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

Chapter 6
Time Series
Now, E[X(t)] is the expected value of X(t).
Covariance(X(t),X(t+h)) = E[(X(t) - E[X(t)]) * (X(t+h) – E[X(t+h)])]
= E[(X(t) - m) * (X(t+h) - m)]
If X(t) is a weak stationary process, then:
E[X(t)] = E[X(t+h)] = m (constant)
= E[X(t) * X(t+h)] – m 2 = c(h)
Here, m is constant, and cov[X(t),X(t+h)] is the function of only h(c(h))
for the weakly stationary process. c(h) is known as autocovariance.
Similarly, the correlation (X(t),X(t+h) = ρ(h) = r(h) = c(h) = ¸ c(0) is
known as autocorrelation.
If X(t) is a stationary process that is realized as an autoregressive
model, then:
X(t) = a1 * X(t-1) + a2 * X(t-2) + ….. + ap * X(t-p) + Z(t)
Correlation(X(t),X(t)) = a1 * correlation (X(t),X(t-1)) + …. +
ap * correlation (X(t),X(t-p))+0
As covariance, (X(t),X(t+h)) is dependent only on h, so:
r0 = a1 * r1 + a2 * r2 + … + ap * rp
r1 = a1 * r0 + a2 * r1 + …. + ap * r(p-1)
So, for an n-order model, you can easily generate the n equation and
from there find the n coefficient by solving the n equation system.
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