NAG Library Function Document

nag_tsa_mean_range (g13auc)

+− 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

1 Purpose

nag_tsa_mean_range (g13auc) calculates the range (or standard deviation) and the mean for groups of successive time series values. It is intended for use in the construction of range-mean plots.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_mean_range (Integer n, const double z[], Integer m, Nag_RangeStat rs, double y[], double mean[], NagError *fail)

3 Description

Let

Z_{1}, Z_{2}, \dots, Z_{n}

denote

n

successive observations in a time series. The series may be divided into groups of

m

successive values and for each group the range or standard deviation (depending on a user-supplied option) and the mean are calculated. If

n

is not a multiple of

m

then groups of equal size

m

are found starting from the end of the series of observations provided, and any remaining observations at the start of the series are ignored. The number of groups used,

k

, is the integer part of

n / m

. If you wish to ensure that no observations are ignored then the number of observations,

n

, should be chosen so that

n

is divisible by

m

The mean,

M_{i}

, the range,

R_{i}

, and the standard deviation,

S_{i}

, for the

i

th group are defined as

\begin{matrix} M_{i} = \frac{1}{m} \sum_{j = 1}^{m} Z_{l + m (i - 1) + j} \\ R_{i} = \max_{1 \leq j \leq m} \{Z_{l + m (i - 1) + j}\} - \min_{1 \leq j \leq m} \{Z_{l + m (i - 1) + j}\} \end{matrix}

and

S_{i} = \sqrt{(\frac{1}{m - 1}) \sum_{j = 1}^{m} {(Z_{l + m (i - 1) + j} - M_{i})}^{2}}

where

l = n - k m

, the number of observations ignored.

For seasonal data it is recommended that

m

should be equal to the seasonal period. For non-seasonal data the recommended group size is

8

A plot of range against mean or of standard deviation against mean is useful for finding a transformation of the series which makes the variance constant. If the plot appears random or the range (or standard deviation) seems to be constant irrespective of the mean level then this suggests that no transformation of the time series is called for. On the other hand an approximate linear relationship between range (or standard deviation) and mean would indicate that a log transformation is appropriate. Further details may be found in either Jenkins (1979) or McLeod (1982).

You have the choice of whether to use the range or the standard deviation as a measure of variability. If the group size is small they are both equally good but if the group size is fairly large (e.g.,

m = 12

for monthly data) then the range may not be as good an estimate of variability as the standard deviation.

4 References

Jenkins G M (1979) Practical Experiences with Modelling and Forecasting Time Series GJP Publications, Lancaster

McLeod G (1982) Box–Jenkins in Practice. 1: Univariate Stochastic and Single Output Transfer Function/Noise Analysis GJP Publications, Lancaster

5 Arguments

1: n – IntegerInput

On entry:

n

, the number of observations in the time series.

Constraint:

n \geq m

2: z[n] – const doubleInput

On entry:

z [t - 1]

must contain the

t

th observation

Z_{t}

, for

t = 1, 2, \dots, n

3: m – IntegerInput

On entry:

m

, the group size.

Constraint:

m \geq 2

4: rs – Nag_RangeStatInput

On entry: indicates whether ranges or standard deviations are to be calculated.

$rs = Nag_UseRange$: Ranges are calculated.
$rs = Nag_UseSD$: Standard deviations are calculated.

Constraint:

rs = Nag_UseRange

Nag_UseSD

5: y[ $int (n / m)$ ] – doubleOutput

On exit:

y [i - 1]

contains the range or standard deviation, as determined by rs, of the

i

th group of observations, for

i = 1, 2, \dots, k

6: mean[ $int (n / m)$ ] – doubleOutput

On exit:

mean [i - 1]

contains the mean of the

i

th group of observations, for

i = 1, 2, \dots, k

7: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 2$ .
NE_INT_2: On entry, $n = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $n \geq m$ .
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.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_tsa_mean_range (g13auc) is approximately proportional to

n

10 Example

The following program produces the statistics for a range-mean plot for a series of

100

observations divided into groups of

8

NAG Library Function Documentnag_tsa_mean_range (g13auc)

+− 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_mean_range (g13auc)