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

scalar (order of the trend component).

trend = 0 implies no trend, trend = 1 implies linear trend with intercept, trend = 2 implies quadratic trend, trend = 3 implies cubic trend.

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

In the paper RPRH to denote the order of the trend symbol A is used. If this field is not present into input structure model, model.trend=2 is used.

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 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.

In the paper RPRH to denote the number of frequencies of the seasonal component symbol B is used, while symbol G is used to denote the order of the trend of the seasonal component.

Therefore, for example, model.seasonal=201 corresponds to B=1 and G=2, while model.seasonal=3 corresponds to B=3 and G=0.

If this field is not present into input structure model, model.seasonal=303 is used.

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

scalar or vector associated to level shift component. lshift=0 (default) implies no level shift component.

If model.lshift is vector of positive integers, then it is associated to the positions of level shifts which have to be considered. The most significant one is included in the fitted model.

For example if model.lshift =[13 20 36] a tentative level shift is imposed in position $t=13$, $t=20$ and $t=36$. The most significant among these positions in included in the final model. In other words, the following extra parameters are added to the final model: $\beta_{LS1}* I(t \geq \beta_{LS2})$ where $\beta_{LS1}$ is a real number (associated with the magnitude of the level shift) and $\beta_{LS2}$ is an integer which assumes values 13, 20 or 36 and and $I$ denotes the indicator function.

As a particular case, if model.lshift =13 then a level shift in position $t=13$ is added to the model. In other words the following additional parameters are added: $\beta_{LS1}* I(t \geq 13)$ where $\beta_{LS1}$ is a real number and $I$ denotes the indicator function.

If lshift = -1 tentative level shifts are considered for positions $p+1,p+2, ..., T-p$ and the most significant one is included in the final model ($p$ is the total number of parameters in the fitted model). Note that lshift=-1 is not supported for C-coder translation.

In the paper RPRH $\beta_{LS1}$ is denoted with symbol $\delta_1$, while, $\beta_{LS2}$ is denoted with symbol $\delta_2$.

vector with non negative integer numbers specifying the autoregressive components. For example: model.ARp=[1 2] means a AR(2) process;

model.ARp=2 means just the lag 2 component;

model.ARp=[1 2 5 8] means AR(2) + lag 5 + lag 8;

model.ARp=0 (default) means no autoregressive component.

matrix of size r-by-2 containing the list of the units declared as outliers (first column) and corresponding fitted values (second column) or empty scalar. If model.ARtentout is not empty, when the autoregressive component is present, the y values which are used to compute the autoregressive component are replaced by model.tentout(:,2) for the units contained in model.tentout(:,1) Remark: the default overparametrized model is for monthly data with a quadratic trend (3 parameters) + seasonal component with three harmonics which grows in a cubic way over time (9 parameters), no additional explanatory variables, no level shift and no AR component that is model=struct;

model.s=12;

model.trend=2;

model.seasonal=303;

model.X='';

model.lshift=0;

model.ARp=0;

Using the notation of the paper RPRH we have A=2, B=3; G=3; and $\delta_1=0$.