Descriptive statistics

Auto-covariances and auto-correlations

When the mean of the sample $\lbrace y_t \rbrace_{t \in [0, n]}$ is known ($=\mu$), we compute the sample auto-covariance of rank k by

$cov_k(y_t)=1/n \sum_{j=k}^{j \lt n} (y_j -\mu) (y_{j-k}-\mu)$

So, when $\mu = 0$, we have that

$cov_k(y_t)=1/n \sum_{j=k}^{j \lt n} y_j y_{j-k}$

When the mean is unknown, we define

$cov_k(y_t)=1/(n-k) \sum_{j=k}^{j \lt n} (y_j-\overline y)( y_{j-k}-\overline y)$

where $\overline y=1/n \sum_{j=0}^{j \lt n} y_j$.

The sample auto-correlations are defined by $\gamma_k(y_t)=cov_k(y_t)/cov_0(y_t)$

Implementations

The properties of samples with unkonwn mean and with 0 mean are provided respectively by the classes demetra.stats.samples.DefaultOrderedSample and demetra.stats.samples.OrderedSampleWithZeroMean.