=''. No constraint is imposed

**Example: **```
'constr',[1 6 3]
```

**Data Types: **`double`

`out`

— description
StructureStructure which contains the following fields

Value | Description |
---|---|

`RES` |
n x (n-init+1) = matrix containing the monitoring of scaled residuals: 1st row = residual for first unit; ...; nth row = residual for nth unit. |

`LEV` |
(n+1) x (n-init+1) = matrix containing the monitoring of leverage: 1st row = leverage for first unit; ...; nth row = leverage for nth unit. |

`BB` |
n x (n-init+1) matrix containing the information about the units belonging to the subset at each step of the forward search: 1st col = indexes of the units forming subset in the initial step; ...; last column = units forming subset in the final step (all units). |

`mdr` |
n-init x 3 matrix which contains the monitoring of minimum deletion residual or (m+1)ordered residual at each step of the forward search: 1st col = fwd search index (from init to n-1); 2nd col = minimum deletion residual; 3rd col = (m+1)-ordered residual. Remark: these quantities are stored with sign, that is the min deletion residual is stored with negative sign if it corresponds to a negative residual. |

`msr` |
n-init+1 x 3 = matrix which contains the monitoring of maximum studentized residual or m-th ordered residual: 1st col = fwd search index (from init to n); 2nd col = maximum studentized residual; 3rd col = (m)-ordered studentized residual. |

`nor` |
(n-init+1) x 4 matrix containing the monitoring of normality test in each step of the forward search: 1st col = fwd search index (from init to n); 2nd col = Asymmetry test; 3rd col = Kurtosis test; 4th col = Normality test. |

`Bgls` |
(n-init+1) x (p+1) matrix containing the monitoring of estimated beta coefficients in each step of the forward search. |

`S2` |
(n-init+1) x 4 matrix containing the monitoring of S2 or R2 in each step of the forward search: 1st col = fwd search index (from init to n); 2nd col = monitoring of unbiased S2; 3rd col = monitoring of R2; 4th col = monitoring of rescaled S2. In this case the estimate of $\sigma^2$ at step m is divided by the consistency factor (to make the estimate asymptotically unbiased). |

`coo` |
(n-init+1) x 3 matrix containing the monitoring of Cook or modified Cook distance in each step of the forward search: 1st col = fwd search index (from init to n); 2nd col = monitoring of Cook distance; 3rd col = monitoring of modified Cook distance. |

`Tgls` |
(n-init+1) x (p+1) matrix containing the monitoring of estimated t-statistics (as specified in option input 'tstat' in each step of the forward search. |

`Un` |
(n-init) x 11 Matrix which contains the unit(s) included in the subset at each step of the fwd search. REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one Un(1,2) for example contains the unit included in step init+1 Un(end,2) contains the units included in the final step of the search |

`betaINT` |
Confidence intervals for the elements of $\beta$. betaINT is a (n-init+1)-by-2*length(confint)-by-p 3D array. Each third dimension refers to an element of beta: betaINT(:,:,1) is associated with first element of beta; ...; betaINT(:,:,p) is associated with last element of beta. The first two columns contain the lower and upper confidence limits associated with conflev(1); Columns three and four contain the lower and upper confidence limits associated with conflev(2); ...; The last two columns contain the lower and upper confidence limits associated with conflev(end). For example betaint(:,3:4,5) contain the lower and upper confidence limits for the fifth element of beta using confidence level specified in the second element of input option conflev. |

`sigma2INT` |
confidence interval for $\sigma^2$. 1st col = fwd search index; 2nd col = lower confidence limit based on conflev(1); 3rd col = upper confidence limit based on conflev(1); 4th col = lower confidence limit based on conflev(2); 5th col = upper confidence limit based on conflev(2); ...; penultimate col = lower confidence limit based on conflev(end); last col = upper confidence limit based on conflev(end). |

`Hetero` |
estimate of coefficients of scedastic equation: 1st col = fwd search index; 2nd col = estimate of first coeff of scedastic equation; ...; (r+1) col = estimate of last coeff of scedastic equation. |

`WEI` |
Matrix which contains in each column the estimate of the weights. n x (n-init+1) matrix which contains information about the weights assigned to each unit to make the regression equation skedastic. More precisely, if: $var (\epsilon)= \sigma^2 \Omega$, where $\Omega=diag(\omega_1, \ldots, \omega_n)$ the weights which are stored are $(\omega_1^{-0.5}, \ldots, \omega_n^{-0.5})$; |

`y` |
A vector with n elements that contains the response variable which has been used. |

`X` |
Data matrix of explanatory variables which has been used (it also contains the column of ones if input option intercept was missing or equal to true). |

`Z` |
Predictor variables in the scedastic equation. |

`class` |
'FSRHeda'. |

`typeH` |
'art' or 'har'; |

Atkinson, A.C. and Riani, M. (2000), "Robust Diagnostic Regression Analysis", Springer Verlag, New York.

Atkinson, A.C., Riani, M. and Torti, F. (2016), Robust methods for heteroskedastic regression, "Computational Statistics and Data Analysis", Vol. 104, p. 209-222, http://dx.doi.org/10.1016/j.csda.2016.07.002 [ART]