void	g01wac (Integer m, Integer nb, const double x[], Nag_Weightstype iwt, const double wt[], Integer pn, double rmean[], double rsd[], double rcomm[], NagError fail)

The function may be called by the names: g01wac, nag_stat_moving_average or nag_moving_average.

3 Description

Given a sample of

n

observations, denoted by

x = {x_{i} : i = 1, 2, \dots, n}

and a set of weights,

w = {w_{j} : j = 1, 2, \dots, m}

, g01wac calculates the mean and, optionally, the standard deviation, in a rolling window of length

m

For the

i

th window the mean is defined as

μ_{i} = \frac{\sum_{j = 1}^{m} w_{j} x_{i + j - 1}}{W}

(1)

and the standard deviation as

σ_{i} = \sqrt{\frac{\sum_{j = 1}^{m} w_{j} {(x_{i + j - 1} - μ_{i})}^{2}}{W - \frac{\sum_{j = 1}^{m} w_{j}^{2}}{W}}}

(2)

with

W = \sum_{j = 1}^{m} w_{j}

Four different types of weighting are possible:

(i)No weights ( $w_{j} = 1$ )

When no weights are required both the mean and standard deviations can be calculated in an iterative manner, with

\begin{matrix} μ_{i + 1} = & μ_{i} + \frac{(x_{i + m} - x_{i})}{m} \\ σ_{i + 1}^{2} = & (m - 1) σ_{i}^{2} + {(x_{i + m} - μ_{i})}^{2} - {(x_{i} - μ_{i})}^{2} - \frac{{(x_{i + m} - x_{i})}^{2}}{m} \end{matrix}

where the initial values

μ_{1}

and

σ_{1}

are obtained using the one pass algorithm of West (1979).

(ii)Each observation has its own weight
In this case, rather than supplying a vector of $m$ weights a vector of $n$ weights is supplied instead, $v = {v_{j} : j = 1, 2, \dots, n}$ and $w_{j} = v_{i + j - 1}$ in (1) and (2).

If the standard deviations are not required then the mean is calculated using the iterative formula:

$\begin{matrix} W_{i + 1} = & W_{i} + (v_{i + m} - v_{i}) \\ μ_{i + 1} = & μ_{i} + W_{i}^{- 1} (v_{i + m} x_{i + m} - v_{i} x_{i}) \end{matrix}$

where $W_{1} = \sum_{i = 1}^{m} v_{i}$ and $μ_{1} = W_{1}^{−1} \sum_{i = 1}^{m} v_{i} x_{i}$ .

If both the mean and standard deviation are required then the one pass algorithm of West (1979) is used in each window.
(iii)Each position in the window has its own weight
This is the case as described in (1) and (2), where the weight given to each observation differs depending on which summary is being produced. When these types of weights are specified both the mean and standard deviation are calculated by applying the one pass algorithm of West (1979) multiple times.
(iv)Each position in the window has a weight equal to its position number ( $w_{j} = j$ )
This is a special case of (iii).

If the standard deviations are not required then the mean is calculated using the iterative formula:

$\begin{matrix} S_{i + 1} = & S_{i} + (x_{i + m} - x_{i}) \\ μ_{i + 1} = & μ_{i} + \frac{2 (m x_{i + m} - S_{i})}{m (m + 1)} \end{matrix}$

where $S_{1} = \sum_{i = 1}^{m} x_{i}$ and $μ_{1} = 2 {(m^{2} + m)}^{−1} S_{1}$ .

If both the mean and standard deviation are required then the one pass algorithm of West is applied multiple times.

For large datasets, or where all the data is not available at the same time,

x

(and if each observation has its own weight,

v

) can be split into arbitrary sized blocks and g01wac called multiple times.

4 References

Chan T F, Golub G H and Leveque R J (1982) Updating Formulae and a Pairwise Algorithm for Computing Sample Variances Compstat, Physica-Verlag

West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

5 Arguments

1: $m$ – Integer Input

On entry:

m

, the length of the rolling window.

pn \neq 0

, m must be unchanged since the last call to g01wac.

Constraint:

m \geq 1

2: $nb$ – Integer Input

On entry:

b

, the number of observations in the current block of data. The size of the block of data supplied in x (and when

iwt = Nag_WeightObs

, wt) can vary;, therefore, nb can change between calls to g01wac.

Constraints:

$nb \geq 0$ ;
if $rcomm is NULL$ , $nb \geq m$ .

3: $x [nb]$ – const double Input

On entry: the current block of observations, corresponding to

x_{i}

, for

i = k + 1, \dots, k + b

, where

k

is the number of observations processed so far and

b

is the size of the current block of data.

4: $iwt$ – Nag_Weightstype Input

On entry: the type of weighting to use.

$iwt = Nag_NoWeights$: No weights are used.
$iwt = Nag_WeightObs$: Each observation has its own weight.
$iwt = Nag_WeightWindow$: Each position in the window has its own weight.
$iwt = Nag_WeightWindowPos$: Each position in the window has a weight equal to its position number.

pn \neq 0

, iwt must be unchanged since the last call to g01wac.

Constraint:

iwt = Nag_NoWeights

Nag_WeightObs

Nag_WeightWindow

Nag_WeightWindowPos

5: $wt [\dim]$ – const double Input

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

$nb$ , when $iwt = Nag_WeightObs$ ;
$m$ , when $iwt = Nag_WeightWindow$ ;
otherwise $wt$ is not referenced and may be NULL.

On entry: the user-supplied weights.

iwt = Nag_WeightObs

wt [i - 1] = ν_{i + k}

, for

i = 1, 2, \dots, b

iwt = Nag_WeightWindow

wt [j - 1] = w_{j}

, for

j = 1, 2, \dots, m

Constraints:

if $iwt = Nag_WeightObs$ , $wt [i - 1] \geq 0$ , for $i = 1, 2, \dots, nb$ ;
if $iwt = Nag_WeightWindow$ , $wt [0] \neq 0$ and $\sum_{j = 1}^{m} wt [j - 1] > 0$ ;
if $iwt = Nag_WeightWindow$ and $rsd is not NULL$ , $wt [j - 1] \geq 0$ , for $j = 1, 2, \dots, m$ .

6: $pn$ – Integer * Input/Output

On entry:

k

, the number of observations processed so far. On the first call to g01wac, or when starting to summarise a new dataset, pn must be set to

0

pn \neq 0

, it must be the same value as returned by the last call to g01wac.

On exit:

k + b

, the updated number of observations processed so far.

Constraint:

pn \geq 0

7: $rmean [\dim]$ – double Output

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

\max (0, nb + \min (0, pn - m + 1))

On exit:

μ_{l}

, the (weighted) moving averages, for

l = 1, 2, \dots, b + \min (0, k - m + 1)

. Therefore,

μ_{l}

is the mean of the data in the window that ends on

x [l + m - \min (k, m - 1) - 2]

If, on entry,

pn \geq m - 1

, i.e., at least one windows worth of data has been previously processed, then

rmean [l - 1]

is the summary corresponding to the window that ends on

x [l - 1]

. On the other hand, if, on entry,

pn = 0

, i.e., no data has been previously processed, then

rmean [l - 1]

is the summary corresponding to the window that ends on

x [m + l - 2]

(or, equivalently, starts on

x [l - 1]

8: $rsd [\dim]$ – double Output

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

$\max (0, nb + \min (0, pn - m + 1))$ , when $rsd is not NULL$ ;
otherwise $rsd$ is not referenced and may be NULL.

if standard deviations are not required then rsd must be NULL.

On exit: if

rsd is not NULL

on entry then

σ_{l}

, the (weighted) standard deviation. The ordering of rsd is the same as the ordering of rmean.

9: $rcomm [\dim]$ – double Communication Array

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

$0$ , when $rcomm is NULL$ ;
$2 m + 20$ , otherwise.

On entry: communication array, used to store information between calls to g01wac. If

rcomm is NULL

then pn must be set to zero and all the data must be supplied in one go.

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_ILLEGAL_COMM: rcomm has been corrupted between calls.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $nb = ⟨ value ⟩$ .
Constraint: $nb \geq 0$ .

On entry, $nb = ⟨ value ⟩$ , $m = ⟨ value ⟩$ .
Constraint: if $rcomm is NULL$ , $nb \geq m$ .

On entry, $pn = ⟨ value ⟩$ .
Constraint: $pn \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NEG_WEIGHT: On entry, $wt [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: $wt [i - 1] \geq 0$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PREV_CALL: if $pn > 0$ , iwt must be unchanged since previous call.

On entry, $m = ⟨ value ⟩$ .
On entry at previous call, $m = ⟨ value ⟩$ .
Constraint: if $pn > 0$ , m must be unchanged since previous call.

On entry, $pn = ⟨ value ⟩$ .
On exit from previous call, $pn = ⟨ value ⟩$ .
Constraint: if $pn > 0$ , pn must be unchanged since previous call.
NE_SUM_WEIGHT: On entry, sum of weights supplied in wt is $⟨ value ⟩$ .
Constraint: if $iwt = Nag_WeightWindow$ , the sum of the weights $> 0$ .
NE_WEIGHT_ZERO: On entry, $wt [0] = ⟨ value ⟩$ .
Constraint: if $iwt = Nag_WeightWindow$ , $wt [0] > 0$ .
NW_POTENTIAL_PROBLEM: On entry, at least one window had all zero weights.

On entry, unable to calculate at least one standard deviation due to the weights supplied.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g01wac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g01wac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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

9 Further Comments

The more data that is supplied to g01wac in one call, i.e., the larger nb is, the more efficient the function will be.

10 Example

This example calculates Spencer's

15

-point moving average for the change in rate of the Earth's rotation between

1821

and

1850

. The data is supplied in three chunks, the first consisting of five observations, the second

10

observations and the last

15

observations.

This example plot shows the smoothing effect of using different length rolling windows on the mean and standard deviation. Two different window lengths,