Time Series Forecasting with ARIMA

We have obtained a model for our time series that can now be used to produce forecasts. We start by comparing predicted values to real values of the time series, which will help us understand the accuracy of our forecasts. The get_prediction() and conf_int() attributes allow us to obtain the values and associated confidence intervals for forecasts of the time series.

pred = results.get_prediction(start=pd.to_datetime('1998-01-01'), dynamic=False)
pred_ci = pred.conf_int()

The code above requires the forecasts to start at January 1998.

The dynamic=False argument ensures that we produce one-step ahead forecasts, meaning that forecasts at each point are generated using the full history up to that point.

We can plot the real and forecasted values of the CO2 time series to assess how well we did. Notice how we zoomed in on the end of the time series by slicing the date index.

ax = y['1990':].plot(label='observed')
pred.predicted_mean.plot(ax=ax, label='One-step ahead Forecast', alpha=.7)

ax.fill_between(pred_ci.index,
                pred_ci.iloc[:, 0],
                pred_ci.iloc[:, 1], color='k', alpha=.2)

ax.set_xlabel('Date')
ax.set_ylabel('CO2 Levels')
plt.legend()

plt.show()

Overall, our forecasts align with the true values very well, showing an overall increasing trend.

It is also useful to quantify the accuracy of our forecasts. We will use the MSE (Mean Squared Error), which summarizes the average error of our forecasts. For each predicted value, we compute its distance to the true value and square the result. The results need to be squared so that positive/negative differences do not cancel each other out when we compute the overall mean.

y_forecasted = pred.predicted_mean
y_truth = y['1998-01-01':]

# Compute the mean square error
mse = ((y_forecasted - y_truth) ** 2).mean()
print('The Mean Squared Error of our forecasts is {}'.format(round(mse, 2)))

Output
The Mean Squared Error of our forecasts is 0.07

The MSE of our one-step ahead forecasts yields a value of 0.07, which is very low as it is close to 0. An MSE of 0 would that the estimator is predicting observations of the parameter with perfect accuracy, which would be an ideal scenario, but it is not typically possible.

However, a better representation of our true predictive power can be obtained using dynamic forecasts. In this case, we only use information from the time series up to a certain point, and after that, forecasts are generated using values from previous forecasted time points.

In the code chunk below, we specify to start computing the dynamic forecasts and confidence intervals from January 1998 onwards.

pred_dynamic = results.get_prediction(start=pd.to_datetime('1998-01-01'), dynamic=True, full_results=True)
pred_dynamic_ci = pred_dynamic.conf_int()

Plotting the observed and forecasted values of the time series, we see that the overall forecasts are accurate even when using dynamic forecasts. All forecasted values (red line) match pretty closely to the ground truth (blue line) and are well within the confidence intervals of our forecast.

ax = y['1990':].plot(label='observed', figsize=(20, 15))
pred_dynamic.predicted_mean.plot(label='Dynamic Forecast', ax=ax)

ax.fill_between(pred_dynamic_ci.index,
                pred_dynamic_ci.iloc[:, 0],
                pred_dynamic_ci.iloc[:, 1], color='k', alpha=.25)

ax.fill_betweenx(ax.get_ylim(), pd.to_datetime('1998-01-01'), y.index[-1],
                 alpha=.1, zorder=-1)

ax.set_xlabel('Date')
ax.set_ylabel('CO2 Levels')

plt.legend()
plt.show()

Once again, we quantify the predictive performance of our forecasts by computing the MSE:

# Extract the predicted and true values of our time series
y_forecasted = pred_dynamic.predicted_mean
y_truth = y['1998-01-01':]

# Compute the mean square error
mse = ((y_forecasted - y_truth) ** 2).mean()
print('The Mean Squared Error of our forecasts is {}'.format(round(mse, 2)))

Output
The Mean Squared Error of our forecasts is 1.01

The predicted values obtained from the dynamic forecasts yield an MSE of 1.01. This is slightly higher than the one step ahead, which is to be expected given that we are relying on less historical data from the time series.

Both the one-step ahead and dynamic forecasts confirm that this time series model is valid. However, much of the interest in time series forecasting is the ability to forecast future values way ahead in time.