autocorr

Sample autocorrelation

Syntax

[acf,lags] = autocorr(y)

ACFTbl = autocorr(Tbl)

[___,bounds] = autocorr(___)

[___] = autocorr(___,Name=Value)

autocorr(___)

autocorr(ax,___)

[___,h] = autocorr(___)

Description

[acf,lags] = autocorr(y) returns the sample autocorrelation function (ACF) acf and associated lags lags of the univariate time series y.

ACFTbl = autocorr(Tbl) returns the table ACFTbl containing variables for the sample ACF and associated lags of the last variable in the input table or timetable Tbl. To select a different variable in Tbl, for which to compute the ACF, use the DataVariable name-value argument.

[___,bounds] = autocorr(___) uses any input-argument combination in the previous syntaxes, and returns the output-argument combination for the corresponding input arguments and the approximate upper and lower confidence bounds bounds on the ACF.

[___] = autocorr(___,Name=Value) uses additional options specified by one or more name-value arguments. For example, autocorr(Tbl,DataVariable="RGDP",NumLags=10,NumSTD=1.96) returns 10 lags of the sample ACF of the table variable "RGDP" in Tbl and 95% confidence bounds.

autocorr(___) plots the sample ACF of the input series with confidence bounds.

autocorr(ax,___) plots on the axes specified by ax instead of the current axes (gca). ax can precede any of the input argument combinations in the previous syntaxes.

[___,h] = autocorr(___) plots the sample ACF of the input series and additionally returns handles to plotted graphics objects. Use elements of h to modify properties of the plot after you create it.

Examples

Return ACF on Vector of Time Series Data

Compute the ACF of a univariate time series. Input the time series data as a numeric vector.

Load the quarterly real GDP series in Data_GDP.mat. Plot the series, which is stored in the numeric vector Data.

load Data_GDP
plot(Data)

The series exhibits exponential growth.

Compute the returns of the series.

ret = price2ret(Data);

ret is a series of real GDP returns; it has one less observation than the real GDP series.

Compute the ACF of the real GDP returns, and return the associated lags.

[acf,lags] = autocorr(ret);
[acf lags]

ans = 21×2

    1.0000         0
    0.3329    1.0000
    0.1836    2.0000
   -0.0216    3.0000
   -0.1172    4.0000
   -0.1632    5.0000
   -0.0870    6.0000
   -0.0707    7.0000
   -0.0380    8.0000
    0.0554    9.0000
      ⋮

Let ${\mathit{y}}_{\mathit{t}}$ be the real GDP return at time $\mathit{t}$. In general, acf(j) = Corr(${\mathit{y}}_{\mathit{t}}$,${\mathit{y}}_{\mathit{t}-\mathrm{lags}\left(\mathit{j}\right)}$). Therefore, acf(1) = Corr(${\mathit{y}}_{\mathit{t}}$,${\mathit{y}}_{\mathit{t}}$) = 1.0000, acf(2) = Corr(${\mathit{y}}_{\mathit{t}}$,${\mathit{y}}_{\mathit{t}-1}$) = 0.3329, and so on.

Compute ACF of Table Variable

Compute the ACF of a time series, which is one variable in a table.

Load the electricity spot price data set Data_ElectricityPrices.mat, which contains the daily spot prices in the timetable DataTimeTable.

load Data_ElectricityPrices.mat
DataTimeTable.Properties.VariableNames

ans = 1x1 cell array
    {'SpotPrice'}

Plot the series.

plot(DataTimeTable.SpotPrice)

The time series plot does not clearly indicate an exponential trend or unit root.

Compute the ACF of the raw spot price series.

ACFTbl = autocorr(DataTimeTable)

ACFTbl=21×2 table
    Lags      ACF  
    ____    _______

      0           1
      1     0.55405
      2     0.38251
      3     0.31713
      4     0.25107
      5     0.21436
      6     0.21275
      7     0.19396
      8     0.18292
      9     0.18826
     10     0.19476
     11     0.19043
     12     0.19963
     13     0.19397
     14     0.19957
     15     0.25495
      ⋮

autocorr returns the results in the table ACFTbl, where variables correspond to the ACF (ACF) and associated lags (Lags).

By default, autocorr computes the ACF of the last variable in the table. To select a variable from an input table, set the DataVariable option.

Return ACF Confidence Bounds

Consider the electricity spot prices in Compute ACF of Table Variable.

Load the electricity spot price data set Data_ElectricityPrices.mat. Compute the ACF and return the ACF confidence bounds.

load Data_ElectricityPrices
[ACFTbl,bounds] = autocorr(DataTimeTable)

ACFTbl=21×2 table
    Lags      ACF  
    ____    _______

      0           1
      1     0.55405
      2     0.38251
      3     0.31713
      4     0.25107
      5     0.21436
      6     0.21275
      7     0.19396
      8     0.18292
      9     0.18826
     10     0.19476
     11     0.19043
     12     0.19963
     13     0.19397
     14     0.19957
     15     0.25495
      ⋮

bounds = 2×1

    0.0532
   -0.0532

Assuming the spot prices follow a Gaussian white noise series, an approximate 95.4% confidence interval on the ACF is (-0.0532, 0.0532).

Compare the ACF for Normalized and Unnormalized Series

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,NumLags=10);
unnormalizedACF = normalizedACF*var(y,1);

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

[lags normalizedACF unnormalizedACF]

ans = 11×3

         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
      ⋮

Plot ACF of Simulated Time Series

Specify the MA(2) model:

$$y_t={\epsilon }_t-0.5{\epsilon }_{t-1}+0.4{\epsilon }_{t-2},$$

where $${\epsilon }_t$$ 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 with properties:

     Description: "ARIMA(0,0,2) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 0
               D: 0
               Q: 2
        Constant: 0
              AR: {}
             SAR: {}
              MA: {-0.5 0.4} at lags [1 2]
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: 1

Simulate 1000 observations from Mdl.

y = simulate(Mdl,1000);

Plot the ACF of the simulated series. Specify that the series is an MA(2) process.

autocorr(y,NumMA=2)

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

Specify Additional Lags in ACF Plot

Specify the multiplicative seasonal ARMA $$(2,0,1)\times (3,0,0{)}_{12}$$ model:

$$(1-0.75L-0.15L^2)(1-0.9L^{12}+0.5L^{24}-0.5L^{36})y_t=2+{\epsilon }_t-0.5{\epsilon }_{t-1},$$

where $${\epsilon }_t$$ 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,NumLags=40)

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

Input Arguments

y — Observed univariate time series

numeric vector

Observed univariate time series for which autocorr computes or plots the ACF, specified as a numeric vector.

Data Types: double

Tbl — Time series data

table | timetable

Time series data, specified as a table or timetable. Each row of Tbl contains contemporaneous observations of all variables.

Specify a single series (variable) by using the DataVariable argument. The selected variable must be numeric.

ax — Axes on which to plot

Axes object

Axes on which to plot, specified as an Axes object.

By default, autocorr plots to the current axes (gca).

Note

Specify missing observations using NaN. The autocorr function treats missing values as missing completely at random.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: autocorr(Tbl,DataVariable="RGDP",NumLags=10,NumSTD=3) plots 10 lags of the sample ACF of the variable "RGDP" in Tbl, and displays confidence bounds consisting of 3 standard errors away from 0.

NumLags — Number of lags

positive integer

Number of lags in the sample ACF, specified as a positive integer. autocorr uses lags 0:NumLags to estimate the ACF.

The default is min([20,T – 1]), where T is the effective sample size of the input time series.

Example: autocorr(y,NumLags=10) plots the sample ACF of y for lags 0 through 10.

Data Types: double

NumMA — Number of lags in theoretical MA model

0 (default) | nonnegative integer

Number of lags in a theoretical MA model of the input time series, specified as a nonnegative integer less than NumLags.

autocorr uses NumMA to estimate confidence bounds.

• For lags > NumMA, autocorr uses Bartlett’s approximation [1] to estimate the standard errors under the model assumption.

• If NumMA = 0, then autocorr assumes that the input time series is a Gaussian white noise process with a standard error of approximately $$1/\sqrt{T},$$ where T is the effective sample size of the input time series.

Example: autocorr(y,NumMA=10) specifies that y is an MA(10) process and plots confidence bounds for all lags greater than 10.

Data Types: double

NumSTD — Number of standard errors in confidence bounds

2 (default) | nonnegative scalar

Number of standard errors in the confidence bounds, specified as a nonnegative scalar. For all lags greater than NumMA, the confidence bounds are 0 ±NumSTD*$$\widehat{\sigma }$$, where $$\widehat{\sigma }$$ is the estimated standard error of the sample autocorrelation.

The default yields the approximate 95% confidence bounds.

Example: autocorr(y,NumSTD=1.5) plots the ACF of y with confidence bounds 1.5 standard errors away from 0.

Data Types: double

DataVariable — Variable in Tbl

last variable (default) | string scalar | character vector | integer | logical vector

Variable in Tbl for which autocorr computes the ACF, specified as a string scalar or character vector containing a variable name in Tbl.Properties.VariableNames, or an integer or logical vector representing the index of a name. The selected variable must be numeric.

Example: DataVariable="GDP"

Example: DataVariable=[false true false false] or DataVariable=2 selects the second table variable.

Data Types: double | logical | char | string

Output Arguments

acf — Sample ACF

numeric vector

Sample ACF, returned as a numeric vector of length NumLags + 1. autocorr returns acf only when you supply the input y.

The elements of acf correspond to lags 0,1,2,..., NumLags (that is, elements of lags). For all time series, the lag 0 autocorrelation acf(1) = 1.

lags — ACF lags

numeric vector

ACF lags, returned as a numeric vector with elements 0:NumLags. autocorr returns lags only when you supply the input y.

ACFTbl — Sample ACF

table

Sample ACF, returned as a table with variables for the outputs acf and lags. autocorr returns ACFTbl when you supply the input Tbl.

bounds — Approximate upper and lower confidence bounds

numeric vector

Approximate upper and lower confidence bounds assuming the input series is an MA(NumMA) process, returned as a two-element numeric vector. The NumSTD option specifies the number of standard errors in the confidence bounds.

h — Handles to plotted graphics objects

graphics array

Handles to plotted graphics objects, returned as a graphics array. h contains unique plot identifiers, which you can use to query or modify properties of the plot.

More About

Autocorrelation Function

The autocorrelation function measures the correlation between the univariate time series y_t and y_{t + k}, where k = 0,...,K and y_t is a stochastic process.

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

$$r_k=\frac{c_k}{c_0},$$

where

• $$c_k=\frac{1}{T}{\displaystyle \sum _{t=1}^{T-k}(y_t-\overline{y})}(y_{t+k}-\overline{y}).$$

• 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

$$SE(r_k)=\sqrt{\frac{1}{T}\left(1+2{\displaystyle \sum _{j=1}^q{r}_j^2}\right)}.$$

If the series is completely random, then the standard error reduces to $$1/\sqrt{T}$$.

Missing Completely at Random

Observations of a random variable are missing completely at random if the tendency of an observation to be missing is independent of both the random variable and the tendency of all other observations to be missing.

Tips

• To plot the ACF without confidence bounds, set NumSTD=0.

Algorithms

• If the input series is a fully observed series (that is, it does not contain any NaN values), autocorr uses a Fourier transform to compute the ACF in the frequency domain, then converts back to the time domain using an inverse Fourier transform.

• If the input series is not fully observed (that is, it contains at least one NaN value), autocorr computes the ACF at lag k in the time domain, and includes in the sample average only those terms for which the cross product y_ty_t+k exists. Consequently, the effective sample size is a random variable.

• autocorr plots the ACF when you do not return any output or when you return the fourth output h.

References

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

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

Version History

Introduced before R2006a