g05pm simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt–Winters method.

Syntax

C#
public static void g05pm( int mode, int n, int itype, int p, double[] param, double[] init, double var, double[] r, G05..::..G05State g05state, double[] e, int en, double[] x, out int ifail )

Visual Basic
Public Shared Sub g05pm ( _ mode As Integer, _ n As Integer, _ itype As Integer, _ p As Integer, _ param As Double(), _ init As Double(), _ var As Double, _ r As Double(), _ g05state As G05..::..G05State, _ e As Double(), _ en As Integer, _ x As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g05pm ( _
	mode As Integer, _
	n As Integer, _
	itype As Integer, _
	p As Integer, _
	param As Double(), _
	init As Double(), _
	var As Double, _
	r As Double(), _
	g05state As G05..::..G05State, _
	e As Double(), _
	en As Integer, _
	x As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g05pm( int mode, int n, int itype, int p, array<double>^ param, array<double>^ init, double var, array<double>^ r, G05..::..G05State^ g05state, array<double>^ e, int en, array<double>^ x, [OutAttribute] int% ifail )

Visual C++

public:
static void g05pm(
	int mode, 
	int n, 
	int itype, 
	int p, 
	array<double>^ param, 
	array<double>^ init, 
	double var, 
	array<double>^ r, 
	G05..::..G05State^ g05state, 
	array<double>^ e, 
	int en, 
	array<double>^ x, 
	[OutAttribute] int% ifail
)

F#
static member g05pm : mode : int * n : int * itype : int * p : int * param : float[] * init : float[] * var : float * r : float[] * g05state : G05..::..G05State * e : float[] * en : int * x : float[] * ifail : int byref -> unit

static member g05pm : 
        mode : int * 
        n : int * 
        itype : int * 
        p : int * 
        param : float[] * 
        init : float[] * 
        var : float * 
        r : float[] * 
        g05state : G05..::..G05State * 
        e : float[] * 
        en : int * 
        x : float[] * 
        ifail : int byref -> unit

Parameters

mode

Type: System..::..Int32

On entry: indicates if g05pm is continuing from a previous call or, if not, how the initial values are computed.

$mode = 0$: Values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are supplied in init.
$mode = 1$: g05pm continues from a previous call using values that are supplied in r. r is not updated.
$mode = 2$: g05pm continues from a previous call using values that are supplied in r. r is updated.

Constraint:

mode = 0

1

2

n: Type: System..::..Int32
On entry: the number of terms of the time series being generated.

Constraint: $n \geq 0$ .

itype

Type: System..::..Int32

On entry: 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.

Constraint:

itype = 1

2

3

4

5

p: Type: System..::..Int32
On entry: if $itype = 4$ or $5$ , the seasonal order, $p$ , otherwise p is not referenced.

Constraint: if $itype = 4$ or $5$ , $p > 1$ .

param

Type: array<System..::..Double>[]()[][]

An array of size [_lparam]

Note: the dimension of the array param must be at least

1

itype = 1

2

3

itype = 3

and at least

4

itype = 4

5

On entry: the smoothing parameters.

itype = 1

2

param [0] = α

and any remaining elements of param are not referenced.

itype = 3

param [0] = α

param [1] = γ

param [2] = ϕ

and any remaining elements of param are not referenced.

itype = 4

5

param [0] = α

param [1] = γ

param [2] = β

and

param [3] = ϕ

and any remaining elements of param are not referenced.

Constraints:

if $itype = 1$ , $0.0 \leq α \leq 1.0$ ;
if $itype = 2$ , $0.0 < α \leq 1.0$ ;
if $itype = 3$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $ϕ \geq 0.0$ ;
if $itype = 4$ or $5$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $0.0 \leq β \leq 1.0$ and $ϕ \geq 0.0$ .

init: Type: array<System..::..Double>[]()[][]
An array of size [_linit]
Note: the dimension of the array init must be at least $1$ if $itype = 1$ , $2$ if $itype = 2$ or $3$ and at least $2 + p$ if $itype = 4$ or $5$ .
On entry: if $mode = 0$ , the initial values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , used to initialize the smoothing.
If $itype = 1$ , $init [0] = m_{0}$ and any remaining elements of init are not referenced.

If $itype = 2$ or $3$ , $init [0] = m_{0}$ and $init [1] = r_{0}$ and any remaining elements of init are not referenced.

If $itype = 4$ or $5$ , $init [0] = m_{0}$ , $init [1] = r_{0}$ and $init [2]$ to $init [2 + p - 1]$ hold the values for $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ . Any remaining elements of init are not referenced.

var: Type: System..::..Double
On entry: the variance, $σ^{2}$ of the Normal distribution used to generate the errors $ε_{i}$ . If $var \leq 0.0$ then Normally distributed errors are not used.

r: Type: array<System..::..Double>[]()[][]
An array of size [_lr]
Note: the dimension of the array r must be at least $13$ if $itype = 1$ , $2$ or $3$ and at least $13 + p$ if $itype = 4$ or $5$ .
On entry: if $mode = 1$ or $2$ , r must contain the values as returned by a previous call to g05pm, r need not be set otherwise.
On exit: if $mode = 1$ , r is unchanged. Otherwise, r contains the information on the current state of smoothing.

Constraint: if $mode = 1$ or $2$ , r must have been initialized by at least one call to g05pm or g13am with $mode \neq 1$ , and r must not have been changed since that call.

g05state: Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.

e: Type: array<System..::..Double>[]()[][]
An array of size [en]
On entry: if $en > 0$ and $var \leq 0.0$ , a vector from which the errors, $ε_{t}$ are randomly drawn, with replacement.
If $en \leq 0$ , e is not referenced.

en: Type: System..::..Int32
On entry: if $en > 0$ , then the length of the vector e.
If both $var \leq 0.0$ and $en \leq 0$ then $ε_{t} = 0.0$ , for $t = 1, 2, \dots, n$ .

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the generated time series, $x_{t}$ , for $t = 1, 2, \dots, n$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

g05pm 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
$\begin{matrix} x_{t} & = & m_{t - 1} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) m_{t - 1} \end{matrix}$
Brown Double Exponential Smoothing
$\begin{matrix} x_{t} & = & m_{t - 1} + \frac{r_{t - 1}}{α} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) m_{t - 1} \\ r_{t} & = & α (m_{t} - m_{t - 1}) + (1 - α) r_{t - 1} \end{matrix}$
Linear Holt Exponential Smoothing
$\begin{matrix} x_{t} & = & m_{t - 1} + ϕ r_{t - 1} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \end{matrix}$
Additive Holt–Winters Smoothing
$\begin{matrix} x_{t} & = & m_{t - 1} + ϕ r_{t - 1} + s_{t - 1 - p} + ε_{t} \\ m_{t} & = & α (x_{t} - s_{t - p}) + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ s_{t} & = & β (x_{t} - m_{t}) + (1 - β) s_{t - p} \end{matrix}$
Multiplicative Holt–Winters Smoothing
$\begin{matrix} x_{t} & = & (m_{t - 1} + ϕ r_{t - 1}) \times s_{t - 1 - p} + ε_{t} \\ m_{t} & = & α x_{t} / s_{t - p} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ s_{t} & = & β x_{t} / m_{t} + (1 - β) s_{t - p} \end{matrix}$

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

σ^{2}

or randomly sampled, with replacement, from a user-supplied vector.

References

Chatfield C (1980) The Analysis of Time Series Chapman and Hall

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: On entry, $mode \neq 0$ , $1$ or $2$ .

$ifail = 2$: On entry, $itype \neq 1$ , $2$ , $3$ , $4$ or $5$ .

$ifail = 3$

On entry,

itype = 4

5

and

p < 2

$ifail = 4$: On entry, at least one of $α$ , $β$ or $γ > 0.0$ or $> 1.0$ .
On entry, $itype = 2$ and $α = 0.0$ .
On entry, $ϕ < 0.0$ .

$ifail = 5$

On entry,

n < 0

$ifail = 8$

On entry,

mode = 1

2

and the array r has not been initialized correctly.

$ifail = 9$: On entry, the array state has not been initialized correctly.

$ifail = 12$: $itype = 5$ and model is unsuitable for multiplicative Holt–Winter.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

None.

Example

g05pm is then called multiple times to obtain simulated forecast confidence intervals.

Example program (C#): g05pme.cs

Example program data: g05pme.d

Example program results: g05pme.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also