autokovarianciát

Autokovariancia, often translated as autocovariance, is a statistical measure used in time series analysis. It quantifies the linear relationship between observations of a time series at different points in time. Specifically, it measures how the covariance of a random variable with itself changes as the time lag between the two observations increases.

The autocovariance function, denoted by $\gamma_X(h)$, for a stationary time series $X_t$ at lag $h$ is defined as the expected value of the product of the deviation of $X_t$ from its mean $\mu$ and the deviation of $X_{t-h}$ from its mean $\mu$. Mathematically, this is expressed as $\gamma_X(h) = E[(X_t - \mu)(X_{t-h} - \mu)]$. For a mean-zero process, this simplifies to $\gamma_X(h) = E[X_t X_{t-h}]$.

A key property of autocovariance is that $\gamma_X(0)$ represents the variance of the time series, as it measures the covariance of the series with itself at zero lag. As the lag $h$ increases, the autocovariance generally decreases, indicating that observations further apart in time are less linearly related. The pattern of the autocovariance function over different lags provides valuable insights into the underlying structure and dependence of the time series, which is crucial for tasks like forecasting, modeling, and hypothesis testing. It helps in identifying seasonality, trend, and the presence of autoregressive or moving average components.