nag_tsa_exp_smooth (g13amc) : NAG Library, Mark 24

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_exp_smooth (Nag_InitialValues mode, Nag_ExpSmoothType itype, Integer p, const double param[], Integer n, const double y[], Integer k, double init[], Integer nf, double fv[], double fse[], double yhat[], double res[], double dv, double ad, double r[], NagError *fail)

3 Description

Exponential smoothing is a relatively simple method of short term forecasting for a time series. nag_tsa_exp_smooth (g13amc) provides five types of exponential smoothing; single exponential, Brown's double exponential, linear Holt (also called double exponential smoothing in some references), additive Holt–Winters and multiplicative Holt–Winters. The choice of smoothing method used depends on the characteristics of the time series. If the mean of the series is only slowly changing then single exponential smoothing may be suitable. If there is a trend in the time series, which itself may be slowly changing, then double exponential smoothing may be suitable. If there is a seasonal component to the time series, e.g., daily or monthly data, then one of the two Holt–Winters methods may be suitable.

For a time series

y_{t}

, for

t = 1, 2, \dots, n

, the five smoothing functions are defined by the following:

Single Exponential Smoothing
$\begin{matrix} m_{t} & = & α y_{t} + (1 - α) m_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + (f - 1) α^{2}) \end{matrix}$

Brown Double Exponential Smoothing

\begin{matrix} m_{t} & = & α y_{t} + (1 - α) m_{t - 1} \\ r_{t} & = & α (m_{t} - m_{t - 1}) + (1 - α) r_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} + ((f - 1) + 1 / α) r_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 0}^{f - 1} {(2 α + (i - 1) α^{2})}^{2}) \end{matrix}

Linear Holt Smoothing

\begin{matrix} m_{t} & = & α y_{t} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} + \sum_{i = 1}^{f} ϕ^{i} r_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 1}^{f - 1} {(α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)})}^{2}) \end{matrix}

Additive Holt–Winters Smoothing

\begin{matrix} m_{t} & = & α (y_{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 (y_{t} - m_{t}) + (1 - β) s_{t - p} \\ {\hat{y}}_{t + f} & = & m_{t} + (\sum_{i = 1}^{f} ϕ^{i} r_{t}) + s_{t - p} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 1}^{f - 1} ψ_{i}^{2}) \\ ψ_{i} & = & \{\begin{matrix} 0 & if ​ i \geq f \\ α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)} & if ​ i mod p \neq 0 \\ α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)} + β (1 - α) & otherwise \end{matrix} \end{matrix}

Multiplicative Holt–Winters Smoothing

\begin{matrix} m_{t} & = & α y_{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 y_{t} / m_{t} + (1 - β) s_{t - p} \\ {\hat{y}}_{t + f} & = & (m_{t} + \sum_{i = 1}^{f} ϕ^{i} r_{t}) \times s_{t - p} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (\sum_{i = 0}^{\infty} \sum_{j = 0}^{p - 1} {(ψ_{j + i p} \frac{s_{t + f}}{s_{t + f - j}})}^{2}) \end{matrix}

and

ψ

is defined as in the additive Holt–Winters smoothing,

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

f

-step ahead forecasts are given by

{\hat{y}}_{t + f}

and their variances by

var ({\hat{y}}_{t + f})

. The term

var (ε_{t})

is estimated as the mean deviation.

The parameters,

α

β

and

γ

control the amount of smoothing. The nearer these parameters are to one, the greater the emphasis on the current data point. Generally these parameters take values in the range

0.1

0.3

. The linear Holt and two Holt–Winters smoothers include an additional parameter,

ϕ

, which acts as a trend dampener. For

0.0 < ϕ < 1.0

the trend is dampened and for

ϕ > 1.0

the forecast function has an exponential trend,

ϕ = 0.0

removes the trend term from the forecast function and

ϕ = 1.0

does not dampen the trend.

For all methods, values for

α

β

γ

and

ψ

can be chosen by trying different values and then visually comparing the results by plotting the fitted values along side the original data. Alternatively, for single exponential smoothing a suitable value for

α

can be obtained by fitting an

ARIMA (0, 1, 1)

model (see nag_tsa_multi_inp_model_estim (g13bec)). For Brown's double exponential smoothing and linear Holt smoothing with no dampening, (i.e.,

ϕ = 1.0

), suitable values for

α

and

γ

can be obtained by fitting an

ARIMA (0, 2, 2)

model. Similarly, the linear Holt method, with

ϕ \neq 1.0

, can be expressed as an

ARIMA (1, 2, 2)

model and the additive Holt–Winters, with no dampening, (

ϕ = 1.0

), can be expressed as a seasonal ARIMA model with order

p

of the form

ARIMA (0, 1, p + 1) (0, 1, 0)

. There is no similar procedure for obtaining parameter values for the multiplicative Holt–Winters method, or the additive Holt–Winters method with

ϕ \neq 1.0

. In these cases parameters could be selected by minimizing a measure of fit using one of the nonlinear optimization functions in Chapter e04.

In addition to values for

α

β

γ

and

ψ

, initial values,

m_{0}

r_{0}

and

s_{- j}

, for

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

, are required to start the smoothing process. You can either supply these or they can be calculated by nag_tsa_exp_smooth (g13amc) from the first

k

observations. For single exponential smoothing the mean of the observations is used to estimate

m_{0}

. For Brown double exponential smoothing and linear Holt smoothing, a simple linear regression is carried out with the series as the dependent variable and the sequence

1, 2, \dots, k

as the independent variable. The intercept is then used to estimate

m_{0}

and the slope to estimate

r_{0}

. In the case of the additive Holt–Winters method, the same regression is carried out, but a separate intercept is used for each of the

p

seasonal groupings. The slope gives an estimate for

r_{0}

and the mean of the

p

intercepts is used as the estimate of

m_{0}

. The seasonal parameters

s_{- j}

, for

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

, are estimated as the

p

intercepts –

m_{0}

. A similar approach is adopted for the multiplicative Holt–Winter's method.

One step ahead forecasts,

{\hat{y}}_{t + 1}

are supplied along with the residuals computed as

(y_{t + 1} - {\hat{y}}_{t + 1})

. In addition, two measures of fit are provided. The mean absolute deviation,

\frac{1}{n} \sum_{t = 1}^{n} |y_{t} - {\hat{y}}_{t}|

and the square root of the mean deviation

\sqrt{\frac{1}{n} \sum_{t = 1}^{n} {(y_{t} - {\hat{y}}_{t})}^{2}} .

5 Arguments

1: mode – Nag_InitialValuesInput

On entry: indicates if nag_tsa_exp_smooth (g13amc) is continuing from a previous call or, if not, how the initial values are computed.

$mode = Nag_InitialValuesSupplied$: Required values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are supplied in init.
$mode = Nag_ContinueAndUpdate$: nag_tsa_exp_smooth (g13amc) continues from a previous call using values that are supplied in r.
$mode = Nag_EstimateInitialValues$: Required values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are estimated using the first $k$ observations.

Constraint:

mode = Nag_InitialValuesSupplied

Nag_ContinueAndUpdate

Nag_EstimateInitialValues

2: itype – Nag_ExpSmoothTypeInput

On entry: the smoothing function.

$itype = Nag_SingleExponential$: Single exponential.
$itype = Nag_BrownsExponential$: Brown double exponential.
$itype = Nag_LinearHolt$: Linear Holt.
$itype = Nag_AdditiveHoltWinters$: Additive Holt–Winters.
$itype = Nag_MultiplicativeHoltWinters$: Multiplicative Holt–Winters.

Constraint:

itype = Nag_SingleExponential

Nag_BrownsExponential

Nag_LinearHolt

Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

3: p – IntegerInput

On entry: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

, the seasonal order,

p

, otherwise p is not referenced.

Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p > 1

4: param[

\dim

] – const doubleInput

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

$1$ when $itype = Nag_SingleExponential$ or $Nag_BrownsExponential$ ;
$3$ when $itype = Nag_LinearHolt$ ;
$4$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: the smoothing parameters.

itype = Nag_SingleExponential

Nag_BrownsExponential

param [0] = α

and any remaining elements of param are not referenced.

itype = Nag_LinearHolt

param [0] = α

param [1] = γ

param [2] = ϕ

and any remaining elements of param are not referenced.

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

param [0] = α

param [1] = γ

param [2] = β

and

param [3] = ϕ

Constraints:

if $itype = Nag_SingleExponential$ , $0.0 \leq α \leq 1.0$ ;
if $itype = Nag_BrownsExponential$ , $0.0 < α \leq 1.0$ ;
if $itype = Nag_LinearHolt$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $ϕ \geq 0.0$ ;
if $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $0.0 \leq β \leq 1.0$ and $ϕ \geq 0.0$ .

5: n – IntegerInput

On entry: the number of observations in the series.

Constraint:

n \geq 0

6: y[n] – const doubleInput

On entry: the time series.

7: k – IntegerInput

On entry: if

mode = Nag_EstimateInitialValues

, the number of observations used to initialize the smoothing.

mode \neq Nag_EstimateInitialValues

, k is not referenced.

Constraints:

if $mode = Nag_EstimateInitialValues$ and $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ , $2 \times p \leq k \leq n$ ;
if $mode = Nag_EstimateInitialValues$ and $itype = Nag_SingleExponential$ , $Nag_BrownsExponential$ or $Nag_LinearHolt$ , $1 \leq k \leq n$ .

8: init[

\dim

] – doubleInput/Output

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

$1$ when $itype = Nag_SingleExponential$ ;
$2$ when $itype = Nag_BrownsExponential$ or $Nag_LinearHolt$ ;
$2 + p$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: if

mode = Nag_InitialValuesSupplied

, the initial values for

m_{0}

r_{0}

and

s_{- j}

, for

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

, used to initialize the smoothing.

itype = Nag_SingleExponential

init [0] = m_{0}

and the remaining elements of init are not referenced.

itype = Nag_BrownsExponential

Nag_LinearHolt

init [0] = m_{0}

and

init [1] = r_{0}

and the remaining elements of init are not referenced.

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

init [0] = m_{0}

init [1] = r_{0}

and

init [2]

init [p + 1]

hold the values for

s_{- j}

, for

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

. The remaining elements of init are not referenced.

On exit: if

mode \neq Nag_ContinueAndUpdate

, the values used to initialize the smoothing. These are in the same order as described above.

9: nf – IntegerInput

On entry: the number of forecasts required beyond the end of the series. Note, the one step ahead forecast is always produced.

Constraint:

nf \geq 0

10: fv[nf] – doubleOutput

On exit:

{\hat{y}}_{t + f}

, for

f = 1, 2, \dots, nf

, the next nf step forecasts. Where

t = n

, if

mode \neq Nag_ContinueAndUpdate

, else

t

is the total number of smoothed and forecast values already produced.

11: fse[nf] – doubleOutput

On exit: the forecast standard errors for the values given in fv.

12: yhat[n] – doubleOutput

On exit:

{\hat{y}}_{t + 1}

, for

t = 1, 2, \dots, n

, the one step ahead forecast values, with

yhat [i - 1]

being the one step ahead forecast of

y [i - 2]

13: res[n] – doubleOutput

On exit: the residuals,

(y_{t + 1} - {\hat{y}}_{t + 1})

, for

t = 1, 2, \dots, n

14: dv – double *Output

On exit: the square root of the mean deviation.

15: ad – double *Output

On exit: the mean absolute deviation.

16: r[

\dim

] – doubleInput/Output

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

$13$ when $itype = Nag_SingleExponential$ , $Nag_BrownsExponential$ or $Nag_LinearHolt$ ;
$13 + p$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: if

mode = Nag_ContinueAndUpdate

, r must contain the values as returned by a previous call to nag_rand_exp_smooth (g05pmc) or nag_tsa_exp_smooth (g13amc), r need not be set otherwise.

itype = Nag_SingleExponential

Nag_BrownsExponential

Nag_LinearHolt

, only the first

13

elements of r are referenced, otherwise the first

13 + p

elements are referenced.

On exit: the information on the current state of the smoothing.

Constraint: if

mode = Nag_ContinueAndUpdate

, r must have been initialized by at least one previous call to nag_rand_exp_smooth (g05pmc) or nag_tsa_exp_smooth (g13amc) with

mode \neq Nag_ContinueAndUpdate

, and r should not have been changed since the last call to nag_rand_exp_smooth (g05pmc) or nag_tsa_exp_smooth (g13amc).

17: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_ENUM_INT

On entry,

itype = ⟨value⟩

and

p = ⟨value⟩

.
Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p > 1

On entry,

p = ⟨value⟩

.
Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p > 1

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 0

On entry,

nf = ⟨value⟩

.
Constraint:

nf \geq 0

NE_INT_2

On entry,

k = ⟨value⟩

and

n = ⟨value⟩

.
Constraint: if

mode = Nag_EstimateInitialValues

and

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

1 \leq k \leq n

NE_INT_3

On entry,

k = ⟨value⟩

2 \times p = ⟨value⟩

.
Constraint: if

mode = Nag_EstimateInitialValues

and

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

2 \times p \leq k

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_MODEL_PARAMS

A multiplicative Holt–Winters model cannot be used with the supplied data.

NE_REAL_ARRAY

On entry,

param [⟨value⟩] = ⟨value⟩

.
Constraint:

0.0 \leq param [⟨value⟩] \leq 1.0

On entry,

param [⟨value⟩] = ⟨value⟩

.
Constraint: if

itype = Nag_BrownsExponential

0.0 < param [⟨value⟩] \leq 1.0

On entry,

param [⟨value⟩] = ⟨value⟩

.
Constraint:

param [⟨value⟩] \geq 0.0

On entry, the array r has not been initialized correctly.

NAG Library Function Document

nag_tsa_exp_smooth (g13amc)

+− 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 Library Function Documentnag_tsa_exp_smooth (g13amc)

+− 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 Library Function Document

nag_tsa_exp_smooth (g13amc)