ARIMA (Auto-Regressive Integrated Moving Average) is a statistical modeling technique for time series analysis and forecasting. Compared to machine learning, ARIMA is a classical modeling approach that is particularly powerful when the time series being analyzed follows a clear pattern. This is, for example, the case when the time series exhibits seasonality. Due to its effectiveness, ARIMA is frequently used in various economic and scientific applications. Examples are forecasting weather data, predicting order quantities, and sales forecasting. This tutorial explains the basics of ARIMA modeling and shows how to develop your own ARIMA time series model in Python that predicts beer sales quantities.

This tutorial proceeds in two parts: The first part covers the concepts behind ARIMA. You will learn how ARIMA works, what stationarity means, and when it is appropriate to use ARIMA. The second part is a Python hands-on tutorial that applies auto-ARIMA to the Sales Forecasting domain. We’ll be working with a time series of beer sales, and our goal is to predict how the beer sales quantities will evolve in the coming years. First, we check if the time series is stationary. Then we train an ARIMA forecasting model. Finally, we use the model to produce a sales forecast and measure the model performance.

## Introduction to ARIMA Time Series Modelling

ARIMA models provide an alternative approach to time series forecasting that differs significantly from machine learning methods. Working with ARIMA requires a good understanding of stationarity and knowledge of the transformations used to make time-series data stationary. The concept of stationarity is, therefore, first on our schedule.

### The Concept of Stationarity

Stationarity is an essential concept in stochastic processes that describes the nature of a time series. We consider a time series strictly stationary if its statistical properties do not change over time. In this case, summary statistics such as the mean and variance do not change over time. However, the time-series data we encounter in the real world often show a trend or significant irregular fluctuations, making them non-stationary or weakly stationary.

So why is stationarity such an essential concept for ARIMA? If a time series is stationary, we can assume that the past values of the time series are predictive of future development. In other words, a stationary time series exhibits consistent behavior that makes it predictable. On the other hand, a non-stationary time series is characterized by a kind of random behavior that will be difficult to capture in modeling. Namely, if random movements characterized the past, there is a high probability that the future will be no different.

Fortunately, in many cases, it is possible to transform a time series that is non-stationary into a stationary form and, in this way, build better prediction models.

### How to Test Whether a Time Series is Stationary

The first step in the ARIMA modeling approach is to determine whether a time series is stationary. There are different ways to determine whether a time series is stationary:

**Plotting:**We can plot the time series and visually check if it shows consistent behavior or changes over a more extended period.**Summary statistics**: We can split the time series into different periods and calculate the summary statistics, such as the variance. If these metrics are subject to significant changes, the time series is non-stationary. However, the results will also depend on the respective periods, leading to false conclusions.**Statistic tests**: There are various tests to determine the stationary of a time series, such as Kwiatkowski–Phillips–Schmidt–Shin, Augmented Dickey-Fuller, or Phillips–Perron. These tests systematically check a time series and measure the results against the null hypothesis, providing an indicator of the trustworthiness of the results.

### What is an (S)ARIMA Model?

As the name implies, ARIMA uses autoregression (AR), integration (differencing), and moving averages (MA) to fit a linear regression model to a time series.

#### ARIMA Parameters

The default notation for ARIMA is a model with parameters p, d, and q, whereby each parameter takes an integer value:

**d (differencing):**In the case of a non-stationary time series, there is a chance to remove a trend from the data by differencing once or several times, thus bringing the data to a stationary state. The model parameter d determines the order of the differentiation. A value of d = 0 simplifies the ARIMA model to an ARMA model, lacking the integration aspect. If this is the case, we do not need to integrate the function because the time series is already stationary.**p (order of the AR terms):**The autoregressive process describes the dependent relationship between an observation and several lagged observations (lags). Predictions are then based on past data from the same time series using linear functions. p = 1 means that the model uses values that lag by one period.**q (order of the MA terms):**The parameter q determines the number of lagged forecast errors in the prediction equation. In contrast to the AR process, the MA process assumes that values at a future point in time depend on the errors made by predictions at current and past points in time. This means that it is not previous events that determine the predictions but rather the previous estimation or prediction errors used to calculate the following time series value.

#### SARIMA

In the real world, many time series have seasonal effects. Examples are monthly retail sales figures, temperature reports, weekly airline passenger data, etc. To take this into account, we can specify a seasonal range (e.g., m=12 for monthly data) and additional seasonal AR or MA components for our model that deal with the seasonality. Such a model is also called a SARIMA model, and we can define it as a model(p, d, q)(P, D, Q)[m].

#### Auto-(S)ARIMA

When working with ARIMA, we can set the model parameters manually or use auto-ARIMA and let the model search for the optimal parameters. We do this by varying the parameters and then testing against stationarity. With the seasonal option enabled, the process tries to identify the optimal hyperparameters for the seasonal components of the model. Auto-ARIMA works by conducting differencing tests to determine the order of differencing, `d`

and then fitting models with parameters in defined ranges, e.g., start_`p`

, `max_p`

as well as `start_q`

, `max_q`

. If our model has a seasonal component, we can also define parameter ranges for the seasonal part of the model.

## Creating a Sales Forecast with ARIMA in Python

Now that you are familiar with the concepts behind ARIMA, we can turn to the hands-on part and create a sales forecasting model in Python. Forecasting sales data is a common application for ARIMA because seasonal changes with longer trends often characterize sales data. The dataset we will be working with contains monthly beer sales in the United States (in millions of US $) between 1992 and 2018. Our goal is to model the time series with ARIMA and predict the next steps. In terms of technology, we will build our ARIMA sales model using the statsmodels Python library and pmdarima.

The code is available on the GitHub repository.

### Prerequisites

Before we start coding, make sure that you have set up your Python 3 environment and required packages. If you don’t have an environment set up yet, you can follow this tutorial to set up the Anaconda environment.

Also, make sure you install all required packages. In this tutorial, we will be working with the following standard packages:

In addition, we will be using the statsmodels library and pmdarima.

You can install packages using console commands:

*pip install <package name>**conda install <package name>*(if you are using the anaconda packet manager)

### Step #1 Load the Sales Data to Our Python Project

We begin by loading the data. Running the Python code below will make all required imports and load the sales data into our project, where we can then further prepare it for ARIMA.

# A tutorial for this file is available at www.relataly.com # Tested with Python 3.88 # Setting up packages for data manipulation and machine learning import pandas as pd import numpy as np import matplotlib.pyplot as plt import pmdarima as pm from statsmodels.graphics.tsaplots import plot_acf from statsmodels.tsa.seasonal import seasonal_decompose import seaborn as sns sns.set_style('white', { 'axes.spines.right': False, 'axes.spines.top': False}) # Link to the dataset: # https://www.kaggle.com/bulentsiyah/for-simple-exercises-time-series-forecasting path = "https://raw.githubusercontent.com/flo7up/relataly_data/main/alcohol_sales/BeerWineLiquor.csv" df = pd.read_csv(path) df.head()

As shown above, the sales figures in this dataset stem from the first day of each month.

### Step #2 Visualize the Time Series and Check it for Stationarity

Before modeling the sales data, we visualize the time series and test it for stationarity. Visualization helps us choose the parameters for our ARIMA model, thus making it an essential step.

First, we will look at the different components of the time series. We do this by using the seasonal_decompose function of the statsmodels library.

# Decompose the time series plt.rcParams["figure.figsize"] = (10,6) result = seasonal_decompose(df['beer'], model='multiplicative', period = 12) result.plot() plt.show()

To test for stationarity, we use the ADFuller test. It is common to run this test multiple times throughout a data science project. Therefore, we create a function that we can then reuse later.

def check_stationarity(df_sales, title_string, labels): # Visualize the data fig, ax = plt.subplots(figsize=(16, 8)) plt.title(title_string, fontsize=14) if df_sales.index.size > 12: df_sales['ma_12_month'] = df_sales['beer'].rolling(window=12).mean() df_sales['ma_25_month'] = df_sales['beer'].rolling(window=25).mean() sns.lineplot(data=df_sales[['beer', 'ma_25_month', 'ma_12_month']], palette=sns.color_palette("mako_r", 3)) plt.legend(title='Smoker', loc='upper left', labels=labels) else: sns.lineplot(data=df_sales[['beer']]) plt.show() sales = df_sales['beer'].dropna() # Perform an Ad Fuller Test # the default alpha = .05 stands for a 95% confidence interval adf_test = pm.arima.ADFTest(alpha = 0.05) print(adf_test.should_diff(sales)) df_sales = pd.DataFrame(df['beer'], columns=['beer']) df_sales.index = pd.to_datetime(df['date']) title = "Beer sales in the US between 1992 and 2018 in million US$/month" labels = ['beer', 'ma_12_month', 'ma_25_month'] check_stationarity(df_sales, title, labels)

The data does not appear to be stationary. We can see that our time series is steadily increasing and shows annual seasonality. The steady increase indicates a continuous growth in beer consumption over the last decades. The seasonality in the sales data likely results from people drinking more beer in summer than in other seasons.

### Step #3 Exemplary Differencing and Autocorrelation

The chart from the previous section shows that our time series is non-stationary. The reason is that it follows a clear upward trend. We also know that the time series has a seasonal component. Therefore, we need to define additional parameters and construct a SARIMA model.

Before we use auto-correlation to determine the optimal parameters, we will try manual differencing to make the time series stationary. There is no guarantee that differencing works. It is essential to remember that differencing can sometimes also worsen prediction performance. So be careful, not to overdifference! We could also trust that the auto-ARIMA model chooses the best parameters for us. However, we should always validate the selected parameters.

The ideal differencing parameter is the least number of differencing steps to achieve a stationary time series. We will monitor the results with autocorrelation plots to check whether differencing was successful.

We print the autocorrelation for the original time series and after the first and second-order differencing.

# 3.1 Non-seasonal part def auto_correlation(df, prefix, lags): plt.rcParams.update({'figure.figsize':(7,7), 'figure.dpi':120}) # Define the plot grid fig, axes = plt.subplots(3,2, sharex=False) # First Difference axes[0, 0].plot(df) axes[0, 0].set_title('Original' + prefix) plot_acf(df, lags=lags, ax=axes[0, 1]) # First Difference df_first_diff = df.diff().dropna() axes[1, 0].plot(df_first_diff) axes[1, 0].set_title('First Order Difference' + prefix) plot_acf(df_first_diff, lags=lags - 1, ax=axes[1, 1]) # Second Difference df_second_diff = df.diff().diff().dropna() axes[2, 0].plot(df_second_diff) axes[2, 0].set_title('Second Order Difference' + prefix) plot_acf(df_second_diff, lags=lags - 2, ax=axes[2, 1]) plt.tight_layout() plt.show() auto_correlation(df_sales['beer'], '', 10)

(0.019143247561160443, False)

The charts above show that the time series becomes stationary after one order differencing. However, we can see that the lag goes into the negative very quickly, which indicates overdifferencing.

Next, we perform the same procedure for the seasonal part of our time series.

# 3.2 Seasonal part # Reduce the timeframe to a single seasonal period df_sales_s = df_sales['beer'][0:12] # Autocorrelation for the seasonal part auto_correlation(df_sales_s, '', 10)

# Check if the first difference of the seasonal period is stationary df_diff = pd.DataFrame(df_sales_s.diff()) df_diff.index = pd.date_range(df_sales_s.diff().iloc[1], periods=12, freq='MS') check_stationarity(df_diff, "First Difference (Seasonal)", ['difference'])

(0.99, True)

After first order differencing, the seasonal part of the time series is stationary. The autocorrelation plot shows that the values go into the negative but remain within acceptable boundaries. Second-order differencing does not seem to improve these values. Consequently, we conclude that first-order differencing is a good choice for the D parameter.

### Step #4 Finding an Optimal Model with Auto-ARIMA

Next, we auto-fit an ARIMA model to our time series. In this way, we ensure that we can later measure the performance of our model against a fresh set of data that the model has not seen so far. We will split our dataset into train and test in preparation for this.

Once we have created the train and test data sets, we can configure the parameters for the auto_arima stepwise optimization. By setting max_d = 1, we tell the model to test no-differencing and first-order differencing. Also, we set max_p and max_q to 3.

To deal with the seasonality in our time series, we set the “seasonal” parameter to True and the “m” parameter to 12 data points. We turn our model into a SARIMA model that allows us to configure additional D, P, and Q parameters. We define a max value for Q and P of 3. Previously we have already seen that further differencing does not improve the stationarity. Therefore, we can set the value of D to 1.

After configuring the parameters, we next fit the model to the time series. The model will try to find the optimal parameters and choose the model with the least AIC.

# split into train and test pred_periods = 30 split_number = df_sales['beer'].count() - pred_periods # corresponds to a prediction horizion of 2,5 years df_train = pd.DataFrame(df_sales['beer'][:split_number]).rename(columns={'beer':'y_train'}) df_test = pd.DataFrame(df_sales['beer'][split_number:]).rename(columns={'beer':'y_test'}) # auto_arima model_fit = pm.auto_arima(df_train, test='adf', max_p=3, max_d=3, max_q=3, seasonal=True, m=12, max_P=3, max_D=2, max_Q=3, trace=True, error_action='ignore', suppress_warnings=True, stepwise=True) # summarize the model characteristics print(model_fit.summary())

Auto-ARIMA has determined that the best model is (3,0,0)(3,1,1). These results match well with the results from section 3, in which we manually performed differencing steps.

### Step #5 Simulate the Time Series using in-sample Forecasting

Now that we have trained our model, we want to use it to simulate the entire time series. We will do this by calling the predict method in the sample function. The prediction will match the same period as the original time series with which we trained the model. Because the model predicts one step, the prediction results will naturally be close to the actual values.

# Generate in-sample Predictions # The parameter dynamic=False means that the model makes predictions upon the lagged values. # This means that the model is trained until a point in the time-series and then tries to predict the next value. pred = model_fit.predict_in_sample(dynamic=False) # works only with auto-arima df_train['y_train_pred'] = pred # Calculate the percentage difference df_train['diff_percent'] = abs((df_train['x_train'] - pred) / df_train['x_train'])* 100 # Print the predicted time-series fig, ax1 = plt.subplots(figsize=(16, 8)) plt.title("In Sample Sales Prediction", fontsize=14) sns.lineplot(data=df_train[['x_train', 'y_train_pred']], linewidth=1.0) # Print percentage prediction errors on a separate axis (ax2) ax2 = ax1.twinx() ax2.set_ylabel('Prediction Errors in %', color='purple', fontsize=14) ax2.set_ylim([0, 50]) ax2.bar(height=df_train['diff_percent'][20:], x=df_train.index[20:], width=20, color='purple', label='absolute errors') plt.legend() plt.show()

Next, we take a look at the prediction errors.

### Step #6 Generate and Visualize a Sales Forecast

Now that we have trained an optimal model, we are ready to generate a sales forecast. First, we specify the number of periods that we want to predict. In addition, we create an index from the number of predictions adjacent to the original time series and continue it (prediction_index).

# Generate prediction for n periods, # Predictions start from the last date of the training data test_pred = model_fit.predict(n_periods=pred_periods, dynamic=False) df_test['y_test_pred'] = test_pred df_union = pd.concat([df_train, df_test]) df_union.rename(columns={'beer':'y_test'}, inplace=True) # Print the predicted time-series fig, ax = plt.subplots(figsize=(16, 8)) plt.title("Test/Pred Comparison", fontsize=14) sns.despine(); sns.lineplot(data=df_union[['y_train', 'y_train_pred', 'y_test', 'y_test_pred']], linewidth=1.0, dashes=False, palette='muted') ax.set_xlim([df_union.index[150],df_union.index.max()]) plt.legend() plt.show()

As shown above, our model’s forecast continues the seasonal pattern of the beer sales time series. On the one hand, this indicates that US beer sales will continue to rise and, on the other hand, that our model works just fine 🙂

### Step #7 Measure the Performance of the Sales Forecasting Model

In this section, we will measure the performance of our ARIMA model. If you want to learn more about this topic, check out this relataly article measuring regression performance.

The previous section’s simulation chart shows a few outliers among the prediction errors. Therefore, we focus our analysis on the percentage errors. Two helpful metrics are the mean absolute error (MAPE) and the mean absolute percentage error (MDAPE).

# Mean Absolute Percentage Error (MAPE) MAPE = np.mean((np.abs(np.subtract(df_test['y_test'], df_test['y_test_pred'])/ df_test['y_test']))) * 100 print(f'Mean Absolute Percentage Error (MAPE): {np.round(MAPE, 2)} %') # Median Absolute Percentage Error (MDAPE) MDAPE = np.median((np.abs(np.subtract(df_test['y_test'], df_test['y_test_pred'])/ df_test['y_test'])) ) * 100 print(f'Median Absolute Percentage Error (MDAPE): {np.round(MDAPE, 2)} %')

Mean Absolute Percentage Error (MAPE): 3.94 % Median Absolute Percentage Error (MDAPE): 3.49 %

The percent errors show that our ARIMA model achieves a decent predictive performance.

## Summary

This Python tutorial has shown how to analyze a seasonal time series with (S)ARIMA. In the first part, we have learned how ARIMA works, what stationarity is and how to check if a time series is stationary. In the second part, we developed an ARIMA model in Python to create a forecast for US beer sales. For this purpose, we created an in-sample forecast and used Auto-tARIMA to find the optimal parameters for our sales forecasting model.

If you have any questions or suggestions, please let me know in the comments, and I will do my best to answer.

Want to learn more about time series analysis and prediction?

Check out these recent relataly tutorials: