Autocorrelation

10. Autocorrelation#

10.1. Autocorrelation#

In time series data, you can use autocorrelation (Box and Jenkins, 1976) calculations to:

  • To detect non-randomness in data.

  • To identify an appropriate time series model if the data are not random.

Non-randomness is common in time series data and other non-linear models. Random data is an assumption of many statistical tools. In this case, randomness is \(Y_i = A_0(model) + E_i\) where \(E_i\) is the error/noise term.

10.1.1. Autocorrelation in Practice#

Individual autocorrelation

import numpy as np
import pandas as pd
rng = np.random.default_rng()

ran = rng.random(size=(50,))
time = np.sin(np.arange(50))


ran = pd.Series(ran)
time = pd.Series(time)

ran.autocorr()
time.autocorr()

ran.autocorr(lag=10)

From the pd.Series.autocorr documentation:

This method computes the Pearson correlation between the Series and its shifted self.

The full collection of lags

import numpy as np
rng = np.random.default_rng()

ran = rng.random(size=(50,))
time = np.sin(np.arange(50))

import statsmodels.api as sm
sm.tsa.acf(ran, nlags = 10)
sm.tsa.acf(time, nlags = 10)

10.1.2. Autocorrelation of Random vs Time-Dependent Variables#

import numpy as np
import pandas as pd
rng = np.random.default_rng()

ran = pd.Series(rng.random(size=(50,)))
time = pd.Series(np.sin(np.arange(50)))


for i in range(len(ran)):
    print(ran.autocorr(lag=i))

for i in range(len(time)):  
    print(time.autocorr(lag=i))


from statsmodels.graphics.tsaplots import plot_acf
import matplotlib.pyplot as plt

plot_acf(ran, lags=20)
plt.show()

plot_acf(time, lags=20)
plt.show()

10.2. Suggested Reading#

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