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: 0.03 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');

Total estimated time to complete LMS: 0.12 seconds Warning: interchange greater than 10 when m=30 Warning: interchange greater than 10 when m=31 Warning: Rank deficient, rank = 2, tol = 8.814907e+02. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.205886e-19. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.253335e-16. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 5.330781e-19. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.851063e-19. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 2.658769e-19. 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 = 1.826966e-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,'modeltype','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: 0.45 seconds ------------------------------ Warning: Number of subsets without full rank equal to 32.4% 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 = 2.539797e-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 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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
```

,```
'modeltype','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`

`modeltype`

—Parametric function to be used in the skedastic equation.string.If modeltype is 'arc' (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 modeltype 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: **```
'modeltype','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 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'. |

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]