g13ab computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation function is zero.

Syntax

C#
public static void g13ab( double[] x, int nx, int nk, out double xm, out double xv, double[] r, out double stat, out int ifail )

Visual Basic
Public Shared Sub g13ab ( _ x As Double(), _ nx As Integer, _ nk As Integer, _ <OutAttribute> ByRef xm As Double, _ <OutAttribute> ByRef xv As Double, _ r As Double(), _ <OutAttribute> ByRef stat As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13ab ( _
	x As Double(), _
	nx As Integer, _
	nk As Integer, _
	<OutAttribute> ByRef xm As Double, _
	<OutAttribute> ByRef xv As Double, _
	r As Double(), _
	<OutAttribute> ByRef stat As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13ab( array<double>^ x, int nx, int nk, [OutAttribute] double% xm, [OutAttribute] double% xv, array<double>^ r, [OutAttribute] double% stat, [OutAttribute] int% ifail )

F#
static member g13ab : x : float[] * nx : int * nk : int * xm : float byref * xv : float byref * r : float[] * stat : float byref * ifail : int byref -> unit

Parameters

x: Type: array<System..::..Double>[]()[][]
An array of size [nx]
On entry: the time series, $x_{i}$ , for $i = 1, 2, \dots, n$ .

nx: Type: System..::..Int32
On entry: $n$ , the number of values in the time series.

Constraint: $nx > 1$ .

nk: Type: System..::..Int32
On entry: $K$ , the number of lags for which the autocorrelations are required. The lags range from $1$ to $K$ and do not include zero.

Constraint: $0 < nk < nx$ .

xm: Type: System..::..Double%
On exit: the sample mean of the input time series.

xv: Type: System..::..Double%
On exit: the sample variance of the input time series.

r: Type: array<System..::..Double>[]()[][]
An array of size [nk]
On exit: the sample autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .

stat: Type: System..::..Double%
On exit: the statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The data consists of

n

observations

x_{i}

, for

i = 1, 2, \dots, n

from a time series.

The quantities calculated are

(a)

The sample mean

\bar{x} = \frac{\sum_{i = 1}^{n} x_{i}}{n} .

(b)

The sample variance (for

n \geq 2

)

s^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{(n - 1)} .

(c)

The sample autocorrelation coefficients of lags

k = 1, 2, \dots, K

, where

K

is a user-specified maximum lag, and

K < n

n > 1

The coefficient of lag

k

is defined as

r_{k} = \frac{\sum_{i = 1}^{n - k} (x_{i} - \bar{x}) (x_{i + k} - \bar{x})}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}} .

See page 496 of Box and Jenkins (1976) for further details.

(d)

A test statistic defined as

stat = n \sum_{k = 1}^{K} r_{k}^{2},

which can be used to test the hypothesis that the true autocorrelation function is identically zero.

n

is large and

K

is much smaller than

n

, stat has a

χ_{K}^{2}

distribution under the hypothesis of a zero autocorrelation function. Values of stat in the upper tail of the distribution provide evidence against the hypothesis; g01ec can be used to compute the tail probability.

Section 8.2.2 of Box and Jenkins (1976) provides further details of the use of stat.

References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$nx \leq nk$ ,
or	$nx \leq 1$ ,
or	$nk \leq 0$ .

$ifail = 2$: On entry, all values of x are practically identical, giving zero variance. In this case r and stat are undefined on exit.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The computations are believed to be stable.

Parallelism and Performance

None.

Further Comments

n < 100

, or

K < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by g13ab is approximately proportional to

n K

, otherwise the autocorrelations are calculated by utilizing fast fourier transforms (FFTs) and the time taken is approximately proportional to

n \log (n)

. If FFTs are used then g13ab internally allocates approximately

4 n

real elements.

If the input series for g13ab was generated by differencing using g13aa, ensure that only the differenced values are input to g13ab, and not the reconstituting information.