Machine learning notes - Introduction to autocorrelation and partial autocorrelation

1、 summary  

         Autocorrelation and partial autocorrelation graphs are widely used in time series analysis and prediction .

         These charts graphically summarize the strength of the relationship with observations in the time series and observations in previous time steps . For beginners of time series prediction , The difference between autocorrelation and partial autocorrelation can be confusing .

         We will learn how to use Python Calculate and draw autocorrelation and partial correlation diagrams . And understand the following

         How to draw and view the autocorrelation function of time series .

         How to plot and view the partial autocorrelation function of time series .

         Difference between autocorrelation function and partial autocorrelation function used in time series analysis .

2、 Minimum daily temperature data set

         This data set describes the city of Melbourne, Australia 10 year （1981-1990 year ） Minimum daily temperature of .

         The unit is centigrade , Yes 3,650 An observation . The data source is Australian meteorological agency .

        Dataset download address

 link ：https://pan.baidu.com/s/19G9YsOKtRDXNYAKdOwBq6A 
 Extraction code ：ateb

        Now load the minimum daily temperature and plot the time series .

from pandas import read_csv
from matplotlib import pyplot
series = read_csv('daily-minimum-temperatures.csv', header=0, index_col=0)
series.plot()
pyplot.show()

         Run this example to load the dataset as Pandas Series and create a line graph of the time series .

Minimum daily temperature data set diagram

3、 Correlation and autocorrelation

         Correlation summarizes the strength of the relationship between two variables . We can assume that the distribution of each variable is Gaussian （ A bell curve ） Distribution . If that's the case , We can use Pearson correlation coefficient to summarize the correlation between variables .

        Pearson The correlation coefficient is between -1 and 1 Number between , Describe negative or positive correlation respectively . A value of zero indicates no correlation .

         We can calculate the correlation between time series observations and previous time step observations , It is called hysteresis . Because the correlation of time series observations is calculated using the values of previous time series of the same time , So this is called sequence correlation or autocorrelation .

         The lag graph of time series autocorrelation is called AutoCorrelation Function, Or acronyms ACF. This graph is sometimes called a correlation graph or an autocorrelation graph .

         Here's how to use statsmodels In the library plot_acf() Function to calculate and plot the autocorrelation diagram of the minimum daily temperature .

from pandas import read_csv
from matplotlib import pyplot
from statsmodels.graphics.tsaplots import plot_acf
series = read_csv('daily-minimum-temperatures.csv', header=0, index_col=0)
plot_acf(series)
pyplot.show()

         Running the example creates a 2D chart , Display edge x The hysteresis value of the shaft and at -1 and 1 Between y Correlation on the axis .

         The confidence interval is plotted as a cone . By default , This is set to 95% The confidence interval of , This indicates that correlation values outside this code are likely to be correlations rather than statistical flukes .

Autocorrelation plot of minimum daily temperature data set

          By default , All hysteresis values will be printed , This makes the drawing noisy . We can x The number of lags on the shaft is limited to 50, To make the drawing easier to read .

Autocorrelation chart with less lag for the minimum daily temperature data set

4、 Partial autocorrelation function

         Partial autocorrelation is a summary of the relationship between observations in time series and observations in previous time steps , The relationship between intervention and observation is deleted .

         The autocorrelation between the observed value and the observed value of the previous time step consists of direct correlation and indirect correlation . These indirect correlations are linear functions of observed correlations , Observe on the intervention time step .

         It is these indirect correlations that the partial autocorrelation function attempts to eliminate .

         The following example uses statsmodels In the library plot_pacf() The minimum daily temperature data set is calculated and plotted 50 Partial autocorrelation function with time delay .

from pandas import read_csv
from matplotlib import pyplot
from statsmodels.graphics.tsaplots import plot_pacf
series = read_csv('daily-minimum-temperatures.csv', header=0, index_col=0)
plot_pacf(series, lags=50)
pyplot.show()

Partial autocorrelation of the lowest daily temperature data set