Time Series prediction is a difficult problem both to frame and to address with machine learning.

In this post, you will discover how to develop neural network models for time series prediction in Python using the Keras deep learning library.

After reading this post you will know:

• About the airline passengers univariate time series prediction problem.
• How to phrase time series prediction as a regression problem and develop a neural network model for it.
• How to frame time series prediction with a time lag and develop a neural network model for it.

Let’s get started.

• Update Oct/2016: Replaced graphs with more accurate versions, commented on the limited performance of the first method.
• Update Mar/2017: Updated example for Keras 2.0.2, TensorFlow 1.0.1 and Theano 0.9.0.

Problem Description

The problem we are going to look at in this post is the international airline passengers prediction problem.

This is a problem where given a year and a month, the task is to predict the number of international airline passengers in units of 1,000. The data ranges from January 1949 to December 1960 or 12 years, with 144 observations.

The dataset is available for free from the DataMarket webpage as a CSV download with the filename “international-airline-passengers.csv“.

Below is a sample of the first few lines of the file.

We can load this dataset easily using the Pandas library. We are not interested in the date, given that each observation is separated by the same interval of one month. Therefore when we load the dataset we can exclude the first column.

The downloaded dataset also has footer information that we can exclude with the skipfooter argument to pandas.read_csv() set to 3 for the 3 footer lines. Once loaded we can easily plot the whole dataset. The code to load and plot the dataset is listed below.

You can see an upward trend in the plot.

You can also see some periodicity to the dataset that probably corresponds to the northern hemisphere summer holiday period.

Plot of the Airline Passengers Dataset

We are going to keep things simple and work with the data as-is.

Normally, it is a good idea to investigate various data preparation techniques to rescale the data and to make it stationary.

Multilayer Perceptron Regression

We want to phrase the time series prediction problem as a regression problem.

That is, given the number of passengers (in units of thousands) this month, what is the number of passengers next month.

We can write a simple function to convert our single column of data into a two-column dataset. The first column containing this month’s (t) passenger count and the second column containing next month’s (t+1) passenger count, to be predicted.

Before we get started, let’s first import all of the functions and classes we intend to use. This assumes a working SciPy environment with the Keras deep learning library installed.

Before we do anything, it is a good idea to fix the random number seed to ensure our results are reproducible.

We can also use the code from the previous section to load the dataset as a Pandas dataframe. We can then extract the NumPy array from the dataframe and convert the integer values to floating point values which are more suitable for modeling with a neural network.

After we model our data and estimate the skill of our model on the training dataset, we need to get an idea of the skill of the model on new unseen data. For a normal classification or regression problem we would do this using cross validation.

With time series data, the sequence of values is important. A simple method that we can use is to split the ordered dataset into train and test datasets. The code below calculates the index of the split point and separates the data into the training datasets with 67% of the observations that we can use to train our model, leaving the remaining 33% for testing the model.

Now we can define a function to create a new dataset as described above. The function takes two arguments, the dataset which is a NumPy array that we want to convert into a dataset and the look_back which is the number of previous time steps to use as input variables to predict the next time period, in this case, defaulted to 1.

This default will create a dataset where X is the number of passengers at a given time (t) and Y is the number of passengers at the next time (t + 1).

It can be configured and we will look at constructing a differently shaped dataset in the next section.

Let’s take a look at the effect of this function on the first few rows of the dataset.

If you compare these first 5 rows to the original dataset sample listed in the previous section, you can see the X=t and Y=t+1 pattern in the numbers.

Let’s use this function to prepare the train and test datasets ready for modeling.

We can now fit a Multilayer Perceptron model to the training data.

We use a simple network with 1 input, 1 hidden layer with 8 neurons and an output layer. The model is fit using mean squared error, which if we take the square root gives us an error score in the units of the dataset.

I tried a few rough parameters and settled on the configuration below, but by no means is the network listed  optimized.

Once the model is fit, we can estimate the performance of the model on the train and test datasets. This will give us a point of comparison for new models.

Finally, we can generate predictions using the model for both the train and test dataset to get a visual indication of the skill of the model.

Because of how the dataset was prepared, we must shift the predictions so that they aline on the x-axis with the original dataset. Once prepared, the data is plotted, showing the original dataset in blue, the predictions for the train dataset in green the predictions on the unseen test dataset in red.

We can see that the model did a pretty poor job of fitting both the training and the test datasets. It basically predicted the same input value as the output.

Naive Time Series Predictions With Neural Network
Blue=Whole Dataset, Green=Training, Red=Predictions

For completeness, below is the entire code listing.