FSRtsmdr computes minimum deletion residual for time series models in each step of the search

linear trend + just one harmonic for seasonal Common part to all examples: load airline dataset.

% 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(:)); % Monitor minimum deletion residual in each step of the forward search. % Start from a random subset. The final part of the trajectory is % completely unaffected by the starting point. % Plot the trajectory of mdr. mdr=FSRtsmdr(y,0,'plots',1);

Compute minimum deletion residual and start monitoring it from step m=80.

% Set up a personalized model. model=struct; model.trend=1; % linear trend model.s=12; % monthly time series model.seasonal=104; % four harmonics with time varying seasonality % Choose step to start monitoring. init=80; out1=FSRtsmdr(y,0,'model',model,'init',80,'plots',1);

Common part to all examples: load airline dataset.

% 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 y=(y(:)); % Compute minimum deletion residual and analyze the units entering % subset in each step of the fwd search (matrix Un). As is well known, % the FS provides an ordering of the data from those most in agreement % with a suggested model (which enter the first steps) to those least in % agreement with it (which are included in the final steps). % Set up a personalized model. model=struct; model.trend=1; % linear trend model.s=12; % monthly time series model.seasonal=104; % four harmonics with time varying seasonality % Choose step to start monitoring. init=80; [mdr,Un,BB,Bols,S2,Exflag]=FSRtsmdr(y,0,'model',model,'init',80,'plots',1); % Check if there was convergence in all step which were monitored. if min(Exflag(:,2))<1 disp('Warning: in some steps there was not convergence') else disp('Convergence obtained in all steps') end % Check the last two units which are included in the last two steps. disp(Un(end-1:end,:))

Convergence obtained in all steps 143 139 NaN NaN NaN NaN NaN NaN NaN NaN NaN 144 142 NaN NaN NaN NaN NaN NaN NaN NaN NaN

In this example the units forming subset are stored just for selected steps.

model=struct; model.trend=1; % linear trend model.s=12; % monthly time series model.seasonal=104; % four harmonics with time varying seasonality init=80; [mdr,Un,BB,Bols,S2] =FSRtsmdr(y,0,'model',model,'init',80,'bsbsteps',[90 120]); % BB has just two columns % First column contains information about units forming subset at step m=90 % sum(~isnan(BB(:,1))) is 90 % Second column contains information about units forming subset at step m=120 % sum(~isnan(BB(:,2))) is 120 disp(sum(~isnan(BB(:,1)))) disp(sum(~isnan(BB(:,2))))

90 120

Set up the model.

model=struct; model.trend=1; % linear trend model.s=12; % monthly time series model.seasonal=104; % four harmonics with time varying seasonality % Call LTSts out=LTSts(y,'model',model'); % Extract best initial subset from LTSts. [~,indres]=sort(abs(out.residuals)); bs=indres(1:50); [mdr,Un,BB,Bols,S2,Exflag] =FSRtsmdr(y,bs,'model',model,'init',length(bs)+1,'plots',1);

`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.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). p is the total number of regression parameters which have to be estimated.

**
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
```

.

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

,```
'model', model
```

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

,```
'nocheck',1
```

,```
'msg',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`

`plots`

—Plot on the screen.scalar.If equal to one a plot of minimum deletion residual appears on the screen with 1%, 50% and 99% confidence bands else (default) no plot is shown.

Remark: the plot which is produced is very simple. In order to control a series of options in this plot and in order to connect it dynamically to the other forward plots it is necessary to use function mdrplot.

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

**Data Types: **`double`

`nocheck`

—Check input arguments inside structure model.as default nocheck=0.
**Example: **```
'nocheck',1
```

**Data Types: **`double`

`msg`

—Level of output to display.scalar.It controls whether to display or not messages about great interchange on the screen If msg==1 (default) messages are displayed on the screen else no message is displayed on the screen

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

**Data Types: **`double`

`constr`

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

**Example: **```
'constr',[1:10] forces the first 10 units to join
```

the subset in the last 10 steps

**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`

`bsbsteps`

—Save the units forming subsets.vector.It specifies for which steps of the fwd search it is necessary to save the units forming subsets. 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 T<=5000, else to store the units forming subset at steps init and steps which are multiple of 100. For example, as default, if T=753 and init=6, units forming subset are stored for m=init, 100, 200, 300, 400, 500 and 600.

**Example: **```
'bsbsteps',[100 200] stores the unis forming
```

subset in steps 100 and 200.

**Data Types: **`double`

`mdr`

—Monitoring of minimum
deletion residual at each step of the forward search.
`T -init -by- 2 `

matrix1st col = fwd search index (from init to T-1).

2nd col = minimum deletion residual.

REMARK: if in a certain step of the search matrix is singular, this procedure checks how many observations produce a singular matrix. In this case mdr is a column vector which contains the list of units for which matrix X is non singular.

`Un`

—Units included in each step.
` `

Matrix(T-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.
` `

MatrixT x (T-init+1) or T-by-length(bsbsteps) matrix (depending on input option bsbsteps) which contains information about the units belonging to the subset at each step of the forward search. BB has the following structure:

1-st row has number 1 in correspondence of the steps in which unit 1 is included inside subset and a missing value for the other steps;

......

(T-1)-th row has number T-1 in correspondence of the steps in which unit T-1 is included inside subset and a missing value for the other steps;

T-th row has number Tn in correspondence of the steps in which unit T is included inside subset and a missing value for the other steps The size of matrix BB is T x (T-init+1) if option input bsbsteps is 0 else the size is T-by-length(bsbsteps).

`Bols`

—beta coefficents.
` `

Matrix(T-init+1) x (p+1) matrix containing the monitoring of estimated beta coefficients in each step of the forward search.

`S2`

—S2 and R2.
` `

Matrix(T-init+1) x 3 matrix containing the monitoring of S2 (2nd column)and R2 (third column) in each step of the forward search.

`Exflag`

—Reason nlinfit stopped.
` Integer `

matrix(T-init+1) x 2 matrix containing information about the result of the maximization procedure.

If the model is non linear out.Exflag(i,2) is equal to 1 if at step out.Exflag(i,1) the maximization procedure did not produce warnings or the warning was different from "ILL Conditiioned Jacobian". For any other warning which is produced (for example, "Overparameterized", "IterationLimitExceeded", 'MATLAB:rankDeficientMatrix") out.Exflag(i,2) is equal to -1;

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]