# Moving Average Smoothing for Data Preparation and Time Series Forecasting in Python

Last Updated on August 15, 2020

Moving average smoothing is a naive and effective technique in time series forecasting.

It can be used for data preparation, feature engineering, and even directly for making predictions.

In this tutorial, you will discover how to use moving average smoothing for time series forecasting with Python.

After completing this tutorial, you will know:

• How moving average smoothing works and some expectations of your data before you can use it.
• How to use moving average smoothing for data preparation and feature engineering.
• How to use moving average smoothing to make predictions.

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• Updated Apr/2019: Updated the link to dataset.

Moving Average Smoothing for Data Preparation, Feature Engineering, and Time Series Forecasting with Python
Photo by Bureau of Land Management, some rights reserved.

## Moving Average Smoothing

Smoothing is a technique applied to time series to remove the fine-grained variation between time steps.

The hope of smoothing is to remove noise and better expose the signal of the underlying causal processes. Moving averages are a simple and common type of smoothing used in time series analysis and time series forecasting.

Calculating a moving average involves creating a new series where the values are comprised of the average of raw observations in the original time series.

A moving average requires that you specify a window size called the window width. This defines the number of raw observations used to calculate the moving average value.

The “moving” part in the moving average refers to the fact that the window defined by the window width is slid along the time series to calculate the average values in the new series.

There are two main types of moving average that are used: Centered and Trailing Moving Average.

### Centered Moving Average

The value at time (t) is calculated as the average of raw observations at, before, and after time (t).

For example, a center moving average with a window of 3 would be calculated as:

This method requires knowledge of future values, and as such is used on time series analysis to better understand the dataset.

A center moving average can be used as a general method to remove trend and seasonal components from a time series, a method that we often cannot use when forecasting.

### Trailing Moving Average

The value at time (t) is calculated as the average of the raw observations at and before the time (t).

For example, a trailing moving average with a window of 3 would be calculated as:

Trailing moving average only uses historical observations and is used on time series forecasting.

It is the type of moving average that we will focus on in this tutorial.

### Data Expectations

Calculating a moving average of a time series makes some assumptions about your data.

It is assumed that both trend and seasonal components have been removed from your time series.

This means that your time series is stationary, or does not show obvious trends (long-term increasing or decreasing movement) or seasonality (consistent periodic structure).

There are many methods to remove trends and seasonality from a time series dataset when forecasting. Two good methods for each are to use the differencing method and to model the behavior and explicitly subtract it from the series.

Moving average values can be used in a number of ways when using machine learning algorithms on time series problems.

In this tutorial, we will look at how we can calculate trailing moving average values for use as data preparation, feature engineering, and for directly making predictions.

Before we dive into these examples, let’s look at the Daily Female Births dataset that we will use in each example.

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## Daily Female Births Dataset

This dataset describes the number of daily female births in California in 1959.

The units are a count and there are 365 observations. The source of the dataset is credited to Newton (1988).

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

Below is a plot of the entire dataset.

Daily Female Births Dataset

This dataset is a good example for exploring the moving average method as it does not show any clear trend or seasonality.

### Load the Daily Female Births Dataset

Download the dataset and place it in the current working directory with the filename “daily-total-female-births.csv“.

The snippet below loads the dataset as a Series, displays the first 5 rows of the dataset, and graphs the whole series as a line plot.

Running the example prints the first 5 rows as follows:

Below is the displayed line plot of the loaded data.

Daily Female Births Dataset Plot

## Moving Average as Data Preparation

Moving average can be used as a data preparation technique to create a smoothed version of the original dataset.

Smoothing is useful as a data preparation technique as it can reduce the random variation in the observations and better expose the structure of the underlying causal processes.

The rolling() function on the Series Pandas object will automatically group observations into a window. You can specify the window size, and by default a trailing window is created. Once the window is created, we can take the mean value, and this is our transformed dataset.

New observations in the future can be just as easily transformed by keeping the raw values for the last few observations and updating a new average value.

To make this concrete, with a window size of 3, the transformed value at time (t) is calculated as the mean value for the previous 3 observations (t-2, t-1, t), as follows:

For the Daily Female Births dataset, the first moving average would be on January 3rd, as follows:

Below is an example of transforming the Daily Female Births dataset into a moving average with a window size of 3 days, chosen arbitrarily.

Running the example prints the first 10 observations from the transformed dataset.

We can see that the first 2 observations will need to be discarded.

The raw observations are plotted (blue) with the moving average transform overlaid (red).

Moving Average Transform

To get a better idea of the effect of the transform, we can zoom in and plot the first 100 observations.

Zoomed Moving Average Transform

Here, you can clearly see the lag in the transformed dataset.

Next, let’s take a look at using the moving average as a feature engineering method.

## Moving Average as Feature Engineering

The moving average can be used as a source of new information when modeling a time series forecast as a supervised learning problem.

In this case, the moving average is calculated and added as a new input feature used to predict the next time step.

First, a copy of the series must be shifted forward by one time step. This will represent the input to our prediction problem, or a lag=1 version of the series. This is a standard supervised learning view of the time series problem. For example:

Next, a second copy of the series needs to be shifted forward by one, minus the window size. This is to ensure that the moving average summarizes the last few values and does not include the value to be predicted in the average, which would be an invalid framing of the problem as the input would contain knowledge of the future being predicted.

For example, with a window size of 3, we must shift the series forward by 2 time steps. This is because we want to include the previous two observations as well as the current observation in the moving average in order to predict the next value. We can then calculate the moving average from this shifted series.

Below is an example of how the first 5 moving average values are calculated. Remember, the dataset is shifted forward 2 time steps and as we move along the time series, it takes at least 3 time steps before we even have enough data to calculate a window=3 moving average.

Below is an example of including the moving average of the previous 3 values as a new feature, as wellas a lag-1 input feature for the Daily Female Births dataset.

Running the example creates the new dataset and prints the first 10 rows.

We can see that the first 3 rows cannot be used and must be discarded. The first row of the lag1 dataset cannot be used because there are no previous observations to predict the first observation, therefore a NaN value is used.

The next section will look at how to use the moving average as a naive model to make predictions.

## Moving Average as Prediction

The moving average value can also be used directly to make predictions.

It is a naive model and assumes that the trend and seasonality components of the time series have already been removed or adjusted for.

The moving average model for predictions can easily be used in a walk-forward manner. As new observations are made available (e.g. daily), the model can be updated and a prediction made for the next day.

We can implement this manually in Python. Below is an example of the moving average model used in a walk-forward manner.

Running the example prints the predicted and expected value each time step moving forward, starting from time step 4 (1959-01-04).

Finally, the mean squared error (MSE) is reported for all predictions made.

The example ends by plotting the expected test values (blue) compared to the predictions (red).

Moving Average Predictions

Again, zooming in on the first 100 predictions gives an idea of the skill of the 3-day moving average predictions.

Note the window width of 3 was chosen arbitrary and was not optimized.

Zoomed Moving Average Predictions

This section lists some resources on smoothing moving averages for time series analysis and time series forecasting that you may find useful.

## Summary

In this tutorial, you discovered how to use moving average smoothing for time series forecasting with Python.

Specifically, you learned:

• How moving average smoothing works and the expectations of time series data before using it.
• How to use moving average smoothing for data preparation in Python.
• How to use moving average smoothing for feature engineering in Python.
• How to use moving average smoothing to make predictions in Python.

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### 56 Responses to Moving Average Smoothing for Data Preparation and Time Series Forecasting in Python

1. Petr S. February 6, 2018 at 12:12 pm #

Hello,
thank you for all the articles and books, they are great.

Here I am not sure about one thing. Is the shift(width - 1) correct? In this code:

width = 3
lag1 = df.shift(1)
lag3 = df.shift(width – 1)

I think it should be always shift(2), no matter what is the width, because we do the shift to skip only one single current value (so it is not included in the window later).

For example for an input series df=[1,2,3,4,5,6,7] and width=4 there will be shift(3) that leads to:

t t+1 shift(3) mean
– 1 – –
1 2 – –
2 3 – –
3 4 1 –
4 5 2 –
5 6 3 –
6 7 4 2,5

but the mean 2,5 belongs to t=5, not t=6, right?

But of course maybe I just didn’t understand it correctly.

2. Vishal Jadhav March 13, 2018 at 2:53 pm #

What if dataset does not continuous sequences, meaning observations for some days are missing. In that case will rolling function be able to identify discontinuous sequences and create mean accordingly?

3. joshua wojnas April 18, 2018 at 7:54 am #

I am surprised you do not have pd.rolling_apply because some people might want rolling standard deviation or time domain statistics. “A robust, real-time control scheme for multifunction myoelectric control”

4. Matt February 26, 2019 at 8:59 am #

I would like to piggyback onto Petr’s comment and say that I believe what is desired is a change from this

lag1 = df.shift(1)

to this

lag1 = df.shift(window – 2)

I also agree with him in that your tutorial’s are great.

5. CSPAN March 6, 2019 at 6:57 pm #

Great article. I am trying to use it with a CPU utilization of a node. I am interested in finding out how long usage stayed at 90% or higher. How can I achieve it. Can you please provide some pointer.

6. Kalim April 27, 2019 at 12:03 am #

Great article….

if ynew == 2:
print (“a movement is Identified”)

else:
print (“Stable Condition”)

time.sleep(0.5)

how can I take moving average on (ynew== 2) in this condition.

• Jason Brownlee April 27, 2019 at 6:33 am #

I don’t follow sorry, what do you mean exactly?

7. Fahim May 18, 2019 at 4:34 pm #

In the case of trailing moving average calculation, why do we need to shift the data? I mean, the rolling window will always be based on the current and past value, not the future values. So, even without shifting the data, wouldn’t we get valid results? In the case above, we would get valid mean from index 2 (in the feature engineering section).
Am I missing something?

• Jason Brownlee May 19, 2019 at 8:00 am #

Shifting has to do with transforming a sequence into a supervised learning problem:
https://machinelearningmastery.com/time-series-forecasting-supervised-learning/

• Claire Taylor May 31, 2019 at 12:24 pm #

Speaking of shifting-time, I think you can help my current work about forecasting time series with Generalized Linear Model. The reason why I use GLM method is I consider temperature (T(t),T(t-1),…), rainfall (R(t),R(t-1),…), humidity (H(t),H(t-1),…), and past event (Y(t-1),…) as the explanatory variables to current event (Y(t)). Let’s say Y(t)~T(t)+R(t)+H(t)+Y(t-1)+T(t-1)+R(t-1)+H(t-1) for lag-1. By comparing plots, I can see that my prediction at time t consistently more looks alike the real obervation at time t-1. I say it’s like shifting (delay) one time-step. I have no idea why this is happening. I think it’s related to “time series” as random walk process. Can you explain me why? Thank you in advance, Jason. I’m a student by the way, so I do really appreciate your help.

8. Tester123 May 25, 2019 at 5:53 am #

Hey, how do you decide on the window size in a time series data? Is there any publication on this front that I can refer to?

• Jason Brownlee May 25, 2019 at 7:55 am #

Empirical evaluation.

Test a suite of window sizes and see what works well.best on your specific dataset.

9. isaac Augustus May 30, 2019 at 8:23 pm #

Hi professor thanks for this priceless knowledge you impart on here . For a moving average as one of the features to an LSTM model. Do I need to still remove trend or seasonality?
And does this work
MA30 = data[‘close’].rolling(30).mean().
I thought that should work for moving average and also for .var.
Assuming there was already a dataset with close as a column for closing price.
Thank you

• Jason Brownlee May 31, 2019 at 7:46 am #

Yes, it would be a good idea to make the series stationary first.

10. Anand January 17, 2020 at 4:08 am #

1 doubt from the above article. Kindly reply.

> A center moving average can be used as a general method to remove trend and seasonal components from a time series, a method that we often cannot use when forecasting.

> It is a naive model (for prediction) and assumes that the trend and seasonality components of the time series have already been removed or adjusted for.

1) Why to remove trend and seasonality components for prediction?

2) If seasonality has to be removed before forecasting, then why center moving average cannot be used? As per the above article, center moving average would remove seasonality!

11. jokonug February 2, 2020 at 2:05 am #

Thanks for the article, it’s so enlighthening
just interesting the smoothing with moving average for the data preparation.
let say later on, it will use certain algorithm to make prediction.
whether the smoothing applied to the whole raw data or for training dataset only?

• Jason Brownlee February 2, 2020 at 6:27 am #

All input to the model, training and any new data that is fed in.

12. Alex April 8, 2020 at 12:37 am #

When I average the dataset every 60 seconds, it becomes a dataset every minute. When some important information is hidden in seconds, but it disappears after averaging every minute, how can I use the information of seconds to improve the prediction every minute? Thank you

• Jason Brownlee April 8, 2020 at 7:54 am #

It can smooth out the variance seen at the “seconds” level of detail. It may or may not help – depends on your data and choice of model.

13. Leo May 23, 2020 at 3:04 am #

Hi Jason, great tutorials for an introduction to time series. However, this time I don’t follow the logic behind shifting the time series for feature engineering. Surely if you want to make use of all the data you have at time t (to predict something and compare it with what you have at the non-shifted time t+1 ). Therefore, I would expect that if you fix your window size to say, n, whenever you have the same number of usable rows at time t, you would be able to create an average and use it to predict. This would mean that you only need to shift the lag_mean by 1 and the .rolling function will do it’s thing, fill the column with nans until you have n measurements at time t and we could calculate their mean.

Am I missing something obvious?

• Jason Brownlee May 23, 2020 at 6:28 am #

We may have to drop some obs in order to construct the summary input feature.

Not sure I follow the problem you see, sorry.

14. Arunkumar July 6, 2020 at 9:42 pm #

i want in r code

• Jason Brownlee July 7, 2020 at 6:36 am #

Sorry, I don’t have an example of this tutorial in R.

15. user1 July 14, 2020 at 8:14 am #

Dear Jason,
Thank you for the great post.

If I use MA for smoothing before training an MLP model with lags, should I reverse my smoothing before calculating the MSE and compare the predicted value to the original value before smoothing?

for example, the way we would when we transform the data and reverse the transformation before calculating the MSE?

If so, how could one inverse a smoothing MA?

I really appreciate your response and opinion

• Jason Brownlee July 14, 2020 at 1:38 pm #

Yes, any transformed applied to the target variable must be reversed before evaluating or using predictions from the model.

A moving average is not reversible. It could be used if you prefer to operate on a moving average as input only, as a supplement input, or if you prefer to operate with smoothed version of the data.

16. user1 July 14, 2020 at 10:40 pm #

Jason, the only way my MLP model would give low (good enough) MSE is if I include MA data smoothing as part of data preparation.

But, I am concerned that this no longer represents my actual data.

• Jason Brownlee July 15, 2020 at 8:19 am #

That is correct, you will be choosing to model a diffrent version or framing of the problem.

Start with a strong definition of your problem/project then consider if the new framing of the problem can solve it.

17. Fauzan July 20, 2020 at 10:14 am #

18. Hammad Malik December 24, 2020 at 10:17 pm #

hi Jason. thanks for nice post.
i am using your code to calculate the mean using sliding window for Moving Average as Feature Engineering. now i also wants to calculate variance and standard deviation using sliding window. could you guide me how i can go with that. thanks

df = DataFrame(series.values)
width = 3
lag1 = df.shift(1)
lag3 = df.shift(width – 1)
window = lag3.rolling(window=width)
means = window.mean()
dataframe = concat([means, lag1, df], axis=1)
dataframe.columns = [‘mean’, ‘t-1’, ‘t+1’]

• Jason Brownlee December 25, 2020 at 5:22 am #

You’re welcome.

Perhaps you can adapt the examples in the above tutorial directly?

19. Alghamdi February 12, 2021 at 8:56 am #

Dear Jason,
Thank you for the great post.

Can I use the moving average as data preparation before making stationary?

Thank you.

20. Misha March 3, 2021 at 8:45 am #

Hi I am trying to use moving average for 15 days by including previous 15 days data and prediction on the next day.

21. DS April 10, 2021 at 8:49 pm #

I have one question reg using MA for making a prediction.
How would I make a prediction on a test (out of sample data) say 12 months out of sample data using training data (say 24 months before the test data).

I cant use the test data to create a MA and predict on the test data. I am only allowed to use the training data.How can then I make predictions using MA?

• Jason Brownlee April 11, 2021 at 4:52 am #

It does not sound like you have enough data.

Perhaps fit your model on the first 12 observations and prediction the next 12 observations.

22. Erik Nugi Ginanjar April 17, 2021 at 4:12 pm #

Hi Jason,

I’ve read your sample, and as other sample. how i can actual get predicted values with actual input. example i want to predict next 30 days from last current data.

23. Om May 4, 2021 at 10:29 am #

Hello Jason,
Great article. I follow your emails regularly. Seems like R has a very concise method for calculating the rolling moving average of a moving average. Is there such a method in python and how would one implement it. Thanks for comments

• Jason Brownlee May 5, 2021 at 6:05 am #

There is the method described in the above tutorial, there may be other methods. I’m not sure of an equivalent for a method you’re describing in R.

24. Evelyn May 26, 2021 at 5:47 pm #

Hi Jason, thanks for this amazing post. I have a question Do you have any post where you explain how estimate seasonality with moving average filter. I am reading the book Introduction to Time Series and Forecasting and the authors explains mathematically how use the moving average filter for seasonality. However, it is a little complicated understand what is happenning

25. Neha June 16, 2021 at 6:15 pm #

Hi Jason,

Thanks for an informative post!!! I have a question related to a different dataset. It is a store sales data with yearly seasonal pattern. After de-seasonalizing, how to decide on window size if rolling window average is to be used for prediction?

Thanks

• Jason Brownlee June 17, 2021 at 6:15 am #

Perhaps evaluate different window sizes and use whatever works best for your data and model.

26. Saeideh August 10, 2021 at 12:10 am #

I have two questions.
1. How can I use this method for multi-step prediction? Should I continue predicting for next time steps based on my predictions which obviously contain some errors?

2. Is this logical to use SMA for preprocessing (to remove trend and/or seasonality) and also for prediction?

• Jason Brownlee August 10, 2021 at 5:30 am #

You can extrapolate the method beyond the dataset as a type of prediction, perhaps you can adapt the above code for this purpose.

Perhaps try it and compare to using the raw dataset.

27. Mourad January 5, 2022 at 12:19 am #

‘A center moving average can be used as a general method to remove trend and seasonal components from a time series, a method that we often cannot use when forecasting.’

I don’t understand how it can help remove trend and seasonal component. Doesnt the average will also be impacted by the trend and seasonality?