ARIMA Model
November 24, 2020
[2]: import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
import statsmodels.api as sm
from statsmodels.graphics.tsaplots import plot_acf,plot_pacf
from statsmodels.tsa.arima_model import ARIMA
from statsmodels.tsa.stattools import adfuller
from pandas.tseries.offsets import DateOffset
[3]: df=pd.read_csv(r'C:\Users\Asus\Desktop\OPM_Assignment Data.csv')
df = df.rename(columns={ "Date": "date", ' Sales ': "sales"})
[4]: df['sales'] = df['sales'].astype(int)
df['date']=pd.to_datetime(df['date'])
[5]: df.set_index('date',inplace=True)
[6]: df.plot()
[6]: <matplotlib.axes._subplots.AxesSubplot at 0x24358badb20>
1
Testing For Stationarity
[7]: test_result = adfuller(df['sales'])
[8]: #Ho: It is non stationary
#H1: It is stationary
def adfuller_test(sales):
result=adfuller(sales)
labels = ['ADF Test Statistic','p-value','#Lags Used','Number of␣
,→Observations Used']
for value,label in zip(result,labels):
print(label+' : '+str(value) )
if result[1] <= 0.05:
print("strong evidence against the null hypothesis(Ho), reject the null␣
,→hypothesis. Data has no unit root and is stationary")
else:
print("weak evidence against null hypothesis, time series has a unit␣
,→root, indicating it is non-stationary ")
[9]: adfuller_test(df['sales'])
ADF Test Statistic : -3.5804829030383365
p-value : 0.0061453039375673
2
#Lags Used : 5
Number of Observations Used : 27
strong evidence against the null hypothesis(Ho), reject the null hypothesis.
Data has no unit root and is stationary
Identification of an AR model is often best done with the PACF. For an AR model, the
theoretical PACF “shuts off” past the order of the model. The phrase “shuts off” means that in
theory the partial autocorrelations are equal to 0 beyond that point. Put another way, the number
of non-zero partial autocorrelations gives the order of the AR model. By the “order of the model”
we mean the most extreme lag of x that is used as a predictor. Identification of an MA model is
often best done with the ACF rather than the PACF.
Identification of an MA model is often best done with the ACF. For an MA model, the
theoretical PACF does not shut off, but instead tapers toward 0 in some manner. A clearer pattern
for an MA model is in the ACF. The ACF will have non-zero autocorrelations only at lags involved
in the model.
p,d,q p AR model lags d differencing q MA lags
[10]: fig
ax1
fig
ax2
fig
=
=
=
=
=
plt.figure(figsize=(12,8))
fig.add_subplot(211)
sm.graphics.tsa.plot_acf(df['sales'],ax=ax1) #.iloc[13:],lags=40
fig.add_subplot(212)
sm.graphics.tsa.plot_pacf(df['sales'],ax=ax2) #.iloc[13:],lags=40
3
[11]: # For non-seasonal data
#p=1, d=1, q= 0 or 1
ARIMA Model
[12]: model=ARIMA(df['sales'],order=(1,1,0))
model_fit=model.fit()
C:\Users\Asus\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:159:
ValueWarning: No frequency information was provided, so inferred frequency MS
will be used.
warnings.warn('No frequency information was'
C:\Users\Asus\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:159:
ValueWarning: No frequency information was provided, so inferred frequency MS
will be used.
warnings.warn('No frequency information was'
[13]: model_fit.summary()
[13]: <class 'statsmodels.iolib.summary.Summary'>
"""
ARIMA Model Results
==============================================================================
Dep. Variable:
D.sales
No. Observations:
32
Model:
ARIMA(1, 1, 0)
Log Likelihood
-453.176
Method:
css-mle
S.D. of innovations
342079.278
Date:
Tue, 24 Nov 2020
AIC
912.352
Time:
22:38:31
BIC
916.749
Sample:
02-01-2011
HQIC
913.810
- 09-01-2013
================================================================================
=
coef
std err
z
P>|z|
[0.025
0.975]
-------------------------------------------------------------------------------2.544e+04
5.91e+04
0.667
-9.04e+04
const
0.430
1.41e+05
ar.L1.D.sales
-0.0289
0.187
-0.154
0.878
-0.396
0.339
Roots
=============================================================================
Real
Imaginary
Modulus
Frequency
----------------------------------------------------------------------------AR.1
-34.6596
+0.0000j
34.6596
0.5000
----------------------------------------------------------------------------"""
4
[14]: df['forecast']=model_fit.predict(start=24,end=33,dynamic=True)
df[['sales','forecast']].plot(figsize=(12,8))
[14]: <matplotlib.axes._subplots.AxesSubplot at 0x24359871580>
ARIMA yields unsatisfatory result. So, we try with SARIMA
SARIMA Model
[15]: model=sm.tsa.statespace.SARIMAX(df['sales'],order=(1, 1, 1),seasonal_order=(1,␣
,→1, 1,12))
results=model.fit()
C:\Users\Asus\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:159:
ValueWarning: No frequency information was provided, so inferred frequency MS
will be used.
warnings.warn('No frequency information was'
C:\Users\Asus\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:159:
ValueWarning: No frequency information was provided, so inferred frequency MS
will be used.
warnings.warn('No frequency information was'
C:\Users\Asus\anaconda3\lib\sitepackages\statsmodels\tsa\statespace\sarimax.py:866: UserWarning: Too few
5
observations to estimate starting parameters for seasonal ARMA. All parameters
except for variances will be set to zeros.
warn('Too few observations to estimate starting parameters%s.'
[35]: df['forecast']=results.predict(start=24,end=33,dynamic=True)
df[['sales','forecast']].plot(figsize=(12,8))
[35]: <matplotlib.axes._subplots.AxesSubplot at 0x2435b5d4160>
[17]: future_dates=[df.index[-1]+ DateOffset(months=x)for x in range(0,24)]
[18]: future_datest_df=pd.DataFrame(index=future_dates[1:],columns=df.columns)
[19]: future_df=pd.concat([df,future_datest_df])
[30]: future_df['forecast'] = results.predict(start = 24, end = 38, dynamic= True)
future_df[['sales', 'forecast']].plot(figsize=(12, 8))
[30]: <matplotlib.axes._subplots.AxesSubplot at 0x2435b508b80>
6
[31]: future_df[:-17]
[31]:
2011-01-01
2011-02-01
2011-03-01
2011-04-01
2011-05-01
2011-06-01
2011-07-01
2011-08-01
2011-09-01
2011-10-01
2011-11-01
2011-12-01
2012-01-01
2012-02-01
2012-03-01
2012-04-01
2012-05-01
2012-06-01
2012-07-01
2012-08-01
sales
3240325
2741349
2987427
3456892
3740738
3979178
4160454
4162013
3809132
3794419
3719219
3812981
3480451
3183133
3764529
3500189
3389811
4348789
4442455
4593383
forecast
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
7
2012-09-01
2012-10-01
2012-11-01
2012-12-01
2013-01-01
2013-02-01
2013-03-01
2013-04-01
2013-05-01
2013-06-01
2013-07-01
2013-08-01
2013-09-01
2013-10-01
2013-11-01
2013-12-01
2014-01-01
2014-02-01
2014-03-01
4029783
4211211
3854682
3554831
3488309
3270444
3709943
3655530
4097990
4472583
4531711
4504466
4054338
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
3.703144e+06
3.455306e+06
3.884572e+06
3.753771e+06
3.708422e+06
4.393622e+06
4.477484e+06
4.581709e+06
4.157130e+06
4.282700e+06
4.036895e+06
3.864442e+06
4.218167e+06
3.985965e+06
4.339405e+06
[32]: df.to_excel("Sarima_output.xlsx")
[33]: future_df[:-17].to_excel("future_forecast.xlsx")
8