Basic Time Series Algorithms and Statistical Assumptions in R

Time series algorithms are extensively used for analyzing and forecasting time-based data. These algorithms are built on underlying statistical assumptions. In this guide, you will learn the underlying statistical assumptions and the basic time series algorithms and how to implement them in R.

Let's begin with the problem statement and data.

Unemployment is a major socio-economic and political issue for any country, and managing it is a primary task for any government. In this guide, you'll forecast unemployment levels for a twelve-month period.

The data used in this guide is from US economic time series data available from http://research.stlouisfed.org/fred2h. The data contains 574 rows and 6 variables, as described below:

The focus will be on the date and unemploy variables, as the area of interest is univariate time-series forecasting.

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

Output:

For model validation, create the training and test datasets.

Output:

In order to run forecasting models in R, you'll have to convert the data into a time series object, which is done in the first line of code below. The start and end arguments specify the time of the first and the last observations, respectively. The argument frequency specifies the number of observations per unit of time.

You'll also need a utility function for calculating the mean absolute percentage error (MAPE), which will be used to evaluate the performance of the forecasting models. The lower the MAPE value, the better the forecasting model performance. This is done in the second to fourth lines of code.

With the data and the MAPE function prepared, you are ready to move to the forecasting techniques in the subsequent sections. However, before moving to forecasting, it's important to understand the important statistical concepts of white noise and stationarity in time series.

A white noise series is a time series that is purely random, and the variables are independent and identically distributed with a mean of zero. This means that the observations have the same variance and there is no autocorrelation. The lines of code below set the seed for reproducibility and generate the plot of the series.

Output:

There seems to be information in the data, and it is not a purely random series. In a white noise series, it is expected that the autocorrelation will be zero. To visualize this, use the ggAcf() function, as shown in the code below.

Output:

If this is a white noise time series, 95 percent or more of the lags will lie between the bounds on a graph of the ACF. This is shown with the blue dashed lines above. In this case, you can see that majority of the lines are above the blue dashed line, which indicates that this is not a white noise time series.

Instead of the visualization above, you can also use the Ljung-Box test to find out if the series is a white noise series. A test of significance, indicated by a small p-value, confirms that the series is probably not white noise. The code below performs this test.

Output:

The p-value is less than 0.05, indicating that the series is not a white noise series. That means there is information in the data that can be used to forecast future values.

One of the popular time series algorithm is the Auto Regressive Integrated Moving Average (ARIMA), which is defined for stationary series. A stationary series is one where the properties do not change over time. In simple terms, the level and variance of the series stays roughly constant over time. You can visualize the series with the code below.

Output:

The above plot shows that the series is not stationary, which is a required for building an ARIMA model. To make the series stationary, perform the statistical operation differencing using the diff() function in R. This is done below.

Output:

The above plot shows that the time series of first differences does appear to be roughly stationary in mean and variance. Thus, it appears that we have an ARIMA(p,1,q) model. Having understood the basic statistical concepts of time series, you'll now build some time series forecasting models.

In exponential smoothing methods, forecasts are produced using weighted averages of past observations, with the weights decaying exponentially as the observations get older. The value of the smoothing parameter for the level is decided by the parameter alpha. The code below creates the simple exponential smoothing model and prints the summary.

Output:

The output above shows that the simple exponential smoothing has the same value for all the forecasts. Because the alpha value is close to 1, the forecasts are closer to the most recent observations. Now, evaluate the model performance on the test data.

The code below stores the output of the model in a data frame and adds a new variable, simplexp, in the test data which contains the forecasted value from the simple exponential model. Finally, the mape function is used to produce the MAPE error on the test data, which comes out to be 8.5 percent.

Output:

This is an extension of the simple exponential smoothing method that takes into account the trend component while generating forecasts. This method involves two smoothing equations, one for the level and one for the trend component.

The code below creates the holt's model and prints the summary.

Output:

The output above shows that the MAPE for the training data is 2.1 percent. Evaluate the model performance on the test data using the lines of code below. The MAPE error on the test data comes out to be 6.6 percent, which is an improvement over the previous models.

Output:

While exponential smoothing models are based on a description of the trend and seasonality, ARIMA models aim to describe the auto-correlations in the data. The auto.arima() function in R is used to build an ARIMA model. The code below creates the model and prints the summary.

Output:

The output above shows that the MAPE for the training data is 1.94 percent. The next step is to evaluate the model performance on the test data, which is done in the lines of code below. The MAPE error on the test data comes out to be 2.1 percent, which is an improvement over all the previous models.

Output:

In this guide, you learned about the underlying statistical concepts of white noise and stationarity in time series data. You also learned how to implement basic time series forecasting models using R. The performance of the models on the test data is summarized below:

The Simple Exponential Smoothing model did well to achieve a lower MAPE of 8.5 percent. The other two models outperformed it by producing an even lower MAPE. The ARIMA model emerged as the winner based on its lowest MAPE of 2.1 percent.