Integer, Intent (In)	::	m, nb, iwt, lrsd, lrcomm
Integer, Intent (Inout)	::	pn, ifail
Real (Kind=nag_wp), Intent (In)	::	x(nb), wt(*)
Real (Kind=nag_wp), Intent (Inout)	::	rcomm(lrcomm)
Real (Kind=nag_wp), Intent (Out)	::	rmean(max(0,nb+min(0,pn-m+1))), rsd(lrsd)

C Header Interface

#include <nag.h>

void	g01waf_ (const Integer m, const Integer nb, const double x[], const Integer iwt, const double wt[], Integer pn, double rmean[], double rsd[], const Integer lrsd, double rcomm[], const Integer lrcomm, Integer *ifail)

The routine may be called by the names g01waf or nagf_stat_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}

, g01waf 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 g01waf 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 g01waf.

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 = 1

, wt) can vary;, therefore, nb can change between calls to g01waf.

Constraints:

$nb \geq 0$ ;
if $lrcomm = 0$ , $nb \geq m$ .

3: $x (nb)$ – Real (Kind=nag_wp) array 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: $iwt$ – Integer Input

On entry: the type of weighting to use.

$iwt = 0$: No weights are used.
$iwt = 1$: Each observation has its own weight.
$iwt = 2$: Each position in the window has its own weight.
$iwt = 3$: 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 g01waf.

Constraint:

iwt = 0

1

2

3

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

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

nb

iwt = 1

m

iwt = 2

On entry: the user-supplied weights.

iwt = 1

wt (i) = ν_{i + k}

, for

i = 1, 2, \dots, b

iwt = 2

wt (j) = w_{j}

, for

j = 1, 2, \dots, m

Constraints:

if $iwt = 1$ , $wt (i) \geq 0$ , for $i = 1, 2, \dots, nb$ ;
if $iwt = 2$ , $wt (1) \neq 0$ and $\sum_{j = 1}^{m} wt (j) > 0$ ;
if $iwt = 2$ and $lrsd \neq 0$ , $wt (j) \geq 0$ , 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 g01waf, 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 g01waf.

On exit:

k + b

, the updated number of observations processed so far.

Constraint:

pn \geq 0

7: $rmean (\max (0, nb + \min (0, pn - m + 1)))$ – Real (Kind=nag_wp) array Output

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) - 1)

If, on entry,

pn \geq m - 1

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

rmean (l)

is the summary corresponding to the window that ends on

x (l)

. On the other hand, if, on entry,

pn = 0

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

rmean (l)

is the summary corresponding to the window that ends on

x (m + l - 1)

(or, equivalently, starts on

x (l)

8: $rsd (lrsd)$ – Real (Kind=nag_wp) array Output

On exit: if

lrsd \neq 0

then

σ_{l}

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

lrsd = 0

, rsd is not referenced.

9: $lrsd$ – Integer Input

On entry: the dimension of the array rsd as declared in the (sub)program from which g01waf is called. If the standard deviations are not required then lrsd should be set to zero.

Constraint:

lrsd = 0

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

10: $rcomm (lrcomm)$ – Real (Kind=nag_wp) array Communication Array

On entry: communication array, used to store information between calls to g01waf. If

lrcomm = 0

, rcomm is not referenced and all the data must be supplied in one go.

11: $lrcomm$ – Integer Input

On entry: the dimension of the array rcomm as declared in the (sub)program from which g01waf is called.

Constraint:

lrcomm = 0

lrcomm \geq 2 m + 20

12: $ifail$ – 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:

$ifail = 11$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

$ifail = 12$: On entry, $m = ⟨ value ⟩$ .
On entry at previous call, $m = ⟨ value ⟩$ .
Constraint: if $pn > 0$ , m must be unchanged since previous call.

$ifail = 21$: On entry, $nb = ⟨ value ⟩$ .
Constraint: $nb \geq 0$ .

$ifail = 22$: On entry, $nb = ⟨ value ⟩$ , $m = ⟨ value ⟩$ .
Constraint: if $lrcomm = 0$ , $nb \geq m$ .

$ifail = 41$: On entry, $iwt = ⟨ value ⟩$ .
Constraint: $iwt = 0$ , $1$ , $2$ or $3$ .

$ifail = 42$: On entry, $iwt = ⟨ value ⟩$ .
On entry at previous call, $iwt = ⟨ value ⟩$ .
Constraint: if $pn > 0$ , iwt must be unchanged since previous call.

$ifail = 51$: On entry, $wt (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $wt (i) \geq 0$ .

$ifail = 52$: On entry, $wt (1) = ⟨ value ⟩$ .
Constraint: if $iwt = 2$ , $wt (1) > 0$ .

$ifail = 53$: On entry, at least one window had all zero weights.

$ifail = 54$: On entry, unable to calculate at least one standard deviation due to the weights supplied.

$ifail = 55$: On entry, sum of weights supplied in wt is $⟨ value ⟩$ .
Constraint: if $iwt = 2$ , the sum of the weights $> 0$ .

$ifail = 61$: On entry, $pn = ⟨ value ⟩$ .
Constraint: $pn \geq 0$ .

$ifail = 62$: On entry, $pn = ⟨ value ⟩$ .
On exit from previous call, $pn = ⟨ value ⟩$ .
Constraint: if $pn > 0$ , pn must be unchanged since previous call.

$ifail = 91$: On entry, $lrsd = ⟨ value ⟩$ .
Constraint: $lrsd = 0$ or $lrsd \geq ⟨ value ⟩$ .

$ifail = 101$: rcomm has been corrupted between calls.

$ifail = 111$: On entry, $lrcomm = ⟨ value ⟩$ .
Constraint: $lrcomm \geq ⟨ value ⟩$ .

$ifail = - 99$: 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

Background information to multithreading can be found in the Multithreading documentation.

g01waf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g01waf 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 routine. 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 g01waf in one call, i.e., the larger nb is, the more efficient the routine 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,