In [2]:

    

from __future__ import print_function
import os
import pandas as pd
import numpy as np
from statsmodels.tsa import stattools
from statsmodels.tsa import seasonal
%matplotlib inline
from matplotlib import pyplot as plt


    

In [5]:

    

"""
This notebook illustrates time series decomposition by the statsmodels package.
Both additive and multiplicative models are demonstrated.
"""


    

    
Out[5]:





'\nThis notebook illustrates time series decomposition by the statsmodels package.\nBoth additive and multiplicative models are demonstrated.\n'

In [6]:

    

"""
Let us demonstrate the addtive model using Wisconsin Employment
Jan. 1961 – OCt. 1975 dataset.
"""


    

    
Out[6]:





'\nLet us demonstrate the addtive model using Wisconsin Employment\nJan. 1961 – OCt. 1975 dataset.\n'

In [12]:

    

#read the data from into a pandas.DataFrame
wisc_emp = pd.read_csv('datasets/wisconsin-employment-time-series.csv')
wisc_emp.index = wisc_emp['Month']


    

In [13]:

    

#Let's find out the shape of the DataFrame
print('Shape of the DataFrame:', wisc_emp.shape)


    

    




Shape of the DataFrame: (178, 2)

In [14]:

    

#Let's see first 10 rows of it
wisc_emp.head(10)


    

    
Out[14]:







  

    

      

      
Month
      
Employment
    
    

      
Month
      

      

    
  
  

    

      
1961-01
      
1961-01
      
239.6
    
    

      
1961-02
      
1961-02
      
236.4
    
    

      
1961-03
      
1961-03
      
236.8
    
    

      
1961-04
      
1961-04
      
241.5
    
    

      
1961-05
      
1961-05
      
243.7
    
    

      
1961-06
      
1961-06
      
246.1
    
    

      
1961-07
      
1961-07
      
244.1
    
    

      
1961-08
      
1961-08
      
244.2
    
    

      
1961-09
      
1961-09
      
244.8
    
    

      
1961-10
      
1961-10
      
246.6
    
  

In [15]:

    

#Check for missing values and remove the row
missing = (pd.isnull(wisc_emp['Employment'])) | (pd.isnull(wisc_emp['Month']))
print('Number of missing values found:', missing.sum())
wisc_emp = wisc_emp.loc[~missing, :]


    

    




Number of missing values found: 0

In [16]:

    

#Run ADF test on the original time series
adf_result = stattools.adfuller(wisc_emp['Employment'], autolag='AIC')


    

In [17]:

    

print('p-val of the ADF test on irregular variations in employment data:', adf_result[1])


    

    




p-val of the ADF test on irregular variations in employment data: 0.981000018954

In [18]:

    

decompose_model = seasonal.seasonal_decompose(wisc_emp.Employment.tolist(), freq=12,
                                              model='additive')


    

In [29]:

    

#Plot the original time series, trend, seasonal and random components
fig, axarr = plt.subplots(4, sharex=True)
fig.set_size_inches(5.5, 5.5)

wisc_emp['Employment'].plot(ax=axarr[0], color='b', linestyle='-')
axarr[0].set_title('Monthly Employment')

pd.Series(data=decompose_model.trend, index=wisc_emp.index).plot(color='r', linestyle='-', ax=axarr[1])
axarr[1].set_title('Trend component in monthly employment')

pd.Series(data=decompose_model.seasonal, index=wisc_emp.index).plot(color='g', linestyle='-', ax=axarr[2])
axarr[2].set_title('Seasonal component in monthly employment')

pd.Series(data=decompose_model.resid, index=wisc_emp.index).plot(color='k', linestyle='-', ax=axarr[3])
axarr[3].set_title('Irregular variations in monthly employment')

plt.tight_layout(pad=0.4, w_pad=0.5, h_pad=2.0)
plt.xticks(rotation=10)

plt.savefig('plots/ch2/B07887_02_22.png', format='png', dpi=300)


    

In [13]:

    

#Run ADF test on the irregular variations
adf_result = stattools.adfuller(decompose_model.resid[np.where(np.isfinite(decompose_model.resid))[0]],
                                autolag='AIC')


    

In [14]:

    

print('p-val of the ADF test on irregular variations in employment data:', adf_result[1])


    

    




p-val of the ADF test on irregular variations in employment data: 0.00656093163464

In [15]:

    

"""
The additive decompostion has been able to reduce the p-value
from 0.98 in case of the original time series
to 0.066 after decomposing.
"""


    

    
Out[15]:





'\nThe additive decompostion has been able to reduce the p-value\nfrom 0.98 in case of the original time series\nto 0.066 after decomposing.\n'

In [16]:

    

"""
Now we will attempt decomposition of the original time
using a multiplicative model
"""


    

    
Out[16]:





'\nNow we will attempt decomposition of the original time\nusing a multiplicative model\n'

In [17]:

    

decompose_model = seasonal.seasonal_decompose(wisc_emp.Employment.tolist(), freq=12,
                                              model='multiplicative')


    

In [18]:

    

#Plot the original time series, trend, seasonal and random components
fig, axarr = plt.subplots(4, sharex=True)
fig.set_size_inches(5.5, 5.5)

wisc_emp['Employment'].plot(ax=axarr[0], color='b', linestyle='-')
axarr[0].set_title('Monthly Employment')

axarr[1].plot(decompose_model.trend, color='r', linestyle='-')
axarr[1].set_title('Trend component in monthly employment')

axarr[2].plot(decompose_model.seasonal, color='g', linestyle='-')
axarr[2].set_title('Seasonal component in monthly employment')

axarr[3].plot(decompose_model.resid, color='k', linestyle='-')
axarr[3].set_title('Irregular variations in monthly employment')

plt.savefig('plots/ch2/B07887_02_23.png', format='png', dpi=300)


    

In [19]:

    

#Run ADF test on the irregular variations
adf_result = stattools.adfuller(decompose_model.resid[np.where(np.isfinite(decompose_model.resid))[0]],
                                autolag='AIC')


    

In [20]:

    

print('p-val of the ADF test on irregular variations in employment data:', adf_result[1])


    

    




p-val of the ADF test on irregular variations in employment data: 0.00123478372677

In [21]:

    

"""
Voila! The p-val has further reduced 0.00123.
The null hypothesis about non-stationarity of the irregular variations
can be rejected at even a level of confidence of 99 % (alpha=0.01).
This shows that the original time series has been de-stationarized to
the stationary irregular variations. Besides we have estimates of both trend-cycle
and seasonal components.
"""


    

    
Out[21]:





'\nVoila! The p-val has further reduced 0.00123.\nThe null hypothesis about non-stationarity of the irregular variations\ncan be rejected at even a level of confidence of 99 % (alpha=0.01).\nThis shows that the original time series has been de-stationarized to\nthe stationary irregular variations. Besides we have estimates of both trend-cycle\nand seasonal components.\n'