naginterfaces.library.tsa.uni_​autocorr

naginterfaces.library.tsa.uni_autocorr(x, nk)

uni_autocorr computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation function is zero.

For full information please refer to the NAG Library document for g13ab

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13abf.html

Parameters

xfloat, array-like, shape $\left(\text{nx}\right)$

The time series, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

nkint

$K$, the number of lags for which the autocorrelations are required. The lags range from $1$ to $K$ and do not include zero.

Returns

xmfloat

The sample mean of the input time series.

xvfloat

The sample variance of the input time series.

rfloat, ndarray, shape $\left(nk\right)$

The sample autocorrelation coefficient relating to lag $\text{k}$, for $\text{k}=1,2,\dots ,K$.

statfloat

The statistic used to test the hypothesis that the true autocorrelation function of the time series is identically zero.

Raises

NagValueError

(errno $1$)

On entry, $\text{nx}=⟨value⟩$ and $nk=⟨value⟩$.

Constraint: $\text{nx}>nk$.

(errno $1$)

On entry, $\text{nx}=⟨value⟩$.

Constraint: $\text{nx}>1$.

(errno $1$)

On entry, $nk=⟨value⟩$.

Constraint: $nk>0$.

Warns

NagAlgorithmicWarning

(errno $2$)

On entry, all values of $x$ are practically identical.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The data consists of $n$ observations $x_i$, for $\text{i}=1,2,\dots ,n$ from a time series.

The quantities calculated are

1. The sample mean

   $$\overline{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}\text{.}$$

2. The sample variance (for $n\ge 2$)

   $$s^2=\frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}{(n-1)}\text{.}$$

3. The sample autocorrelation coefficients of lags $k=1,2,\dots ,K$, where $K$ is a user-specified maximum lag, and $K<n$, $n>1$.

   The coefficient of lag $k$ is defined as

   $$r_k=\frac{{\sum }_{i=1}^{n-k}(x_i-\overline{x})(x_{i+k}-\overline{x})}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}\text{.}$$

   See page 496 of Box and Jenkins (1976) for further details.

4. A test statistic defined as

   $$stat=n\sum _{k=1}^K{r}_k^2\text{,}$$

   which can be used to test the hypothesis that the true autocorrelation function is identically zero.

   If $n$ is large and $K$ is much smaller than $n$, $stat$ has a ${\chi }_K^2$ distribution under the hypothesis of a zero autocorrelation function. Values of $stat$ in the upper tail of the distribution provide evidence against the hypothesis; stat.prob_chisq can be used to compute the tail probability.

   Section 8.2.2 of Box and Jenkins (1976) provides further details of the use of $stat$.

References

Box, G E P and Jenkins, G M, 1976, Time Series Analysis: Forecasting and Control, (Revised Edition), Holden–Day