Differencing is a popular and widely used data transform for time series.

In this tutorial, you will discover how to apply the difference operation to your time series data with Python.

After completing this tutorial, you will know:

- About the differencing operation, including the configuration of the lag difference and the difference order.
- How to develop a manual implementation of the differencing operation.
- How to use the built-in Pandas differencing function.

Let’s get started.

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

## Why Difference Time Series Data?

Differencing is a method of transforming a time series dataset.

It can be used to remove the series dependence on time, so-called temporal dependence. This includes structures like trends and seasonality.

Differencing can help stabilize the mean of the time series by removing changes in the level of a time series, and so eliminating (or reducing) trend and seasonality.

— Page 215, Forecasting: principles and practice

Differencing is performed by subtracting the previous observation from the current observation.

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difference(t) = observation(t) - observation(t-1) |

In this way, a series of differences can be calculated.

### Lag Difference

Taking the difference between consecutive observations is called a lag-1 difference.

The lag difference can be adjusted to suit the specific temporal structure.

For time series with a seasonal component, the lag may be expected to be the period (width) of the seasonality.

### Difference Order

Temporal structure may still exist after performing a differencing operation, such as in the case of a nonlinear trend.

As such, the process of differencing can be repeated more than once until all temporal dependence has been removed.

The number of times that differencing is performed is called the difference order.

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## Shampoo Sales Dataset

This dataset describes the monthly number of sales of shampoo over a 3 year period.

The units are a sales count and there are 36 observations. The original dataset is credited to Makridakis, Wheelwright, and Hyndman (1998).

The example below loads and creates a plot of the loaded dataset.

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from pandas import read_csv from pandas import datetime from matplotlib import pyplot def parser(x): return datetime.strptime('190'+x, '%Y-%m') series = read_csv('shampoo-sales.csv', header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser) series.plot() pyplot.show() |

Running the example creates the plot that shows a clear linear trend in the data.

## Manual Differencing

We can difference the dataset manually.

This involves developing a new function that creates a differenced dataset. The function would loop through a provided series and calculate the differenced values at the specified interval or lag.

The function below named *difference()* implements this procedure.

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# create a differenced series def difference(dataset, interval=1): diff = list() for i in range(interval, len(dataset)): value = dataset[i] - dataset[i - interval] diff.append(value) return Series(diff) |

We can see that the function is careful to begin the differenced dataset after the specified interval to ensure differenced values can, in fact, be calculated. A default interval or lag value of 1 is defined. This is a sensible default.

One further improvement would be to also be able to specify the order or number of times to perform the differencing operation.

The example below applies the manual *difference()* function to the Shampoo Sales dataset.

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from pandas import read_csv from pandas import datetime from pandas import Series from matplotlib import pyplot def parser(x): return datetime.strptime('190'+x, '%Y-%m') # create a differenced series def difference(dataset, interval=1): diff = list() for i in range(interval, len(dataset)): value = dataset[i] - dataset[i - interval] diff.append(value) return Series(diff) series = read_csv('shampoo-sales.csv', header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser) X = series.values diff = difference(X) pyplot.plot(diff) pyplot.show() |

Running the example creates the differenced dataset and plots the result.

## Automatic Differencing

The Pandas library provides a function to automatically calculate the difference of a dataset.

This *diff()* function is provided on both the Series and DataFrame objects.

Like the manually defined difference function in the previous section, it takes an argument to specify the interval or lag, in this case called the *periods*.

The example below demonstrates how to use the built-in difference function on the Pandas Series object.

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from pandas import read_csv from pandas import datetime from matplotlib import pyplot def parser(x): return datetime.strptime('190'+x, '%Y-%m') series = read_csv('shampoo-sales.csv', header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser) diff = series.diff() pyplot.plot(diff) pyplot.show() |

As in the previous section, running the example plots the differenced dataset.

A benefit of using the Pandas function, in addition to requiring less code, is that it maintains the date-time information for the differenced series.

## Summary

In this tutorial, you discovered how to apply the difference operation to time series data with Python.

Specifically, you learned:

- About the difference operation, including the configuration of lag and order.
- How to implement the difference transform manually.
- How to use the built-in Pandas implementation of the difference transform.

Do you have any questions about differencing, or about this post?

Ask your questions in the comments below.

Hi there, here is a recent work on time series that gives a time series a symbolic representation.

https://arxiv.org/ftp/arxiv/papers/1611/1611.01698.pdf

Thanks for sharing.

Have a question. What if the difference is negative?

Some differences will be positive, some negative.

Hi, which will be the most pythonic way to set the negative difeferece as zero. Let say that I have some bookings for t+1 and a forecast.

My approach is make it work first, then make it readable.

Are difference functions only useful to remove structures like trends and seasonality,

or can they also be used to build features from trends in data sets?

What other techniques are available to use trends and seasonality in a constructive way in time series predictions?

You can use the transformed variables and extracted structures as features, but check that they lift the skill of the model.

See this post on feature engineering in time series forecasting:

http://machinelearningmastery.com/basic-feature-engineering-time-series-data-python/

Thanks for these posts, Dr. Brownlee! I like the picture of the beach

Thanks Chris.

Hi there,I log on to your new stuff named “How to Difference a Time Series Dataset with Python – Machine Learning Mastery” regularly.Your humoristic style is awesome, keep up the good work! And you can look our website about proxy list.

Thanks.

Thank you for valuable insights. Could you please explain how would it be possible to take the third or second difference ?

You apply the difference operation to the already differenced series.

for “value = int(dataset[i])-int(dataset[i-interval])”

why it shows “TypeError: only length-1 arrays can be converted to Python scalars”

thanks in advance！

Perhaps ensure that you have copied all of the code from the example?

Hi Jason, thanks for posting this, but I’m curious what to do about the NAs after using the diff() function? I’m guessing that data should just be removed? Or should they just be imputed?

Removed.

I TRIED TO RUN YOUR CODE, BUT I RECEIVED THIS MASSAGE

(data_string, format))

ValueError: time data ‘190Sales of shampoo over a three year period’ does not match format ‘%Y-%m’

THANK YOU IN ADVANCE

It looks like you might not have deleted the file footer or downloaded the data in a different format.

Here is a direct link to the data file ready to use:

https://raw.githubusercontent.com/jbrownlee/Datasets/master/shampoo.csv

Do you perform differencing on just the output data or do you difference the features if they are time dependent as well?

Both inputs and outputs.

How does one invert the differencing after the residual forecast has been made to get back to a forecast including the trend and seasonality that was differenced out?

Good question, I show how in this post:

https://machinelearningmastery.com/remove-trends-seasonality-difference-transform-python/

Copy Paste ?

https://www.m-asim.com/2018/10/12/how-to-difference-a-time-series-dataset-with-python/

Thanks for this awesome content by the way !

That’s a shame. I’ll ask him to take it down. Google will also penalize him ferociously.

Doing this, I will have no value for the first observation, I mean Yt-Yt-1 will be my first value and I will have an observation less?

Yes.

How to undifference?

Add the values back.

Hi Jason! As always a great tutorial.

I need to know, how to get the forecast values of unseen data if the data were differenced by first_order.

Detail:

I am doing univariate ARIMA forecasting for oil prices 3 times a day. The data was uneven so interpolated with forward-fill with an hourly rate. I did forecasting using first-order-differencing. To compare test_data and predictions, I reversed the predictions and test-data (integration).

Now the question is what I do when I don’t have test data but I have forecast unseen data. How would I integrate the predictions back to normal then the different predictions?

the ARIMA will perform the differencing and inverse-differencing for you via the d parameter.

Otherwise, you can do it manually, here’s code to do it:

https://machinelearningmastery.com/machine-learning-data-transforms-for-time-series-forecasting/

can u please tell me hoe to extract forecasted value in graph.i got predicted value,but not able to extract forecasted value in python using arima model

predictions_ARIMA_diff=pd.Series(results_ARIMA.fittedvalues, copy=True)

print(predictions_ARIMA_diff.head())

predictions_ARIMA_diff_cumsum=predictions_ARIMA_diff.cumsum()

print(predictions_ARIMA_diff_cumsum.head())

predictions_ARIMA_log=pd.Series(ts_log[0],index=ts_log.index)

predictions_ARIMA_log=predictions_ARIMA_log.add(predictions_ARIMA_diff_cumsum, fill_value=0)

predictions_ARIMA_log.head()

# Next -take the exponent of the series from above (anti-log) which will be the predicted value?—?the time series forecast model.

##Now plot the predicted values with the original.

#Find the RMSE

predictions_ARIMA=np.exp(predictions_ARIMA_log)

plt.plot(ts)

plt.plot(predictions_ARIMA)

plt.title(‘RMSE: %.4f’% np.sqrt(sum((predictions_ARIMA-ts)**2)/len(ts)))

#Future Prediction

#Predict for 5 year. We have 144 data points + 60 for next 5 yrs. i.e. predict for 204 data points

results_ARIMA.plot_predict(1,204)

You can plot a forecast using matplotlib, e.g. the plot() function.