Building an ARIMA Model for Time Series Forecasting in Python

Prabhat Pathak 18 Jan, 2024 • 6 min read

Introduction

A popular and widely used statistical method for time series forecasting is the ARIMA model. Exponential smoothing and ARIMA models are the two most widely used approaches to time series forecasting and provide complementary approaches to the problem. While exponential smoothing models are based on a description of the trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data. Before we talk about the ARIMA model, let’s talk about the concept of stationarity and the technique of differencing time series.

What is Autoregressive Integrated Moving Average (ARIMA)?

ARIMA is an acronym that stands for Auto-Regressive Integrated Moving Average. It is a class of model that captures a suite of different standard temporal structures in time series data.

In this tutorial, We will talk about how to develop an ARIMA model for time series forecasting in Python.

An ARIMA model is a class of statistical models for analyzing and forecasting time series data. It is really simplified in terms of using it, Yet this model is really powerful.

ARIMA stands for Auto-Regressive Integrated Moving Average.

The parameters of the ARIMA model are defined as follows:

  • p: The number of lag observations included in the model, also called the lag order.
  • d: The number of times that the raw observations are differenced, also called the degree of difference.
  • q: The size of the moving average window, also called the order of moving average.

Steps to Use ARIMA Model

We construct a linear regression model by incorporating the specified number and type of terms. Additionally, we prepare the data through differencing to achieve stationarity, effectively eliminating trend and seasonal structures that can adversely impact the regression model.

  1. Visualize the Time Series Data

    Visualize the Time Series Data involves plotting the historical data points over time to observe patterns, trends, and seasonality.

  2. Identify if the date is stationary

    Identify if the data is stationary involves checking whether the time series data exhibits a stable pattern over time or if it has any trends or irregularities. Stationary data is necessary for accurate predictions using ARIMA, and various statistical tests can be employed to determine stationarity.

  3. Plot the Correlation and Auto Correlation Charts

    To plot the correlation and auto-correlation charts in the steps of using the ARIMA model online, you analyze the time series data. The correlation chart displays the relationship between the current observation and lagged observations, while the auto-correlation chart shows the correlation of the time series with its own lagged values. These charts provide insights into potential patterns and dependencies within the data.

  4. Construct the ARIMA Model or Seasonal ARIMA based on the data

    To construct an ARIMA (Autoregressive Integrated Moving Average) model or a Seasonal ARIMA model, one analyzes the data to determine the appropriate model parameters, such as the order of autoregressive (AR) and moving average (MA) components. This step involves selecting the optimal values for the model based on the characteristics and patterns observed in the data.

Stationarity

A stationary time series data is one whose properties do not depend on the time, That is why time series with trends, or with seasonality, are not stationary. the trend and seasonality will affect the value of the time series at different times, On the other hand for stationarity it does not matter when you observe it, it should look much the same at any point in time. In general, a stationary time series will have no predictable patterns in the long-term.

Let’s Start

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline

Defining the Dataset

In this tutorial, I am using the below dataset.

df=pd.read_csv('time_series_data.csv')
df.head()

# Updating the header
df.columns=["Month","Sales"]
df.head()
df.describe()
df.set_index('Month',inplace=True)

from pylab import rcParams
rcParams['figure.figsize'] = 15, 7
df.plot()
ARIMA

If we see the above graph then we will able to find a trend that there is a time when sales are high and vice versa. That means we can see data is following seasonality. For ARIMA first thing we do is identify if the data is stationary or non – stationary. if data is non-stationary we will try to make them stationary then we will process further.

Using Adfuller

Let’s check that if the given dataset is stationary or not, For that we use adfuller.

from statsmodels.tsa.stattools import adfuller

I have imported the adfuller by running the above code.

test_result=adfuller(df['Sales'])

To identify the nature of data, we will be using the null hypothesis.

  • H0: The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
  • H1: The alternative hypothesis: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.

#Ho: It is non-stationary
#H1: It is stationary

We will be considering the null hypothesis that data is not stationary and the alternate hypothesis that data is stationary.

def adfuller_test(sales):
    result=adfuller(sales)
    labels = ['ADF Test Statistic','p-value','#Lags Used','Number of Observations']
    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 is stationary")
else:
    print("weak evidence against null hypothesis,indicating it is non-stationary ")

adfuller_test(df['Sales'])

Finding the P-value

After running the above code we will get P-value,

ADF Test Statistic : -1.8335930563276237
p-value : 0.3639157716602447
#Lags Used : 11
Number of Observations : 93

Here P-value is 0.36 which is greater than 0.05, which means data is accepting the null hypothesis, which means data is non-stationary.

Let’s try to see the first difference and seasonal difference:

df['Sales First Difference'] = df['Sales'] - df['Sales'].shift(1)
df['Seasonal First Difference']=df['Sales']-df['Sales'].shift(12)
df.head()
# Again testing if data is stationary
adfuller_test(df['Seasonal First Difference'].dropna())
ADF Test Statistic : -7.626619157213163
p-value : 2.060579696813685e-11
#Lags Used : 0
Number of Observations : 92

Here P-value is 2.06, which means we will be rejecting the null hypothesis. So data is stationary.

df['Seasonal First Difference'].plot()
Arima model

Create Auto-correlation

I am going to create auto-correlation :

from pandas.plotting import autocorrelation_plot
autocorrelation_plot(df['Sales'])
plt.show()
from statsmodels.graphics.tsaplots import plot_acf,plot_pacf
import statsmodels.api as sm
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(df['Seasonal First Difference'].dropna(),lags=40,ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(df['Seasonal First Difference'].dropna(),lags=40,ax=ax2)
arima models python
# For non-seasonal data
#p=1, d=1, q=0 or 1

from statsmodels.tsa.arima_model import ARIMA
model=ARIMA(df['Sales'],order=(1,1,1))
model_fit=model.fit()
model_fit.summary()
Dep. Variable:D.SalesNo. Observations:104
Model:ARIMA(1, 1, 1)Log-Likelihood-951.126
Method:css-mleS.D. of innovations2227.262
Date:Wed, 28 Oct 2020AIC1910.251
Time:11:49:08BIC1920.829
Sample:02-01-1964 to 09-01-1972HQIC1914.536
 coefstd errzP>|z|[0.0250.975]
const22.784512.4051.8370.066-1.52947.098
ar.L1.D.Sales0.43430.0894.8660.0000.2590.609
ma.L1.D.Sales-1.00000.026-38.5030.000-1.051-0.949
 RealImaginaryModulusFrequency
AR.12.3023+0.0000j2.30230.0000
MA.11.0000+0.0000j1.00000.0000
df['forecast']=model_fit.predict(start=90,end=103,dynamic=True)
df[['Sales','forecast']].plot(figsize=(12,8))
ARIMA
import statsmodels.api as sm
model=sm.tsa.statespace.SARIMAX(df['Sales'],order=(1, 1, 1),seasonal_order=(1,1,1,12))
results=model.fit()
df['forecast']=results.predict(start=90,end=103,dynamic=True)
df[['Sales','forecast']].plot(figsize=(12,8))
ARIMA
from pandas.tseries.offsets import DateOffset
future_dates=[df.index[-1]+ DateOffset(months=x)for x in range(0,24)]
future_datest_df=pd.DataFrame(index=future_dates[1:],columns=df.columns)

future_datest_df.tail()

future_df=pd.concat([df,future_datest_df])

future_df['forecast'] = results.predict(start = 104, end = 120, dynamic= True)
future_df[['Sales', 'forecast']].plot(figsize=(12, 8))
ARIMA

Conclusion

Time Series forecasting is really useful when we have to take future decisions or we have to do analysis, we can quickly do that using ARIMA, there are lots of other Models from we can do the time series forecasting but ARIMA model is really easy to understand. If you’re interested in mastering data science techniques like ARIMA, explore enrolling in the renowned Blackbelt Plus Program. This step will enhance your skills and broaden your horizons in this dynamic field, marking the beginning of your data science journey today!

Frequently Asked Questions

Q1. What is ARIMA in Python?

A. ARIMA, or AutoRegressive Integrated Moving Average, is a time series forecasting method implemented in Python for predicting future data points based on historical time series data.

Q2. Is ARIMA better than regression?

A. ARIMA and regression serve different purposes. ARIMA is suitable for time series data, while regression is used for modeling relationships between variables. The choice depends on the data and the problem.

Q3. Is ARIMA the same as regression?

A. No, ARIMA and regression are different. ARIMA is designed for time series forecasting, focusing on temporal patterns, while regression models the relationship between variables, irrespective of time.

Q4. What is ARIMA for time series forecasting?

A. ARIMA is a powerful technique for time series forecasting. It models time-dependent patterns, such as trends and seasonality, making it valuable for predicting future values in sequential data.

Prabhat Pathak 18 Jan 2024

Frequently Asked Questions

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Responses From Readers

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Gaurav Sinha
Gaurav Sinha 30 Oct, 2020

Very nice and detailed article. Thanks for sharing!

AJ
AJ 31 Oct, 2020

Your article started very well explaining what is going on. The second half was just coding and graphs without explaining what is happening at all. It would be great if you could modify this in the future and explain the steps you took, what they mean and how you can use the model in forecasting.

Gaurang
Gaurang 31 Oct, 2020

Nice explanation Prabhat

Abhishek
Abhishek 02 Nov, 2020

Thanks for sharing

Abhishek
Abhishek 02 Nov, 2020

Insightful information , Thanks for sharing!!

Pantina Chandrashekhar
Pantina Chandrashekhar 10 Jan, 2021

I want to know how to find RMSE for above ARIMA model.

David
David 14 Mar, 2021

thanks so much. this is insightful and well explained. Please can u share the link to the dataset you used? would be grateful

david
david 14 Mar, 2021

Great and well explained. please can u share the link to the dataset that u used?

Vin
Vin 05 Apr, 2021

Informative article - Could you please share the link to the Data Set 'time_series_data.csv' that you use above? Thanks

Baildez
Baildez 08 Apr, 2021

Good explanation! Could you please share the link to the data set 'time_series_data.csv' as used in df=pd.read_csv('time_series_data.csv')? Thanks

Pham Huy Anh
Pham Huy Anh 12 Apr, 2021

Great! Could you send me your data. I woud like to try like you. Thank you so much!

pio
pio 16 Apr, 2021

Very useful. Thanks for posting it but it would be nice to get the dataset that you have used to practice it

Bishwarup Dey
Bishwarup Dey 27 May, 2021

I have a doubt regarding the p- value , in the beginning when the p- value was 0.36 we were accepting the null hypothesis since the p- value was greater than the critical value, but when the p value was 2.06 which is again greater than the critical value we are rejecting the hypothesis why?

Bishwarup Dey
Bishwarup Dey 27 May, 2021

Hi, I have a doubt in this article regarding time series forecasting. Please can anyone help me out I’m a bit confused? In this article when he runs the ad-fuller test the 1st time he gets a p-value of 0.36 which is greater than the critical value 0.05 and so we accept the null hypothesis but when he is again testing the data for stationarity he gets a p-value of 2.06 which is again greater than 0.05 but this time he rejects the null hypothesis.

Jose Pauyac
Jose Pauyac 08 Feb, 2022

Thank you! But could anyone help me please? I can't get the last part right. I get error when typing [df.index[-1]+ DateOffSet(months=2)]. But if I use df.iloc[-1] my date frame of future_dates is empty. I don't understand why.

Vishal
Vishal 04 Apr, 2022

future_dates=[df.index[-1]+ DateOffset(months=x)for x in range(0,24)] Why 24 is taken here in range? please explain

Muhammad Usama Baig
Muhammad Usama Baig 30 Jul, 2022

I am Confuse about the P-VALUES to check the stationary or non-stationary please check your paragraph and reply my question . 1) if 0.36>0.05 accept the null hypothesis. 2)if 2.06>0.05 reject the null hypothesis.

Tv schedule
Tv schedule 13 Oct, 2023

This is a great post! I have been working with time series data for a while now and this post has really helped me to understand the ARIMA model better.

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