### k-fold Cross Validation Does Not Work For Time Series Data and

Techniques That You Can Use Instead.

The goal of time series forecasting is to make accurate predictions about the future.

The fast and powerful methods that we rely on in machine learning, such as using train-test splits and k-fold cross validation, do not work in the case of time series data. This is because they ignore the temporal components inherent in the problem.

In this tutorial, you will discover how to evaluate machine learning models on time series data with Python. In the field of time series forecasting, this is called backtesting or hindcasting.

After completing this tutorial, you will know:

- The limitations of traditional methods of model evaluation from machine learning and why evaluating models on out of sample data is required.
- How to create train-test splits and multiple train-test splits of time series data for model evaluation in Python.
- How walk-forward validation provides the most realistic evaluation of machine learning models on time series data.

Let’s get started.

## Model Evaluation

How do we know how good a given model is?

We could evaluate it on the data used to train it. This would be invalid. It might provide insight into how the selected model works, and even how it may be improved. But, any estimate of performance on this data would be optimistic, and any decisions based on this performance would be biased.

Why?

It is helpful to take it to an extreme:

**A model that remembered the timestamps and value for each observation
would achieve perfect performance.**

All real models we prepare will report a pale version of this result.

When evaluating a model for time series forecasting, we are interested in the performance of the model on data that was not used to train it. In machine learning, we call this unseen or out of sample data.

We can do this by splitting up the data that we do have available. We use some to prepare the model and we hold back some data and ask the model to make predictions for that period. The evaluation of these predictions will provide a good proxy for how the model will perform when we use it operationally.

In applied machine learning, we often split our data into a train and a test set: the training set used to prepare the model and the test set used to evaluate it. We may even use k-fold cross validation that repeats this process by systematically splitting the data into k groups, each given a chance to be a held out model.

**These methods cannot be directly used with time series data. **

This is because they assume that there is no relationship between the observations, that each observation is independent.

This is not true of time series data, where the time dimension of observations means that we cannot randomly split them into groups. Instead, we must split data up and respect the temporal order in which values were observed.

In time series forecasting, this evaluation of models on historical data is called backtesting. In some time series domains, such as meteorology, this is called hindcasting, as opposed to forecasting.

We will look at three different methods that you can use to backtest your machine learning models on time series problems. They are:

**Train-Test split**that respect temporal order of observations.**Multiple Train-Test splits**that respect temporal order of observations.**Walk-Forward Validation**where a model may be updated each time step new data is received.

First, let’s take a look at a small, univariate time series data we will use as context to understand these three backtesting methods: the Sunspot dataset.

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## Monthly Sunspot Dataset

This dataset describes a monthly count of the number of observed sunspots for just over 230 years (1749-1983).

The units are a count and there are 2,820 observations. The source of the dataset is credited as Andrews & Herzberg (1985).

Below is a sample of the first 5 rows of data, including the header row.

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"Month","Sunspots" "1749-01",58.0 "1749-02",62.6 "1749-03",70.0 "1749-04",55.7 "1749-05",85.0 |

Below is a plot of the entire dataset taken from Data Market.

The dataset shows seasonality with large differences between seasons.

Download and learn more about the dataset here.

Download the dataset and save it into your current working directory with the filename “*sunspots.csv*“.

## Load Sunspot Dataset

We can load the Sunspot dataset using Pandas.

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# Load sunspot data from pandas import Series from matplotlib import pyplot series = Series.from_csv('sunspots.csv', header=0) print(series.head()) series.plot() pyplot.show() |

Running the example prints the first 5 rows of data.

1 2 3 4 5 6 7 |
Month 1749-01-01 00:00:00 58.0 1749-02-01 00:00:00 62.6 1749-03-01 00:00:00 70.0 1749-04-01 00:00:00 55.7 1749-05-01 00:00:00 85.0 Name: Sunspots, dtype: float64 |

The dataset is also plotted.

## Train-Test Split

You can split your dataset into training and testing subsets.

Your model can be prepared on the training dataset and predictions can be made and evaluated for the test dataset.

This can be done by selecting an arbitrary split point in the ordered list of observations and creating two new datasets. Depending on the amount of data you have available and the amount of data required, you can use splits of 50-50, 70-30 and 90-10.

It is straightforward to split data in Python.

After loading the dataset as a Pandas Series, we can extract the NumPy array of data values. The split point can be calculated as a specific index in the array. All records up to the split point are taken as the training dataset and all records from the split point to the end of the list of observations are taken as the test set.

Below is an example of this in Python using a split of 66-34.

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from pandas import Series series = Series.from_csv('sunspots.csv', header=0) X = series.values train_size = int(len(X) * 0.66) train, test = X[0:train_size], X[train_size:len(X)] print('Observations: %d' % (len(X))) print('Training Observations: %d' % (len(train))) print('Testing Observations: %d' % (len(test))) |

Running the example prints the size of the loaded dataset and the size of the train and test sets created from the split.

1 2 3 |
Observations: 2820 Training Observations: 1861 Testing Observations: 959 |

We can make this visually by plotting the training and test sets using different colors.

1 2 3 4 5 6 7 8 9 10 11 12 |
from pandas import Series from matplotlib import pyplot series = Series.from_csv('sunspots.csv', header=0) X = series.values train_size = int(len(X) * 0.66) train, test = X[0:train_size], X[train_size:len(X)] print('Observations: %d' % (len(X))) print('Training Observations: %d' % (len(train))) print('Testing Observations: %d' % (len(test))) pyplot.plot(train) pyplot.plot([None for i in train] + [x for x in test]) pyplot.show() |

Running the example plots the training dataset as blue and the test dataset as green.

Using a train-test split method to evaluate machine learning models is fast. Preparing the data is simple and intuitive and only one model is created and evaluated.

It is useful when you have a large amount of data so that both training and tests sets are representative of the original problem.

Next, we will look at repeating this process multiple times.

## Multiple Train-Test Splits

We can repeat the process of splitting the time series into train and test sets multiple times.

This will require multiple models to be trained and evaluated, but this additional computational expense will provide a more robust estimate of the expected performance of the chosen method and configuration on unseen data.

We could do this manually by repeating the process described in the previous section with different split points.

Alternately, the scikit-learn library provides this capability for us in the *TimeSeriesSplit* object.

You must specify the number of splits to create and the *TimeSeriesSplit* to return the indexes of the train and test observations for each requested split.

The total number of training and test observations are calculated each split iteration (*i*) as follows:

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training_size = i * n_samples / (n_splits + 1) + n_samples % (n_splits + 1) test_size = n_samples / (n_splits + 1) |

Where *n_samples* is the total number of observations and *n_splits* is the total number of splits.

Let’s make this concrete with an example. Assume we have 100 observations and we want to create 2 splits.

For the first split, the train and test sizes would be calculated as:

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train = i * n_samples / (n_splits + 1) + n_samples % (n_splits + 1) train = 1 * 100 / (2 + 1) + 100 % (2 + 1) train = 33.3 or 33 test = n_samples / (n_splits + 1) test = 100 / (2 + 1) test = 33.3 or 33 |

Or the first 33 records are used for training and the next 33 records are used for testing.

The second split is calculated as follows:

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train = i * n_samples / (n_splits + 1) + n_samples % (n_splits + 1) train = 2 * 100 / (2 + 1) + 100 % (2 + 1) train = 66.6 or 67 test = n_samples / (n_splits + 1) test = 100 / (2 + 1) test = 33.3 or 33 |

Or, the first 67 records are used for training and the remaining 33 records are used for testing.

You can see that the test size stays consistent. This means that performance statistics calculated on the predictions of each trained model will be consistent and can be combined and compared. It provides an apples-to-apples comparison.

What differs is the number of records used to train the model each split, offering a larger and larger history to work with. This may make an interesting aspect of the analysis of results. Alternately, this too could be controlled by holding the number of observations used to train the model consistent and only using the same number of the most recent (last) observations in the training dataset each split to train the model, 33 in this contrived example.

Let’s look at how we can apply the TimeSeriesSplit on our sunspot data.

The dataset has 2,820 observations. Let’s create 3 splits for the dataset. Using the same arithmetic above, we would expect the following train and test splits to be created:

**Split 1**: 705 train, 705 test**Split 2**: 1,410 train, 705 test**Split 3**: 2,115 train, 705 test

As in the previous example, we will plot the train and test observations using separate colors. In this case, we will have 3 splits, so that will be 3 separate plots of the data.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 |
from pandas import Series from sklearn.model_selection import TimeSeriesSplit from matplotlib import pyplot series = Series.from_csv('sunspots.csv', header=0) X = series.values splits = TimeSeriesSplit(n_splits=3) pyplot.figure(1) index = 1 for train_index, test_index in splits.split(X): train = X[train_index] test = X[test_index] print('Observations: %d' % (len(train) + len(test))) print('Training Observations: %d' % (len(train))) print('Testing Observations: %d' % (len(test))) pyplot.subplot(310 + index) pyplot.plot(train) pyplot.plot([None for i in train] + [x for x in test]) index += 1 pyplot.show() |

Running the example prints the number and size of the train and test sets for each split.

We can see the number of observations in each of the train and test sets for each split match the expectations calculated using the simple arithmetic above.

1 2 3 4 5 6 7 8 9 |
Observations: 1410 Training Observations: 705 Testing Observations: 705 Observations: 2115 Training Observations: 1410 Testing Observations: 705 Observations: 2820 Training Observations: 2115 Testing Observations: 705 |

The plot also shows the 3 splits and the growing number of total observations in each subsequent plot.

Using multiple train-test splits will result in more models being trained, and in turn, a more accurate estimate of the performance of the models on unseen data.

A limitation of the train-test split approach is that the trained models remain fixed as they are evaluated on each evaluation in the test set.

This may not be realistic as models can be retrained as new daily or monthly observations are made available. This concern is addressed in the next section.

## Walk Forward Validation

In practice, we very likely will retrain our model as new data becomes available.

This would give the model the best opportunity to make good forecasts at each time step. We can evaluate our machine learning models under this assumption.

There are few decisions to make:

1. **Minimum Number of Observations**. First, we must select the minimum number of observations required to train the model. This may be thought of as the window width if a sliding window is used (see next point).

2. **Sliding or Expanding Window**. Next, we need to decide whether the model will be trained on all data it has available or only on the most recent observations. This determines whether a sliding or expanding window will be used.

After a sensible configuration is chosen for your test-setup, models can be trained and evaluated.

- Starting at the beginning of the time series, the minimum number of samples in the window is used to train a model.
- The model makes a prediction for the next time step.
- The prediction is stored or evaluated against the known value.
- The window is expanded to include the known value and the process is repeated (go to step 1.)

Because this methodology involves moving along the time series one-time step at a time, it is often called Walk Forward Testing or Walk Forward Validation. Additionally, because a sliding or expanding window is used to train a model, this method is also referred to as Rolling Window Analysis or a Rolling Forecast.

This capability is currently not available in scikit-learn, although you could contrive the same effect with a carefully configured TimeSeriesSplit.

Below is an example of how to split data into train and test sets using the Walk Forward Validation method.

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from pandas import Series from matplotlib import pyplot series = Series.from_csv('sunspots.csv', header=0) X = series.values n_train = 500 n_records = len(X) for i in range(n_train, n_records): train, test = X[0:i], X[i:i+1] print('train=%d, test=%d' % (len(train), len(test))) |

Running the example simply prints the size of the training and test sets created. We can see the train set expanding teach time step and the test set fixed at one time step ahead.

Within the loop is where you would train and evaluate your model.

1 2 3 4 5 6 7 8 9 10 11 |
train=500, test=1 train=501, test=1 train=502, test=1 train=503, test=1 train=504, test=1 ... train=2815, test=1 train=2816, test=1 train=2817, test=1 train=2818, test=1 train=2819, test=1 |

You can see that many more models are created.

This has the benefit again of providing a much more robust estimation of how the chosen modeling method and parameters will perform in practice. This improved estimate comes at the computational cost of creating so many models.

This is not expensive if the modeling method is simple or dataset is small (as in this example), but could be an issue at scale. In the above case, 2,820 models would be created and evaluated.

As such, careful attention needs to be paid to the window width and window type. These could be adjusted to contrive a test harness on your problem that is significantly less computationally expensive.

Walk-forward validation is the gold standard of model evaluation. It is the k-fold cross validation of the time series world and is recommended for your own projects.

## Further Reading

- sklearn.model_selection.TimeSeriesSplit API Documentation
- Rolling-Window Analysis of Time-Series Models for more on rolling windows.
- Backtesting on Wikipedia to learn more about backtesting.

## Summary

In this tutorial, you discovered how to backtest machine learning models on time series data with Python.

Specifically, you learned:

- About the importance of evaluating the performance of models on unseen or out-of-sample data.
- How to create train-test splits of time series data, and how to create multiple such splits automatically.
- How to use walk-forward validation to provide the most realistic test harness for evaluating your models.

Do you have any questions about evaluating your time series model or about this tutorial?

Ask your questions in the comments below and I will do my best to answer.

Jason,

second link from “Further Reading” should probably point to mathworks.com instead of amathworks.com, which is not found

Thanks Michael, fixed!

Many thanks, it is short and full of information.

I’m glad to hear you found it useful.

For walking forward validation it will consume a lot of time to validate after each single interation and even results won’t be much different between each iteration. Better way would be to increase h steps in each iteration and divide train and test data in that manner. Train data could be added for each h steps and test data could be for h steps for each iteration rather than single observation. This is just my sugestion from my point of view. No hard rules here.

Hi Shreyak,

Yes, that would be a sampled version of walk-forward validation, a subset.

This is pretty much what the multiple train-test splits provides in the sklearn TimeSeriesSplit object – if I understood you correctly.

My query is related to walk forward validation:

Suppose a time series forecasting model is trained with a set of data and gives a good evaluation with test-set in time_range-1 and model produces a function F1. For time_range-2 and another set of training and testing data model generates function F2. Similarly for time_range-N the model generate Function FN. How the different models when combined and implemented forecast the result based on forecasting function based of local model and not the combined model of all time range model, which may possibly be producing error in forecasting.

Hi Saurabh,

Sorry, I don’t quite understand the last part of your question. Are you able to restate it?

I am just going through your posts on Time Series. Are you using any particular resource as a reference material for these things ?

A shelf of textbooks mainly 🙂

Hi Jason

Thanks so much for this in-depth post. My question is:

Which performance measure should we use in selecting the model?

For example, if I add one test subset at a time in a binary(1, 0) classification problem, the accuracy would be either 1 or 0.

In this case, how should I select a model? Should I use other measures instead?

I am building my model as stock price classification where 1 represents up, and 0 means down. I use TimeSeriesSplit and divide into T (sample size) – m (rolling window) + 1.

Thanks a lot and I look forward to listening your insights!

Hi Ian,

This is a problem specific question.

Perhaps classification accuracy on the out of sample dataset would be a good way to pick a model in your case?

Jason,

Thanks so much for answering.

If we walk one step forward every time just like what you illustrate in the Walk Forward Validation, doesn’t that mean the test dataset come from out of sample?

Hope this is not too problem specific, and thanks again in advance.

Hi Ian,

Walk forward validation is a method for estimating the skill of the model on out of sample data. We contrive out of sample and each time step one out of sample observation becomes in-sample.

We can use the same model in ops, as long as the walk-forward is performed each time a new observation is received.

Does that make sense?

Thanks Jason for an informative post!

If the time series is very long, e.g. minute values for 10 years, it will take a very long time to train. As I understand you, another way to do this is to fix the length of the training set, e.g. 2 years, but just move it, like this:

Split 1: year 1+2 train, year 3 test

Split 2: year 2+3 train, year 4 test

…

Split 8: year 8+9 train, year 10 test

Is this correct and valid?

Sounds good to me.

Also consider how valuable the older data is to fit the model. It is possible data from 10 years ago is not predictive of today, depends on the problem of course.

Thank you for your post Jason.

I would like to ask you which model we will chose if we have implementation purpose.

In fact, for example if the time series is hour values of 3 years, walk forward could be applied in this way:

Split 1: year 1 train, year 2 test and we will get model1, error of prediction 1

Split 2: year 1+2 train, year 3 test and we will get model2, error of prediction 2

which model should we then choose ?

Great question.

Pick the model that best represents the performance/capability required for your application.

Jason,

I think that when Marwa mentioned ‘models’, she meant applying the same model (such as ARMA) on different data (corresponding to the expanding window).

I think that the walk-forward method, just like k-fold CV, gives an array of metrics whose mean somehow corresponds to the true skill of the model.

I think that when this mean is evaluated, the model should be trained on the entire dataset (check Practical Time Series Forecasting with R- Shmueli ) just like with K-fold CV.

Please correct me if I am wrong.

Regards

Walk forward validation will give a mean estimate of the skill of the model.

Walk forward validation requires some portion of the data be used to fit the model and some to evaluate it, and the portion for evaluation is stepped to be made available to training as we “walk forward”. We do not train on the entire training dataset, if we did and made a prediction, what would we compare the prediction to in order to estimate the skill of the model?

Dear Jason,

Thanks so much for this in-depth post. My question is:

If my time series are discontinuous(such as two weeks in March and two weeks in September), How should I divide the data set?

If I use time series as supervised learning, it could lead to a sample containing data for March and September.

This question has puzzled me for a long time and I look forward to hearing from you.

I don’t have a good answer.

Perhaps try to fill in the missing time with 0s or nans.

Perhaps try to ignore the missing blocks.

Perhaps focus on building a model at a lower scale (month-wise).

Hey Jason, can you comment on Rob Hyndman’s paper stating that CV can, in fact, be used for time-series data (https://robjhyndman.com/papers/cv-wp.pdf)?

As a follow-up, would this code work in a time-series context: http://machinelearningmastery.com/use-keras-deep-learning-models-scikit-learn-python/

Thank you in advance for your guidance!

Try it and see.

I hope to when I have a pocket of time.

Is there a way to store the model fit values in such a way that we can update the model after every iteration instead of recreate an entirely new one?

My dataset has 55,000 samples and I want to run a test set of 5,000, but recreating 5,000 models would take roughly 80 hours. Thanks.

Yes, here’s how to save the model:

http://machinelearningmastery.com/save-arima-time-series-forecasting-model-python/

Thanks for responding so quickly! Say I trained a model, saved it, ran it on a test sample x1 and then iterated to test the next test sample x2. Once I load the old model, how would I add sample x1 to update the model, potentially making it perform better?

That way I am always predicting sample n+1 with a train set from 0 to n without always creating a new model for the 5000 iterations.

You would have to fit the model on just the new data or on a combination of the new and old data.

This can be done with a new model or by updating the existing model.

I do not have an example with ARIMA, but I do have examples with LSTMs here:

http://machinelearningmastery.com/update-lstm-networks-training-time-series-forecasting/