autocorr - Autocorrelation

Syntax

autocorr(Series,nLags,M,nSTDs) [ACF,Lags,Bounds] = autocorr(Series,nLags,M,nSTDs)

Description

autocorr(Series,nLags,M,nSTDs) computes and plots the sample ACF of a univariate, stochastic time series with confidence bounds. To plot the ACF sequence without the confidence bounds, set nSTDs = 0.
[ACF,Lags,Bounds] = autocorr(Series,nLags,M,nSTDs) computes and returns the ACF sequence.

Input Arguments

`Series`	Column vector of observations of a univariate time series for which `autocorr` computes or plots the sample autocorrelation function (ACF). The last row of `Series` contains the most recent observation of the time series.
`nLags`	Positive scalar integer indicating the number of lags of the ACF to compute. If `nLags = []` or is unspecified, the default is to compute the ACF at lags 0, 1, 2, ..., T, where T `= min([20,length(Series)-1])`.
`M`	Nonnegative integer scalar indicating the number of lags beyond which the theoretical ACF is effectively `0`. `autocorr` assumes the underlying `Series` is an MA(M) process, and uses Bartlett's approximation to compute the large-lag standard error for lags greater than `M`. If `M = []` or is unspecified, the default is `0`, and `autocorr` assumes that `Series` is Gaussian white noise. If `Series` is a Gaussian white noise process of length N, the standard error is approximately . `M` must be less than `nLags`.
`nSTDs`	Positive scalar indicating the number of standard deviations of the sample ACF estimation error to compute. `autocorr` assumes the theoretical ACF of `Series` is `0` beyond lag `M`. When `M = 0` and `Series` is a Gaussian white noise process of length N, specifying `nSTDs` results in confidence bounds at . If `nSTDs = []` or is unspecified, the default is `2` (that is, approximate 95 percent confidence interval).

Output Arguments

`ACF`	Sample autocorrelation function of `Series`. `ACF` is a vector of length `nLags`+1 corresponding to lags 0, 1, 2, ..., `nLags`. The first element of `ACF` is unity, that is, `ACF`(1) `= 1 =` lag `0` correlation.
`Lags`	Vector of lags corresponding to `ACF(0,1,2,`...`,nLags)`. Since an ACF is symmetric about `0` lag, `autocorr` ignores negative lags.
`Bounds`	Two-element vector indicating the approximate upper and lower confidence bounds, assuming that `Series` is an MA(M) process. Values of `ACF` beyond lag `M` that are effectively `0` lie within these bounds. `autocorr` computes `Bounds` only for lags greater than `M`.

Examples

Example 1

Create an MA(2) time series from a column vector of 1000 Gaussian deviates. Then, assess whether the ACF is effectively zero for lags greater than 2:

strm = RandStream('mt19937ar'); % reproducible
RandStream.setDefaultStream(strm);
x = randn(1000, 1);           % 1000 Gaussian deviates ~ N(0, 1).
y = filter([1 -1 1], 1, x);   % Create an MA(2) process.

% Compute the ACF with 95 percent confidence.
[ACF, Lags, Bounds] = autocorr(y, [], 2);  
[Lags, ACF]
ans =
         0    1.0000
    1.0000   -0.6682
    2.0000    0.3618
    3.0000   -0.0208
    4.0000    0.0146
    5.0000   -0.0311
    6.0000    0.0611
    7.0000   -0.0828
    8.0000    0.0772
    9.0000   -0.0493
   10.0000    0.0323
   11.0000   -0.0294
   12.0000    0.0576
   13.0000   -0.0633
   14.0000    0.0500
   15.0000   -0.0129
   16.0000   -0.0180
   17.0000    0.0368
   18.0000   -0.0459
   19.0000    0.0439
   20.0000   -0.0318

Bounds
    0.0928
   -0.0928


autocorr(y, [], 2)    % Use the same example, but plot the ACF
                      % sequence with confidence bounds.

Example 2

Although various estimates of the sample autocorrelation function exist, the form adopted here follows that of Box, Jenkins, and Reinsel, specifically:

(11-1)

(11-2)

The autocorr function computes the sample ACF by removing the sample mean of the input Series, then normalizing the sequence such that the ACF at lag zero is unity. In certain applications, it is useful to rescale the resulting normalized ACF by the sample variance. In this case, the correct scale factor to use is var(Series,1).

The following commands simulate 1000 standard Gaussian random numbers, then compares the first 10 lags of the sample ACF with and without normalization:

strm = RandStream('mt19937ar'); % reproducible
RandStream.setDefaultStream(strm);
y = randn(1000, 1);
[ACF, Lags] = autocorr(y, 10);
[Lags  ACF  ACF*var(y,1)]
ans =
         0    1.0000    0.9969
    1.0000    0.0360    0.0359
    2.0000    0.0441    0.0439
    3.0000   -0.0271   -0.0271
    4.0000   -0.0452   -0.0451
    5.0000   -0.0173   -0.0172
    6.0000    0.0456    0.0454
    7.0000   -0.0136   -0.0135
    8.0000    0.0332    0.0331
    9.0000   -0.0356   -0.0355
   10.0000   -0.0115   -0.0115

Example 3

See Example: Using the Default Model.

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1994.

[2] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.