Time series forecasting is typically discussed where only a one-step prediction is required.

What about when you need to predict multiple time steps into the future?

Predicting multiple time steps into the future is called multi-step time series forecasting. There are four main strategies that you can use for multi-step forecasting.

In this post, you will discover the four main strategies for multi-step time series forecasting.

After reading this post, you will know:

- The difference between one-step and multiple-step time series forecasts.
- The traditional direct and recursive strategies for multi-step forecasting.
- The newer direct-recursive hybrid and multiple output strategies for multi-step forecasting.

Let’s get started.

## Multi-Step Forecasting

Generally, time series forecasting describes predicting the observation at the next time step.

This is called a one-step forecast, as only one time step is to be predicted.

There are some time series problems where multiple time steps must be predicted. Contrasted to the one-step forecast, these are called multiple-step or multi-step time series forecasting problems.

For example, given the observed temperature over the last 7 days:

1 2 3 4 5 6 7 8 |
Time, Temperature 1, 56 2, 50 3, 59 4, 63 5, 52 6, 60 7, 55 |

A single-step forecast would require a forecast at time step 8 only.

A multi-step may require a forecast for the next two days, as follows:

1 2 3 |
Time, Temperature 8, ? 9, ? |

There are at least four commonly used strategies for making multi-step forecasts.

They are:

- Direct Multi-step Forecast Strategy.
- Recursive Multi-step Forecast Strategy.
- Direct-Recursive Hybrid Multi-step Forecast Strategies.
- Multiple Output Forecast Strategy.

Let’s take a closer look at each method in turn.

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## 1. Direct Multi-step Forecast Strategy

The direct method involves developing a separate model for each forecast time step.

In the case of predicting the temperature for the next two days, we would develop a model for predicting the temperature on day 1 and a separate model for predicting the temperature on day 2.

For example:

1 2 |
prediction(t+1) = model1(obs(t-1), obs(t-2), ..., obs(t-n)) prediction(t+2) = model1(obs(t-2), obs(t-3), ..., obs(t-n)) |

HavingĀ one model for each time step is an added computational and maintenance burden, especially as the number of time steps to be forecasted increases beyond the trivial.

Because separate models are used, it means that there is no opportunity to model the dependencies between the predictions, such as the prediction on day 2 being dependent on the prediction in day 1, as is often the case in time series.

## 2. Recursive Multi-step Forecast

The recursive strategy involves using a one-step model multiple times where the prediction for the prior time step is used as an input for making a prediction on the following time step.

In the case of predicting the temperature for the next two days, we would develop a one-step forecasting model. This model would then be used to predict day 1, then this prediction would be used as an observation input in order to predict day 2.

For example:

1 2 |
prediction(t+1) = model(obs(t-1), obs(t-2), ..., obs(t-n)) prediction(t+2) = model(prediction(t+1), obs(t-1), ..., obs(t-n)) |

Because predictions are used in place of observations, the recursive strategy allows prediction errors to accumulate such that performance can quickly degrade as the prediction time horizon increases.

## 3. Direct-Recursive Hybrid Strategies

The direct and recursive strategies can be combined to offer the benefits of both methods.

For example, a separate model can be constructed for each time step to be predicted, but each model may use the predictions made by models at prior time steps as input values.

We can see how this might work for predicting the temperature for the next two days, where two models are used, but the output from the first model is used as an input for the second model.

For example:

1 2 |
prediction(t+1) = model1(obs(t-1), obs(t-2), ..., obs(t-n)) prediction(t+2) = model2(prediction(t+1), obs(t-1), ..., obs(t-n)) |

Combining the recursive and direct strategies can help to overcome the limitations of each.

## 4. Multiple Output Strategy

The multiple output strategy involves developing one model that is capable of predicting the entire forecast sequence in a one-shot manner.

In the case of predicting the temperature for the next two days, we would develop one model and use it to predict the next two days as one operation.

For example:

1 |
prediction(t+1), prediction(t+2) = model(obs(t-1), obs(t-2), ..., obs(t-n)) |

Multiple output models are more complex as they can learn the dependence structure between inputs and outputs as well as between outputs.

Being more complex may mean that they are slower to train and require more data to avoid overfitting the problem.

## Further Reading

See the resources below for further reading on multi-step forecasts.

- Machine Learning Strategies for Time Series Forecasting, 2013
- Recursive and direct multi-step forecasting: the best of both worlds, 2012 [PDF]

## Summary

In this post, you discovered strategies that you can use to make multiple-step time series forecasts.

Specifically, you learned:

- How to train multiple parallel models in the direct strategy or reuse a one-step model in the recursive strategy.
- How to combine the best parts of the direct and recursive strategies in the hybrid strategy.
- How to predict the entire forecast sequence in a one-shot manner using the multiple output strategy.

Do you have any questions about multi-step time series forecasts, or about this post? Ask your questions in the comments below and I will do my best to answer.

Thanks Jason for a wonderful post. Why does your model skips the value at “t”?

Just a choice of terminology, think of t+1 as t.

I could have made it clearer, thanks for the note.

Hi Jason, it is always helpful to read your post. I have some confusion related to Time Series Forecasting.

There is traffic data (1440 pieces in total, and 288 pieces each day) I collected to predict traffic flow. The data is collected every 5 min in five consecutive working days. I am going to use the traffic data of the first four day to train the prediction model, while the traffic data of the fifth day is used to test the model.

Here is my question, if I want to predict the traffic flow of the fifth day, do I only need to treat my prediction as one-step forecast or do I have to predict 288-step?

Look forward to your advice.

Thanks for your post again.

Hi Dylan,

If you want to predict an entire day in advance (288 observations), this sounds like a multi-step forecast.

You could use a recursive one-step strategy or something like a neural net to predict the entire sequence one a one-shot manner.

Predicting so many steps in advance is hard work (a hard problem) and results may be poor. You will do better if you can use data as it comes in to continually refine your forecast.

Does that help?

Yes, your response is very helpful. Thank you very much. Now I realize my prediction is a multi-step forecast.

Could you recommend me some more detailed materials related to the multi-step forecast, like the recursive one-step strategy or the neural net?

Now I am reading your post, it is great.

Thank you for your advice.

Best regards

I am working on this type of material at the moment, it should be on the blog in coming weeks/months.

You can use an ARIMA recursively by taking forecasts as inputs to make the next prediction.

You can use a neural network to make a multi-step forecast by setting the output sequence length as the number of neurons in the output layer.

I hope that helps as a start.

Thank you for your advice. I will keep digging into this puzzle. Hope to discuss with you again. It is very helpful. Thank you for your time.

Best regards.

Let me know how you go.

Yes, I will. Discussing with you is always helpful. Look forward to reading your new post on Time Series Forecast.

Hi Jason, just another brilliant post. Can you show up a working example for first or second method like you have always shown in other tutorials. It would be immense help to a novice like me. Thanks…

I do hope to have many examples on the blog in the coming weeks.

Thank you Jason for your wonderful articles ! you are a life saver!

But I suppose you did a mistake in the example for number2 and 3. both has the same value as

prediction(t+1) = model1(obs(t-1), obs(t-2), …, obs(t-n))

prediction(t+2) = model2(prediction(t-1), obs(t-2), …, obs(t-n))

I believe that one of them should be

prediction(t+2) = model2(prediction(t+1), obs(t-1), …, obs(t-n-1))

Hmmm, I guess you’re right. I was thinking from the frame of the second prediction rather than the frame of both predictions.

Fixed.

Hello Jason,

What kind of Multiple output models would you recommend if we are opting for the fourth strategy?

Neural networks, such as MLPs.