Descriptive statistics
Auto-covariances and auto-correlations
We consider the sample $\lbrace y_t \rbrace_{t \in [0, n]}$ from a population $P$
When the mean of the population 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 simply have that
\[cov_k(y_t)=1/n \sum_{j=k}^{j \lt n} y_j y_{j-k}\]When the mean of the population 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 $\overline y$ is the sample mean:
\[\overline y=1/n \sum_{j=0}^{j \lt n} y_j\]The sample auto-correlations are defined by $\gamma_k(y_t)=cov_k(y_t)/cov_0(y_t)$
Implementations in JD+
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.