Introduction

25

Time series algorithms are used extensively for analyzing and forecasting time-based data. One set of popular and powerful time series algorithms is the ARIMA class of models, which are based on describing autocorrelations in the data.

ARIMA stands for *Autoregressive Integrated Moving Average* and has three components, `p`

, `d`

, and `q`

, that are required to build the ARIMA model. These three components are:

`p`

: Number of autoregressive lags

`d`

: Order of differencing required to make the series stationary

`q`

: Number of moving average lags

In this guide, you will learn the core concepts of ARIMA modeling and how to implement it in Python. Let's begin with understanding and loading the data.

This guide uses the fictitious monthly sales data of a supermarket chain containing 564 observations and three variables, as described below:

`Date`

: the first date of every month`Sales`

: daily sales, in thousands of dollars`Class`

: the variable denoting the training and test data set partition

The lines of code below import the required libraries and the data.

`1 2 3 4 5 6 7`

`import pandas as pd import numpy as np # Reading the data df = pd.read_csv("data.csv") print(df.shape) print(df.info())`

python

Output:

`1 2 3 4 5 6 7 8 9 10 11`

`(564, 3) <class 'pandas.core.frame.DataFrame'> RangeIndex: 564 entries, 0 to 563 Data columns (total 3 columns): Date 564 non-null object Sales 564 non-null int64 Class 564 non-null object dtypes: int64(1), object(2) memory usage: 13.3+ KB None`

The next step is to create the training and test datasets for model building and evaluation.

`1 2 3 4`

`train = df[df["Class"] == "Train"] test = df[df["Class"] == "Test"] print(train.shape) print(test.shape)`

python

Output:

`1 2`

`(552, 3) (12, 3)`

You should also create train and test arrays with the code below.

`1 2 3 4 5`

`train_array = train["Sales"] print(train_array.shape) test_array = test["Sales"] print(test_array.shape)`

python

Output:

`1 2 3`

`(552,) (12,)`

With the data prepared, you are ready to move to the forecasting techniques in the subsequent sections. However, before building ARIMA models, it's important to understand the statistical concept of stationarity.

One of the requirements for ARIMA is that the time series should be stationary. A *stationary series* is one where the properties do not change over time. There are several methods to check the stationarity of a series. The one you’ll use in this guide is the Augmented Dickey-Fuller test.

The *Augmented Dickey-Fuller test* is a type of statistical unit root test. The test uses an autoregressive model and optimizes an information criterion across multiple different lag values.

The null hypothesis of the test is that the time series is not stationary, while the alternate hypothesis (rejecting the null hypothesis) is that the time series is stationary.

The first step is to import the `adfuller`

module from the `statsmodels`

package. This is done in the first line of code below. The second line performs and prints the p-value of the test.

`1 2 3`

`from statsmodels.tsa.stattools import adfuller print("p-value:", adfuller(train_array.dropna())[1])`

python

Output:

`1`

`p-value: 0.3440379665909026`

The output above shows that the p-value is greater than the significance level of 0.05, so we fail to reject the null hypothesis. The series is not stationary and requires differencing.

The series can be differenced using the `diff()`

function. The first line of code below performs the first order differencing, while the second line performs the Augmented Dickey-Fuller Test.

`1 2`

`diff_1 = train_array.diff().dropna() print("p-value:", adfuller(diff_1.dropna())[1])`

python

Output:

`1`

`p-value: 0.001`

The p-value now is below the significance level, indicating that the series is stationary.

You are now ready to build the ARIMA model and make predictions. You will be using the `auto_arima`

function in Python, which automatically discovers the optimal order for an ARIMA model. In simple terms, the function will automatically determine the parameters `p`

, `d`

’, and `q`

of the ARIMA model.

The important parameters of the function are:

- The time-series to which you fit the ARIMA model.

`start_p`

: the starting value of`p`

, the order of the auto-regressive (AR) model. This must be a positive integer.

`start_q`

: the starting value of`q`

, the order of the moving-average (MA) model. This must be a positive integer.

`d`

: the order of first-differencing. The default setting is none, and then the value is selected automatically based on the results of the test, in this case the Augmented Dickey-Fuller test.

`test`

: type of unit root test to use in order to detect stationarity if stationary is False and`d`

is none.

You will now build the ARIMA estimator. The first step is to import the `pmdarima`

library that contains the `auto_arima`

function. The second step is to define a function that takes in the time series array and returns the auto-arima model. These steps are done in the code below.

`1 2 3 4 5 6 7 8 9`

`import pmdarima as pmd def arimamodel(timeseriesarray): autoarima_model = pmd.auto_arima(timeseriesarray, start_p=1, start_q=1, test="adf", trace=True) return autoarima_model`

python

The next step is to use the function defined above and build the ARIMA estimator on the training data.

`1 2`

`arima_model = arimamodel(train_array) arima_model.summary()`

python

Output:

`1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21`

`Fit ARIMA: order=(1, 1, 1); AIC=7974.318, BIC=7991.565, Fit time=0.425 seconds Fit ARIMA: order=(0, 1, 0); AIC=7975.310, BIC=7983.934, Fit time=0.011 seconds Fit ARIMA: order=(1, 1, 0); AIC=7973.112, BIC=7986.047, Fit time=0.177 seconds Fit ARIMA: order=(0, 1, 1); AIC=7973.484, BIC=7986.419, Fit time=0.084 seconds Fit ARIMA: order=(2, 1, 0); AIC=7974.012, BIC=7991.259, Fit time=0.274 seconds Fit ARIMA: order=(2, 1, 1); AIC=7973.626, BIC=7995.185, Fit time=0.989 seconds Total fit time: 2.000 seconds ARIMA Model Results Dep. Variable: D.y No. Observations: 551 Model: ARIMA(1, 1, 0) Log Likelihood -3983.556 Method: css-mle S.D. of innovations 333.866 Date: Wed, 27 May 2020 AIC 7973.112 Time: 11:37:46 BIC 7986.047 Sample: 1 HQIC 7978.166 coef std err z P>|z| [0.025 0.975] const 9.0108 13.085 0.689 0.491 -16.636 34.658 ar.L1.D.y -0.0871 0.042 -2.053 0.041 -0.170 -0.004 Roots Real Imaginary Modulus Frequency AR.1 -11.4797 +0.0000j 11.4797 0.5000`

The output above shows that the final model fitted was an ARIMA(1,1,0) estimator, where the values of the parameters `p`

, `d`

, and `q`

were one, one, and zero, respectively. The `auto_arima`

functions tests the time series with different combinations of `p`

, `d`

, and `q`

using `AIC`

as the criterion. AIC stands for *Akaike Information Criterion*, which estimates the relative amount of information lost by a given model. In simple terms, a lower AIC value is preferred. In the above output, the lowest AIC value of 7973.112 was obtained for the ARIMA(1, 1, 0) model, and that is used as the final estimator.

You have trained the model and will now use it make predictions on the test data and perform model evaluation. One step before doing this is to create a utility function that will be used as an evaluation metric. The code below creates a utility function for calculating the mean absolute percentage error (MAPE), which is the metric to be used. The lower the MAPE value, the better the forecasting model performance.

`1 2 3`

`def mean_absolute_percentage_error(y_true, y_pred): y_true, y_pred = np.array(y_true), np.array(y_pred) return np.mean(np.abs((y_true - y_pred) / y_true)) * 100`

python

The next step is to make predictions on the test data, which is done using the code below. The second line prints the first five observations.

`1 2 3`

`test['ARIMA'] = automodel.predict(len(test)) test.head(5)`

python

Output:

`1 2 3 4 5 6`

`Date Sales Class ARIMA 552 01-01-2014 6785 Test 6882.9 553 01-02-2014 6856 Test 6889.8 554 01-03-2014 6853 Test 6898.9 555 01-04-2014 6400 Test 6907.9 556 01-05-2014 6442 Test 6916.9`

The final step is to evaluate the predictions on the test data using the utility function as shown below.

`1`

`mean_absolute_percentage_error(test.Sales, test.ARIMA)`

python

Output:

`1`

`9.7846`

The output above shows that the MAPE for the test data is 9.8%. The low value means that the model results are good.

In this guide, you learned about forecasting time series data using ARIMA. You learned about the stationarity requirement of time series and how to make a non-stationary series stationary through differencing. Finally, you learned how to build and interpret the ARIMA estimator for forecasting using Python.

To learn more about data science using Python, please refer to the following guides.

25