A **correlogram** is a graph used to interpret a set of **autocorrelation coefficients** in which $r_k$ is plotted against the $log k$. A **correlogram** is often very helpful for visual inspection.

### Some general advice to interpret the **correlogram** are:

**A Random Series:**If a**time series**is completely random, then for large $N$, $r_k \cong 0$ for all non-zero values of $k$. A random time series $r_k$ is approximately $N\left(0, \frac{1}{N}\right)$. If a time series is random, 19 out of 20 of the values of $r_k$ can be expected to lie between $\pm \frac{2}{\sqrt{N}}$. However, plotting the first 20 values of $r_k$, one can expect to find one significant value on average even when the time series is random.**Short-term Correlation: Stationary series**often exhibit short-term correlation characterized by a fairly large value of $r_1$ followed by 2 or 3 more coefficients (significantly greater than zero) tend to get successively smaller values of $r_k$ for larger lags tend to get be approximately zero. A time series that gives rise to such a correlogram is one for which an observation above the mean tends to be followed by one or more further observations above the mean and similarly for observation below the mean. A model called an autoregressive model may be appropriate for a series of this type.**Alternating Series:**If a time series tends to alternate with successive observations on different sides of the overall mean, then the correlogram also tends to alternate. The value of $r_1$ will be negative, however, the value of $r_2$ will be positive as observation at lag 2 will tend to be on the same side of the mean.**Non-Stationary Series:**If a time series contains a trend, then the value of $r_k$ will not come down to zero except for very large values of the lags. This is because of a large number of further observations on the same side of the mean because of the trend. The sample autocorrelation function $\{ r_k \}$ should only be calculated for stationary time series and no trend should be removed before calculating $\{ r_k\}$.**Seasonal Fluctuations:**If a time series contains a**seasonal fluctuation**then the**correlogram**will also exhibit an oscillation at the same frequency. If $x_t$ follows a sinusoidal pattern then so does $r_k$.

$x_t=a\, cos\, t\, w, $ where $a$ is constant, $w$ is frequency such that $0 < w < \pi$. Therefore $r_k \cong cos\, k\, w$ for large $N$.

If the**seasonal variation**is removed from seasonal data then the**correlogram**may provide useful information.**Outliers:**If a time series contains one or more*outliers*the**correlogram**may be seriously affected. If there is one*outlier*in the time series and it is not adjusted, then the plot of $x_y$ vs $x_{t+k}$ will contain two extreme points, which will tend to depress the sample*correlation coefficients*towards zero. If there are two*outliers,*this effect is more noticeable.**General Remarks:**Experience is required to interpret*autocorrelation coefficients*. We need to study the probability theory of stationary series and the classes of model too. We also need to know the sampling properties of $x_t$.

AKARUE BLESSINGDr i ran an analysis were some data are stationary while other are not but somebody said i cant use both stationary and no stationary data what can i do, but at levels some data were stationary. what is your advice