look at decompositional element.

ARIMA stands for AutoregRessive Integrated Moving Average, and it's a common method to model time series data where there is dependence among temporal values. SARIMA is a similar method that adds seasonality element to ARIMA. As shown below, user needs to specify some parameters to fit an ARIMA model within the Python statsmodel package (version 0.8). The ARIMA parameters are (p,d,q):

p - the auto-regression term that comprises p number of past values to predict present value. For example, if p=2, then past values y(t-1), y(t-2) would be used to predict y(t).
d - the integrated part of the model. Generally d=1, corresponding to the difference between current value and previous one. If d >1, it means the differencing would be performed more than once (i.e., difference of prior d number of values and present value).
q - the Moving Average terms, which is used to generate the error terms of the model. This results in a linear combination of errors for the prior q data point, where each error is defined as the difference between the moving average value and the actual value at a given time point (t).

For SARIMA model, there's also (P,D,Q,S) parameters specified along with the (p,d,q). The P,D,Q values are similar to the parameters described above but it's applied to the seasonality componenet of the SARIMA model. S is the periodicity of the time series (4 for quarterly, 12 for yearly).

Source of reference: https://www.digitalocean.com/community/tutorials/a-guide-to-time-series-forecasting-with-arima-in-python-3

import warnings
import itertools
import pandas as pd
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
%matplotlib inline

#show datasets avaialble
dir(sm.datasets)

['__builtins__',
 '__cached__',
 '__doc__',
 '__file__',
 '__loader__',
 '__name__',
 '__package__',
 '__path__',
 '__spec__',
 'anes96',
 'cancer',
 'ccard',
 'check_internet',
 'china_smoking',
 'clear_data_home',
 'co2',
 'committee',
 'copper',
 'cpunish',
 'elnino',
 'engel',
 'fair',
 'fertility',
 'get_data_home',
 'get_rdataset',
 'grunfeld',
 'heart',
 'longley',
 'macrodata',
 'modechoice',
 'nile',
 'randhie',
 'scotland',
 'spector',
 'stackloss',
 'star98',
 'statecrime',
 'strikes',
 'sunspots',
 'utils',
 'webuse']

nile= sm.datasets.nile.load_pandas()

#load the dataset into Dataframe
data=nile.data

data.volume.values

array([ 1120.,  1160.,   963.,  1210.,  1160.,  1160.,   813.,  1230.,
        1370.,  1140.,   995.,   935.,  1110.,   994.,  1020.,   960.,
        1180.,   799.,   958.,  1140.,  1100.,  1210.,  1150.,  1250.,
        1260.,  1220.,  1030.,  1100.,   774.,   840.,   874.,   694.,
         940.,   833.,   701.,   916.,   692.,  1020.,  1050.,   969.,
         831.,   726.,   456.,   824.,   702.,  1120.,  1100.,   832.,
         764.,   821.,   768.,   845.,   864.,   862.,   698.,   845.,
         744.,   796.,  1040.,   759.,   781.,   865.,   845.,   944.,
         984.,   897.,   822.,  1010.,   771.,   676.,   649.,   846.,
         812.,   742.,   801.,  1040.,   860.,   874.,   848.,   890.,
         744.,   749.,   838.,  1050.,   918.,   986.,   797.,   923.,
         975.,   815.,  1020.,   906.,   901.,  1170.,   912.,   746.,
         919.,   718.,   714.,   740.])

tp=[]

for i in range(len(data)):
    
        timepoint=data.iloc[i]['year'].astype(int).astype(str) + '-'+'01-01'
        tp.append(timepoint)

vol_data=pd.DataFrame({'Volume':data.volume.values}, index=pd.to_datetime(tp))

vol_data.head()

	Volume
1871-01-01	1120.0
1872-01-01	1160.0
1873-01-01	963.0
1874-01-01	1210.0
1875-01-01	1160.0

vol_data.plot(figsize=(15, 6))
plt.show()

look at decompositional element.

from statsmodels.tsa.seasonal import seasonal_decompose

#take the log of the volumne data to make it more of a normal distribution
decomposition = seasonal_decompose(np.log(vol_data))
trend = decomposition.trend
seasonal = decomposition.seasonal
residual = decomposition.resid

plt.subplot(411)
plt.plot(np.log(vol_data), label='Original')
plt.legend(loc='best')
plt.subplot(412)
plt.plot(trend, label='Trend')
plt.legend(loc='best')
plt.subplot(413)
plt.plot(seasonal,label='Seasonality')
plt.legend(loc='best')
plt.subplot(414)
plt.plot(residual, label='Residuals')
plt.legend(loc='best')
plt.tight_layout()

No seasonality detected. To find optimal parameters for sarimax time series model, set up a grid search.

p = d = q = range(0, 2)

# Generate all different combinations of p, q and q triplets
pdq = list(itertools.product(p, d, q))

# Generate all different combinations of seasonal p, q and q triplets
seasonal_pdq = [(x[0], x[1], x[2], 12) for x in list(itertools.product(p, d, q))]

print('Examples of parameter combinations for Seasonal ARIMA...')
print('SARIMAX: {} x {}'.format(pdq[1], seasonal_pdq[1]))
print('SARIMAX: {} x {}'.format(pdq[1], seasonal_pdq[2]))
print('SARIMAX: {} x {}'.format(pdq[2], seasonal_pdq[3]))
print('SARIMAX: {} x {}'.format(pdq[2], seasonal_pdq[4]))

Examples of parameter combinations for Seasonal ARIMA...
SARIMAX: (0, 0, 1) x (0, 0, 1, 12)
SARIMAX: (0, 0, 1) x (0, 1, 0, 12)
SARIMAX: (0, 1, 0) x (0, 1, 1, 12)
SARIMAX: (0, 1, 0) x (1, 0, 0, 12)

warnings.filterwarnings("ignore") # specify to ignore warning messages

for param in pdq:
    for param_seasonal in seasonal_pdq:
        try:
            mod = sm.tsa.statespace.SARIMAX(np.log(vol_data),
                                            order=param,
                                            seasonal_order=param_seasonal,
                                            enforce_stationarity=False,
                                            enforce_invertibility=False)

            results = mod.fit()

            print('ARIMA{}x{}12 - AIC:{}'.format(param, param_seasonal, results.aic))
        except:
            continue

ARIMA(0, 0, 0)x(0, 0, 1, 12)12 - AIC:500.3253707784356
ARIMA(0, 0, 0)x(0, 1, 1, 12)12 - AIC:-15.002974230655422
ARIMA(0, 0, 0)x(1, 0, 0, 12)12 - AIC:-10.845892328459849
ARIMA(0, 0, 0)x(1, 0, 1, 12)12 - AIC:-30.436467640234355
ARIMA(0, 0, 0)x(1, 1, 0, 12)12 - AIC:-11.555765139808228
ARIMA(0, 0, 0)x(1, 1, 1, 12)12 - AIC:-15.858251913484871
ARIMA(0, 0, 1)x(0, 0, 0, 12)12 - AIC:531.888905163303
ARIMA(0, 0, 1)x(0, 0, 1, 12)12 - AIC:386.81945792953263
ARIMA(0, 0, 1)x(0, 1, 0, 12)12 - AIC:-11.153708233957529
ARIMA(0, 0, 1)x(0, 1, 1, 12)12 - AIC:-24.96722312324776
ARIMA(0, 0, 1)x(1, 0, 0, 12)12 - AIC:-12.820225589303263
ARIMA(0, 0, 1)x(1, 0, 1, 12)12 - AIC:-7.2492131661396435
ARIMA(0, 0, 1)x(1, 1, 0, 12)12 - AIC:-17.44245615456768
ARIMA(0, 0, 1)x(1, 1, 1, 12)12 - AIC:-21.662877329308497
ARIMA(0, 1, 0)x(0, 0, 1, 12)12 - AIC:-37.555668300937036
ARIMA(0, 1, 0)x(0, 1, 1, 12)12 - AIC:-15.491188733740213
ARIMA(0, 1, 0)x(1, 0, 0, 12)12 - AIC:-38.748298067057696
ARIMA(0, 1, 0)x(1, 0, 1, 12)12 - AIC:-35.615560770148
ARIMA(0, 1, 0)x(1, 1, 0, 12)12 - AIC:5.698988977219167
ARIMA(0, 1, 0)x(1, 1, 1, 12)12 - AIC:-7.9120698368811215
ARIMA(0, 1, 1)x(0, 0, 0, 12)12 - AIC:-72.30712907240154
ARIMA(0, 1, 1)x(0, 0, 1, 12)12 - AIC:-59.2018114019037
ARIMA(0, 1, 1)x(0, 1, 0, 12)12 - AIC:-7.1549214144135735
ARIMA(0, 1, 1)x(0, 1, 1, 12)12 - AIC:-31.693469023329676
ARIMA(0, 1, 1)x(1, 0, 0, 12)12 - AIC:-62.09696091264061
ARIMA(0, 1, 1)x(1, 0, 1, 12)12 - AIC:-57.30997339892333
ARIMA(0, 1, 1)x(1, 1, 0, 12)12 - AIC:-15.403997672325985
ARIMA(0, 1, 1)x(1, 1, 1, 12)12 - AIC:-24.344734233928854
ARIMA(1, 0, 0)x(0, 0, 0, 12)12 - AIC:-43.18510110735455
ARIMA(1, 0, 0)x(0, 0, 1, 12)12 - AIC:-36.835292306451684
ARIMA(1, 0, 0)x(0, 1, 0, 12)12 - AIC:-13.837649086824964
ARIMA(1, 0, 0)x(0, 1, 1, 12)12 - AIC:-33.74128886813366
ARIMA(1, 0, 0)x(1, 0, 0, 12)12 - AIC:-36.878147402122934
ARIMA(1, 0, 0)x(1, 0, 1, 12)12 - AIC:-34.640908950086846
ARIMA(1, 0, 0)x(1, 1, 0, 12)12 - AIC:-19.591953600320952
ARIMA(1, 0, 0)x(1, 1, 1, 12)12 - AIC:-27.43260097682949
ARIMA(1, 0, 1)x(0, 0, 0, 12)12 - AIC:-71.71546557549023
ARIMA(1, 0, 1)x(0, 0, 1, 12)12 - AIC:-58.93520789776957
ARIMA(1, 0, 1)x(0, 1, 0, 12)12 - AIC:-13.776962364364945
ARIMA(1, 0, 1)x(0, 1, 1, 12)12 - AIC:-34.53626801988043
ARIMA(1, 0, 1)x(1, 0, 0, 12)12 - AIC:-60.563492106138185
ARIMA(1, 0, 1)x(1, 0, 1, 12)12 - AIC:-54.0016479601604
ARIMA(1, 0, 1)x(1, 1, 0, 12)12 - AIC:-19.47450504132724
ARIMA(1, 0, 1)x(1, 1, 1, 12)12 - AIC:-28.71991461156157
ARIMA(1, 1, 0)x(0, 0, 0, 12)12 - AIC:-61.00886709444421
ARIMA(1, 1, 0)x(0, 0, 1, 12)12 - AIC:-53.55516667463787
ARIMA(1, 1, 0)x(0, 1, 0, 12)12 - AIC:0.3994338987208228
ARIMA(1, 1, 0)x(0, 1, 1, 12)12 - AIC:-24.569456994212924
ARIMA(1, 1, 0)x(1, 0, 0, 12)12 - AIC:-53.76698709887258
ARIMA(1, 1, 0)x(1, 0, 1, 12)12 - AIC:-51.87917735635104
ARIMA(1, 1, 0)x(1, 1, 0, 12)12 - AIC:-5.933834326115097
ARIMA(1, 1, 0)x(1, 1, 1, 12)12 - AIC:-19.112458940846267
ARIMA(1, 1, 1)x(0, 0, 0, 12)12 - AIC:-73.57350719870367
ARIMA(1, 1, 1)x(0, 0, 1, 12)12 - AIC:-60.338781390431066
ARIMA(1, 1, 1)x(0, 1, 0, 12)12 - AIC:-8.168838962391622
ARIMA(1, 1, 1)x(0, 1, 1, 12)12 - AIC:-33.71156153090305
ARIMA(1, 1, 1)x(1, 0, 0, 12)12 - AIC:-59.04358549688331
ARIMA(1, 1, 1)x(1, 0, 1, 12)12 - AIC:-59.17159315884642
ARIMA(1, 1, 1)x(1, 1, 0, 12)12 - AIC:-16.221496009136462
ARIMA(1, 1, 1)x(1, 1, 1, 12)12 - AIC:-25.273482792637516

Pick the model with the lowest AIC value. Lower AIC corresponds a model that fits the data very well while using fewer features. ARIMA(1, 1, 1)x(0, 0, 0, 12)12 - AIC:-73.57350719870367

mod = sm.tsa.statespace.SARIMAX(np.log(vol_data),
                                order=(1,1,1),
                                seasonal_order=(0, 0, 0, 12),
                                enforce_stationarity=False,
                                enforce_invertibility=False)

results = mod.fit()

print(results.summary().tables[1])

==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1          0.2268      0.132      1.717      0.086      -0.032       0.486
ma.L1         -1.1422      0.070    -16.356      0.000      -1.279      -1.005
sigma2         0.0196      0.004      5.559      0.000       0.013       0.026
==============================================================================

results.plot_diagnostics(figsize=(15, 12))
plt.show()

standardized residual seems be stationary with mean centered around 0. In the histogram plot, the KDE should follow closely with the normal distribution N(0,1). Q-Q plot should be linear for normally distributed residuals. The correlogram in the bottom right shows whether or not the time series residuals have correlation with the previous data points. This plot indicates no correlation since all points are within the light blue confidence interval.

In sample one-step-ahead prediction

pred = results.get_prediction(start=str('1930-01-01'), dynamic=False)
pred_ci = pred.conf_int()

vol_data.index=pd.to_datetime(vol_data.index)

predictions_ARIMA=np.exp(pred.predicted_mean)
plt.plot(vol_data,'black')
plt.plot(predictions_ARIMA)

[<matplotlib.lines.Line2D at 0xc823898>]

Try dynamic True. It only uses information from the time series up to a certain point, and after that, forecasts are generated using values from previous forecasted time points.

pred_dynamic = results.get_prediction(start='1950-01-01', dynamic=True, full_results=True)
pred_dynamic_ci = pred_dynamic.conf_int()

predictions_ARIMA=np.exp(pred_dynamic.predicted_mean)

plt.plot(vol_data, 'black')
plt.plot(predictions_ARIMA)

[<matplotlib.lines.Line2D at 0xc8896a0>]

Accumulation of errors in dynamic forecasting makes it a poor choice in predicting future values since it plateaus quickly. ARIMA/SARIMA models don't seem to work too well on this Nile river dataset due to lack of strong trend or seasonality.

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For SARIMA model, there's also (P,D,Q,S) parameters specified along with the (p,d,q). The P,D,Q values are similar to the parameters described above but it's applied to the seasonality componenet of the SARIMA model. S is the periodicity of the time series (4 for quarterly, 12 for yearly).

Source of reference: https://www.digitalocean.com/community/tutorials/a-guide-to-time-series-forecasting-with-arima-in-python-3

look at decompositional element.

No seasonality detected. To find optimal parameters for sarimax time series model, set up a grid search.

In sample one-step-ahead prediction

Try dynamic True. It only uses information from the time series up to a certain point, and after that, forecasts are generated using values from previous forecasted time points.

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For SARIMA model, there's also (P,D,Q,S) parameters specified along with the (p,d,q). The P,D,Q values are similar to the parameters described above but it's applied to the seasonality componenet of the SARIMA model. S is the periodicity of the time series (4 for quarterly, 12 for yearly).

Source of reference: https://www.digitalocean.com/community/tutorials/a-guide-to-time-series-forecasting-with-arima-in-python-3

look at decompositional element.

No seasonality detected. To find optimal parameters for sarimax time series model, set up a grid search.

In sample one-step-ahead prediction

Try dynamic True. It only uses information from the time series up to a certain point, and after that, forecasts are generated using values from previous forecasted time points.

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