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Seasonal sub series plot

For a known periodicity of seasonal variations, seasonal sub series redraws the original series over batches of successive time periods. For example, the periodicity in the CO₂ concentrations is 12 months and based on this a seasonal sub series plots on mean and standard deviation of the residuals are shown in the following figure. To visualize seasonality in the residuals, we create quarterly mean and standard deviations.

A seasonal sub series reveals two properties:

Variations within seasons as within a batch of successive months
Variations between seasons as between batches of successive months

Figure 1.8: Quarterly mean of the residuals from a linear model of the general trend in CO2 readings

Figure 1.9: Quarterly standard deviation of the residuals from a linear model of the general trend in CO2 readings

Let us now describe the code used for generating the preceding plots. First, we need to add the residuals and quarter labels to the CO2 concentrations DataFrame. This is done as follows:

data['Residuals'] = residuals 
month_quarter_map = {1: 'Q1', 2: 'Q1', 3: 'Q1', 
                     4: 'Q2', 5: 'Q2', 6: 'Q2', 
                     7: 'Q3', 8: 'Q3', 9: 'Q3', 
                     10: 'Q4', 11: 'Q4', 12: 'Q4' 
                    } 
data['Quarter'] = data['Month'].map(lambda m: month_quarter_map.get(m))

Next, the seasonal mean and standard deviations are computed by grouping by the data over Year and Quarter:

seasonal_sub_series_data = data.groupby(by=['Year', 'Quarter'])['Residuals']\ 
                           .aggregate([np.mean, np.std])

This creates the new DataFrame as seasonal_sub_series_data, which has quarterly mean and standard deviations over the years. These columns are renamed as follows:

seasonal_sub_series_data.columns = ['Quarterly Mean', 'Quarterly Standard Deviation']

Next, the quarterly mean and standard deviations are plotted by running the following lines of code:

#plot quarterly mean of residuals
plt.figure(figsize=(5.5, 5.5))
seasonal_sub_series_data['Quarterly Mean'].plot(color='b')
plt.title('Quarterly Mean of Residuals')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

#plot quarterly standard deviation of residuals
plt.figure(figsize=(5.5, 5.5))
seasonal_sub_series_data['Quarterly Standard Deviation'].plot(color='b')
plt.title('Quarterly Standard Deviation of Residuals')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)