Autocorrelation

Definition for wide-sense stationary stochastic process

If ${\displaystyle \left\{X_{t}\right\}}$ is a wide-sense stationary process then the mean ${\displaystyle \mu }$ and the variance ${\displaystyle \sigma ^{2}}$ are time-independent, and further the autocovariance function depends only on the lag between ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$: the autocovariance depends only on the time-distance between the pair of values but not on their position in time. This further implies that the autocovariance and autocorrelation can be expressed as a function of the time-lag, and that this would be an even function of the lag ${\displaystyle \tau =t_{2}-t_{1}}$. This gives the more familiar forms for the autocorrelation function^: 395

and the auto-covariance function:

In particular, note that

$${\displaystyle \operatorname {K} _{XX}(0)=\sigma ^{2}.}$$

Normalization

It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.

The definition of the autocorrelation coefficient of a stochastic process is^: 169

$${\displaystyle {\begin{aligned}\rho _{XX}(t_{1},t_{2})&={\frac {\operatorname {K} _{XX}(t_{1},t_{2})}{\sigma _{t_{1}}\sigma _{t_{2}}}}\\&={\frac {\operatorname {E} \left[\left(X_{t_{1}}-\mu _{t_{1}}\right){\overline {\left(X_{t_{2}}-\mu _{t_{2}}\right)}}\right]}{\sigma _{t_{1}}\sigma _{t_{2}}}}.\end{aligned}}}$$

If the function ${\displaystyle \rho _{XX}}$ is well defined, its value must lie in the range ${\displaystyle [-1,1]}$, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For a wide-sense stationary (WSS) process, the definition is

$${\displaystyle \rho _{XX}(\tau )={\frac {\operatorname {K} _{XX}(\tau )}{\sigma ^{2}}}={\frac {\operatorname {E} \left[(X_{t+\tau }-\mu ){\overline {(X_{t}-\mu )}}\right]}{\sigma ^{2}}}.}$$

The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations.

Properties

Properties of the autocorrelation matrix

• The autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors.^: 190
• The autocorrelation matrix is a positive semidefinite matrix,^: 190 i.e. ${\displaystyle \mathbf {a} ^{\mathrm {T} }\operatorname {R} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}$ for a real random vector, and respectively ${\displaystyle \mathbf {a} ^{\mathrm {H} }\operatorname {R} _{\mathbf {Z} \mathbf {Z} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {C} ^{n}}$ in case of a complex random vector.
• All eigenvalues of the autocorrelation matrix are real and non-negative.
• The auto-covariance matrix is related to the autocorrelation matrix as follows:$${\displaystyle {\begin{aligned}\operatorname {K} _{\mathbf {X} \mathbf {X} }&=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}\right]\\&=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{\rm {T}}\end{aligned}}}$$Respectively for complex random vectors:$${\displaystyle {\begin{aligned}\operatorname {K} _{\mathbf {Z} \mathbf {Z} }&=\operatorname {E} \left[(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])^{\rm {H}}\right]\\&=\operatorname {R} _{\mathbf {Z} \mathbf {Z} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ]^{\rm {H}}\end{aligned}}}$$

Autocorrelation of continuous-time signal

Given a signal ${\displaystyle f(t)}$, the continuous autocorrelation ${\displaystyle R_{ff}(\tau )}$ is most often defined as the continuous cross-correlation integral of ${\displaystyle f(t)}$ with itself, at lag ${\displaystyle \tau }$.^: 411

where ${\displaystyle {\overline {f(t)}}}$ represents the complex conjugate of ${\displaystyle f(t)}$. Note that the parameter ${\displaystyle t}$ in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.

Autocorrelation of discrete-time signal

The discrete autocorrelation ${\displaystyle R}$ at lag ${\displaystyle \ell }$ for a discrete-time signal ${\displaystyle y(n)}$ is

The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as

$${\displaystyle {\begin{aligned}R_{ff}(\tau )&=\operatorname {E} \left[f(t){\overline {f(t-\tau )}}\right]\\R_{yy}(\ell )&=\operatorname {E} \left[y(n)\,{\overline {y(n-\ell )}}\right].\end{aligned}}}$$

For processes that are not stationary, these will also be functions of ${\displaystyle t}$, or ${\displaystyle n}$.

For processes that are also ergodic, the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to

$${\displaystyle {\begin{aligned}R_{ff}(\tau )&=\lim _{T\rightarrow \infty }{\frac {1}{T}}\int _{0}^{T}f(t+\tau ){\overline {f(t)}}\,{\rm {d}}t\\R_{yy}(\ell )&=\lim _{N\rightarrow \infty }{\frac {1}{N}}\sum _{n=0}^{N-1}y(n)\,{\overline {y(n-\ell )}}.\end{aligned}}}$$

These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.

Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.)

Definition for periodic signals

If ${\displaystyle f}$ is a continuous periodic function of period ${\displaystyle T}$, the integration from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$ is replaced by integration over any interval ${\displaystyle [t_{0},t_{0}+T]}$ of length ${\displaystyle T}$:

$${\displaystyle R_{ff}(\tau )\triangleq \int _{t_{0}}^{t_{0}+T}f(t+\tau ){\overline {f(t)}}\,dt}$$

which is equivalent to

$${\displaystyle R_{ff}(\tau )\triangleq \int _{t_{0}}^{t_{0}+T}f(t){\overline {f(t-\tau )}}\,dt}$$

Properties

In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for wide-sense stationary processes.

• A fundamental property of the autocorrelation is symmetry, ${\displaystyle R_{ff}(\tau )=R_{ff}(-\tau )}$, which is easy to prove from the definition. In the continuous case,
  • the autocorrelation is an even function ${\displaystyle R_{ff}(-\tau )=R_{ff}(\tau )}$ when ${\displaystyle f}$ is a real function, and
  • the autocorrelation is a Hermitian function ${\displaystyle R_{ff}(-\tau )=R_{ff}^{*}(\tau )}$ when ${\displaystyle f}$ is a complex function.
• The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay ${\displaystyle \tau }$, ${\displaystyle |R_{ff}(\tau )|\leq R_{ff}(0)}$.^: 410 This is a consequence of the rearrangement inequality. The same result holds in the discrete case.
• The autocorrelation of a periodic function is, itself, periodic with the same period.
• The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all ${\displaystyle \tau }$) is the sum of the autocorrelations of each function separately.
• Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation.
• By using the symbol ${\displaystyle *}$ to represent convolution and ${\displaystyle g_{-1}}$ is a function which manipulates the function ${\displaystyle f}$ and is defined as ${\displaystyle g_{-1}(f)(t)=f(-t)}$, the definition for ${\displaystyle R_{ff}(\tau )}$ may be written as:$${\displaystyle R_{ff}(\tau )=(f*g_{-1}({\overline {f}}))(\tau )}$$

Algebraic basis

The identity follows from the fact that the centered observations satisfy

$${\displaystyle \sum _{t=1}^{T}z_{t}=0.}$$

Squaring this relation gives

$${\displaystyle 0=\left(\sum _{t=1}^{T}z_{t}\right)^{2}=\sum _{t=1}^{T}z_{t}^{2}+2\sum _{1\leq t<s\leq T}z_{t}z_{s}.}$$

The second term contains every pairwise product of observations at positive separations. Grouping these products according to their lag yields

$${\displaystyle \sum _{h=1}^{T-1}\sum _{t=1}^{T-h}z_{t}z_{t+h}=-{\frac {1}{2}}\sum _{t=1}^{T}z_{t}^{2}.}$$

Dividing both sides by ${\displaystyle \sum _{t=1}^{T}z_{t}^{2}}$ gives the −1/2 identity.

This derivation also demonstrates why the result does not imply that the theoretical autocorrelations of the underlying process sum to −1/2. The restriction arises because the same data are used both to estimate the sample mean and to construct all of the empirical autocorrelations.

Related finite-sample effects arising from estimation of an unknown mean had previously been discussed by Percival, who showed that conventional estimators of variance and autocovariance can possess unintuitive properties when the population mean is replaced by the sample mean.

Statistical interpretation

A consequence of the theorem is that the complete vector of sample autocorrelations,

$${\displaystyle ({\hat {\rho }}(1),\ldots ,{\hat {\rho }}(T-1)),}$$

is subject to an exact linear constraint. The sample autocorrelations at different lags therefore cannot be mutually independent when the complete collection of lags is considered.

This observation is relevant to standard large-sample approximations in time-series analysis. For certain processes, sample autocorrelations at a fixed and relatively small number of lags may be approximately normally distributed and asymptotically uncorrelated. The −1/2 theorem does not contradict such fixed-lag asymptotic results. Instead, it shows that these approximations cannot be extended indiscriminately to the entire set of ${\displaystyle T-1}$ sample autocorrelations, because the full collection satisfies the exact finite-sample constraint.

The theorem also implies that positive empirical correlations at some lags must be offset by negative contributions at other lags when all positive lags are included. Consequently, the unrestricted sum of sample autocorrelations cannot be interpreted directly as an empirical measure of the overall amount or direction of serial dependence.

Relation to long-memory processes

The distinction between theoretical and sample autocorrelations is particularly important in the study of long-memory processes. A common theoretical characterization of long memory is based on a slowly decaying autocorrelation function whose infinite sum does not converge absolutely, or may diverge.

For a finite sample, however, the sum of the conventionally estimated autocorrelations over all available positive lags remains −1/2, irrespective of whether the underlying process is theoretically short-memory or long-memory. Hassani, Leonenko and Patterson therefore argued that the full sum of sample autocorrelations cannot itself be used as a consistent empirical diagnostic for long-range dependence.

This does not imply that sample autocorrelations contain no information about long memory. Rather, inference generally depends on the rate of decay, selected ranges of lags, spectral behaviour near frequency zero, or dedicated estimators of long-memory parameters instead of the unrestricted sum across every available lag.

Diagnostic testing and the Ljung–Box statistic

The theorem has also been examined in connection with residual diagnostics and portmanteau tests. The Ljung–Box test, for example, is based on a weighted sum of squared sample autocorrelations over a specified number of lags rather than on their simple signed sum.

Hassani and Yeganegi investigated how results of the Ljung–Box procedure can depend on the number of included lags and used the −1/2 identity as motivation for studying the collective behaviour of sample autocorrelations.

Their analysis emphasized that diagnostic conclusions may change with the truncation lag used in a portmanteau statistic and that finite-sample dependence among estimated autocorrelations should be distinguished from asymptotic approximations applying to a fixed number of lags.

Subsequent work has similarly examined the behaviour of sample ACFs for white-noise and fitted-model residuals and has questioned whether standard normal approximations remain satisfactory when increasingly large sets of lags are considered.

Model identification

Sample ACFs are widely used for identifying time-series models, particularly moving-average (MA) processes, for which the theoretical ACF becomes zero beyond the model order. In finite samples, however, estimated autocorrelations fluctuate around their theoretical values.

Recent research has used the −1/2 theorem to investigate the joint behaviour and distribution of sample ACFs in white-noise and MA processes. Hassani and co-authors reported departures from simple independent-normal approximations when a sufficiently large range of sample autocorrelations is considered.

Royer-Carenzi and Hassani subsequently examined the implications for identifying MA(${\displaystyle q}$) and ARMA(${\displaystyle p,q}$) models, including procedures based on the ACF and extended autocorrelation function. Their simulations indicated that conventional identification methods remain useful for sufficiently long samples but may be less reliable in particular parameter configurations, motivating the use of additional diagnostics and predictive comparisons.

Short-memory processes

The −1/2 identity has also been discussed in relation to empirical definitions of short-memory processes. Some definitions characterize short memory through summability or rapid decay of theoretical autocorrelations. Because the complete finite-sample ACF sum is predetermined, directly transferring such a theoretical summability criterion to the unrestricted empirical ACF can be problematic.

Hassani and co-authors revisited commonly used definitions of short memory and argued that empirical identification should distinguish properties of the underlying theoretical ACF from algebraic properties imposed on its finite-sample estimator.

Applications and related results

The −1/2 relation has appeared outside studies devoted specifically to sample-ACF methodology. For example, Di Crescenzo, Martinucci and Paraggio obtained autocorrelation structures for stationary interarrival times in a stochastic model of vessel arrivals and noted a −1/2 autocorrelation-sum relationship with reference to Hassani's earlier result.

Software implementations

Software implementations of the Hassani −1/2 theorem are available for both R and Python.

The R package Hassani.SACF, available through the Comprehensive R Archive Network (CRAN), provides functions related to the sum of the sample autocorrelation function and the sensitivity of the Ljung–Box test to the choice of lag length. The package implements calculations associated with the sample-ACF sum and related diagnostic quantities.

A Python implementation is also available through the Python Package Index (PyPI) as Hassani.SACF. The package provides functions for computing the sample autocorrelation function, the sum of sample autocorrelations over all positive lags, and the periodogram value at zero frequency. Under the common fixed-divisor sample-autocovariance convention, the software can be used to reproduce the two finite-sample identities associated with the theorem: a sample-ACF sum of approximately ${\displaystyle -1/2}$ and a zero-frequency periodogram ordinate of approximately zero for demeaned data.

Scope and limitations

The theorem depends on the definition of the sample ACF. In particular, the exact −1/2 equality applies to the conventional estimator in which the autocovariance numerator at each lag is normalized by the same lag-zero sum of squares after subtraction of the sample mean.

Alternative estimators may use different normalizations, bias corrections, known rather than estimated means, tapering, missing-data adjustments, or restrictions to a subset of lags. Such estimators need not satisfy the exact −1/2 identity.

The theorem should therefore not be interpreted as stating that

the theoretical ACF of every stochastic process sums to −1/2; every possible estimator of an ACF has this property; individual sample autocorrelations cannot be positive; standard fixed-lag asymptotic ACF theory is invalid; or ACF-based methods cannot be used for time-series identification.

Rather, it establishes an exact dependence constraint on the complete set of autocorrelations obtained from a particular finite-sample estimator.

Theoretical versus sample autocorrelation

The distinction between the theoretical and sample ACF is central to interpretation of the theorem. For a stationary stochastic process, the theoretical autocorrelation at lag ${\displaystyle h}$ is a population quantity,

$${\displaystyle \rho (h)={\frac {\operatorname {Cov} (X_{t},X_{t+h})}{\operatorname {Var} (X_{t})}},}$$

and its behaviour depends on the underlying probability model. Its sum may be positive, negative, finite, divergent, or undefined depending on the process.

By contrast, ${\displaystyle {\hat {\rho }}(h)}$ is calculated from one finite realization. When the observations are centered using their own sample mean and the conventional common-denominator estimator is used, the complete finite-sample collection satisfies the deterministic −1/2 constraint. The theorem therefore illustrates a broader principle in time-series statistics: finite-sample estimators can possess structural properties that have no direct counterpart in the population quantity they estimate.