regressts

regressts computes estimates of regression parameters for a time series models

Syntax

Description

example

out =regressts(y) regressTS with all the default options.

example

out =regressts(y, Name, Value) Example of the use of input option model and plots.

Examples

expand all

  • regressTS with all the default options.
  • 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(:));
    % linear trend + just one harmonic for seasonal
    out=regressts(y);

  • Example of the use of input option model and plots.
  • Model with linear trend, two harmonics for seasonal component and varying amplitude using a linear trend.

    % 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(:));
    model=struct;
    model.trend=1;              % linear trend
    model.s=12;                 % monthly time series
    model.seasonal=102;         % two harmonics with time varying seasonality
    out=regressts(y,'model',model,'plots',1);
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

    Related Examples

    expand all

  • Example of the use of input option bsb and dispresults.
  • Model with linear trend, two harmonics for seasonal component and varying amplitude using a linear trend.

    % 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(:));
    model=struct;
    model.trend=1;              % linear trend
    model.s=12;                 % monthly time series
    model.seasonal=102;         % two harmonics with time varying seasonality
    % Fit is based just on the first 40 observations
    out=regressts(y,'model',model,'bsb',[1:40],'dispresults', true, 'plots',1);

  • Example of the use of input option StartDate.
  • 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(:));
    StartDate=[1949,1]
    % Imposed level shift in position t=60 and 4 harmonics
    model=struct;
    model.trend=1;              % linear trend
    model.s=12;                 % monthly time series
    model.seasonal=104;         % four harmonics with time varying seasonality
    model.posLS=60;             % level shift in position t=60
    out=regressts(y,'model',model,'plots',1,'StartDate',StartDate);

  • Compare scaled residuals using or not correction factors.
  • Example of the use of input option model and plots.

    % Model with linear trend, two harmonics for seasonal component and
    % varying amplitude using a linear trend.
    % 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(:));
    model=struct;
    model.trend=1;              % linear trend
    model.s=12;                 % monthly time series
    model.seasonal=104;         % two harmonics with time varying seasonality
    bsini=1:72;
    out=regressts(y,'model',model,'bsb',bsini);    
    outCR=regressts(y,'model',model,'bsb',bsini,'smallsamplecor',true,'asymptcor',true);   
    h1=subplot(2,1,1);
    title('Estimate of the scale without correction factors')
    resindexplot(out.residuals,'h',h1)
    h2=subplot(2,1,2);
    resindexplot(outCR.residuals,'h',h2)
    title('Estimate of the scale with correction factors')

    Input Arguments

    expand all

    y — Response variable. Vector.

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

    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: 'model', model , 'bsb',[3 5 20:30] , 'smallsamplecor',false , 'asymptcor',false , 'plots',1 , 'nocheck',false , 'dispresults',true , 'StartDate',[2016,3]

    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

    bsb —units forming subset.vector.

    m x 1 vector.

    The default value of bsb is 1:length(y), that is all units are used to compute parameter estimates.

    Example: 'bsb',[3 5 20:30]

    Data Types: double

    smallsamplecor —small sample correction factor.boolean.

    Boolean which defines whether to use or not small sample correction factor to inflate the scale estimate just in case option bsb is used and length(bsb)<length(y). If it is true the small sample correction factor is used. The default value of smallsamplecor is 1, that is the correction is used. See http://users.ugent.be/~svaelst/publications/corrections.pdf for further details about the correction factor. The default value of smallsamplecor is false.

    Example: 'smallsamplecor',false

    Data Types: logical

    asymptcor —asymptotic consistency factor.boolean.

    Boolean which defines whether to use or not consistency correction factor to inflate the scale estimate just in case option bsb is used and length(bsb)<length(y). If it is true the asmptotic consistency is used. The default value of asymptcor is false, that is the asymptotic consistency factor is not used.

    Example: 'asymptcor',false

    Data Types: logical

    plots —Plot on the screen.scalar.

    If equal to one a two panel plot appears on the scree. The top panel contains real and fitted value. The bottom panel contains scaled residuals with a 99.9 per cent confidence band, 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.boolean.

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

    Example: 'nocheck',false

    Data Types: logical

    dispresults —Display results of final fit.boolean.

    If dispresults is true, labels of coefficients, estimated coefficients, standard errors, tstat and p-values are shown on the screen in a fully formatted way. The default value of dispresults is false.

    Example: 'dispresults',true

    Data Types: logical

    StartDate —The time of the first observation.numeric vector of length 2.

    Vector with two integers, which specify a natural time unit and a (1-based) number of samples into the time unit. For example, if model.s=12 (that is the data are monthly) and the first observation starts in March 2016, then StartDate=[2016,3]; Similarly, if models.s=4 (that is the data are quarterly) and the first observation starts in the second quarter or year 2014, then StartData=[2014,2]. The information in option StartDate will be used to create in the output the dates inside the time series object.

    Example: 'StartDate',[2016,3]

    Data Types: double

    Output Arguments

    expand all

    out — description Structure

    A structure containing the following fields

    Value Description
    B

    Matrix containing estimated beta coefficients, (including the intercept when options.intercept=true) standard errors, t-stat and p-values The content of matrix B is as follows:

    1st col = beta coefficients The order of the beta coefficients is as follows:

    1) trend elements (if present). If the trend is of order two there are r+1 coefficients if the intercept is present otherwise there are just r components;

    2) linear part of seasonal component 2, 4, 6, ..., s-2, s-1 coefficients (if present);

    3) coefficients associated with the matrix of explanatory variables which have a potential effect on the time series under study (X);

    4) non linear part of seasonal component, that is varying amplitude. If varying amplitude is of order k there are k coefficients (if present);

    5) level shift component (if present). In this case there are two coefficients, the second (which is also the last element of vector beta) is an integer which specifies the time in which level shift takes place and the first (which is also the penultime element of vector beta) is a real number which identifies the magnitude of the upward (downward) level shift;

    2nd col = standard errors;

    3rd col = t-statistics;

    4th col = p values.

    yhat

    vector of fitted values after final (NLS=non linear least squares) step.

    $ (\hat \eta_1, \hat \eta_2, \ldots, \hat \eta_T)'$

    e

    Vector T-by-1 containing the raw residuals.

    residuals

    Vector T-by-1 containing the scaled residuals.

    scale

    scale estimate of the residuals \[ \hat \sigma = cor \times \sum_{i \in S_m} [y_i- \eta(x_i,\hat \beta)]^2/(m-p) \] where $S_m$ is a set of cardinality $m$ which contains the units belonging to bsb and $p$ is the total number of estimated parameters and $cor$ is a correction factor to make the estimator consistent (see input options smallsamplecor and asymptcor).

    invXX

    $cov(\beta)/\hat \sigma^2$. p-by-p, square matrix.

    If the model is linear out.invXX is equal to $(X'X)^{-1}$, else out.invXX is equal to $(A'A)^{-1}$ where $A$ is the matrix of partial derivatives. More precisely:

    \[ a_{i,j}=\frac{\partial \eta_i(x_i, \hat \beta)}{\partial \hat \beta_j} \] where \begin{eqnarray} y_i & = & \eta(x_i,\beta)+ \epsilon_i \\ & = & \eta_i +\epsilon_i \\ & = & \eta(x_i,\hat \beta)+ e_i \\ & = & \hat \eta_i + e_i \end{eqnarray}

    covB

    $cov(\beta)$. p-by-p, square matrix containing variance covariance matrix of estimated coefficients.

    y

    response vector y.

    X

    data matrix X containing trend, seasonal, expl and lshift, if the model is linear or linearized version of $\eta(x_i, \beta)$ if the model is non linear containing in the columns partial derivatives evaluated in correspondence of out.B(:,1) with respect to each parameter. In other words, the $i,j$-th element of out.X is \[ \frac{\partial \eta_i(x_i, \hat \beta)}{\partial \hat \beta_j} \]

    $j=1, 2, \ldots, p$, $i \in S_m$.

    The size of this matrix is:

    n-length(out.outliers)-by-p.

    Exflag

    Reason nlinfit stopped. Integer.

    If the model is non linear out.Exflag is equal to 1 if 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 is equal to -1;

    ExflagALS

    Reason ALS routine stopped. Integer.

    If in the iterations missing or NaN are found out.ExflagALS=-1.

    Note that ALS routine is used just in case there was no convergence inside routine nlinfit in order to provide a better set of initial parameter estimates before retrying nlinfit.

    If there was immediate convergence in nlinfit out.ExflagALS is empty.

    iterALS

    Number of iterations in the ALS routine. Intger between 1 and 10000 (maximum number of iterations).

    Note that ALS routine is used just in case there was no convergence inside routine nlinfit in order to provide a better set of initial parameter estimates before retrying nlinfit.

    If there was immediate convergence in nlinfit out.iterALS is empty.

    References

    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]

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