The IMSL_AUTOCORRELATION function computes the sample autocorrelation function of a stationary time series.
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The IMSL_AUTOCORRELATION function estimates the autocorrelation function of a stationary time series given a sample of n = N_ELEMENTS(X observations {Xt} for t = 1, 2, ..., n.
Let:
be the estimate of the mean μ of the time series {Xt} where:
The autocovariance function σ(k) is estimated by:
where K = lagmax. Note that:
σ(0)
is an estimate of the sample variance. The autocorrelation function σ(k) is estimated by:
Note that:
ρ(0) = 1
by definition.
The standard errors of the sample auto-correlations may be optionally computed according to the keyword Se_Option for the output keyword Seac. One method (Bartlett 1946) is based on a general asymptotic expression for the variance of the sample autocorrelation coefficient of a stationary time series with independent, identically distributed normal errors. The theoretical formula is:
where:
ρ(k)
assumes m is unknown. For computational purposes, the auto-correlations ρ(k) are replaced by their estimates:
ρ(k)
for |k| ≤ K, and the limits of summation are bounded because of the assumption that ρ(k) = 0 for all k such that |k| > K.
A second method (Moran 1947) utilizes an exact formula for the variance of the sample autocorrelation coefficient of a random process with independent, identically distributed normal errors. The theoretical formula is:
where μ is assumed to be equal to zero. Note that this formula does not depend on the autocorrelation function.
Example
Consider the Wolfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. The IMSL_AUTOCORRELATION function computes the estimated auto-covariances, estimated auto-correlations, and estimated standard errors of the auto-correlations.
.RUN
PRO print_results, xm, acv, result, seac
PRINT, ';Mean =', xm
PRINT, ';Variance =', acv(0)
PRINT, '; Lag ACV AC SEAC'
PRINT, '; 0', acv(0), result(0)
FOR j = 1, 20 DO $
PRINT, j, acv(j), result(j), seac(j - 1)
END
lagmax = 20
data = IMSL_STATDATA(2)
x = data(21:120,1)
result = IMSL_AUTOCORRELATION(x, lagmax, ACV = acv, $
SE_OPTION = 1, SEAC = seac, XMEAN_OUT = xm)
print_results, xm, acv, result, seac
IDL prints:
Mean = 46.9760
Variance = 1382.91
Lag ACV AC SEAC
0 1382.91 1.00000
1 1115.03 0.806293 0.0347834
20 592.004 0.428087 0.0962420
30 95.2974 0.0689109 0.156783
40 -235.952 -0.170620 0.205767
50 -370.011 -0.267560 0.230956
60 -294.255 -0.212780 0.228995
70 -60.4423 -0.0437067 0.208622
80 227.633 0.164604 0.178476
90 458.381 0.331462 0.145727
100 567.841 0.410613 0.134406
110 546.122 0.394908 0.150676
120 398.937 0.288477 0.174348
130 197.757 0.143001 0.190619
140 26.8911 0.0194453 0.195490
150 -77.2807 -0.0558828 0.195893
160 -143.733 -0.103935 0.196285
170 202.048 -0.146104 0.196021
180 -245.372 -0.177432 0.198716
190 -230.816 -0.166906 0.205359
200 -142.879 -0.103318 0.209387
Syntax
Result = IMSL_AUTOCORRELATION(x, lagmax [, ACV=variable] [, /DOUBLE] [, SE_OPTION=value] [, SEAC=variable] [, XMEAN_IN=value] [, XMEAN_OUT=variable])
Return Value
One-dimensional array of length Lagmax + 1 containing the auto-correlations of the time series X. The 0-th element of this array is 1. The k-th element of this array contains the autocorrelation of lag k where k = 1, ..., Lagmax
Arguments
Lagmax
Scalar integer containing the maximum lag of autocovariance, auto-correlations, and standard errors of auto-correlations to be computed. Lagmax must be greater than or equal to 1 and less than N_ELEMENTS(X.
X
One-dimensional array containing the time series. N_ELEMENTS(X must be greater than or equal to 2.
Keywords
ACV (optional)
Named variable into which an array of length Lagmax + 1 containing the variance and auto-covariances of the time series X is stored. The 0-th element of this array is the variance of the time series X. The k-th element contains the autocovariance of lag k where k = 1, ..., Lagmax.
DOUBLE (optional)
If present and nonzero, then double precision is used.
SE_OPTION (optional)
Method of computation for standard errors of the auto-correlations. The keywords SE_OPTION and SEAC must be used together.
- 1: Compute the standard errors of autocorrelation using Barlett's formula.
- 2: Compute the standard errors of autocorrelation using Moran's formula.
SEAC (optional)
Named variable into which an array of length Lagmax containing the standard errors of the auto-correlations of the time series X is stored. The keywords SE_OPTION and SEAC must be used together.
XMEAN_IN (optional)
The estimate of the mean of the time series X.
XMEAN_OUT (optional)
Named variable into which the estimate of the mean of the time series X is stored.
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