Multi-step forecasting is the task of predicting multiple values of a time series into the future. This is a challenging task, as it requires the model to capture the dynamics of the time series and to make accurate predictions over a long period of time.
There are two main approaches to multi-step forecasting: the iterated approach and the direct approach.
The iterated approach is a recursive method that uses a single model to predict multiple steps ahead. The model is first trained to predict one step ahead. Then, the predicted value is used as input to the model to predict the next step, and so on. This process is repeated until the desired forecast horizon is reached.
The iterated approach has the advantage of being relatively simple to implement. It also requires only a single model, which can be advantageous if computational resources are limited. However, the iterated approach can be sensitive to errors in the initial predictions. If the first prediction is wrong, it can propagate through the subsequent predictions, leading to increasingly inaccurate forecasts.
The direct approach uses a separate model for each step in the forecast horizon. This means that each model is specifically trained to predict a particular step ahead. This can help to improve the accuracy of the forecasts, as each model is tailored to the dynamics of that particular step. However, the direct approach requires training multiple models, which can be computationally expensive.
The MIMO (Multiple Input, Multiple Output) approach is a hybrid approach that combines the iterated and direct approaches. In the MIMO approach, a single model is trained to predict multiple steps ahead. However, the model is allowed to have multiple inputs, which can be the predicted values from the previous steps. This can help to improve the accuracy of the forecasts, as the model is able to take into account the dynamics of the previous steps.
Choosing the Best Approach
The best approach for multi-step forecasting depends on many factors, including the nature of the time series, the computational resources available, and the importance of forecasting accuracy at various horizons. In general, the iterated approach is a good choice if computational resources are limited and if the forecast horizon is not too long. The direct approach is a good choice if accuracy is essential and if the forecast horizon is long. The MIMO approach is a good compromise between the iterated and direct methods, and it can be a good choice if the time series is relatively noisy.
Additional Concepts and Equations
Here are some additional concepts and equations that are relevant to multi-step forecasting:
- Autoregressive (AR) model: An AR model is a statistical model that predicts the future value of a time series based on its past values. AR models are commonly used for multi-step forecasting.
- Moving average (MA) model: An MA model is a statistical model that predicts the future value of a time series based on its past errors. MA models are also commonly used for multi-step forecasting.
- ARIMA model: An ARIMA model combines an AR model and a MA model. ARIMA models are a powerful tool for multi-step forecasting, but they can be more difficult to fit than AR or MA models.
- Forecast error: The forecast error is the difference between the actual value of a time series and the forecast value. The forecast error is used to evaluate the accuracy of the forecasts.
- Mean squared error (MSE): The MSE measures the average squared forecast error. The MSE is a commonly used metric for evaluating the accuracy of forecasts.
Multi-step forecasting is a challenging task, but it is an important one for many applications. There are a variety of approaches to multi-step forecasting, and the best approach depends on the specific application. The iterated approach, the direct approach, and the MIMO approach are three common approaches that are discussed in this article.