FSRB gives an automatic outlier detection procedure in Bayesian linear regression

Common part to the first examples: load Houses Price Dataset.

load hprice.txt; n=size(hprice,1); y=hprice(:,1); X=hprice(:,2:5); out=FSRB(y,X);

load hprice.txt; n=size(hprice,1); y=hprice(:,1); X=hprice(:,2:5); % set \beta components beta0=0*ones(5,1); beta0(2,1)=10; beta0(3,1)=5000; beta0(4,1)=10000; beta0(5,1)=10000; % \tau s02=1/4.0e-8; tau0=1/s02; % R prior settings R=2.4*eye(5); R(2,2)=6e-7; R(3,3)=.15; R(4,4)=.6; R(5,5)=.6; R=inv(R); % define a Bayes structure with previous data n0=5; bayes=struct; bayes.R=R; bayes.n0=n0; bayes.beta0=beta0; bayes.tau0=tau0; intercept=true; % function call out=FSRB(y,X,'bayes',bayes,'msg',0,'plots',1,'init',round(n/2),'intercept', intercept)

out = struct with fields: ListOut: [1×18 double] outliers: [1×18 double] mdr: [273×2 double] Un: [273×11 double] nout: [2×5 double] beta: [5×1 double] scale: 1.4951e+04 class: 'FSRB'

nsamp is the number of subsamples to use in the frequentist analysis of first year, in order to find initial subset using LMS.

close all nsamp=3000; % threshold to be used to increase susbet of good units threshold=300; bonflev=0.99; % Bonferroni confidence level to be used for first year bonflevB=0.99; % Bonferroni confidence level to be used for subsequent years % Load 2002 Fishery dataset Fishery2002=load('Fishery2002.txt'); y02=Fishery2002(:,3); X02=Fishery2002(:,2); n02=length(y02); seq02=1:n02; % frequentist Forward Search, 1st year [out02]=FSR(y02,X02,'nsamp',nsamp,'plots',1,'msg',0,'init',round(n02*3/4),'bonflev',bonflev); % In what follows % g stands for good units % i stand for intermediate units (i.e. units whose raw residual is smaller % than threshold) % o stands for outliers % gi stands for good +intermediate units % u02g = good units % n02g = number of good units u02g=setdiff(seq02,out02.ListOut); n02g=length(u02g); X02g=[ones(length(u02g),1) X02(u02g,:)]; y02g=y02(u02g); % b02g = regression coefficients just using g units b02g=X02g\y02g; % res02 = squared raw residuals for all units using b02g res02=(y02-[ones(length(X02),1) X02]*b02g).^2; res02o=res02(out02.ListOut); % sel= boolean vector which is true for the intermediate units % (units whose squared residual is below the threshold) sel=res02o<threshold^2; % u02i = vector containing intermediate units (that is outliers whose % residual is smaller than threshold) u02i=out02.ListOut(sel); % u02o = vector containing outliers whose residual is out of the threshold u02o=out02.ListOut(~sel); % u02gi = g + i units if ~isempty(u02i) u02gi=[u02g u02i]; else u02gi=u02g; end % n02gi = number of good + intermediate units n02gi=length(u02gi); % plotting section hold('off') % good units, plotted as (+) plot(X02(u02g)',y02(u02g)','Marker','+','LineStyle','none','Color','b') hold('on') % intermediate units plotted as (X) plot(X02(u02i)',y02(u02i)','Marker','X','MarkerSize',9,'LineWidth',2,'LineStyle','none','Color','m') % outliers, plotted as (O) plot(X02(u02o)',y02(u02o)','Marker','o','LineStyle','none','Color','r') xlabel('Quantity'); ylabel('Price'); title('Frequentist - 2002'); % S202gi = estimated of sigma^2 using g+i units S202gi=sum(res02(u02gi))/(n02gi-2); % X02gi = X matrix referred to good + intermediate units X02gi=[ones(n02gi,1) X02(u02gi,:)]; % y02gi = y vector referred to good + intermediate units y02gi=y02(u02gi); % bayes = structure which contains prior information to be used in year % 2003 bayes=struct; bayes.beta0=b02g; % beta prior is beta based on g units tau0=1/S202gi; % tau0 is based on g + i units bayes.tau0=tau0; R=X02g'*X02g; % R is based on g units bayes.n0=n02gi; % n0 is based on g + i units bayes.R=R; % 2003 % Load 2003 Fishery dataset Fishery2003=load('Fishery2003.txt'); y03=Fishery2003(:,3); X03=Fishery2003(:,2); n03=length(y03); seq03=1:n03; % Bayesian Forward Search, 2nd year out03=FSRB(y03,X03,'bayes',bayes,'msg',0,'plots',1,'init',round(n03/2),'bonflev',bonflevB); u03g=setdiff(seq03,out03.ListOut); n03g=length(u03g); % compute beta coefficient for year 2003 just using good units X03g=[ones(n03g,1) X03(u03g,:)]; y03g=y03(u03g); b03g=X03g\y03g; res03=(y03-[ones(length(X03),1) X03]*b03g).^2; res03o=res03(out03.ListOut); sel=res03o<threshold^2; % u03i = units to add to the good units subset (intermediate units) u03i=out03.ListOut(sel); % u03o = outliers out of the threshold u03o=out03.ListOut(~sel); if ~isempty(u03i) u03gi=[u03g u03i]; else u03gi=u03g; end n03gi=length(u03gi); X03gi=[ones(n03gi,1) X03(u03gi,:)]; y03gi=y03(u03gi,:); % plotting section hold('off') % good units, plotted as (+) plot(X03(u03g)',y03(u03g)','Marker','+','LineStyle','none','Color','b') hold('on') % units below the threshold, plotted as (X) plot(X03(u03i)',y03(u03i)','Marker','X','MarkerSize',9,'LineWidth',2,'LineStyle','none','Color','m') % outliers, plotted as (O) plot(X03(u03o)',y03(u03o)','Marker','o','LineStyle','none','Color','r') set(gca,'FontSize',14) xlabel('QUANTITY in tons') ylabel('VALUE in 1000 euro') title('2003'); % Definition of bayes structure (based on 2002 and 2003) bayes=struct; X02gX03g=[X02g; X03g]; y02gy03g=[y02g; y03g]; n02gn03g=n02g+n03g; % b0203g prior estimate of beta for year 2004 is computed using good units % for years 2002 and 2003 b0203g=X02gX03g\y02gy03g; bayes.beta0=b0203g; % R is just referred to good units for years 2002 and 2003 R=X02gX03g'*X02gX03g; bayes.R=R; % n0 is referred to g + i units in 2002 and 2003 bayes.n0=n02gi+n03gi; X02giX03gi=[X02gi; X03gi]; y02giy03gi=[y02gi; y03gi]; % n02gin03gi = number of g+i units in 2002 and 2003 n02gin03gi=n02gi+n03gi; % res = residuals for g+i units using b0203g res=y02giy03gi-X02giX03gi*b0203g; S203gi=sum(res.^2)/(n02gin03gi-2); % estimate of tau is based on g + i units tau0=1/S203gi; bayes.tau0=tau0; % Load 2004 Fishery dataset Fishery2004=load('Fishery2004.txt'); y04=Fishery2004(:,3); X04=Fishery2004(:,2); n04=length(y04); seq04=1:n04; % Bayesian Forward Search, 3rd year out04=FSRB(y04,X04,'bayes',bayes,'msg',0,'plots',1,'init',round(n04/2),'bonflev',bonflevB); u04g=setdiff(seq04,out04.ListOut); n04g=length(u04g); X04g=[ones(n04g,1) X04(u04g,:)]; y04g=y04(u04g); % b04g = beta based on good units for year 2004 b04g=X04g\y04g; res04=(y04-[ones(length(X04),1) X04]*b04g).^2; % res04o squared residuals for the tentative outliers res04o=res04(out04.ListOut); % we keep statistical units below the threshold sel=res04o<threshold^2; % u04i = units to add to the good units subset (intermediate units) u04i=out04.ListOut(sel); % u04o = units outliers out of the threshold u04o=out04.ListOut(~sel); if ~isempty(u04i) u04gi=[u04g u04i]; else u04gi=u04g; end n04gi=length(u04gi); % plotting section hold('off') % good units, plotted as (+) plot(X04(u04g)',y04(u04g)','Marker','+','LineStyle','none','Color','b') hold('on') % units below the treshold, plotted as (X) plot(X04(u04i)',y04(u04i)','Marker','X','MarkerSize',9,'LineWidth',2,'LineStyle','none','Color','m') % outliers, plotted as (O) plot(X04(u04o)',y04(u04o)','Marker','o','LineStyle','none','Color','r') set(gca,'FontSize',14) xlabel('QUANTITY in tons') ylabel('VALUE in 1000 euro') % frequentist Forward Search, 3rd year out04=FSR(y04,X04,'nsamp',nsamp,'plots',1,'msg',0,'init',round(n04/2),'bonflev',bonflev); xlabel('QUANTITY in tons') ylabel('VALUE in 1000 euro') title('Frequentist - 2004');

close all % nsamp is the number of subsamples to use in the frequentist analysis of first % year, in order to find initial subset using LMS. nsamp=3000; % threshold to be used to increase susbet of good units threshold=300; bonflev=0.99; % Bonferroni confidence level % Load 2002 Fishery dataset Fishery2002=load('Fishery2002.txt'); y02=Fishery2002(:,3); X02=Fishery2002(:,2); n02=length(y02); seq02=1:n02; % frequentist Forward Search, 1st year (regression without intercept) [out02]=FSR(y02,X02,'intercept',false,'nsamp',nsamp,'plots',1,'msg',0,'init',round(n02*3/4),'bonflev',bonflev); % In what follows % g stands for good units % i stand for intermediate units (i.e. units whose raw residual is smaller % than threshold) % o stands for outliers % gi stands for good +intermediate units % u02g = good units % n02g = number of good units u02g=setdiff(seq02,out02.ListOut); X02g=X02(u02g,:); y02g=y02(u02g); % b02g = regression coefficients just using g units % Note that b02g is a scalar because the intercept has not been added b02g=X02g\y02g; % res02 = squared raw residuals for all units using b02g res02=(y02-X02*b02g).^2; res02o=res02(out02.ListOut); % sel= boolean vector which is true for the intermediate units % (units whose squared residual is below the threshold) sel=res02o<threshold^2; % u02i = vector containing intermediate units (that is outliers whose % residual is smaller than threshold) u02i=out02.ListOut(sel); % u02gi = g + i units if ~isempty(u02i) u02gi=[u02g u02i]; else u02gi=u02g; end % n02gi = number of good + intermediate units n02gi=length(u02gi); % S202gi = estimated of sigma^2 using g+i units S202gi=sum(res02(u02gi))/(n02gi-1); % bayes = structure which contains prior information to be used in year % 2003 bayes=struct; bayes.beta0=b02g; % beta prior is beta based on g units tau0=1/S202gi; % tau0 is based on g + i units bayes.tau0=tau0; R=X02g'*X02g; % R is based on g units bayes.n0=n02gi; % n0 is based on g + i units bayes.R=R; % 2003 % Load 2003 Fishery dataset Fishery2003=load('Fishery2003.txt'); y03=Fishery2003(:,3); X03=Fishery2003(:,2); n03=length(y03); % Run Bayesian Forward Search for the 2nd year using the prior based on % the first year. out03=FSRB(y03,X03,'bayes',bayes,'msg',0,'plots',1,'init',round(n03/2),'bonflev',bonflev,'intercept',false);

XX=load('BankProfit.txt'); R=load('BankProfitR.txt'); X=XX(:,1:end-1); y=XX(:,end); % Load prior information beta0=zeros(10,1); beta0(1,1)=-0.5; beta0(2,1)=9.1; % Number of products (NUMPRO) beta0(3,1)=0.001; % direct revenues (DIRREV) beta0(4,1)=0.0002; % indirect revenues (INDREV) beta0(5,1)=0.002; % savings accounts SAVACC beta0(6,1)=0.12; % number of operations NUMOPE beta0(7,1)=0.0004; % total amount of operations TOTOPE beta0(8,1)=-0.0004; % Bancomat POS beta0(9,1)=1.3; % Number of cards NUMCAR beta0(10,1)=0.00004; % Amount in cards TOTCAR % \tau s02=10000; tau0=1/s02; % number of obs in which prior was based n0=1500; bayes=struct; bayes.R=R; bayes.n0=n0; bayes.beta0=beta0; bayes.tau0=tau0; intercept=true; n=length(y); out=FSRB(y,X,'bayes',bayes,'msg',1,'plots',1,... 'init',round(n/2),'xlim',[1700 1905],'ylim',[2 4]); % Plot the outliers with a different symbol using a 3x3 layout. selout=out.ListOut; selin=setdiff(1:n,selout); close all % just in case user has additional function subtightplot % http://www.mathworks.com/matlabcentral/fileexchange/39664-subtightplot make_it_tight = false; if make_it_tight == true && exist('subtightplot','file') ==2 subplot = @(m,n,p) subtightplot (m, n, p, [0.05 0.025], [0.1 0.01], [0.1 0.01]); else clear subplot; end % sel = panels in which yticks do not have to be removed sel=[1 4 7]; miny=min(y); maxy=max(y); for j=1:9 subplot(3,3,j) hold('on') plot(X(selin,j),y(selin),'+') plot(X(selout,j),y(selout),'ro') ylim([miny maxy]) xlim([min(X(:,j)) max(X(:,j))]) if isempty(intersect(j,sel)) set(gca,'YTickLabel','') end % Add on the plot the variable name text(0.75,0.1,['x' num2str(j)],'Units','normalized','FontSize',16) end

Observed curve of r_min is at least 10 times greater than 99.99% envelope -------------------------------------------------- ------------------------- Signal detection loop Tentative signal in central part of the search: step m=1830 because rmin(1830,1903)>99.999% ------------------- Signal validation exceedance of upper envelopes Validated signal ------------------------------- Start resuperimposing envelopes from step m=1829 Superimposition stopped because r_{min}(1851,1859)>99.9% envelope Subsample of 1858 units is homogeneous ---------------------------- Final output Number of units declared as outliers=45 Summary of the exceedances 1 99 999 9999 99999 838 77 76 73 73

`y`

— Response variable.
Vector.Response variable, specified as a vector of length n1, where n1 is the number of observations. Each entry in y is the response for the corresponding row of X.

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`

— Predictor variables.
Matrix.Matrix of explanatory variables (also called 'regressors') of dimension n1 x (p-1) where p denotes the number of explanatory variables including the intercept.

Rows of X represent observations, and columns represent variables. By default, there is a constant term in the model, unless you explicitly remove it using input option intercept, so do not include a column of 1s in X. 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.

Remark: note that here we use symbol n1 instead of traditional symbol n because we want to better separate sample information coming from n1 values to prior information coming from n0 previous experiments.

**
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
```

,```
bayes=struct;bayes.R=R;bayes.n0=n0;bayes.beta0=beta0;bayes.tau0=tau0;
```

,```
'plots',1
```

,```
'tag',{'plmdr' 'plyXplot'};
```

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

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

,```
'bivarfit',2
```

,```
'multivarfit','1'
```

,```
'labeladd','1'
```

,```
'nameX',{'NameVar1','NameVar2'}
```

,```
'namey','NameOfResponse'
```

,```
'bonflev',0.99
```

,```
'msg',1
```

`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`

`bayes`

—Prior information.structure.It contains the following fields

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

`beta0` |
p-times-1 vector containing prior mean of \beta |

`R` |
p-times-p positive definite matrix which can be interpreted as X0'X0 where X0 is a n0 x p matrix coming from previous experiments (assuming that the intercept is included in the model. The prior distribution of $\tau_0$ is a gamma distribution with parameters $a_0$ and $b_0$, that is \[ p(\tau_0) \propto \tau^{a_0-1} \exp (-b_0 \tau) \qquad E(\tau_0) = a_0/b_0 \] |

`tau0` |
scalar. Prior estimate of \[ \tau=1/ \sigma^2 =a_0/b_0 \] |

`n0` |
scalar. Sometimes it helps to think of the prior information as coming from n0 previous experiments. Therefore we assume that matrix X0 (which defines R), was made up of n0 observations. REMARK if structure bayes is not supplied the default values which are used are. beta0= zeros(p,1): Vector of zeros. R=eye(p): Identity matrix. tau0=1/1e+6: Very large value for the prior variance, that is a very small value for tau0. n0=1: just one prior observation. $\beta$ is assumed to have a normal distribution with mean $\beta_0$ and (conditional on $\tau_0$) covariance $(1/\tau_0) (X_0'X_0)^{-1}$. $\beta \sim N( \beta_0, (1/\tau_0) (X_0'X_0)^{-1} )$ |

**Example: **```
bayes=struct;bayes.R=R;bayes.n0=n0;bayes.beta0=beta0;bayes.tau0=tau0;
```

**Data Types: **`double`

`plots`

—Plot on the screen.scalar.If plots=1 (default) the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced.

If plots=2 the user can also monitor the intermediate plots based on envelope superimposition.

Else no plot is produced.

**Example: **```
'plots',1
```

**Data Types: **`double`

`tag`

—tags to the plots which are created.character | cell array of characters.This option enables to add a tag to the plots which are created. The default tag names are:

fsr_mdrplot for the plot of mdr based on all the observations;

fsr_yXplot for the plot of y against each column of X with the outliers highlighted;

fsr_resuperplot for the plot of resuperimposed envelopes. The first plot with 4 panel of resuperimposed envelopes has tag fsr_resuperplot1, the second fsr_resuperplot2 ...

If tag is character or a cell of characters of length 1, it is possible to specify the tag for the plot of mdr based on all the observations;

If tag is a cell of length 2 it is possible to control both the tag for the plot of mdr based on all the observations and the tag for the yXplot with outliers highlighted.

If tag is a cell of length 3 the third element specifies the names of the plots of resuperimposed envelopes.

**Example: **```
'tag',{'plmdr' 'plyXplot'};
```

**Data Types: **`char or cell`

`init`

—Search initialization.scalar.scalar which specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it 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`

`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.

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

**Data Types: **`boolean`

`bivarfit`

—Superimpose bivariate least square lines.character.This option adds one or more least square lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi.

bivarfit = '' is the default: no line is fitted.

bivarfit = '1' fits a single ols line to all points of each bivariate plot in the scatter matrix y|X.

bivarfit = '2' fits two ols lines: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted.

bivarfit = '0' fits one ols line to each group. This is useful for the purpose of fitting mixtures of regression lines.

bivarfit = 'i1' or 'i2' or 'i3' etc.

fits an ols line to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

**Example: **```
'bivarfit',2
```

**Data Types: **`char`

`multivarfit`

—Superimpose multivariate least square lines.character.This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi.

multivarfit = '' is the default: no line is fitted.

multivarfit = '1' fits a single ols line to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline{y}'C)) multivarfit = '2' equal to multivarfit ='1' but this time we also add the line based on the group of unselected observations (i.e. the normal units).

**Example: **```
'multivarfit','1'
```

**Data Types: **`char`

`labeladd`

—Add outlier labels in plot.character.If this option is '1', we label the outliers with the unit row index in matrices X and y. The default value is labeladd='', i.e. no label is added.

**Example: **```
'labeladd','1'
```

**Data Types: **`char`

`nameX`

—Add variable labels in plot.cell array of strings.cell array of strings of length p containing the labels of the variables of the regression dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

**Example: **```
'nameX',{'NameVar1','NameVar2'}
```

**Data Types: **`cell`

`namey`

—Add response label.character.character containing the label of the response

**Example: **```
'namey','NameOfResponse'
```

**Data Types: **`char`

`ylim`

—Control y scale in plot.vector.vector with two elements controlling minimum and maximum on the y axis. Default value is '' (automatic scale)

**Example: **```
'ylim','[0,10]' sets the minim value to 0 and the
```

max to 10 on the y axis

**Data Types: **`double`

`xlim`

—Control x scale in plot.vector.vector with two elements controlling minimum and maximum on the x axis. Default value is '' (automatic scale)

**Example: **```
'xlim','[0,10]' sets the minim value to 0 and the
```

max to 10 on the x axis

**Data Types: **`double`

`bonflev`

—Signal to use to identify outliers.scalar.option to be used if the distribution of the data is strongly non normal and, thus, the general signal detection rule based on consecutive exceedances cannot be used. In this case bonflev can be:

- a scalar smaller than 1 which specifies the confidence level for a signal and a stopping rule based on the comparison of the minimum MD with a Bonferroni bound. For example if bonflev=0.99 the procedure stops when the trajectory exceeds for the first time the 99% bonferroni bound.

- A scalar value greater than 1. In this case the procedure stops when the residual trajectory exceeds for the first time this value.

Default value is '', which means to rely on general rules based on consecutive exceedances.

**Example: **```
'bonflev',0.99
```

**Data Types: **`double`

`msg`

—Level of output to display.scalar.scalar which controls whether to display or not messages on the screen If msg==1 (default) messages are displayed on the screen about step in which signal took place and ....

else no message is displayed on the screen

**Example: **```
'msg',1
```

**Data Types: **`double`

`out`

— description
StructureStructure which contains the following fields

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

`ListOut` |
k x 1 vector containing the list of the units declared as outliers or NaN if the sample is homogeneous. This field in future releases will be deleted bacause it will be replaced by out.outliers. |

`outliers` |
k x 1 vector containing the list of the units declared as outliers or NaN if the sample is homogeneous. |

`beta` |
p-by-1 vector containing the posterior mean of $\beta$ (regression coefficents), $\beta = (c*R + X'X)^{-1} (c*R*\beta_0 + X'y)$ in step $n-k$ |

`scale` |
scalar. This is the reciprocal of the square root of the posterior estimate of $\tau$ in step $n-k$ |

`mdr` |
(n-init) x 2 matrix: 1st col = fwd search index; 2nd col = value of Bayesian minimum deletion residual in each step of the fwd 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. |

`nout` |
2 x 5 matrix containing the number of times mdr went out of particular quantiles. First row contains quantiles 1 99 99.9 99.99 99.999. Second row contains the frequency distribution. |

`constr` |
This output is produced only if the search found at a certain step a non singular matrix X. In this case the search runs in a constrained mode, that is including the units which produced a singular matrix in the last n-constr steps. out.constr is a vector which contains the list of units which produced a singular X matrix |

`class` |
'FSRB'. |

Chaloner, K. and Brant, R. (1988), A Bayesian Approach to Outlier Detection and Residual Analysis, "Biometrika", Vol. 75, pp. 651-659.

Riani, M., Corbellini, A. and Atkinson, A.C. (2018), Very Robust Bayesian Regression for Fraud Detection, "International Statistical Review", http://dx.doi.org/10.1111/insr.12247

Atkinson, A.C., Corbellini, A. and Riani, M., (2017), Robust Bayesian Regression with the Forward Search: Theory and Data Analysis, "Test", Vol. 26, pp. 869-886, DOI 10.1007/s11749-017-0542-6