g05pmf simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt–Winters method.

2 Specification

Fortran Interface

Subroutine g05pmf (

mode, n, itype, p, param, init, var, r, state, e, en, x, ifail)

Integer, Intent (In)	::	mode, n, itype, p, en
Integer, Intent (Inout)	::	state(*), ifail
Real (Kind=nag_wp), Intent (In)	::	param(), init(), var, e(en)
Real (Kind=nag_wp), Intent (Inout)	::	r(*)
Real (Kind=nag_wp), Intent (Out)	::	x(n)

C Header Interface

#include <nag.h>

void	g05pmf_ (const Integer mode, const Integer n, const Integer itype, const Integer p, const double param[], const double init[], const double var, double r[], Integer state[], const double e[], const Integer en, double x[], Integer *ifail)

The routine may be called by the names g05pmf or nagf_rand_times_smooth_exp.

3 Description

g05pmf 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.

4 References

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

5 Arguments

1: $mode$ – Integer Input

On entry: indicates if g05pmf 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.
$mode = 1$: g05pmf continues from a previous call using values that are supplied in r. r is not updated.
$mode = 2$: g05pmf continues from a previous call using values that are supplied in r. r is updated.

Constraint:

mode = 0

1

2

2: $n$ – Integer Input

On entry: the number of terms of the time series being generated.

Constraint:

n \geq 0

3: $itype$ – Integer Input

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

4: $p$ – Integer Input

On entry: if

itype = 4

5

, the seasonal order,

p

, otherwise p is not referenced.

Constraint: if

itype = 4

5

p > 1

5: $param (*)$ – Real (Kind=nag_wp) array Input

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 (1) = α

and any remaining elements of param are not referenced.

itype = 3

param (1) = α

param (2) = γ

param (3) = ϕ

and any remaining elements of param are not referenced.

itype = 4

5

param (1) = α

param (2) = γ

param (3) = β

and

param (4) = 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 $ϕ \geq 0.0$ .

6: $init (*)$ – Real (Kind=nag_wp) array Input

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

1

itype = 1

2

itype = 2

3

and at least

2 + p

itype = 4

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.

itype = 1

init (1) = m_{0}

and any remaining elements of init are not referenced.

itype = 2

3

init (1) = m_{0}

and

init (2) = r_{0}

and any remaining elements of init are not referenced.

itype = 4

5

init (1) = m_{0}

init (2) = r_{0}

and

init (3)

init (2 + p)

hold the values for

s_{- j}

, for

j = 0, 1, \dots, p - 1

. Any remaining elements of init are not referenced.

7: $var$ – Real (Kind=nag_wp) Input

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.

8: $r (*)$ – Real (Kind=nag_wp) array Communication Array

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

13

itype = 1

2

3

and at least

13 + p

itype = 4

5

On entry: if

mode = 1

2

, r must contain the values as returned by a previous call to g05pmf, 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

2

, r must have been initialized by at least one call to g05pmf or g13amf with

mode \neq 1

, and r must not have been changed since that call.

9: $state (*)$ – Integer array Communication Array

Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

10: $e (en)$ – Real (Kind=nag_wp) array Input

On entry: if

en > 0

and

var \leq 0.0

, a vector from which the errors,

ε_{t}

are randomly drawn, with replacement.

en \leq 0

, e is not referenced.

11: $en$ – Integer Input

On entry: if

en > 0

, 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

12: $x (n)$ – Real (Kind=nag_wp) array Output

On exit: the generated time series,

x_{t}

, for

t = 1, 2, \dots, n

13: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode = 0$ , $1$ or $2$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = 3$: On entry, $itype = ⟨ value ⟩$ .
Constraint: $itype = 1$ , $2$ , $3$ , $4$ or $5$ .

$ifail = 4$: On entry, $p = ⟨ value ⟩$ .
Constraint: if $itype = 4$ or $5$ , $p \geq 2$ .

$ifail = 5$: On entry, $param (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $0 \leq param (i) \leq 1$ .

On entry, $param (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: if $itype = 2$ , $0 < param (i) \leq 1$ .

On entry, $param (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $param (i) \geq 0$ .

$ifail = 8$: On entry, some of the elements of the array r have been corrupted or have not been initialized.

$ifail = 9$: On entry, state vector has been corrupted or not initialized.

$ifail = 12$: Model unsuitable for multiplicative Holt–Winter, try a different set of parameters.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05pmf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example reads

11

observations from a time series relating to the rate of the earth's rotation about its polar axis and fits an exponential smoothing model using g13amf.

g05pmf is then called multiple times to obtain simulated forecast confidence intervals.

g05pm: FL CL CPP AD

NAG FL Interfaceg05pmf (times_​smooth_​exp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g05pmf (times_smooth_exp)