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# autocorr

Sample autocorrelation

## Syntax

``autocorr(y)``
``autocorr(y,numLags)``
``autocorr(y,numLags,numMA,numSTD)``
``acf = autocorr(y)``
``acf = autocorr(y,numLags)``
``acf = autocorr(y,numLags,numMA,numSTD)``
``````[acf,lags,bounds] = autocorr(___)``````

## Description

example

````autocorr(y)` plots the sample autocorrelation function (ACF) of the univariate, stochastic time series `y` with confidence bounds.```

example

````autocorr(y,numLags)` plots the ACF, where `numLags` indicates the number of lags in the sample ACF.```

example

````autocorr(y,numLags,numMA,numSTD)` plots the ACF, where `numMA` specifies the number of lags beyond which the theoretical ACF is effectively 0, and `numSTD` specifies the number of standard deviations of the sample ACF estimation error.```

example

````acf = autocorr(y)` returns the sample ACF of the univariate, stochastic time series `y`.```

example

````acf = autocorr(y,numLags)` returns the ACF, where `numLags` specifies the number of lags in the sample ACF.```

example

````acf = autocorr(y,numLags,numMA,numSTD)` returns the ACF, where `numMA` specifies the number of lags beyond which the theoretical ACF is effectively 0, and `numSTD` specifies the number of standard deviations of the sample ACF estimation error. ```

example

``````[acf,lags,bounds] = autocorr(___)``` additionally returns the lags (`lags`) corresponding to the ACF and the approximate upper and lower confidence bounds (`bounds`), using any of the input arguments in the previous syntaxes.```

## Examples

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Specify the MA(2) model:

where is Gaussian with mean 0 and variance 1.

```rng(1); % For reproducibility Mdl = arima('MA',{-0.5 0.4},'Constant',0,'Variance',1)```
```Mdl = ARIMA(0,0,2) Model: -------------------- Distribution: Name = 'Gaussian' P: 0 D: 0 Q: 2 Constant: 0 AR: {} SAR: {} MA: {-0.5 0.4} at Lags [1 2] SMA: {} Variance: 1 ```

Simulate 1000 observations from `Mdl`.

`y = simulate(Mdl,1000);`

Compute the ACF.

```[ACF,lags,bounds] = autocorr(y,[],2); bounds```
```bounds = 0.0843 -0.0843 ```

`[]` tells the software to return the default number of lags (20). `numMA = 2` indicates that the ACF is effectively 0 after the second lag. `bounds` displays (-0.0843, 0.0843), which are the upper and lower confidence bounds.

Plot the ACF.

`autocorr(y)`

The ACF cuts off after the second lag. This behavior indicataes an MA(2) process.

Specify the mutliplicative seasonal ARMA model:

where is Gaussian with mean 0 and variance 1.

```Mdl = arima('AR',{0.75,0.15},'SAR',{0.9,-0.5,0.5},... 'SARLags',[12,24,36],'MA',-0.5,'Constant',2,... 'Variance',1);```

Simulate data from `Mdl`.

```rng(1); % For reproducibility y = simulate(Mdl,1000);```

Plot the default autocorrelation function (ACF).

```figure autocorr(y)```

The default correlogram does not display the dependence structure for higher lags.

Plot the ACF for 40 lags.

```figure autocorr(y,40,[],3)```

The correlogram shows the larger correlations at lags 12, 24, and 36.

Although various estimates of the sample autocorrelation function exist, `autocorr` uses the form in Box, Jenkins, and Reinsel, 1994. In their estimate, they scale the correlation at each lag by the sample variance (`var(y,1)`) so that the autocorrelation at lag 0 is unity. However, certain applications require rescaling the normalized ACF by another factor.

Simulate 1000 observations from the standard Gaussian distribution.

```rng(1); % For reproducibility y = randn(1000, 1);```

Compute the normalized and unnormalized sample ACF.

```[normalizedACF, lags] = autocorr(y, 10); unnormalizedACF = normalizedACF*var(y,1);```

Compare the first 10 lags of the sample ACF with and without normalization.

`[lags normalizedACF unnormalizedACF]`
```ans = 0 1.0000 0.9960 1.0000 -0.0180 -0.0180 2.0000 0.0536 0.0534 3.0000 -0.0206 -0.0205 4.0000 -0.0300 -0.0299 5.0000 -0.0086 -0.0086 6.0000 -0.0108 -0.0107 7.0000 -0.0116 -0.0116 8.0000 0.0309 0.0307 9.0000 0.0341 0.0340 ```

## Input Arguments

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Observed univariate time series for which the software computes or plots the ACF, specified as a vector. The last element of `y` contains the most recent observation.

Data Types: `double`

Number of lags of the ACF that the software returns or plots, specified as a positive integer.

For example, `autocorr(y,10)` plots the ACF for lags 0 through 10.

Data Types: `double`

MA order that specifies the number of lags beyond which the theoretical ACF is effectively 0, specified as a nonnegative integer.

• `numMA` must be less than `numLags`.

• Specify `numMA` to assess whether the ACF is effectively zero beyond lag `numMA` [1]. The software uses Bartlett’s approximation to estimate the large-lag standard error for lags that are greater than `numMA`.

• If `numMA = 0`, then the software assumes that `y` is a length T Gaussian white noise process. In this case, the standard error is approximately $\frac{1}{\sqrt{T}}.$

Example: `[~,~,bounds] = autocorr(y,[],5)`

Data Types: `double`

Number of standard deviations for the sample ACF estimation error assuming the theoretical ACF is 0 beyond lag `numMA`, specified as a positive scalar. For example, `autocorr(y,[],[],1.5)` plots the ACF with estimation error bounds 1.5 standard deviations away from 0.

If `numMA = 0` and `y` a lengthT Gaussian process, then the confidence bounds are:

`$±\frac{numSTD}{\sqrt{T}}.$`

The default (`numSTD = 2`) corresponds to approximate 95% confidence bounds.

Data Types: `double`

## Output Arguments

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Sample ACF of the univariate time series `y`, returned as a vector of length `numLags + 1`.

The elements of `acf` correspond to lags 0,1,2,...,`numLags`. The first element, which corresponds to lag 0, is unity (i.e., ```acf(1) = 1```).

Sample ACF lags, returned as a vector. Specifically, ```lags = 0:numLags```.

Approximate confidence bounds of the ACF assuming `y` is an MA(`numMA`) process, returned as a two-element vector. `bounds` is approximate for `lags` > `numMA`.

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### Autocorrelation Function

Measures the correlation between yt and yt + k, where k = 0,...,K and yt is a stochastic process.

According to [1], the formula for the autocorrelation for lag k is

`${r}_{k}=\frac{{c}_{k}}{{c}_{0}},$`
where

• ${c}_{k}=\frac{1}{T}\sum _{t=1}^{T-k}\left({y}_{t}-\overline{y}\right)\left({y}_{t+k}-\overline{y}\right).$

• c0 is the sample variance of the time series.

Suppose that q is the lag beyond which the theoretical ACF is effectively 0. Then, the estimated standard error of the autocorrelation at lag k > q is

`$SE\left({r}_{k}\right)=\sqrt{\frac{1}{T}\left(1+2\sum _{j=1}^{q}{r}_{j}^{2}\right)}.$`
If the series is completely random, then the standard error reduces to $1/\sqrt{T}$ [1].

## Tips

To plot the ACF without confidence bounds, set `numSTD` to 0.

## References

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

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