naginterfaces.library.rand.times_​smooth_​exp

naginterfaces.library.rand.times_smooth_exp(mode, n, itype, p, param, init, var, comm, statecomm, e)

times_smooth_exp simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt–Winters method.

For full information please refer to the NAG Library document for g05pm

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05pmf.html

Parameters

modeint

Indicates if times_smooth_exp is continuing from a previous call or, if not, how the initial values are computed.

$mode=0$

Values for $m_0$, $r_0$ and $s_{-\text{j}}$, for $\text{j}=0,1,\dots ,p-1$, are supplied in $init$.

$mode=1$

times_smooth_exp continues from a previous call using values that are supplied in $comm$[‘r’]. $comm$[‘r’] is not updated.

$mode=2$

times_smooth_exp continues from a previous call using values that are supplied in $comm$[‘r’]. $comm$[‘r’] is updated.

nint

The number of terms of the time series being generated.

itypeint

The smoothing function.

$itype=1$

Single exponential.

$itype=2$

Brown’s double exponential.

$itype=3$

Linear Holt.

$itype=4$

Additive Holt–Winters.

$itype=5$

Multiplicative Holt–Winters.

pint

If $itype=4$ or $5$, the seasonal order, $p$, otherwise $p$ is not referenced.

paramfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $itype\text{in}(1,2)$: $1$; if $itype=3$: $3$; if $itype\text{in}(4,5)$: $4$; otherwise: $0$.

The smoothing parameters.

If $itype=1$ or $2$, $param\left[0\right]=\alpha$ and any remaining elements of $param$ are not referenced.

If $itype=3$, $param\left[0\right]=\alpha$, $param\left[1\right]=\gamma$, $param\left[2\right]=\varphi$ and any remaining elements of $param$ are not referenced.

If $itype=4$ or $5$, $param\left[0\right]=\alpha$, $param\left[1\right]=\gamma$, $param\left[2\right]=\beta$ and $param\left[3\right]=\varphi$ and any remaining elements of $param$ are not referenced.

initfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $itype=1$: $1$; if $itype\text{in}(2,3)$: $2$; if $itype\text{in}(4,5)$: $2+p$; otherwise: $0$.

If $mode=0$, the initial values for $m_0$, $r_0$ and $s_{-\text{j}}$, for $\text{j}=0,1,\dots ,p-1$, used to initialize the smoothing.

If $itype=1$, $init\left[0\right]=m_0$ and any remaining elements of $init$ are not referenced.

If $itype=2$ or $3$, $init\left[0\right]=m_0$ and $init\left[1\right]=r_0$ and any remaining elements of $init$ are not referenced.

If $itype=4$ or $5$, $init\left[0\right]=m_0$, $init\left[1\right]=r_0$ and $init\left[2\right]$ to $init[2+p-1]$ hold the values for $s_{-\text{j}}$, for $\text{j}=0,1,\dots ,p-1$.

Any remaining elements of $init$ are not referenced.

varfloat

The variance, ${\sigma }^2$ of the Normal distribution used to generate the errors ${ϵ}_i$. If $var\le 0.0$ then Normally distributed errors are not used.

commdict, communication object, modified in place

Communication structure for the reference vector.

If $mode=1$ or $2$, this argument must have been initialized by a prior call to times_smooth_exp.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

efloat, array-like, shape $\left(\text{en}\right)$

If $\text{en}>0$ and $var\le 0.0$, a vector from which the errors, ${ϵ}_t$ are randomly drawn, with replacement.

If $\text{en}\le 0$, $e$ is not referenced.

Returns

xfloat, ndarray, shape $\left(n\right)$

The generated time series, $x_{\text{t}}$, for $\text{t}=1,2,\dots ,n$.

Raises

NagValueError

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode=0$, $1$ or $2$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $3$)

On entry, $itype=⟨value⟩$.

Constraint: $itype=1$, $2$, $3$, $4$ or $5$.

(errno $4$)

On entry, $p=⟨value⟩$.

Constraint: if $itype=4$ or $5$, $p\ge 2$.

(errno $5$)

On entry, $param[⟨value⟩]=⟨value⟩$.

Constraint: $0\le param\left[i\right]\le 1$.

(errno $5$)

On entry, $param[⟨value⟩]=⟨value⟩$.

Constraint: if $itype=2$, $0<param\left[i\right]\le 1$.

(errno $5$)

On entry, $param[⟨value⟩]=⟨value⟩$.

Constraint: $param\left[i\right]\ge 0$.

(errno $8$)

On entry, some of the elements of the array $comm$[‘r’] have been corrupted or have not been initialized.

(errno $9$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

(errno $12$)

Model unsuitable for multiplicative Holt–Winter, try a different set of parameters.

Notes

times_smooth_exp returns $\{x_t:t=1,2,\dots ,n\}$, a realization of a time series from an exponential smoothing model defined by one of five smoothing functions:

Single Exponential Smoothing

$$\begin{array}{c}\begin{array}{ccc}{x}_t& =& m_{t-1}+{ϵ}_t\\ m_t& =& \alpha x_t+(1-\alpha )m_{t-1}\end{array}\end{array}$$

Brown Double Exponential Smoothing

$$\begin{array}{c}\begin{array}{ccc}{x}_t& =& m_{t-1}+\frac{r_{t-1}}{\alpha }+{ϵ}_t\\ m_t& =& \alpha x_t+(1-\alpha )m_{t-1}\\ r_t& =& \alpha (m_t-m_{t-1})+(1-\alpha )r_{t-1}\end{array}\end{array}$$

Linear Holt Exponential Smoothing

$$\begin{array}{c}\begin{array}{ccc}{x}_t& =& m_{t-1}+\varphi r_{t-1}+{ϵ}_t\\ m_t& =& \alpha x_t+(1-\alpha )(m_{t-1}+\varphi r_{t-1})\\ r_t& =& \gamma (m_t-m_{t-1})+(1-\gamma )\varphi r_{t-1}\end{array}\end{array}$$

Additive Holt–Winters Smoothing

$$\begin{array}{c}\begin{array}{ccc}{x}_t& =& m_{t-1}+\varphi r_{t-1}+s_{t-1-p}+{ϵ}_t\\ m_t& =& \alpha (x_t-s_{t-p})+(1-\alpha )(m_{t-1}+\varphi r_{t-1})\\ r_t& =& \gamma (m_t-m_{t-1})+(1-\gamma )\varphi r_{t-1}\\ s_t& =& \beta (x_t-m_t)+(1-\beta )s_{t-p}\end{array}\end{array}$$

Multiplicative Holt–Winters Smoothing

$$\begin{array}{c}\begin{array}{ccc}{x}_t& =& (m_{t-1}+\varphi r_{t-1})\times s_{t-1-p}+{ϵ}_t\\ m_t& =& \alpha x_t/s_{t-p}+(1-\alpha )(m_{t-1}+\varphi r_{t-1})\\ r_t& =& \gamma (m_t-m_{t-1})+(1-\gamma )\varphi r_{t-1}\\ s_t& =& \beta x_t/m_t+(1-\beta )s_{t-p}\end{array}\end{array}$$

where $m_t$ is the mean, $r_t$ is the trend and $s_t$ is the seasonal component at time $t$ with $p$ being the seasonal order. The errors, ${ϵ}_t$ are either drawn from a normal distribution with mean zero and variance ${\sigma }^2$ or randomly sampled, with replacement, from a user-supplied vector.

References

Chatfield, C, 1980, The Analysis of Time Series, Chapman and Hall