# FSRtsbsb

FSRtsbsb returns the units belonging to the subset in each step of the forward search

## Syntax

• Un=FSRtsbsb(y,bsb)example
• Un=FSRtsbsb(y,bsb,Name,Value)example
• [Un,BB]=FSRtsbsb(___)example

## Description

 Un =FSRtsbsb(y, bsb) FSRtsbsb with all default options.

 Un =FSRtsbsb(y, bsb, Name, Value) FSRtsbsb with optional arguments.

 [Un, BB] =FSRtsbsb(___) Monitoring the units belonging to subset.

## Examples

expand all

### FSRtsbsb with all default options.

load('fishery');
y=fishery{:,2};
bsbini=[97    77    12     2    26    95    10    60    94   135     7    61   114];
[Un,BB]=FSRtsbsb(y,bsbini);

### FSRtsbsb with optional arguments.

Load airline data 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960.

y = [112  115  145  171  196  204  242  284  315  340  360  417    % Jan
118  126  150  180  196  188  233  277  301  318  342  391    % Feb
132  141  178  193  236  235  267  317  356  362  406  419    % Mar
129  135  163  181  235  227  269  313  348  348  396  461    % Apr
121  125  172  183  229  234  270  318  355  363  420  472    % May
135  149  178  218  243  264  315  374  422  435  472  535    % Jun
148  170  199  230  264  302  364  413  465  491  548  622    % Jul
148  170  199  242  272  293  347  405  467  505  559  606    % Aug
136  158  184  209  237  259  312  355  404  404  463  508    % Sep
119  133  162  191  211  229  274  306  347  359  407  461    % Oct
104  114  146  172  180  203  237  271  305  310  362  390    % Nov
118  140  166  194  201  229  278  306  336  337  405  432 ]; % Dec
% Source:
% http://datamarket.com/data/list/?q=provider:tsdl
y=(y(:));
% Define the model  and show the monitoring units plots.
model=struct;
model.trend=1;              % linear trend
model.s=12;                 % monthly time series
model.seasonal=104;         % four harmonics with time varying seasonality
bsbini=[97    77    12     2    26    95    10    60    94   135     7    61   114];
[Un,BB]=FSRtsbsb(y,bsbini,'model',model,'plots',1);

### Monitoring the units belonging to subset.

%   1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960
y = [112  115  145  171  196  204  242  284  315  340  360  417    % Jan
118  126  150  180  196  188  233  277  301  318  342  391    % Feb
132  141  178  193  236  235  267  317  356  362  406  419    % Mar
129  135  163  181  235  227  269  313  348  348  396  461    % Apr
121  125  172  183  229  234  270  318  355  363  420  472    % May
135  149  178  218  243  264  315  374  422  435  472  535    % Jun
148  170  199  230  264  302  364  413  465  491  548  622    % Jul
148  170  199  242  272  293  347  405  467  505  559  606    % Aug
136  158  184  209  237  259  312  355  404  404  463  508    % Sep
119  133  162  191  211  229  274  306  347  359  407  461    % Oct
104  114  146  172  180  203  237  271  305  310  362  390    % Nov
118  140  166  194  201  229  278  306  336  337  405  432 ]; % Dec
% Source:
% http://datamarket.com/data/list/?q=provider:tsdl
y=(y(:));
% Contaminates units 31:40
y(31:40)=y(31:40)+200;
% Define the model  and show the monitoring units plots.
model=struct;
model.trend=1;              % linear trend
model.s=12;                 % monthly time series
model.seasonal=104;         % four harmonics with time varying seasonality
bsbini=[97    77    12     2    26    95    10    60    94   135     7    61   114];
[Un,BB]=FSRtsbsb(y,0,'model',model,'plots',1);
% Create the 'monitoring units plot'
figure;
seqr=[Un(1,1)-1; Un(:,1)];
plot(seqr,BB','bx');
xlabel('Subset size m');
ylabel('Monitoring units plot');
% The plot, which monitors the units belonging to subset in each step of
% the forward search shows that independently of the initial starting
% point the contaminated units (31:40) are always the last to enter the
% forward search.

## Input Arguments

### y — Time series to analyze. Vector.

A row or a column vector with T elements, which contains the time series.

Data Types: single| double

### bsb — list of units forming the initial subset. Vector | 0.

If bsb=0 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

### Name-Value Pair Arguments

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.

Example:  'init',100 starts monitoring from step m=100 , 'model', model , 'nocheck',false , 'plots',1 , 'bsbmfullrank',1 

### init —Start of monitoring point.scalar.

It specifies the point where to initialize the search and start monitoring required diagnostics. If it is not specified it is set equal floor(0.5*(T+1))

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

Data Types: double

### model —model type.structure.

A structure which specifies the model which will be used. The model structure contains the following fields:

Value Description
s

scalar (length of seasonal period). For monthly data s=12 (default), for quartely data s=4, ...

trend

scalar (order of the trend component).

trend = 1 implies linear trend with intercept (default), trend = 2 implies quadratic trend ...

Admissible values for trend are, 0, 1, 2 and 3.

seasonal

scalar (integer specifying number of frequencies, i.e. harmonics, in the seasonal component. Possible values for seasonal are $1, 2, ..., [s/2]$, where $[s/2]=floor(s/2)$.

For example:

if seasonal =1 (default) we have:

$\beta_1 \cos( 2 \pi t/s) + \beta_2 sin ( 2 \pi t/s)$;

if seasonal =2 we have:

$\beta_1 \cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s) + \beta_3 \cos(4 \pi t/s) + \beta_4 \sin (4 \pi t/s)$.

Note that when $s$ is even the sine term disappears for $j=s/2$ and so the maximum number of trigonometric parameters is $s-1$.

If seasonal is a number greater than 100 then it is possible to specify how the seasonal component grows over time.

For example, seasonal =101 implies a seasonal component which just uses one frequency which grows linearly over time as follows:

$(1+\beta_3 t)\times ( \beta_1 cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s))$.

For example, seasonal =201 implies a seasonal component which just uses one frequency which grows in a quadratic way over time as follows:

$(1+\beta_3 t + \beta_4 t^2)\times( \beta_1 \cos( 2 \pi t/s) + \beta_2 \sin ( 2 \pi t/s))$.

seasonal =0 implies a non seasonal model.

X

matrix of size T-by-nexpl containing the values of nexpl extra covariates which are likely to affect y.

posLS

positive integer which specifies to position to include the level shift component.

For example if model.posLS =13 then the explanatory variable $I(t \geq 13})$ is created.

If this field is not present or if it is empty, the level shift component is not included.

B

column vector or matrix containing the initial values of parameter estimates which have to be used in the maximization procedure. If model.B is a matrix, then initial estimates are extracted from the first colum of this matrix. If this field is empty or if this field is not present, the initial values to be used in the maximization procedure are referred to the OLS parameter estimates of the linear part of the model. The parameters associated to time varying amplitude are initially set to 0.

Remark: the default model is for monthly data with a linear trend (2 parameters) + seasonal component with just one harmonic (2 parameters), no additional explanatory variables and no level shift that is model=struct;

model.s=12;

model.trend=1;

model.seasonal=1;

model.X='';

model.posLS='';

Example:  'model', model 

Data Types: struct

### nocheck —Check input arguments.boolean.

If nocheck is equal to true no check is performed on supplied structure model

Example:  'nocheck',false 

Data Types: logical

### bsbsteps —Save the units forming subsets in selected steps.vector.

It specifies for which steps of the fwd search it is necessary to save the units forming subset. If bsbsteps is 0 we store the units forming subset in all steps. The default is store the units forming subset in all steps if n<=5000, else to store the units forming subset at steps init and steps which are multiple of 100. For example, as default, if n=7530 and init=6, units forming subset are stored for m=init, 100, 200, ..., 7500.

Example:  'bsbsteps',[100 200] stores the unis forming  subset in steps 100 and 200.

Data Types: double

### plots —Plot on the screen.scalar.

If plots=1 the monitoring units plot is displayed on the screen. The default value of plots is 0 (that is no plot is produced on the screen).

Example:  'plots',1 

Data Types: double

### bsbmfullrank —What to do in case subset at step m (say bsbm) produces a non singular X.scalar.

This options controls what to do when rank(X(bsbm,:)) is smaller then number of explanatory variables.

If bsbmfullrank = 1 (default is 1) these units (whose number is say mnofullrank) are constrained to enter the search in the final n-mnofullrank steps else the search continues using as estimate of beta at step m the estimate of beta found in the previous step.

Example:  'bsbmfullrank',1 

Data Types: double

## Output Arguments

### Un —Units included in each step.  Matrix

(n-init) x 11 Matrix which contains the unit(s) included in the subset at each step of the 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.

### BB —Units belonging to subset in each step or selected steps.  Matrix

n-by-(n-init+1) or n-by-length(bsbsteps) matrix which contains the units belonging to the subset at each step (or in selected steps as specified by optional vector bsbsteps) of the forward search.

More precisely:

BB(:,1) contains the units forming subset in step bsbsteps(1);

....;

BB(:,end) contains the units forming subset in step bsbsteps(end);

Row 1 of matrix BB is referred to unit 1;

......;

Row n of matrix BB is referred to unit n;

Units not belonging to subset are denoted with NaN.

## References

Atkinson, A.C. and Riani, M. (2006), Distribution theory and simulations for tests of outliers in regression, "Journal of Computational and Graphical Statistics", Vol. 15, pp. 460-476.

Riani, M. and Atkinson, A.C. (2007), Fast calibrations of the forward search for testing multiple outliers in regression, "Advances in Data Analysis and Classification", Vol. 1, pp. 123-141.

Rousseeuw, P.J., Perrotta D., Riani M. and Hubert, M. (2018), Robust Monitoring of Many Time Series with Application to Fraud Detection, "Econometrics and Statistics". [RPRH]