Advanced Data Analytics Using Python_ With Machine Learning, Deep Learning and NLP Examples ( 2023)

Autocorrelation and the Correlogram
Chapter 6
Time Series
Quantities called sample autocorrelation coefficients act as an important
guide to the properties of a time series. They evaluate the correlation,
if any, between observations at different distances apart and provide
valuable descriptive information. You will see that they are also an
important tool in model building and often provide valuable clues for
a suitable probability model for a given set of data. The quantity lies in
the range [-1,1] and measures the forcefulness of the linear association
between the two variables. It can be easily shown that the value does
not depend on the units in which the two variables are measured; if the
variables are independent, then the ideal correlation is zero.
A helpful supplement in interpreting a set of autocorrelation
coefficients is a graph called a correlogram. The correlogram may be
alternatively called the sample autocorrelation function.
Suppose a stationary stochastic process X(t) has a mean μ, variance σ 2 ,
auto covariance function (acv.f.) γ(t), and auto correlation function (ac.f.) ρ(τ).
g t
rt ( )= ( ) g t s
g ( 0) = ( ) /
2
Estimating Autocovariance and Autocorrelation
Functions
In the stochastic process, the autocovariance is the covariance of the
process with itself at pairs of time points. Autocovariance is calculated as
follows:
n-
h
1
g ( h)= å ( x -x x x n h n
t h )( t
- ),
- < <
+
n
t = 1
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