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

Useful Methods
Let’s start with a moving average.
Moving Average Process
Chapter 6
Time Series
Suppose that {Z t } is a purely random process with mean 0 and variance σ z2 .
Then a process {X t } is said to be a moving average process of order q.
X = b Z + b Z + + b Z
t 0 t 1 t-1 q t-q
Here, {β i } are constants. The Zs are usually scaled so that β 0 = 1.
E( X )= 0
t
2 2
Var( X )= s b
t
q
å
Z
i=
0
i
The Zs are independent.
g ( k)= Cov ( X , X )
t t+
k
( t q t- q t+ k q t+ k-q)
= Cov b0Z + + b Z , b0Z + +
b Z
ì 0 k > q
q-k
ï 2
= ísZ
å bibi+
k
k= 01 ,, ¼,
q
ï i=
0
î
ï g (-k) k<
0
ìs Z
Cov ( 2
Zs,
Zt)=
í
î 0
s=
t
s¹
t
As γ(k) is not dependent on t and the mean is constant, the process is
second-order stationary for all values of {β i }.
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