Types of Autocorrelation

Autocorrelation: 

As we discussed in this article, Autocorrelation is defined as the measure of the degree of similarity between a given time series and the lagged version of that time series over successive time periods. Autocorrelation measures the degree of similarity between a time series and the lagged version of that time series at different intervals.

Autocorrelation Function:

Suppose we have a time series {X_t} which has the following mean:

and the autocovariance functions

at t=0,

and the autocorrelation function is defined as:

The value of autocorrelation varies from -1 for perfectly negative autocorrelation and 1 for perfectly positive autocorrelation. The value closer to 0 is referred to as no autocorrelation.

Positive Autocorrelation:

Positive autocorrelation occurs when an error of a given sign between two values of time series lagged by k followed by an error of the same sign.

Below is the graph of the dataset that represents positive autocorrelation at lag=1:

Negative Autocorrelation:

Negative autocorrelation occurs when an error of a given sign between two values of time series lagged by k followed by an error of the different sign.

Below is the graph of time series that represents negative autocorrelation at lag=1:

Strong Autocorrelation

We can conclude that the data have strong autocorrelation if the autocorrelation plot has similar to the following plots:

The autocorrelation plot starts with a very high autocorrelation at lag 1 but slowly declines until it becomes negative and starts showing an increasing negative autocorrelation. This type of pattern indicates a strong autocorrelation, which can be helpful in predicting future trends

 The next step would be to estimate the parameters for the autoregressive model:

The randomness assumption for least-squares fitting applies to the residuals of the model. That is, even though the original data exhibit non-randomness, the residuals after fitting Y_i against Y_i-1 should result in random residuals.

Weak Autocorrelation

We can conclude that the data have weak autocorrelation if the autocorrelation plot has similar to the following plot at lag = 1:

Lag plot at Lag =1

The above plot shows that there is some autocorrelation at lag=1 because if there is no autocorrelation the plot will be similar to this plot on random values with lag=1

The conclusion can be drawn from the above plot

• An underlying autoregressive model with moderate positive/negative autocorrelation.
• There were very few outliers.

The above weak autocorrelation plot have some autoregressive model that can be represented in such a form 

at Y_i =0, we can obtain the residual of estimators.

It is easy to perform estimation on the lag plot because of the Y_i+1 and Y_i as their axes.

Implementation

• In this implementation, we will be looking on how to generate correlation plots and lag plots. We will use the flicker dataset and some randomly generated samples for this purpose.

python3

# Necessary imports
import numpy as np
from numpy.random import random_sample
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
from statsmodels.graphics.tsaplots import plot_acf

# Generate Autocorrelation plot at different lags
# with a given level of significance.
weak_Corr_df = pd.read_csv('flicker.csv', sep ='\n', header=None)
plot_acf(weak_Corr_df, alpha = 0.05)

# Generate Lag plots for a particular lag value
pd.plotting.lag_plot(weak_Corr_df, lag = 1)

# Generate 200 random numbers and plot lag plot and autocorrelation plot for that
random_Series = pd.Series(random_sample(200))
pd.plotting.lag_plot(random_Series, lag = 1)
plot_acf(random_Series, alpha = 0.05)

• For Flicker dataset, the plots are as follows:

Autocorrelation Flicker data

Lag plot for flicker data

• For Random Normal datasets, the plots are as follows

Autocorrelation plot

Lag plot for random data at lag=1