Last Updated on August 19, 2020

A popular and widely used statistical method for time series forecasting is the ARIMA model.

ARIMA is an acronym that stands for AutoRegressive Integrated Moving Average. It is a class of model that captures a suite of different standard temporal structures in time series data.

In this tutorial, you will discover how to **develop an ARIMA model for time series** forecasting in Python.

After completing this tutorial, you will know:

- About the ARIMA model the parameters used and assumptions made by the model.
- How to fit an ARIMA model to data and use it to make forecasts.
- How to configure the ARIMA model on your time series problem.

**Kick-start your project** with my new book Time Series Forecasting With Python, including *step-by-step tutorials* and the *Python source code* files for all examples.

Let’s get started.

**Updated Apr/2019**: Updated the link to dataset.**Updated Sept/2019**: Updated examples to use latest API.

## Autoregressive Integrated Moving Average Model

An ARIMA model is a class of statistical models for analyzing and forecasting time series data.

It explicitly caters to a suite of standard structures in time series data, and as such provides a simple yet powerful method for making skillful time series forecasts.

ARIMA is an acronym that stands for AutoRegressive Integrated Moving Average. It is a generalization of the simpler AutoRegressive Moving Average and adds the notion of integration.

This acronym is descriptive, capturing the key aspects of the model itself. Briefly, they are:

**AR**:*Autoregression*. A model that uses the dependent relationship between an observation and some number of lagged observations.**I**:*Integrated*. The use of differencing of raw observations (e.g. subtracting an observation from an observation at the previous time step) in order to make the time series stationary.**MA**:*Moving Average*. A model that uses the dependency between an observation and a residual error from a moving average model applied to lagged observations.

Each of these components are explicitly specified in the model as a parameter. A standard notation is used of ARIMA(p,d,q) where the parameters are substituted with integer values to quickly indicate the specific ARIMA model being used.

The parameters of the ARIMA model are defined as follows:

**p**: The number of lag observations included in the model, also called the lag order.**d**: The number of times that the raw observations are differenced, also called the degree of differencing.**q**: The size of the moving average window, also called the order of moving average.

A linear regression model is constructed including the specified number and type of terms, and the data is prepared by a degree of differencing in order to make it stationary, i.e. to remove trend and seasonal structures that negatively affect the regression model.

A value of 0 can be used for a parameter, which indicates to not use that element of the model. This way, the ARIMA model can be configured to perform the function of an ARMA model, and even a simple AR, I, or MA model.

Adopting an ARIMA model for a time series assumes that the underlying process that generated the observations is an ARIMA process. This may seem obvious, but helps to motivate the need to confirm the assumptions of the model in the raw observations and in the residual errors of forecasts from the model.

Next, let’s take a look at how we can use the ARIMA model in Python. We will start with loading a simple univariate time series.

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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).

Download the dataset and place it in your current working directory with the filename “*shampoo-sales.csv*“.

Below is an example of loading the Shampoo Sales dataset with Pandas with a custom function to parse the date-time field. The dataset is baselined in an arbitrary year, in this case 1900.

1 2 3 4 5 6 7 8 9 10 11 |
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) print(series.head()) series.plot() pyplot.show() |

Running the example prints the first 5 rows of the dataset.

1 2 3 4 5 6 7 |
Month 1901-01-01 266.0 1901-02-01 145.9 1901-03-01 183.1 1901-04-01 119.3 1901-05-01 180.3 Name: Sales, dtype: float64 |

The data is also plotted as a time series with the month along the x-axis and sales figures on the y-axis.

We can see that the Shampoo Sales dataset has a clear trend.

This suggests that the time series is not stationary and will require differencing to make it stationary, at least a difference order of 1.

Let’s also take a quick look at an autocorrelation plot of the time series. This is also built-in to Pandas. The example below plots the autocorrelation for a large number of lags in the time series.

1 2 3 4 5 6 7 8 9 10 11 |
from pandas import read_csv from pandas import datetime from matplotlib import pyplot from pandas.plotting import autocorrelation_plot 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) autocorrelation_plot(series) pyplot.show() |

Running the example, we can see that there is a positive correlation with the first 10-to-12 lags that is perhaps significant for the first 5 lags.

A good starting point for the AR parameter of the model may be 5.

## ARIMA with Python

The statsmodels library provides the capability to fit an ARIMA model.

An ARIMA model can be created using the statsmodels library as follows:

- Define the model by calling ARIMA() and passing in the
*p*,*d*, and*q*parameters. - The model is prepared on the training data by calling the fit() function.
- Predictions can be made by calling the predict() function and specifying the index of the time or times to be predicted.

Let’s start off with something simple. We will fit an ARIMA model to the entire Shampoo Sales dataset and review the residual errors.

First, we fit an ARIMA(5,1,0) model. This sets the lag value to 5 for autoregression, uses a difference order of 1 to make the time series stationary, and uses a moving average model of 0.

When fitting the model, a lot of debug information is provided about the fit of the linear regression model. We can turn this off by setting the *disp* argument to 0.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 |
from pandas import read_csv from pandas import datetime from pandas import DataFrame from statsmodels.tsa.arima_model import ARIMA 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) # fit model model = ARIMA(series, order=(5,1,0)) model_fit = model.fit(disp=0) print(model_fit.summary()) # plot residual errors residuals = DataFrame(model_fit.resid) residuals.plot() pyplot.show() residuals.plot(kind='kde') pyplot.show() print(residuals.describe()) |

Running the example prints a summary of the fit model. This summarizes the coefficient values used as well as the skill of the fit on the on the in-sample observations.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 |
ARIMA Model Results ============================================================================== Dep. Variable: D.Sales No. Observations: 35 Model: ARIMA(5, 1, 0) Log Likelihood -196.170 Method: css-mle S.D. of innovations 64.241 Date: Mon, 12 Dec 2016 AIC 406.340 Time: 11:09:13 BIC 417.227 Sample: 02-01-1901 HQIC 410.098 - 12-01-1903 ================================================================================= coef std err z P>|z| [95.0% Conf. Int.] --------------------------------------------------------------------------------- const 12.0649 3.652 3.304 0.003 4.908 19.222 ar.L1.D.Sales -1.1082 0.183 -6.063 0.000 -1.466 -0.750 ar.L2.D.Sales -0.6203 0.282 -2.203 0.036 -1.172 -0.068 ar.L3.D.Sales -0.3606 0.295 -1.222 0.231 -0.939 0.218 ar.L4.D.Sales -0.1252 0.280 -0.447 0.658 -0.674 0.424 ar.L5.D.Sales 0.1289 0.191 0.673 0.506 -0.246 0.504 Roots ============================================================================= Real Imaginary Modulus Frequency ----------------------------------------------------------------------------- AR.1 -1.0617 -0.5064j 1.1763 -0.4292 AR.2 -1.0617 +0.5064j 1.1763 0.4292 AR.3 0.0816 -1.3804j 1.3828 -0.2406 AR.4 0.0816 +1.3804j 1.3828 0.2406 AR.5 2.9315 -0.0000j 2.9315 -0.0000 ----------------------------------------------------------------------------- |

First, we get a line plot of the residual errors, suggesting that there may still be some trend information not captured by the model.

Next, we get a density plot of the residual error values, suggesting the errors are Gaussian, but may not be centered on zero.

The distribution of the residual errors is displayed. The results show that indeed there is a bias in the prediction (a non-zero mean in the residuals).

1 2 3 4 5 6 7 8 |
count 35.000000 mean -5.495213 std 68.132882 min -133.296597 25% -42.477935 50% -7.186584 75% 24.748357 max 133.237980 |

Note, that although above we used the entire dataset for time series analysis, ideally we would perform this analysis on just the training dataset when developing a predictive model.

Next, let’s look at how we can use the ARIMA model to make forecasts.

## Rolling Forecast ARIMA Model

The ARIMA model can be used to forecast future time steps.

We can use the predict() function on the ARIMAResults object to make predictions. It accepts the index of the time steps to make predictions as arguments. These indexes are relative to the start of the training dataset used to make predictions.

If we used 100 observations in the training dataset to fit the model, then the index of the next time step for making a prediction would be specified to the prediction function as *start=101, end=101*. This would return an array with one element containing the prediction.

We also would prefer the forecasted values to be in the original scale, in case we performed any differencing (*d>0* when configuring the model). This can be specified by setting the *typ* argument to the value *‘levels’*: *typ=’levels’*.

Alternately, we can avoid all of these specifications by using the forecast() function, which performs a one-step forecast using the model.

We can split the training dataset into train and test sets, use the train set to fit the model, and generate a prediction for each element on the test set.

A rolling forecast is required given the dependence on observations in prior time steps for differencing and the AR model. A crude way to perform this rolling forecast is to re-create the ARIMA model after each new observation is received.

We manually keep track of all observations in a list called history that is seeded with the training data and to which new observations are appended each iteration.

Putting this all together, below is an example of a rolling forecast with the ARIMA model in Python.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 |
from pandas import read_csv from pandas import datetime from matplotlib import pyplot from statsmodels.tsa.arima_model import ARIMA from sklearn.metrics import mean_squared_error def parser(x): return datetime.strptime('190'+x, '%Y-%m') X = series.values size = int(len(X) * 0.66) train, test = X[0:size], X[size:len(X)] history = [x for x in train] predictions = list() for t in range(len(test)): model = ARIMA(history, order=(5,1,0)) model_fit = model.fit(disp=0) output = model_fit.forecast() yhat = output[0] predictions.append(yhat) obs = test[t] history.append(obs) print('predicted=%f, expected=%f' % (yhat, obs)) error = mean_squared_error(test, predictions) print('Test MSE: %.3f' % error) # plot pyplot.plot(test) pyplot.plot(predictions, color='red') pyplot.show() |

Running the example prints the prediction and expected value each iteration.

We can also calculate a final mean squared error score (MSE) for the predictions, providing a point of comparison for other ARIMA configurations.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 |
predicted=349.117688, expected=342.300000 predicted=306.512968, expected=339.700000 predicted=387.376422, expected=440.400000 predicted=348.154111, expected=315.900000 predicted=386.308808, expected=439.300000 predicted=356.081996, expected=401.300000 predicted=446.379501, expected=437.400000 predicted=394.737286, expected=575.500000 predicted=434.915566, expected=407.600000 predicted=507.923407, expected=682.000000 predicted=435.483082, expected=475.300000 predicted=652.743772, expected=581.300000 predicted=546.343485, expected=646.900000 Test MSE: 6958.325 |

A line plot is created showing the expected values (blue) compared to the rolling forecast predictions (red). We can see the values show some trend and are in the correct scale.

The model could use further tuning of the p, d, and maybe even the q parameters.

## Configuring an ARIMA Model

The classical approach for fitting an ARIMA model is to follow the Box-Jenkins Methodology.

This is a process that uses time series analysis and diagnostics to discover good parameters for the ARIMA model.

In summary, the steps of this process are as follows:

**Model Identification**. Use plots and summary statistics to identify trends, seasonality, and autoregression elements to get an idea of the amount of differencing and the size of the lag that will be required.**Parameter Estimation**. Use a fitting procedure to find the coefficients of the regression model.**Model Checking**. Use plots and statistical tests of the residual errors to determine the amount and type of temporal structure not captured by the model.

The process is repeated until either a desirable level of fit is achieved on the in-sample or out-of-sample observations (e.g. training or test datasets).

The process was described in the classic 1970 textbook on the topic titled Time Series Analysis: Forecasting and Control by George Box and Gwilym Jenkins. An updated 5th edition is now available if you are interested in going deeper into this type of model and methodology.

Given that the model can be fit efficiently on modest-sized time series datasets, grid searching parameters of the model can be a valuable approach.

For an example of how to grid search the hyperparameters of the ARIMA model, see the tutorial:

## Summary

In this tutorial, you discovered how to develop an ARIMA model for time series forecasting in Python.

Specifically, you learned:

- About the ARIMA model, how it can be configured, and assumptions made by the model.
- How to perform a quick time series analysis using the ARIMA model.
- How to use an ARIMA model to forecast out of sample predictions.

**Do you have any questions about ARIMA, or about this tutorial?**

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

Many thank

You’re welcome.

Hi Jason! Great tutorial.

Just a reeal quick question ..how can I fit and run the last code for multiple varialbles?..the dataset that looks like this:

Date,CO,NO2,O3,PM10,SO2,Temperature

2016-01-01 00:00:00,0.615,0.01966,0.00761,49.92,0.00055,18.1

You can model the target variable alone.

Alternately you can provide the other variables as exog variables, such as SARIMAX.

https://machinelearningmastery.com/time-series-forecasting-methods-in-python-cheat-sheet/

Finally, you could use a neural network:

https://machinelearningmastery.com/start-here/#deep_learning_time_series

Hey,

Nice article, it helped me a lot.

I have a question as to how to make predictions in a scenario where you are attempting to make new predictions not included in the dataset.

For each item in the test set, after a prediction is made, the correct data point, taken from test, is added to the history.

How can I make predictions when I don’t have a test set to extract the right data points from?

Good question, see this tutorial:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

have a question am doing a project concerning data analytics insights for retail company sales case study certain supermarket in my area and am proposing to use ARIMA can it be appropriate and how can i apply it

Perhaps start by modeling one product?

Hi Jason! Great Tutorial!!

I have a usecase of timeseries forecasting where I have to predict sales of different products out of the electronics store. There are around 300 types of different products. And I have to predict the sales on the next day for each of the product based on previous one year data. But not every product is being sold each day.

My guess is I have to create a tsa for each product. but the data quality for each product is low as not each product is being sold each day. And my use case is that I have to predict sales of each product.

Any way I can use time series on whole data without using tsa on each product individually?

Good question, I have some suggestions here (replace “sites” with “products”):

https://machinelearningmastery.com/faq/single-faq/how-to-develop-forecast-models-for-multiple-sites

Hi Jason! Great tutorial.

I got a question that needs your kind help.

For some reason, I need to calculate residuals of a fitted ARMA-GARCH model manually, but found that the calculated residuals are different of those directly from the R package such rugarh. I put the estimated parameters back to the model and use the training data to back out the residuals. How to get the staring residuals at t=0, t=-1 etc. Should I treat the fitted ARMA-GARCH just as an fitted ARMA model? In that case why we need to fit an ARMA-GARCH model to the training data.

Sorry, I’m not familiar wit the “rugarh” package or how it functions.

Hi Jason,

Could you do a GaussianProcess example with the same data. And compare the two- those two methods seem to be applicable to similar problems- I would love to see your insights.

Thanks for the great suggestion. I hope to cover Gaussian Processes in the future.

Thanks. If you also did a comparative study of the two, that would be great- I realize that might be out of the regular, thought I’d still ask. Also can I sign up for email notification?

Thanks.

You can sign-up for notification about all new tutorials here:

https://machinelearningmastery.com/newsletter/

Hello,

I have climate change data for the past 8 years and I need to do a regression model using climate as a factor so I need at least climate data for 30 years which I can’t find online. Is it possible to get the previous 22 years climate change using ARIMA based on the last 8 years data.

Thank you

No, that would be way too much data. ARIMA is for small datasets – or at least the python implementation cannot handle much data.

Perhaps explore using a linear regression or other ML methods as a first step.

ARIMA model can be used for any number of observations, yes its performance is more better if one used it for short-term forecasting.

Generally, yes.

Much appreciated, Jason. Keep them coming, please.

Sure thing! I’m glad you’re finding them useful.

What else would you like to see?

Hi Jason ,can you suggest how one can solve time series problem if the target variable is categorical having around 500 categories.

Thanks

That is a lot of categories.

Perhaps moving to a neural network type model with a lot of capacity. You may also require a vast amount of data to learn this problem.

What if there are multiple columns in dataset. For example: Instead of only 1 items like the shampoo, there could be a column with item numbers ranging from 1 – 20 and a column with number of stores and finally a column with respective sales?

If you have parallel input time series, you can use the other variables as exogenous variables. If you want to predict all variables, you can use VAR.

If you want to support multiple series generally as input, you can use ML methods, this will help as a start:

https://machinelearningmastery.com/convert-time-series-supervised-learning-problem-python/

OMG. Searching for weeks, never found an article like this one. Thank a lot.

I need your advice please,

I need to predict Retail sales data with variables like weather, sales Discount’, holiday etc.

Which is the best model is to use? And why?

How can decide the best fit model?

(Can I use SARIMAX for this?)

Love from Sri Lanka

Sorry for bad English

You’re welcome.

Perhaps test a few diffrent models and discover what works best for your dataset.

But I’m your suggestion tour pointed out, we can’t use arimax for multivariate forecasting.

What is your suggestion??

Any link I could follow to find a solution

Thanks again

Perhaps try some of the techniques listed here:

https://machinelearningmastery.com/start-here/#deep_learning_time_series

Hi Jason,

Recently I am working on time series prediction, but my research is a little bit complicated for me to understand how to fix a time series models to predict future values of multi targets.

Recently I read your post in multi-step and multivariate time series prediction with LSTM. But my problem have a series input values for every time (for each second we have recorded more than 500 samples). We have 22 inputs and 3 targets. All the data has been collected during 600 seconds and then predict 3 targets for 600 next seconds. Please help me how can solve this problem?

It is noticed we have trend and seasonality pulses for targets during the time.

do you find a solution to this problem?

good,Has been paid close attention to your blog.

Thanks!

Gives me loads of errors:

Traceback (most recent call last):

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 2276, in converter

date_parser(*date_cols), errors=’ignore’)

File “/Users/kevinoost/PycharmProjects/ARIMA/main.py”, line 6, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

TypeError: strptime() argument 1 must be str, not numpy.ndarray

During handling of the above exception, another exception occurred:

Traceback (most recent call last):

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 2285, in converter

dayfirst=dayfirst),

File “pandas/src/inference.pyx”, line 841, in pandas.lib.try_parse_dates (pandas/lib.c:57884)

File “pandas/src/inference.pyx”, line 838, in pandas.lib.try_parse_dates (pandas/lib.c:57802)

File “/Users/kevinoost/PycharmProjects/ARIMA/main.py”, line 6, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

File “/Users/kevinoost/anaconda/lib/python3.5/_strptime.py”, line 510, in _strptime_datetime

tt, fraction = _strptime(data_string, format)

File “/Users/kevinoost/anaconda/lib/python3.5/_strptime.py”, line 343, in _strptime

(data_string, format))

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

During handling of the above exception, another exception occurred:

Traceback (most recent call last):

File “/Users/kevinoost/PycharmProjects/ARIMA/main.py”, line 8, in

series = read_csv(‘shampoo-sales.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 562, in parser_f

return _read(filepath_or_buffer, kwds)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 325, in _read

return parser.read()

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 815, in read

ret = self._engine.read(nrows)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 1387, in read

index, names = self._make_index(data, alldata, names)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 1030, in _make_index

index = self._agg_index(index)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 1111, in _agg_index

arr = self._date_conv(arr)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/parsers.py”, line 2288, in converter

return generic_parser(date_parser, *date_cols)

File “/Users/kevinoost/anaconda/lib/python3.5/site-packages/pandas/io/date_converters.py”, line 38, in generic_parser

results[i] = parse_func(*args)

File “/Users/kevinoost/PycharmProjects/ARIMA/main.py”, line 6, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

File “/Users/kevinoost/anaconda/lib/python3.5/_strptime.py”, line 510, in _strptime_datetime

tt, fraction = _strptime(data_string, format)

File “/Users/kevinoost/anaconda/lib/python3.5/_strptime.py”, line 343, in _strptime

(data_string, format))

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

Process finished with exit code 1

Help would be much appreciated.

It looks like there might be an issue with your data file.

Open the csv in a text editor and confirm the header line looks sensible.

Also confirm that you have no extra data at the end of the file. Sometimes the datamarket files download with footer data that you need to delete.

Hi Jason,

I’m getting this same error. I checked the data and looks fine. I not sure what else to do, still learning. Please help.

Data

“Month”;”Sales of shampoo over a three year period”

“1-01”;266.0

“1-02”;145.9

“1-03”;183.1

“1-04”;119.3

“1-05”;180.3

“1-06”;168.5

“1-07”;231.8

“1-08”;224.5

“1-09”;192.8

“1-10”;122.9

“1-11”;336.5

“1-12”;185.9

“2-01”;194.3

“2-02”;149.5

“2-03”;210.1

“2-04”;273.3

“2-05”;191.4

“2-06”;287.0

“2-07”;226.0

“2-08”;303.6

“2-09”;289.9

“2-10”;421.6

“2-11”;264.5

“2-12”;342.3

“3-01”;339.7

“3-02”;440.4

“3-03”;315.9

“3-04”;439.3

“3-05”;401.3

“3-06”;437.4

“3-07”;575.5

“3-08”;407.6

“3-09”;682.0

“3-10”;475.3

“3-11”;581.3

“3-12”;646.9

The data you have pasted is separated by semicolons, not commas as expected.

Hi Kevin,

the last line of the data set, at least in the current version that you can download, is the text line “Sales of shampoo over a three year period”. The parser barfs on this because it is not in the specified format for the data lines. Try using the “nrows” parameter in read_csv.

series = read_csv(‘~/Downloads/shampoo-sales.csv’, header=0, nrows=36, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

worked for me.

Great tip!

Thanks for your excellent tip

Thanks, had the same problem, worked!

Let say I have a time series data with many attribute. For example a row will have (speed, fuel, tire_pressure), how could we made a model out of this ? the value of each column may affect each other, so we cannot do forecasting on solely 1 column. I google a lot but all the example I’ve found so far only work on time series of 1 attribute.

This is called multivariate time series forecasting. Linear models like ARIMA were not designed for this type of problem.

generally, you can use the lag-based representation of each feature and then apply a standard machine learning algorithm.

I hope to have some tutorials on this soon.

Wanted to check in on this, do you have any tutorials on multivariate time series forecasting?

Also, when you say standard machine learning algorithm, would a random forest model work?

Thanks!

Update: the

`statsmodels.tsa.arima_model.ARIMA()`

function documentation says it takes the optional parameter`exog`

, which is described in the documentation as ‘an optional array of exogenous variables’. This sounds like multivariate analysis to me, would you agree?I am trying to predict number of cases of a mosquito-borne disease, over time, given weather data. So I believe the ARIMA model should work for this, correct?

Thank you!

I have not experimented with this argument.

No multivariate examples at this stage.

Yes, any supervised learning method.

Hello Ng,

Your problem fits what VAR (Vector Autoregression) models is designed for. See the following links for more information. I hope this helps your work.

https://en.wikipedia.org/wiki/Vector_autoregression

http://statsmodels.sourceforge.net/devel/vector_ar.html

Hi, would you have a example for the seasonal ARIMA post? I have installed latest statsmodels module, but there is an error of import the SARIMAX. Do help if you manage to figure it out. Thanks.

Hi Kelvid, I don’t have one at the moment. I ‘ll prepare an example of SARIMAX and post it soon.

It is so informative..thankyou

I’m glad to hear that Muhammad.

Great post Jason!

I have a couple of questions:

– Just to be sure. model_fit.forecast() is single step ahead forecasts and model_fit.predict() is for multiple step ahead forecasts?

– I am working with a series that seems at least quite similar to the shampoo series (by inspection). When I use predict on the training data, I get this zig-zag pattern in the prediction as well. But for the test data, the prediction is much smoother and seems to saturate at some level. Would you expect this? If not, what could be wrong?

Hi Sebastian,

Yes, forecast() is for one step forecasts. You can do one step forecasts with predict() also, but it is more work.

I would not expect prediction beyond a few time steps to be very accurate, if that is your question?

Thanks for the reply!

Concerning the second question. Yes, you are right the prediction is not very accurate. But moreover, the predicted time series has a totally different frequency content. As I said, it is smooth and not zig-zaggy as the original data. Is this normal or am I doing something wrong. I also tried the multiple step prediction (model_fit.predict()) on the training data and then the forecast seem to have more or less the same frequency content (more zig-zaggy) as the data I am trying to predict.

Hi Sebastian, I see.

In the case of predicting on the training dataset, the model has access to real observations. For example, if you predict the next 5 obs somewhere in the training dataset, it will use obs(t+4) to predict t+5 rather than prediction(t+4).

In the case of predicting beyond the end of the model data, it does not have obs to make predictions (unless you provide them), it only has access to the predictions it made for prior time steps. The result is the errors compound and things go off the rails fast (flat forecast).

Does that make sense/help?

That helped!

Thanks!

Glad to hear it Sebastian.

Hi Jason,

suppose my training set is 1949 to 1961. Can I get the data for 1970 with using Forecast or Predict function

Thanks

Satya

Yes, you would have to predict 10 years worth of data though. The predictions after 10 years would likely have a lot of error.

Hi Jason,

Continuing on this note, how far ahead can you forecast using something like ARIMA or AR or GARCH in Python? I’m guessing most of these utilize some sort of Kalman filter forecasting mechanism?

To give you a sense of my data, given between 60k and 80k data points, how far ahead in terms of number of predictions can we make reliably? Similar to Sebastian, I have pretty jagged predictions in-sample, but essentially as soon as the valid/test area begins, I have no semblance of that behavior and instead just get a pretty flat curve. Let me know what you think. Thanks!

The skill of AR+GARH (or either) really depends on the choice of model parameters and on the specifics of the problem.

Perhaps you can try grid searching different parameters?

Perhaps you can review ACF/PACF plots for your data that may suggest better parameters?

Perhaps you can try non-linear methods?

Perhaps your problem is truly challenging/not predictable?

I hope that helps as a start.

So this is building a model and then checking it off of the given data right?

-How can I predict what would come next after the last data point? Am I misunderstanding the code?

Hi Elliot,

You can predict the next data point at the end of the data by training on all of the available data then calling model.forecast().

I have a post on how to make predictions here:

http://machinelearningmastery.com/make-predictions-time-series-forecasting-python/

Does that help?

I tried the model.forecast at the end of the program.

“AttributeError: ‘ARIMA’ object has no attribute ‘forecast'”

Also on your article: http://machinelearningmastery.com/make-predictions-time-series-forecasting-python/

In step 3, when it says “Prediction: 46.755211”, is that meaning after it fit the model on the dataset, it uses the model to predict what would happen next from the dataset, right?

Hi Elliot, the forecast() function is on the ARIMAResults object. You can learn more about it here:

http://statsmodels.sourceforge.net/stable/generated/statsmodels.tsa.arima_model.ARIMAResults.forecast.html

Thanks Jason for this post!

It was really useful. And your blogs are becoming a must read for me because of the applicable and piecemeal nature of your tutorials.

Keep up the good work!

You’re welcome, I’m glad to hear that.

Hi,

This is not the first post on ARIMA, but it is the best so far. Thank you.

I’m glad to hear you say that Kalin.

Hey Jason,

thank you very much for the post, very good written! I have a question: so I used your approach to build the model, but when I try to forecast the data that are out of sample, I commented out the obs = test[t] and change history.append(obs) to history.append(yhat), and I got a flat prediction… so what could be the reason? and how do you actually do the out-of-sample predictions based on the model fitted on train dataset? Thank you very much!

Hi james,

Each loop in the rolling forecast shows you how to make a one-step out of sample forecast.

Train your ARIMA on all available data and call forecast().

If you want to perform a multi-step forecast, indeed, you will need to treat prior forecasts as “observations” and use them for subsequent forecasts. You can do this automatically using the predict() function. Depending on the problem, this approach is often not skillful (e.g. a flat forecast).

Does that help?

Hi Jason,

thank you for you reply! so what could be the reason a flat forecast occurs and how to avoid it?

Hi James,

The model may not have enough information to make a good forecast.

Consider exploring alternate methods that can perform multi-step forecasts in one step – like neural nets or recurrent neural nets.

Hi Jason,

thanks a lot for your information! still need to learn a lot from people like you! 😀 nice day!

I’m here to help James!

when i calculate train and test error , train rmse is greater than test rmse.. why is it so?

I see this happen sometimes Supriya.

It suggests the model may not be well suited for the data.

Hello Jason, thanks for this amazing post.

I was wondering how does the “size” work here. For example lets say i want to forecast only 30 days ahead. I keep getting problems with the degrees of freedom.

Could you please explain this to me.

Thanks

Hi Matias, the “size” in the example is used to split the data into train/test sets for model evaluation using walk forward validation.

You can set this any way you like or evaluate your model different ways.

To forecast 30 days ahead, you are going to need a robust model and enough historic data to evaluate this model effectively.

I get it. Thanks Jason.

I was thinking, in this particular example, ¿will the prediction change if we keep adding data?

Great question Matias.

The amount of history is one variable to test with your model.

Design experiments to test if having more or less history improves performance.

Dear Jason,

Thank you for explaining the ARIMA model in such clear detail.

It helped me to make my own model to get numerical forrcasts and store it in a database.

So nice that we live in an era where knowledge is de-mystified .

I’m glad to here it!

Hi Jason. Very good work!

It would be great to see how forecasting models can be used to detect anomalies in time series. thanks.

Great suggestion, thanks Jacques.

Hi there. Many thanks. I think you need to change the way you parse the datetime to:

datetime.strptime(’19’+x, ‘%Y-%b’)

Many thanks

Are you sure?

See this list of abbreviations:

https://docs.python.org/3/library/datetime.html#strftime-and-strptime-behavior

The “%m” refers to “Month as a zero-padded decimal number.” which is exactly what we have here.

See a sample of the raw data file:

The “%b” refers to “Month as locale’s abbreviated name.” which we do not have here.

Hi Jason,

Lucky i found this at the begining of my project.. Its a great start point and enriching.

Keep it coming :).

This can also be used for non linear time series as well?

Thanks,

niri

Glad to hear it Niirkshith.

Try and see.

Dear Dr Jason,

In the above example of the rolling forecast, you used the rmse of the predicted and the actual value.

Another way of getting the residuals of the model is to get the std devs of the residuals of the fitted model

Question, is the std dev of the residuals the same as the root_mean_squared(actual, predicted)?

Thank you

Anthony of Sydney NSW

what is the difference between measuring the std deviation of the residuals of a fitted model and the rmse of the rolling forecast will

No, they are not the same.

See this post on performance measures:

http://machinelearningmastery.com/time-series-forecasting-performance-measures-with-python/

The RMSE is like the average residual error, but not quite because of the square and square root that makes the result positive.

Hi Jason,

Great writeup, had a query, when u have a seasonal data and do seasonal differencing. i.e for exy(t)=y(t)-y(t-12) for yearly data. What will be the value of d in ARIMA(p,d,q).

typo, ex y(t)=y(t)-y(t-12) for monthly data not yearly

Great question Niirkshith.

ARIMA will not do seasonal differencing (there is a version that will called SARIMA). The d value on ARIMA will be unrelated to the seasonal differencing and will assume the input data is already seasonally adjusted.

Thanks for getting back.

Hi, Jason

thanks for this example. My question how is chosen the parameter q ?

best Ivan

You can use ACF and PACF plots to help choose the values for p and q.

See this post:

http://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Hi Jason, I am wondering if you did a similar tutorial on multi-variate time series forecasting?

Not yet, I am working on some.

Hi Jason,

any updates on the same

Hi Jason,

Nice post.

Can you please suggest how should I resolve this error: LinAlgError: SVD did not converge

I have a univariate time series.

Sounds like the data is not a good fit for the method, it may have all zeros or some other quirk.

Hi Jason,

Thanks for the great post! It was very helpful. I’m currently trying to forecast with the ARIMA model using order (4, 1, 5) and I’m getting an error message “The computed initial MA coefficients are not invertible. You should induce invertibility, choose a different model order, or you can pass your own start_params.” The model works when fitting, but seems to error out when I move to model_fit = model.fit(disp=0). The forecast works well when using your parameters of (0, 1, 5) and I used ACF and PACF plots to find my initial p and q parameters. Any ideas on the cause/fix for the error? Any tips would be much appreciated.

i have the same problem as yours, i use ARIMA with order (5,1,2) and i have been searching for a solution, but still couldn’t find it.

Hi, I have exactly the same problem. Have you already found any solution to that?

Thank you for any information,

Vit

Perhaps try a different model configuration?

Sorry, it is difficult for (3,1,3) as well.

It worked for prediction for the first step of the test data, but gave out the error on the second prediction step.

My code is as follow:

It’s a great blog that you have, but the PACF determines the AR order not the ACF.

Thanks Tom.

I believe ACF and PACF both inform values for q and p:

http://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Good afternoon!

Is there an analog to the function auto.arima in the package for python from the package of the language R.

For automatic selection of ARIMA parameters?

Thank you!

Yes, you can grid search yourself, see how here:

http://machinelearningmastery.com/grid-search-arima-hyperparameters-with-python/

Hi. Great one. Suppose I have multiple airlines data number of passengers for two years recorded on daily basis. Now I want to predict for each airline number of possible passangers on next few months. How can I fit these time series models. Separate model for each airline or one single model?

Try both approaches and double down on what works best.

Hi Jason, if in my dataset, my first column is date (YYYYMMDD) and second column is time (hhmmss) and third column is value at given date and time. So could I use ARIMA model for forecasting such type of time series ?

Yes, use a custom parse function to combine the date and time into one index column.

I have very similar data set. So how to train arima/sarima single model with above kind of data, i.e.. multiple data points at each timestep?

I’m not sure these models can support data of that type.

Perhaps start here:

https://machinelearningmastery.com/start-here/#deep_learning_time_series

Hi Sir, Do you have tutorial about vector auto regression model (for multi-variate time series forecasting?)

Not at the moment.

Thanks a lot, Dr. Jason. This tutorial explained a lot. But I tried to run it on an oil prices data set from Bp and I get the following error:

SVD did not converge

I used (p,d,q) = (5, 1, 0)

Would you please help me on solving or at least understanding this error?

Perhaps consider rescaling your input data and explore other configurations?

Hi Jason,

I have a general question about ARIMA model in the case of multiple Time Series:

suppose you have not only one time series but many (i.e. the power generated per hour at 1000 different wind farms). So you have a dataset of 1000 time series of N points each and you want to predict the next N+M points for each of the time series.

Analyzing each time series separately with the ARIMA could be a waste. Maybe there are similarities in the time evolution of these 1000 different patterns which could help my predictions. What approach would you suggest in this case?

You could not use ARIMA.

For linear models, you could use vector autoregressions (VAR).

For nonlinear methods, I’d recommend a neural network.

I hope that helps as a start.

Hi Jeson, it’s possible to training the ARIMA with more files? Thanks!

Do you mean multiple series?

See VAR:

http://www.statsmodels.org/dev/vector_ar.html

“First, we get a line plot of the residual errors, suggesting that there may still be some trend information not captured by the model.”

So are you looking for a smooth flat line in the curve?

No, the upward trend that appears to exist in the plot of residuals.

At the end of the code, when I tried to print the predictions, it printed as the array, how do I convert it to the data points???

print(predictions)

[array([ 309.59070719]), array([ 388.64159699]), array([ 348.77807261]), array([ 383.60202178]), array([ 360.99214813]), array([ 449.34210105]), array([ 395.44928401]), array([ 434.86484106]), array([ 512.30201612]), array([ 428.59722583]), array([ 625.99359188]), array([ 543.53887362])]

Never mind.. I figured it out…

forecasts = numpy.array(predictions)

[[ 309.59070719]

[ 388.64159699]

[ 348.77807261]

[ 383.60202178]

[ 360.99214813]

[ 449.34210105]

[ 395.44928401]

[ 434.86484106]

[ 512.30201612]

[ 428.59722583]

[ 625.99359188]

[ 543.53887362]]

Keep up the good work Jason.. Your blogs are extremely helpful and easy to follow.. Loads of appreciation..

Glad to hear it.

Hi Jason and thank you for this post, its really helpful!

I have one question regarding ARIMA computation time.

I’m working on a dataset of 10K samples, and I’ve tried rolling and “non rolling” (where coefficients are only estimated once or at least not every new sample) forecasting with ARIMA :

– rolling forecast produces good results but takes a big amount of time (I’m working with an old computer, around 3/6h depending on the ARMA model);

– “non rolling” doesn’t forecast well at all.

Re-estimating the coefficients for each new sample is the only possibility for proper ARIMA forecasting?

Thanks for your help!

I would focus on the approach that gives the best results on your problem and is robust. Don’t get caught up on “proper”.

Dear Respected Sir, I have tried to use ARIMA model for my dataset, some samples of my dataset are following,

YYYYMMDD hhmmss Duration

20100916 130748 18

20100916 131131 99

20100916 131324 214

20100916 131735 72

20100916 135342 37

20100916 144059 250

20100916 150148 87

20100916 150339 0

20100916 150401 180

20100916 154652 248

20100916 183403 0

20100916 210148 0

20100917 71222 179

20100917 73320 0

20100917 81718 25

20100917 93715 15

But when I used ARIMA model for such type of dataset, the prediction was very bad and test MSE was very high as well, My dataset has irregular pattern and autocorrelation is also very low. so could ARIMA model be used for such type of dataset ? or I have to do some modification in my dataset for using ARIMA model?

Looking forward.

Thanks

Perhaps try data transforms?

Perhaps try other algorithms?

Perhaps try gathering more data.

Here are more ideas:

http://machinelearningmastery.com/machine-learning-performance-improvement-cheat-sheet/

Hi Jason,

def parser(x):

return datetime.strptime(‘190’+x, ‘%Y-%m’)

series = read_csv(‘/home/administrator/Downloads/shampoo.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

print(series.head())

for these lines of code, I’m getting the following error

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

Please help.

Thanks

Check that you have deleted the footer in the raw data file.

Hi Jason

Does ARIMA have any limitations for size of the sample. I have a dataset with 18k rows of data, ARIMA just doesn’t complete.

Thanks

Kushal

Yes, it does not work well with lots of data (linalg methods under the covers blow up) and it can take forever as you see.

You could fit the model using gradient descent, but not with statsmodels, you may need to code it yourself.

Love this. The code is very straightforward and the explanations are nice.

I would like to see a HMM model on here. I have been struggling with a few different packages (pomegranate and hmmlearn) for some time now. would like to see what you can do with it! (particularly a stock market example)

Thanks Olivia, I hope to cover HMMs in the future.

Good evening,

In what I am doing, I have a training set and a test set. In the training set, I am fitting an ARIMA model, let’s say ARIMA(0,1,1) to the training set. What I want to do is use this model and apply it to the test set to get the residuals.

So far I have:

model = ARIMA(data,order = (0,1,1))

model_fit = model.fit(disp=0)

res = model_fit.resid

This gives me the residuals for the training set. So I want to apply the ARIMA model in ‘model’ to the test data.

Is there a function to do this?

Thank you

Hi Ben,

You could use your fit model to make a prediction for the test dataset then compare the predictions vs the real values to calculate the residual errors.

Could you give me an example of the syntax? I understand that idea, but when I would try the results were very poor.

I provide a suite of examples, please search the blog for ARIMA or start here:

http://machinelearningmastery.com/start-here/#timeseries

Hi Jason,

In your example, you append the real data set to the history list- aren’t you supposed to append the prediction?

history.append(obs), where obs is test[t].

in a real example, you don’t have access to the real “future” data. if you were to continue your example with dates beyond the data given in the csv, the results are poor. Can you elaborate?

We are doing walk-forward validation.

In this case, we are assuming that the real ob is made available after the prediction is made and before the next prediction is required.

Hi,

How i do fix following error ?

—————————————————————————

ImportError Traceback (most recent call last)

in ()

6 #fix deprecated – end

7 from pandas import DataFrame

—-> 8 from statsmodels.tsa.arima_model import ARIMA

9

10 def parser(x):

ImportError: No module named ‘statsmodels’

i have already install the statsmodels module.

(py_env) E:\WinPython-64bit-3.5.3.1Qt5_2\virtual_env\scikit-learn>pip3 install –

-upgrade “E:\WinPython\packages\statsmodels-0.8.0-cp35-cp35m-win_amd64.whl”

Processing e:\winpython\packages\statsmodels-0.8.0-cp35-cp35m-win_amd64.whl

Installing collected packages: statsmodels

Successfully installed statsmodels-0.8.0

http://www.lfd.uci.edu/~gohlke/pythonlibs/

problem fixed,

from statsmodels.tsa.arima_model import ARIMA

#this must come after statsmodels.tsa.arima_model, not before

from matplotlib import pyplot

Glad to hear it.

It looks like statsmodels was not installed correctly or is not available in your current environment.

You installed using pip3, are you running a python3 env to run the code?

interestingly, under your Rolling Forecast ARIMA Model explanation, matplotlib was above statsmodels.

from matplotlib import pyplot

from statsmodels.tsa.arima_model import ARIMA

i am using jupyter notebook from WinPython-64bit-3.5.3.1Qt5 to run your examples. i keep getting ImportError: No module named ‘statsmodels’ if i declare import this way in ARIMA with Python explanation

from matplotlib import pyplot

from pandas import DataFrame

from statsmodels.tsa.arima_model import ARIMA

i think it could be i need to restart the virtual environment to let the environment recognize it, today i re-test the following declarations it is ok.

from matplotlib import pyplot

from pandas import DataFrame

from statsmodels.tsa.arima_model import ARIMA

thanks for the replies. case close

Glad to hear it.

You will need to install statsmodels.

Great explanation

can anyone help me to write code in R about forecasting such as (50,52,50,55,57) i need to forecasting the next 3 hour, kindly help me to write code using R with ARIMA and SARIMA Model

thanks in advance

I have a good list of books to help you with ARIMA in R here:

http://machinelearningmastery.com/books-on-time-series-forecasting-with-r/

Dear :sir

i hope all of you fine

could any help me to analysis my data I will pay for him

if u can help me plz contact me fathi_nias@yahoo.com

thanks

Consider hiring someone on upwork.com

Can the ACF be shown using bars so you can look to see where it drops off when estimating order of MA model? Or have you done a tutorial on interpreting ACF/PACF plots please elsewhere?

Yes, consider using the blog search. Here it is:

http://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Hi Jason

I am getting the error when trying to run the code:

from matplotlib import pyplot

from pandas import DataFrame

from pandas.core import datetools

from pandas import read_csv

from statsmodels.tsa.arima_model import ARIMA

series = read_csv(‘sales-of-shampoo-over-a-three-year.csv’, header=0, parse_dates=[0], index_col=0)

# fit model

model = ARIMA(series, order=(0, 0, 0))

model_fit = model.fit(disp=0)

print(model_fit.summary())

# plot residual errors

residuals = DataFrame(model_fit.resid)

residuals.plot()

pyplot.show()

residuals.plot(kind=’kde’)

pyplot.show()

print(residuals.describe())

Error Mesg on Console :

C:\Python36\python.exe C:/Users/aamrit/Desktop/untitled1/am.py

C:/Users/aamrit/Desktop/untitled1/am.py:3: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.

from pandas.core import datetools

Traceback (most recent call last):

File “C:\Python36\lib\site-packages\pandas\core\tools\datetimes.py”, line 444, in _convert_listlike

values, tz = tslib.datetime_to_datetime64(arg)

File “pandas\_libs\tslib.pyx”, line 1810, in pandas._libs.tslib.datetime_to_datetime64 (pandas\_libs\tslib.c:33275)

TypeError: Unrecognized value type:

During handling of the above exception, another exception occurred:

Traceback (most recent call last):

File “C:\Python36\lib\site-packages\statsmodels\tsa\base\tsa_model.py”, line 56, in _init_dates

dates = to_datetime(dates)

File “C:\Python36\lib\site-packages\pandas\core\tools\datetimes.py”, line 514, in to_datetime

result = _convert_listlike(arg, box, format, name=arg.name)

File “C:\Python36\lib\site-packages\pandas\core\tools\datetimes.py”, line 447, in _convert_listlike

raise e

File “C:\Python36\lib\site-packages\pandas\core\tools\datetimes.py”, line 435, in _convert_listlike

require_iso8601=require_iso8601

File “pandas\_libs\tslib.pyx”, line 2355, in pandas._libs.tslib.array_to_datetime (pandas\_libs\tslib.c:46617)

File “pandas\_libs\tslib.pyx”, line 2538, in pandas._libs.tslib.array_to_datetime (pandas\_libs\tslib.c:45511)

File “pandas\_libs\tslib.pyx”, line 2506, in pandas._libs.tslib.array_to_datetime (pandas\_libs\tslib.c:44978)

File “pandas\_libs\tslib.pyx”, line 2500, in pandas._libs.tslib.array_to_datetime (pandas\_libs\tslib.c:44859)

File “pandas\_libs\tslib.pyx”, line 1517, in pandas._libs.tslib.convert_to_tsobject (pandas\_libs\tslib.c:28598)

File “pandas\_libs\tslib.pyx”, line 1774, in pandas._libs.tslib._check_dts_bounds (pandas\_libs\tslib.c:32752)

pandas._libs.tslib.OutOfBoundsDatetime: Out of bounds nanosecond timestamp: 1-01-01 00:00:00

During handling of the above exception, another exception occurred:

Traceback (most recent call last):

File “C:/Users/aamrit/Desktop/untitled1/am.py”, line 9, in

model = ARIMA(series, order=(0, 0, 0))

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 997, in __new__

return ARMA(endog, (p, q), exog, dates, freq, missing)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 452, in __init__

super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)

File “C:\Python36\lib\site-packages\statsmodels\tsa\base\tsa_model.py”, line 44, in __init__

self._init_dates(dates, freq)

File “C:\Python36\lib\site-packages\statsmodels\tsa\base\tsa_model.py”, line 58, in _init_dates

raise ValueError(“Given a pandas object and the index does ”

ValueError: Given a pandas object and the index does not contain dates

Process finished with exit code 1

Ensure you have removed the footer data from the CSV data file.

Hi Jason

Please help me to resolve the error

I am getting error :

Traceback (most recent call last):

File “C:/Users/aamrit/Desktop/untitled1/am.py”, line 10, in

model_fit = model.fit(disp=0)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 1151, in fit

callback, start_ar_lags, **kwargs)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 956, in fit

start_ar_lags)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 578, in _fit_start_params

start_params = self._fit_start_params_hr(order, start_ar_lags)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 508, in _fit_start_params_hr

endog -= np.dot(exog, ols_params).squeeze()

TypeError: Cannot cast ufunc subtract output from dtype(‘float64’) to dtype(‘int64’) with casting rule ‘same_kind’

Code :

import pandas as pd

import numpy as np

import matplotlib.pylab as plt

from datetime import datetime

from statsmodels.tsa.arima_model import ARIMA

data = pd.read_csv(‘AirPassengers.csv’, header=0, parse_dates=[0], index_col=0)

model = ARIMA(data, order=(1,1,0),exog=None, dates=None, freq=None, missing=’none’)

model_fit = model.fit(disp=0)

print(model_fit.summary())

Sorry, I have not seen this error before, consider posting to stack overflow.

It is a bug in statsmodels. You should convert the integer values in ‘data’ to float first (e.g., by using np.float()).

Great tip.

@kyci is correct as you can check in https://github.com/statsmodels/statsmodels/issues/3504.

I was following this tutorial for my dataset, and what fixed my problem was just converting to float, like this:

X = series.values

X = X.astype(‘float32’)

How can I add multiple EXOG variales in the model?

Jason, I am able to implement the model but the results are very vague for the predicted….

how to find the exact values for p,d and q ?

My best advice is to use a grid search for those parameters:

https://machinelearningmastery.com/grid-search-arima-hyperparameters-with-python/

Thanks a lot Jason…. if value of d=0 then we should not bother about using differncing methods ?

It depends.

The d only does a 1-step difference. You may still want to perform a seasonal difference.

Jason, Can I get a link to understand it in a better way ? I am a bit confused on this.

You can get started with time series here:

https://machinelearningmastery.com/start-here/#timeseries

Hi Jason

I am trying to predict values for the future. I am facing issue.

My data is till 31st July and I want to have prediction of 20 days…..

My Date format in excel file for the model is 4/22/17 –MM-DD-YY

output = model_fit.predict(start=’2017-01-08′,end=’2017-20-08′)

Error :

Traceback (most recent call last):

File “C:/untitled1/prediction_new.py”, line 31, in

output = model_fit.predict(start=’2017-01-08′,end=’2017-20-08′)

File “C:\Python36\lib\site-packages\statsmodels\base\wrapper.py”, line 95, in wrapper

obj = data.wrap_output(func(results, *args, **kwargs), how)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 1492, in predict

return self.model.predict(self.params, start, end, exog, dynamic)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 733, in predict

start = self._get_predict_start(start, dynamic)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 668, in _get_predict_start

method)

File “C:\Python36\lib\site-packages\statsmodels\tsa\arima_model.py”, line 375, in _validate

start = _index_date(start, dates)

File “C:\Python36\lib\site-packages\statsmodels\tsa\base\datetools.py”, line 52, in _index_date

date = dates.get_loc(date)

AttributeError: ‘NoneType’ object has no attribute ‘get_loc’

Can you please help ?

Sorry, I’m not sure about the cause of this error. Perhaps try predicting one day and go from there?

Not working … can you please help ?

Hi Sir

Please help me to resolve this error

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(‘E:/data/csv/shampoo-sales.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

print(series.head())

series.plot()

pyplot.show()

ERROR is

runfile(‘C:/Users/kashi/Desktop/prog/Date_time.py’, wdir=’C:/Users/kashi/Desktop/prog’)

Traceback (most recent call last):

File “”, line 1, in

runfile(‘C:/Users/kashi/Desktop/prog/Date_time.py’, wdir=’C:/Users/kashi/Desktop/prog’)

File “C:\Users\kashi\Anaconda3\lib\site-packages\spyder\utils\site\sitecustomize.py”, line 866, in runfile

execfile(filename, namespace)

File “C:\Users\kashi\Anaconda3\lib\site-packages\spyder\utils\site\sitecustomize.py”, line 102, in execfile

exec(compile(f.read(), filename, ‘exec’), namespace)

File “C:/Users/kashi/Desktop/prog/Date_time.py”, line 10, in

series = read_csv(‘E:/data/csv/shampoo-sales.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 562, in parser_f

return _read(filepath_or_buffer, kwds)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 325, in _read

return parser.read()

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 815, in read

ret = self._engine.read(nrows)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1387, in read

index, names = self._make_index(data, alldata, names)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1030, in _make_index

index = self._agg_index(index)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1111, in _agg_index

arr = self._date_conv(arr)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 2288, in converter

return generic_parser(date_parser, *date_cols)

File “C:\Users\kashi\Anaconda3\lib\site-packages\pandas\io\date_converters.py”, line 38, in generic_parser

results[i] = parse_func(*args)

File “C:/Users/kashi/Desktop/prog/Date_time.py”, line 8, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

File “C:\Users\kashi\Anaconda3\lib\_strptime.py”, line 510, in _strptime_datetime

tt, fraction = _strptime(data_string, format)

File “C:\Users\kashi\Anaconda3\lib\_strptime.py”, line 343, in _strptime

(data_string, format))

ValueError: time data ‘1901-Jan’ does not match format ‘%Y-%m’

I have already removed the footer note from the dataset and I also open dataset in text editor. But I couldn’t remove this error. But when I comment ”date_parser=parser” my code runs but doesn’t show years,

How to resolve it?

Thanks

Perhaps %m should be %b?

Getting this problem:

File “/shampoo.py”, line 6, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

TypeError: ufunc ‘add’ did not contain a loop with signature matching types dtype(‘<U32') dtype('<U32') dtype('<U32')

I've tried '%Y-%b' but that only gives me the "does not match format" error.

Any ideas?

/ Thanks

Hi Alex, sorry to hear that.

Confirm that you downloaded the CSV version of the dataset and that you have deleted the footer information from the file.

Hey,

I got it to work right after I wrote the post…

The header in the .csv was written as “Month,””Sales” and that caused the error, so I just changed it to “month”, “sales” and it worked.

Thanks for putting in the effort to follow up on posts!

Glad to hear that Alec!

Hey,

I’ve two years monthly data of different products and their sales at different stores. How can I perform Time series forecasting on each product at each location?

Thanks in advance.

You could explore modeling products separately, stores separately, and try models that combine the data. See what works best.

Hey Jason,

You mentioned that since the residuals doesn’t have mean = 0, there is a bias. I have same situation. But the spread of the residuals is in the order of 10^5. So i thought it is okay to have non-zero mean. Your thoughts please?

Btw my mean is ~400

For those who came with an error of ValueError: time data ‘1901-Jan’ does not match format ‘%Y-%m’

please replace the month column with following:

Month

1-1

1-2

1-3

1-4

1-5

1-6

1-7

1-8

1-9

1-10

1-11

1-12

2-1

2-2

2-3

2-4

2-5

2-6

2-7

2-8

2-9

2-10

2-11

2-12

3-1

3-2

3-3

3-4

3-5

3-6

3-7

3-8

3-9

3-10

3-11

3-12

Dear Jason,

Firstly, I would like to thanks about your sharing

Secondly, I have a small question about ARIMA with Python. I have about 700 variables need to be forecasted with ARIMA model. How Python supports this issuse Jason

For example, I have data of total orders in a country, and it will be contributte to each districts

So I need to forecast for each districts (about 700 districts)

Thanks you so much

Generally, ARIMA only supports univariate time series, you may need to use another method.

That is a lot of variables, perhaps you could explore a multilayer perceptron model?

The result of model_fit.forecast() is like (array([ 242.03176448]), array([ 91.37721802]), array([[ 62.93570815, 421.12782081]])). The first number is yhat, can you explain what the other number means in the result? thank you!

It may be the confidence interval:

https://machinelearningmastery.com/time-series-forecast-uncertainty-using-confidence-intervals-python/

Great blogpost Jason!

Had a follow up question on the same topic.

Is it possible to do the forecast with the ARIMA model at a higher frequency than the training dataset?

For instance, let’s say the training dataset is sampled at 15min interval and after building the model, can I forecast at 1second level intervals?

If not directly as is, any ideas on what approaches can be taken? One approach I am entertaining is creating a Kernel Density Estimator and sampling it to create higher frequency samples on top of the forecasts.

Thanks, much appreciate your help!

Hmm, it might not be the best tool. You might need something like a neural net so that you can design a one-to-many mapping function for data points over time.

Hi Jason,

Your tutorial was really helpful to understand the concept of solving time series forecasting problem. But I have small doubt regarding the steps you followed at the very end. I’m pasting your code down below-

X = series.values

size = int(len(X) * 0.66)

train, test = X[0:size], X[size:len(X)]

history = [x for x in train]

predictions = list()

for t in range(len(test)):

model = ARIMA(history, order=(5,1,0))

model_fit = model.fit(disp=0)

output = model_fit.forecast()

yhat = output[0]

predictions.append(yhat)

obs = test[t]

history.append(obs)

print(‘predicted=%f, expected=%f’ % (yhat, obs))

error = mean_squared_error(test, predictions)

Note:1) here in the above for each iteration you’re adding the elements from the “test” and the forecasted value because in real forecasting we don’t have future data to include in test, isn’t it? Or is it that your’re trying to explain something and I’m not getting it.

2) Second doubt, aren’t you suppose to perform “reverse difference” for that you have used first order differencing in the model?

Kindly, please clear my doubt

Note: I have also went through one of your other tutorial where you have forecasted the average daily temperature in Australia.

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

here the steps you followed were convincing, also you have performed “inverse difference” step to scale the prediction to original scale.

I have followed the steps from the one above but I m unable to forecast correctly.

In this case, we are assuming the real observation is available after prediction. This is often the case, but perhaps over days, weeks, months, etc.

The differencing and reverse differencing were performed by the ARIMA model itself.

Hi Jason,

Recently I am working on time series prediction, but my research is a little bit complicated for me to understand how to fix a time series models to predict future values of multi targets.

Recently I read your post in multi-step and multivariate time series prediction with LSTM. But my problem have a series input values for every time (for each second we have recorded more than 500 samples). We have 22 inputs and 3 targets. All the data has been collected during 600 seconds and then predict 3 targets for 600 next seconds. Please help me how can solve this problem?

It is noticed we have trend and seasonality pulses for targets during the time.

Perhaps here would be a good place to start:

https://machinelearningmastery.com/start-here/#timeseries

Hey just a quick check with you regarding the prediction part. I need to do some forecast of future profit based on the data from past profit. Let’s say I got the data for the past 3 years, and then I wanted to perform a forecast on the next 12 months in next year. Does the model above applicable in this case?

Thanks!

This post will help you make predictions that are out of sample:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Hey Jason thanks so much for the clarification! But just to clarify, when I run the example above, my inputs are the past records for the past 3 years grouped by month. Then, how the code actually plot out the forecasted graph is basically takes in those input and plot, am I right? So, can I assumed that the graph that plotted out is meant for the prediction of next year?

I don’t follow, sorry. You can plot anything you wish.

Sorry but what does the expected and predicted means actually?

The expected value is the real observation from your dataset. The predicted value is the value predicted by your model.

Also, why the prediction has 13 points (start from 0 to 12) when each year only have 12 months? Looking forward to hear from you soon and thanks!

I arbitrarily chose to make predictions for 33% of the data which turned out to be 13 months.

You’re right, it would have been clearer if I only predicted the final year.

Hey Jason, thanks so much for the replies! But just to check with you, which line of the code should I modify so that it will only predict for the next 12 months instead of 13?

Also, just to be sure, if I were to predict for the profit for next year, the value that I should take should be the predicted rather than expected, am I right?

Thanks!!

Sorry, I cannot prepare a code example for you, the URLs I have provided show you exactly what to do.

Hey Jason, thanks so much but I am still confused as I am new to data analytic. The model above aims to make a prediction on what you already have or trying to forecast on what you do not have?

Also, may I check with you on how it works? Because I downloaded the sample dataset and the dataset contains the values of past 3 years grouped by months. So, can I assume the prediction takes all the values from past years into account in order to calculate for the prediction value? Or it simply takes the most recent one and calculate for the prediction?

Thanks!

Hey Jason, I am so sorry for the spams. But just a quick check with you again, let’s say I have some zero value for the profit, will it break the forecast function? Or the forecast function must take in all non-zero value. Because sometimes I am getting “numpy.linalg.linalg.LinAlgError: SVD did not converge” error message and I not sure if it is the zero values that is causing the problem. 🙂

Good question, it might depend on the model.

Perhaps spot check some values and see how the model behaves?

May I know what kind of situation will cause the error above? Is it because of drastic up and down from 3 different dataset?

Hi Jason,

Thanks for this post. I am getting following error while running the very first code:

ValueError: time data ‘1901-Jan’ does not match format ‘%Y-%m’

Ensure your data is in CSV format and that the footer was removed.

Hi Jason, thanks so much for the share! The tutorial was good! However, when I am using my own data set, I am getting the same error message as one of the guy above. The error message is ‘numpy.linalg.linalg.LinAlgError: SVD did not converge’.

I tried to crack my head out trying to observe the data sets that caused the error message but I could not figure out anything. I tried with 0 value and very very very drastic drop or increase in the data, some seems okay but at some point, some data set will just fail and return the error message.

May I know what kind of data or condition will trigger the error above so I can take extra precaution when preparing the data?

Perhaps try manually differencing the data first?

Perhaps there are a lot of 0 values in your data that the model does not like?

I tried with multiple set of data without a single zero. I realized a problem but I not sure if my observation is correct as I am still trying to figure out how the code above works, for that part I might need your enlightenment.

Let’s say the data is 1000, 100, 10000 respectively to first, second and third year. This kind of data will throw out the error message above. So can I assume that, as long as there is a big drastic drop/increase in the data set, in this case from 100 to 10000, this kind of condition will execute with error?

Sorry Denise, I’m not sure I follow.

Hey Denise, i got the same issue. did you get any solution for this problem??

Hi Jason,

Thank you for the tutorial, it’s great! I have a question about stationarity and differencing. If time series is non stationary but is made stationary with simple differencing, are you required to have d=1 in your selected model? Can I choose a Model with no differencing for this data if it gives me a better root mean square error and there is no evidence of autocorrelation?

Yes, you can let the ARIMA difference or perform it yourself.

But ARIMA will do it automatically for you which might be easier.

@Jason, This article has helped me a lot for the training set predictions which i had managed to do earlier too, but could you help me with the future forecasting, let say your date data is till 10th November, 2017 and i want to predict the values for the next one week or next 3 days..

If we get help for this, that would be amazing 🙂

See this post on how to make predictions:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

@Jason,

For future predictions, let say i have data till 10th November, and based on your analysis as shown above, can you help me with the future predictions for a week or so, need an idea of how to predict future data..

Yes, see this post:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Great post Jason!

I have a question:

– We need to ensure that the residuals of our model are uncorrelated and normally distributed with zero mean.

What if the residuals are not normally distributed?

It would be very grateful if you could explain how to approach in such scenario.

Thanks

Shariq

It may mean that you could improve your model with some data transform, perhaps something like a boxcox?

@Jason, What if we don’t want Rolling forecast, which means, my forecast should only be based on the training data, and it should predict the test data..

I am using the below code:

X = ts.values

size = int(len(X) * 0.75)

train, test = X[0:size], X[size:len(X)]

model = ARIMA(train, order=(4, 1, 2))

results_AR = model.fit(disp=0)

preds=results_AR.predict(size+1,size+16)

pyplot.plot(test[0:17])

pyplot.plot(preds, color=’red’)

pyplot.show()

This prediction is giving me really bad results, need urgent help on this.

This is called a multi-step forecast and it is very challenging. You may need a different model.

More here:

https://machinelearningmastery.com/multi-step-time-series-forecasting/

Hi Jason, I have two questions.

1. Let’s say I want to estimate an AR model like this: x(t)=a*x(t-2) + e. If I use ARIMA(2,0,0), it will add the term x(t-1) as well, which I don’t want. In SAS I would use p=(2) on the estimate statement of proc arima rather than p=2.

2. How do I incorporate covariates? For example, a simple model like this: x(t)=a*x(t-2) + b*f(t) + e, where f(t) e.g. is 1 if it’s the month of January and 0 otherwise.

Thanks.

Re the first question, it’s good. I don’t know how to do this with statsmodels off the cuff, some google searchers are needed.

Re multivariates, you may need to use ARIMAX or SARIMAX or similar method.

Hi,

I am getting the following error when loading the series dataframe in python

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

Ive just copy pasted the code from this website but its not working. Any suggestions? Im using Sypder

Ensure you remove the footer from the data file.

Hi, may I know what are the yhat, obs and error variable are for? As for the error, is it better with greater value or the other way around? Thanks!

yhat are the predictions. obs are the observations or the actual real data.

Thanks! Then what about the MSE? Is it the greater the better or the other way around?

A smaller error is better.

Could you please have a blog on Anomaly detection using timeseries data, may be from the above example itself.

Thanks for the suggestion.

hey sir , thanks for that , Is ARIMA good for predictions of currencies exchange rate or not ?

I don’t know about currency exchange problems sorry. Try it and see.

Hello,

Is it possible to predict hourly temperature for upcoming 5 years based on hourly temperature data of last 5 years ?

I am trying this out with ARIMA model, its giving me vrey bad output ( attenuating curve ).

You could model that, but I expect the skill to be very poor. The further in the future you want to predict, the worse the skill.

if the time series corresponds to brownian motion time series generated with different Hurst value (let’s say H1 = 0.6 and H2 = 0.7), is this model a good fit to classify if it is H1 or H2?

Hi Jason,

I have followed all of your posts related to Time Series to do my first data science project. I have done the parameter optimization also. The same code is working in my laptop but when i ran in Kaggle it shows “The computed initial AR coefficients are not stationary

You should induce stationarity, choose a different model order, or you can

pass your own start_params”. The python version is same in my environment and in Kaggle. Is this common?

Sorry, I don’t know about “running code in kaggle”.

I get the same error when I run the code in my local PC. Not for every p and q though, but for higher values.

Perhaps try using a “d” term to make the data stationary.

Hello, may I know what is the purpose for these two lines?

size = int(len(X) * 0.66)

train, test = X[0:size], X[size:len(X)]

Thanks!

Also, just to double confirm with you on my understanding, basically what the algorithm does is, take in all input in csv and fit into model, perform a forecast, append the forecast value into the model, then go thru the for loop again to recreate a new ARIMA model, forecast then append new forecast value, then go thru the for loop again?

In addition, the next row prediction is always depends on the past prediction values?

Yes, I believe so. Note, this is just one framing of the problem.

To split the dataset into train and test sets.

Is there a specific reason for you to multiply with 0.66? Thanks!

No reason, just an arbitrarily chosen 66%/37% split of the data.

I need to forecasting the next x hour. How can i do this?

This post might help:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Thanks Jason for making it simple. I run the program but getting error

1st error :

TypeError: Cannot cast ufunc subtract output from dtype(‘float64’) to dtype(‘int64’) with casting rule ‘same_kind’

After changing code , i got 2nd error

model = ARIMA(series.astype(float), order=(5,1,0))

I m getting following error

LinAlgError: SVD did not converge

Looks like the data might have some issues. Perhaps calculate some summary stats, visualizations and look at the raw data to see if there is anything obvious.

Thanks Jason for the quick response. Now i tried for Sampoo dataset, getting following error :

ValueError: time data ‘1901-Jan’ does not match format ‘%d-%m’

Code :

def parser(x): return datetime.strptime(‘190’+x, ‘%d-%m’)

series = read_csv(‘shampoo-sales.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

print(series.head())

series.plot()

pyplot.show()

Perhaps your data contains the footer. Here is a clean version of the data ready to go:

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

When we use a recursive model for ARIMA, let say like saw in one of your examples:

Why my final test vs predicted graph is coming as if, the predictions are following the test values, it’s like if test is following a pattern, predictions is following similar pattern, hence ultimately our ARIMA predictions isn’t working properly, i hope you got my point.

For example: if test[0] keeps increasing till test[5] and decreases, then prediction[1] keeps increasing till predictions[5] and decreases..

It suggests the model is not skilful and is acting like a persistence model.

It may also be possible that persistence is the best that can be achieved on your problem.

Does that mean, ARIMA isn’t giving good results for my problem?

What are different ways of solving this problem by ARIMA, can differencing or Log approach be a good solution?

You can use ACF/PACF plots to help choose ARIMA parameters, or you can grid search ARIMA parametres on your test set.

Hello! Thank you for this great tutorial. It’d be a great help if you guide me through one of my problems.

I want to implement a machine learning model to predict(forecast) points scored by each player in the upcoming game week.

Say I have values for a player (Lukaku) for 28 game weeks and I train my model based on some selected features for those 28 weeks. How do I predict the outcome of the 29th week?

I am trying to predict total points to be scored by every player for the coming game week.

So basically what should be the input to my model for 29th game week? Since the game assigns points as per live football games happening during the week, I wont have any input data for 29th week.

Thank you 🙂

I would recommend looking into rating systems:

https://en.wikipedia.org/wiki/Elo_rating_system

Hi Jason,

Great tutorial once again!

I have a question on your Rolling Forecast ARIMA model.

When your are appending obs (test(t)) on each step to history, aren’t we getting data leakage?

The test set is supposed to be unseen data, right? Or are you using the test set as a validation set?

In this case no, we are assuming the real observation is available at the end of each iteration.

You can change the assumptions and therefore the test setup if you like.

oh I see, i misunderstood this assumption, sorry. But how can I predict multiple steps? I used the predict() method from ARIMA model but the results were weird.

Yes, you can use the predict() function. Performance may be poor as predicting multiple steps into the future is very challenging.

Hi,

In case we try to introduce more than one input, then how can fit the model and make prediction?

Thanks

We don’t fit one point, we fit a series of points.

Hi Jason,

Very nice introduction! Thank you very much for always bringing us excellent ML knowledge.

Can you further explain why you chose (p,d,q) = (5,1,0)? Or you did gird search (which you show in other blogs) using training/test sets to find minimum msg appears at (5,1,0)? Did you know any good reference for diagnostic plots for the hyper-parameters grid searching?

Meanwhile, I am interested in both time-series book and LSTM book. If I purchased both, any further deal?

I recommend using both a PACF/ACF interpretation and grid searching approaches. I have tutorials on both.

Sorry, I cannot create custom bundles of books, you can see the full catalog here:

https://machinelearningmastery.com/products

Hi Jason,

Thank you for your answer. I have purchased time series book.

I still have few more questions on ARIMA model:

(1) The shampoo sale data obviously shows non-stationary; strictly speaking, we should transform data until it becomes stationary data by taking logarithm and differencing (Box-Cox transformation), and then apply to ARIMA model. Is it correct?

(2) Does the time series data with first-order differencing on ARIMA (p,0,q) give the similar results to the time series data without differencing on ARIMA(p,1,q)? i.e. d = 1 in ARIMA(p,d,q)

equivalently process data with first-order difference?

(3) In this example, we chose ARIMA (5,1,0) and p=5 came from the autocorrelation plot. However, what I read from the book https://www.otexts.org/fpp/8/5 said to judge value of p, we should check PACF plot, instead ACF. Are there any things I missed or misunderstood?

The shampoo data is non stationary and should be differenced, this can happen before modeling or as part of the ARIMA.

No, 0 and 1 for d mean no differencing and first order differencing respectively.

Yes, you can check ACF and PACF for configuring the p and q variables, see this post:

https://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Hi Jason,

In your code you use :

yhat=output[0]

So you take the first element of output, what are the other elements of output represent?

Thank you

You can see all of the returned elements here:

http://www.statsmodels.org/dev/generated/statsmodels.tsa.arima_model.ARMAResults.forecast.html

Thank you for your efforts … i have question

i’m using the following code as mentioned above

def parser(x):

return datetime.strptime(‘190’ +x, ‘%Y-%m’)

but the error appears :

ValueError: time data ‘1902-Jan’ does not match format ‘%Y-%m’

could you please help me ….

It looks like you downloaded the dataset in a different format.

You can get the correct dataset here:

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

Hey Jason,

Best article I have ever seen. Currently I am working on data driven time series forecasting with PYTHON by ARIMA model. I have data of appliance energy which depends on 26 variables over period of 4 months. My question is how can I use 26 variables to forecast the future value?

Thanks.

Sorry, I don’t have an example of ARIMA with multiple input variables.

Hello Jason,

Thanks for your reply.

Can I solve my problem with ARIMA model?

Perhaps a variant that supports multiple series.

Hey Jason, I am new to data analytics. From the chart, may I know how you determined it is stationary or non-stationary as well as how do you see whether it has a lagged value?

Thanks!

Yes, you can learn more about ACF and PACF and their interpretation here:

https://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Hello Jason,

can Autoregression model be used for forecasting stock price ?

Yes, but it will likely do worse than a persistence model.

Learn more here:

https://machinelearningmastery.com/gentle-introduction-random-walk-times-series-forecasting-python/

Hello! I think you may have made a mistake in the following paragraph.

“If we used 100 observations in the training dataset to fit the model, then the index of the next time step for making a prediction would be specified to the prediction function as start=101, end=101. This would return an array with one element containing the prediction.”

Since python is zero-indexed, the index of the next time step for making a prediction should be 100, I think.

Not in this case. Try it and see.

Hello Jason!

I’m stuck at this error when i execute these lines of code:

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_time_series.csv’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

print(series.head())

series.plot()

pyplot.show().

Error:-

time data ‘19001-Jan’ does not match format ‘%Y-%m’

Perhaps you downloaded a different version of the dataset. Here is a direct link:

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

Does that help?

hi dear,

can ask you please what is the meaning of the arrow that cant be copied, thank you.

Sorry, what arrow?

Hi Jason,

great tutorial, as always! Thank you very much for providing your excellent knowledge to the vast community! You really helped me to get a better understanding of this ARIMA type of models.

Do you plan to make a tutorial on nonlinear time-series models such as SETAR? Would be great, because I could not really find anything in this region.

Thanks for the suggestion.

I do hope to cover more methods for nonlinear time series in the future.

Hi Jason

I tried the code with my data. ACF, PACF plots aren’t showing me any significant correlations. Is there anything by which I can still try the forecast? What should be one’s steps on encounter of such data?

Perhaps try a grid search on ARIMA parameters and see what comes up?

Hi Jason,

Is it possible to make a forecast with xgboost for a time series data with categorical variables?

Yes.

Hello Jason, I have been following your articles and it has been very helpful.

I am running the same code above and get following error:

ValueError Traceback (most recent call last)

in ()

7 pred=list()

8 for i in range(len(test)):

—-> 9 model=ARIMA(history,order=(5,1,0))

10 model_fit=model.fit(disp=0)

11 output=model_fit.forecast()

~\AppData\Local\Continuum\anaconda3\lib\site-packages\statsmodels\tsa\arima_model.py in __new__(cls, endog, order, exog, dates, freq, missing)

998 else:

999 mod = super(ARIMA, cls).__new__(cls)

-> 1000 mod.__init__(endog, order, exog, dates, freq, missing)

1001 return mod

1002

~\AppData\Local\Continuum\anaconda3\lib\site-packages\statsmodels\tsa\arima_model.py in __init__(self, endog, order, exog, dates, freq, missing)

1013 # in the predict method

1014 raise ValueError(“d > 2 is not supported”)

-> 1015 super(ARIMA, self).__init__(endog, (p, q), exog, dates, freq, missing)

1016 self.k_diff = d

1017 self._first_unintegrate = unintegrate_levels(self.endog[:d], d)

~\AppData\Local\Continuum\anaconda3\lib\site-packages\statsmodels\tsa\arima_model.py in __init__(self, endog, order, exog, dates, freq, missing)

452 super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)

453 exog = self.data.exog # get it after it’s gone through processing

–> 454 _check_estimable(len(self.endog), sum(order))

455 self.k_ar = k_ar = order[0]

456 self.k_ma = k_ma = order[1]

~\AppData\Local\Continuum\anaconda3\lib\site-packages\statsmodels\tsa\arima_model.py in _check_estimable(nobs, n_params)

438 def _check_estimable(nobs, n_params):

439 if nobs 440 raise ValueError(“Insufficient degrees of freedom to estimate”)

441

442

ValueError: Insufficient degrees of freedom to estimate

the code used

from sklearn.metrics import mean_squared_error

size = int(len(df) * 0.66)

train,test=df[0:size],df[size:len(df)]

print(train.shape)

print(test.shape)

history=[x for x in train]

pred=list()

for i in range(len(test)):

model=ARIMA(history,order=(5,1,0))

model_fit=model.fit(disp=0)

output=model_fit.forecast()

yhat=output[0]

pred.append(yhat)

obs=test[i]

history.append(obs)

print(‘predicted = %f,expected = %f’,(yhat,obs))

error=mean_squared_error(test,pred)

print(‘Test MSE: %.3f’ % error)

plt.plot(test)

plt.plot(pred,color=’red’)

plt.show()

On;ly change I have made in code is date index. I have done something like this for dates

dt=pd.date_range(“2015-01-01”, “2017-12-1″, freq=”MS”)

Can you explain what is wrong?

also,

I was under impression that you use auto_corr function to determine Q parameter in ARIMA model. then in your code when you call ARIMA why have you used (5,1,0) assuming it is (p,d,q)? i thought it was suppose to be (0,1,5)?

I have more on the ACF/PACF plots and how to interpret them here:

https://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

Hello Jason, I posted a problem earlier today that I have successfully resolved. thanks for your help.

Glad to hear it.

Hello Jason,

Thanks for the helpful article.

My question is :

“A rolling forecast is required given the dependence on observations in prior time steps for differencing and the AR model.”

can you please elaborate?

How do we decide when to use Rolling forecast and when not to use rolling forecast?

what are the factors do you consider?

Thanks

I believe I mean a walk-forward validation. More here:

https://machinelearningmastery.com/backtest-machine-learning-models-time-series-forecasting/

Hello,

My company is supermaket , which have 30 stores and over 2000 products. My boss want me to predict each product sale number in next 7 days.

I think below features would affect sales count much

1. a day is festival

2. a day is weekend

3. a day’s weather

4. a day is coupon day

But I don’t know how to embed above features with ARIMA model.

And also our data is from 2017-12 to now, there is no history season data。

Could you please give me a some advice?

Thank you.

They could be exogenous binary variables that the statsmodels ARIMA does support.

Great article! But I have a question. I have a daily time series, and I am following the steps from the time series forecasting book. How do I obtain the acf and pacf visually (for the Manually Congured ARIMA)? because I will have more than 1000 lag values (as my dataset is for many years), and after this I will need to search for the hyperparameters. I will really appreciate the help

An ARIMA might not be appropriate for 1000 lags.

Great

Thanks.

thank you very much, Jason.

However. I have some problem. Whenever I adopt your code for forcasting when no validation data is available,

`for t in range(93):`

model = ARIMA(history, order=(5,1,0))

model_fit = model.fit(disp=0)

output = model_fit.forecast()

yhat = output[0]

predictions.append(yhat)

history.append(yhat)

print('predicted=%f' % (yhat))

my series converge to a constant number after a certain number of iterations, which is not right. What is the mistake?

You can fit a final model and make a prediction by calling forecast().

Here’s an example:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Hi Jason,

Your articles are great to read as they give just the right amount of background and detail and are practical oriented. Please continue writing.

I have a question though, being not from the statistical background, i am having difficulty in interpreting the output that is displayed after the summary of the fit model under the heading of “ARIMA model results”. This summarizes the coefficient values used as well as the skill of the fit on the on the in-sample observations.

Can you please provide some explanation on their attributes and how the information assists us in the interpretation of the results

Thanks.

Perhaps focus on the skill of the model and using the forecast of the model?

Hi Jason,

Thanks a lot for this awesome tutorial.

I am training on a dataset where I have to predict Traffic and Revenue during a campaign (weeks 53,54,55) driven by this marketing campaigns. I think I can only use data preceding the campaigns (weeks 1 to 52) even though I have the numbers for campaign and post campaign.

I have a file as follows:

week// campaign-period // TV-traffic // Revenue Trafiic

1 //pre-campaign // 108567 // 184196,63

2 //pre-campaign // 99358 // 166628,38

…

53 // Campaign // 135058 //240163,25

54 // Campaign // 129275 //238369,88

…

56 // post-campaign //94062 // 141284,88

…

62 // post-campaign // 86695 // 130153,38

It seems like a statistical problem and I don’t know whether ARIMA is suitable for this use case (very few data, only 52 values to predict the following one). Do you think I can give it a shot with ARIMA or do you think there are other models that could be more suitable for such a use case please?

Thanks a lot for your help.

Perhaps list out 10 or more different framings of the problem, then try fitting models to a few to see what works best?

Hi Jason,

Thanks a lot for this awesome tutorial.

I am training on a dataset where I have to predict Traffic and Revenue during a campaign (weeks 53,54,55) driven by this marketing campaigns. I think I can only use data preceding the campaigns (weeks 1 to 52) even though I have the numbers for campaign and post campaign.

I have a file as follows:

week// campaign-period // TV-traffic // Revenue Trafiic

1 //pre-campaign // 108567 // 184196,63

2 //pre-campaign // 99358 // 166628,38

…

53 // Campaign // 135058 //240163,25

54 // Campaign // 129275 //238369,88

…

56 // post-campaign //94062 // 141284,88

…

62 // post-campaign // 86695 // 130153,38

It seems like a statistical problem and I don’t know whether ARIMA is suitable for this use case (very few data, only 52 values to predict the following one). Do you think I can give it a shot with ARIMA or do you think there are other models that could be more suitable for such a use case please?

Thanks a lot for your help.

Thank you for your help

Perhaps try it and see how you go?

Hi Jason, the constant updates are great and very helpful. I need a bit of help with my work. Im trying to forecast solid waste generation in using ANN. But I’m finding challenges with data and modeling my problem. If you could at least get me a headway that can help me produce something in 2weeks I will be grateful. I want to consider variables such as already generated solid waste, population, income levels, educational levels, etc. I hope to hear from you soon.

This is a good place to start for deep learning:

https://machinelearningmastery.com/start-here/#deeplearning

Many thanks Jason, it’s really helpful!

Just one question, my data set contains some sales value = 0, would that affect the performance of ARIMA model? if there will be issues, anyway I can deal with the zero values in my data set? Thanks in advance for your advice!

It can deal with zero values.

Hello Jason,

Any idea why I am having issues with datetime?

This is the error that I have received

Traceback (most recent call last):

File “/Library/Frameworks/Python.framework/Versions/3.7/lib/python3.7/site-packages/pandas/io/parsers.py”, line 3021, in converter

date_parser(*date_cols), errors=’ignore’)

File “/Users/Brian/PycharmProjects/MachineLearningMasteryTimeSeries1/ARIMA.py”, line 9, in parser

return datetime.strptime(‘190’ + x, ‘%Y-%m’)

TypeError: strptime() argument 1 must be str, not numpy.ndarray

During handling of the above exception, another exception occurred:

Thank You

Brian

Perhaps your data file, try this one instead:

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

The formating of csv seems different for everyone who downloads it, here’s the format that is used by Jason (just copy pasted this into a shampoo-sales.csv file and save)

– thanks to the person above for the tip

1-1,266

1-2,145.9

1-3,183.1

1-4,119.3

1-5,180.3

1-6,168.5

1-7,231.8

1-8,224.5

1-9,192.8

1-10,122.9

1-11,336.5

1-12,185.9

2-1,194.3

2-2,149.5

2-3,210.1

2-4,273.3

2-5,191.4

2-6,287

2-7,226

2-8,303.6

2-9,289.9

2-10,421.6

2-11,264.5

2-12,342.3

3-1,339.7

3-2,440.4

3-3,315.9

3-4,439.3

3-5,401.3

3-6,437.4

3-7,575.5

3-8,407.6

3-9,682

3-10,475.3

3-11,581.3

3-12,646.9

It is also available on my github:

https://github.com/jbrownlee/Datasets

Hello Jason

I’m trying to divide time series dataset into several dataset and select the best one as preprocessing dataset.I would like to use RMSE to evaluate each subset.In other word to select the window size and frame size before I do the training . Please let me know if you have any article on rows selection not column selection

Yes, this post will help tune the parameters of ARIMA that will include tuning the size of the window for each aspect of the ARIMA model:

https://machinelearningmastery.com/grid-search-arima-hyperparameters-with-python/

Hello Jason

Many thanks for your reply. I have tried the code on the following data set and got “Best ARIMANone MSE=inf”

date price

0 20160227 427.1

1 20161118 750.9

2 20160613 690.9

3 20160808 588.7

4 20170206 1047.3

RangeIndex: 657 entries, 0 to 656

Data columns (total 2 columns):

date 657 non-null int64

price 657 non-null float64

dtypes: float64(1), int64(1)

memory usage: 10.3 KB

Hello Jason

Just to clarify my previous question that i have 700 rows of date and price and I would like select the best 70(window size) rows for prediction and decide on the frame size , frame step and extent of prediction.

Sounds great, let me know how you go!

Hi Jason

Please let me know if you have an article help on specifying frame size , frame step and extent of prediction as data pre-processing step using RMSE and SEP.

I do, the grid search of the ARIMA algorithm I linked to above does that.

Perhaps try working through it first?

Thanks Jason. Your post in Grid search is great. I have already applied the Grid Search and got best Arima model .

Now I want to use the result and train the window in LSTM

RIMA(1, 0, 0) MSE=39.723

ARIMA(1, 0, 1) MSE=39.735

ARIMA(1, 1, 0) MSE=36.148

ARIMA(3, 0, 0) MSE=39.749

ARIMA(3, 1, 0) MSE=36.141

ARIMA(3, 1, 1) MSE=36.131

ARIMA(6, 0, 0) MSE=39.806

ARIMA(6, 1, 0) MSE=36.134

ARIMA(6, 1, 1) MSE=36.128

Best ARIMA(6, 1, 1) MSE=36.128

An LSTM is a very different algorithm. Perhaps difference the series and use at least 6 time steps as input?

I have 5 years of time series data .Will 6 time steps (6 days) be enough as window size.I want to get the best optimal window as input to LSTM !

Appreciate your feedback.

Test many different sized subsequence lengths and see what works best.

Can I use Gridsearch for the testing purpose to specify the window size for LSTM?And if yes what would be the paramerters equal to 60/90/120 days ?

I would recommend running the grid search yourself with a for-loop.

Try time periods that might make sense for your problem.

So I did the for-loop and manage to get different windows.

Now to calculate the RMSE do I need to do linear regiression prediction for each window in order to calculate the RMSE or is there any other way around?

I would expect that you would fit a model for different sized windows and compare the RMSE of the models. The models could be anything you wish, try a few diffrent approaches even.

I got the following as example for two window size 360 days and 180 days

For 360 days

Window start after 0 days with windwo size 360 and step 100 have RMSE 734.1743876097737

Window start after 100 days with windwo size 360 and step 100 have RMSE 369.94549420288877

Window start after 200 days with windwo size 360 and step 100 have RMSE 105.70778076287142

For 180 days

Window start after 0 days with windwo size 180 and step 90 have RMSE 653.9070358902835

Window start after 90 days with windwo size 180 and step 90 have RMSE 326.7832188924093

Window start after 180 days with windwo size 180 and step 90 have RMSE 135.01118940666115

Window start after 270 days with windwo size 180 and step 90 have RMSE 38.422587695965746

Window start after 360 days with windwo size 180 and step 90 have RMSE 60.73374764651785

Window start after 450 days with windwo size 180 and step 90 have RMSE 52.386817309349176

Well done!

Thanks Jason

Appreciate your support.

Your posts are really great and well organized.

I’m excited to ready your publications 🙂

Thanks for your support!

Hi Jason! Here client and time series forecaster!

When forecasting, I very often get this error:

LinAlgError: SVD did not converge

Any ideas how to solve this in general?

Thanks!

This is common.

Sounds like the linear algebra library used to solve the linear regression equation for a given configuration failed.

Try other configurations?

Try fitting a linear regression model manually to the lag obs?

Try normalizing the data beforehand?

Let me know how you go.

Hey Jason, what model i can use to equipment fault detection and prediction? So have some variables that correlate with others and i need to identification which are. See you soon.

Try a suite of methods in order to discover what works best for your specific problem.

Hello Jason,

There is something that I struggle to understand, it would awesome if you could give me a hand.

In ARIMA models, the optimization fits the MA and AR parameters. Which can be summed up as parameters of linear combination of previous terms for the AR and previous errors for the MA. A quick math formula could be :

X_t – a_1 X_t-1 … – a_p X_t-p … = e_t + b_1 e_t-1 + … + b_q e_t-q

When the fit method is used, it takes the train values of the signal to fit the parameters (a and b)

When the forecast method is used, it forecast the next value of the signal using the fitted model and the train values

When the predict method is used, it forecast the next values of the signal from start to stop.

Let’s say I fit a model on n steps in the train set. Now I want to make predictions. I can predict step n+1. Now I am days n+1 and I have the exact signal value. I would like to actualize the model to predict n+2.

In the rolling forecast part of your code, you fit again the model with the expanded train set (up to n+1). But in that case the model parameters are changed. It’s not the same model anymore.

Is it possible to train one model and then actualize the signal values (the x and e) without changing the parameters (a and b)?

It seems to me that it is important to keep one unique model and evaluate it against different time steps instead of training n different models for each new time steps we get.

I hope I was clear enough. I miss probably a key to understand the problem.

Thanks

Romain

The model will use the prediction as the input to predict t+2.

Hi Jason – Very helpful post here, thanks for sharing. I’m curious why parameter ‘p’ should be equal to the number of significant lags from the auto correlation plot? Just was wondering if you could give any more context to this part of the problem. Thanks.

Generally, we want to know how may lag observations have a measurable relationship with the next step so that the model can work on using them effectively.

I used your code to forecast daily temperature (it has a lag of 365). The forecast is always a day behind, i.e. learning history cannot accurately forecast next day’s temperature. I’ve played with the params with AIC.

Perhaps try alternate configurations?

Perhaps try alternate algorithms?

Perhaps try additional transforms to the data?

This might help:

https://machinelearningmastery.com/how-to-develop-a-skilful-time-series-forecasting-model/

How to use ARIMA model in SPSS with few sample as 6 years data and according to this data for how many years we can forecast the future.

Sorry, I don’t have examples of SPSS.

Hi Jason,

Thanks for sharing! Very helpful post.

Recently I am writing the methodology of ARIMA, but I can not find any reference (for example, some ARIMA formulas contain constant but some don’t have ). So could you please give me some reference (or ARIMA formula information) of “statsmodels.tsa.arima_model import ARIMA” used in Python?

Thank you in advance.

The best textbook on ARIMA is:

https://amzn.to/2MD9lKw

Hi Qianqian,

Prof. Hyndman’s textbook: https://otexts.com/fpp2/arima.html

Hope this helps.

If one has a time series where the time steps are not uniform, what should be done while fitting a model such as ARIMA? I have price data for a commodity for about 4 years. The prices are available only for days that a purchase was made. This is often, but not always, every day. So sometimes purchases are made after 2, 3 or even more days and the prices are therefore available only for those days I need to forecast the price for the next week.

Thanks for any advice on this.

Perhaps try modeling anyway?

Perhaps try an alternative model?

Perhaps try imputing the missing values?

Thank you, Dr Jason!

Hi Jason,

Thanks for this post.

I am working on finding an anomaly using arima. Will I be able to find from the difference in actual & predicted value shown above ?

Thanks,

Kruthika

Sorry, I don’t have examples of using ARIMA for anomaly detection.

Hi Jason,

I have couple of questions.

1. is it necessary that we need to have always uni variate data set to predict for time series using ARIMA? What if i have couple of features that i want to pass along with the date time?

2. is it also necessary that we have a non-stationary data to use time series for modelling? what if the data is already stationary? can i still do the modelling using time series?

Thanks

Bhadri

ARIMA can support exogenous variables, this is called ARIMAX.

If the data is already stationary, you can begin modeling without transforms.

Thanks Jason!! do u have any examples related to ARIMAX or point me to some articles..

Yes, there are examples here:

https://machinelearningmastery.com/time-series-forecasting-methods-in-python-cheat-sheet/

Hello sir,

This is a great article. But sir I have couple of questions?

1. Assume that if we have three inputs and one output with time period. Then how do we predict the next future value according to the past values to next time period using ARIMA model? (if we need to predict value next time interval period is 120min)

as a example

6:00:00 63 0 0 63

7:00:00 63 0 2 104

8:00:00 104 11 0 93

9:00:00 93 0 50 177

2. To predict value should I have to do time forecast according to the data that I mentioned earlier?

You could treat the other inputs as exogenous variables and use ARIMAX, or you could use another method like a machine learning algorithm or neural network that supports multivariate inputs.

This is a great post, thank you very much.

I’m new in this field, and I look for simple introduction to ARIMA models in general then an article about multivariate ARIMA.

Could you please help me.

Thanks.

I don’t think I have an example of a multivariate ARIMA, maybe ARIMAX/SARIMAX would be useful as a start:

https://machinelearningmastery.com/time-series-forecasting-methods-in-python-cheat-sheet/

Hey Jason,

I was wondering if you are aware of any auto arima functions to fine tune p,d,q parameters. I am aware that R has an auto.arima function to fine tune those parameters but was wondering if you’re familiar with any Python library.

Yes, I wrote one in Python here:

https://machinelearningmastery.com/how-to-grid-search-sarima-model-hyperparameters-for-time-series-forecasting-in-python/

Hi Jaosn.

Thanks a lot for the great tutorial!

Have followed your post : “How to Grid Search ARIMA Model Hyperparameters with Python” to fine tune the p,q and d value. Have come across the below point in the post.

“The first is to ensure the input data are floating point values (as opposed to integers or strings), as this can cause the ARIMA procedure to fail.”

My initial data is in the below format. Month and #Sales

2014-11 4504794

2014-12 7656479

2015-01 9340428

2015-02 7229578

2015-03 7092866

2015-04 14514074

2015-05 9995460

2015-06 8593406

2015-07 8774430

2015-08 8448562

I applied a log transofrmation on the above data set to convert the numbers to flot as below:-

dateparse = lambda dates: pd.datetime.strptime(dates, ‘%Y-%m’)

salessataparsed = pd.read_csv(‘sales.csv’, parse_dates=[‘Month’], index_col=’Month’,date_parser=dateparse)

salessataparsed.head()

ts_log = np.log(salessataparsed[‘#Sales’])

Below is the ts_log.head() output.

2014-11-01 15.320654

2014-12-01 15.851037

2015-01-01 16.049873

2015-02-01 15.793691

2015-03-01 15.774600

2015-04-01 16.490560

2015-05-01 16.117632

2015-06-01 15.966517

With this log value, applied the grid search approach to decide the best value of p,q and d.

Howver, I got Best ARIMA(0, 1, 0) MSE=0.023. Looks good ? is it acceptable? Wondering if p=0 and q=0 is acceptable. Please confirm.

Next, I have 37 Observations from Nov 2014 to 31-Dec-2017. I want to do future predictions for 2018, 2019 etc.How to do this?

Also, do you have any Youtube videos explaining each of the steps in grid approach, how to make future forecatsts available ? It would be great if you can share the Youtube link. 🙂

Once again thanks a lot for the article and your help!

You can discover if your model is skillful by comparing its performance to a naive model:

https://machinelearningmastery.com/faq/single-faq/how-to-know-if-a-model-has-good-performance

Perhaps try standardizing or normalizing the data as well.

I don’t make videos, only text-based tutorials, sorry.

I show how to use an ARIMA model to make forecasts here:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Great ! Thanks a ton Jason.

Kindly confirm if the p,q value is 0 is an acceptable scenario.

Perhaps try standardizing or normalizing the data as well : I am not sure how to proceed with this?

It would be great if you can share related article if you have any. 🙂

For now, I am going to implement the future forecasting using the above link with this ARIMA(0,1,0) and will check how it behaves. 🙂

See this post:

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

Hi Jason, thanks for the tutorial i am new to the world of predictive analysis but i have a project to predict when a customer is likely to make next purchase. I have dataset which include historical transactions and amount.

Will this tutorial help me or is there any suggestion on material/resource i can use.

Could you please advice

I recommend following this process:

https://machinelearningmastery.com/start-here/#process

Hi Jason,

Used your epic tutorial to forecast bookings.

I used the whole of 2017 as my data set and after applying everything in your post the predicted graph seems to be one day off i.e. prediction graph looks spot on with each data point very close to the what it should be, the only thing is is that it’s a day late…is this normal? Is there something within the code that causes something like this?

Thanks

This is a common problem, I explain more here:

https://machinelearningmastery.com/faq/single-faq/why-is-my-forecasted-time-series-right-behind-the-actual-time-series

Hi, i have had a question for a while, now this might be silly but I can’t figure out whats wrong here…

So I have a timeseries data and when i used order=(0,1,0) that is, differencing is 1 then i get a timeseries that is ahead of time by one.

example:

input: 10, 12, 11, 15

output: 8, 9.9, 12.02, 11.3, 14.9

Now if I shift the resulting series by one timeperiod, it’ll match quite well.

Also, similar output can be seen is (0,2,1) that is, differencing is 2 and MA is 1.

Could someone explain why is this happening and what am i missing here.

[numbers in example are representative not actual]

It suggest that the model is using the input as the output, this is called a persistence model:

https://machinelearningmastery.com/faq/single-faq/why-is-my-forecasted-time-series-right-behind-the-actual-time-series

Thanks Jason, I went through the link and it helps me see a clear picture which should have been obvious to notice but i missed it.

If you please, could also share some thoughts on…

– My model uses order(0,1,0). i.e. differencing is 1. Do such model makes sense for a practical scenario where we are trying to predict inventory requirement for a part(based on past consumption) that may fail in coming future(where failing of a part is totally a random act of nature).

– Also, (0,2,1) and (0,1,0) gives very similar results. Is this expected in some sense. Is there any concept that i am missing here.

Thanks a lot again, for your help.

I generally recommend using the model that gives the best performance and is the simplest.

Hello Jason!

Thank you for the tutorial. It’s a good start to implementing an ARIMA model in Python. I have a question: You have used the actual data samples to update your training dataset after each prediction as given in “history.append(obs)”. Now let’s take a real life example when you don’t have any further actual data and you use your predictions only to update your training dataset which looks like “history.append(yhat)”. What will happen in this case? I am working on air quality prediction and in my case, the former scenario keeps the seasonal pattern in the test set but the latter does not show any seasonal pattern at all. Please let me know what’s your take on this.

Regards,

Dhananjai

—

You can re-fit the model using predictions as obs and/or predictions as inputs for subsequent predictions (recursive).

Perhaps evaluate a few approaches on your dataset and see how it impacts model performance.

Hi Jason ,

Thank you for the tutorial.

I have two questions :

first : why you set moving average “q” Parameter by 0 ?

second : why you set Lag value To 5 not 7 for example?

Thanks.

They are an arbitrary configuration.

Perhaps try other configurations and compare results.

Thank you for your great tutorial.

I know that the third output from model_fit.forecast() consists of the confidence interval. But how can I plot the confidence interval on the whole range automatically?

Thanks

I believe this tutorial will help:

https://machinelearningmastery.com/time-series-forecast-uncertainty-using-confidence-intervals-python/

What’s the difference of predicted and expected? Sorry I’m a just a novice.

“Predicted” is what is output by the model.

“Expected” or “actual” are the true observations.

Hey Jason,

Amazing blog, subscribed and loving it. I had a question about how you would send the output of the model to a data table in CSV?

Ramy

You can save a Numpy array as a csv directly:

https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.savetxt.html

Hi Jason, man I love this blog.

I’m running this with a separate data set. I’ve shaped my dataset, but when I run the error line, I’m getting this:

ValueError: Found array with dim 3. Estimator expected <= 2.

What are you thoughts?

Thanks,

Benny

Shaping:

X_train = np.reshape(X_train, (len(X_train), 1, X_train.shape[1]))

X_test = np.reshape(X_test, (len(X_test), 1, X_test.shape[1]))

Code:

history = [x for x in X_train]

predictions = list()

for t in range(len(X_test)):

model = ARIMA(history, order=(10,0,3))

model_fit = model.fit(disp=0)

output = model_fit.forecast()

yhat = output[0]

predictions.append(yhat)

obs = X_test[t]

history.append(obs)

print('predicted=%f, expected=%f' % (yhat, obs))

error = mean_squared_error(X_test, predictions)

print('Test MSE: %.3f' % error)

You’re data has too many dimensions. It should be 2D, but you have given it 3D data, perhaps change it to 2d!

Oh. I thought that’s what I did with reshaping. Whoops =)

I’ll hunt up some code. Thank you.

Hi Jason,

Thanks for this great work!

If you allow me, I have a question: how was the confidence interval calculated in the above example? I know its equation, but I do not know what are the values to be used for (sigma) and (number of samples).

Thank you once more.

You can review the statsmodels source code to see exactly how it was calculated. The API documentation may also be helpful.

Thanks a lot Jason!

I am preparing a time series model for my capstone project, i have around 500 items and the p,d,q value is different for each item, how can i deploy this as a tool? do i have to create model each time for different items?

Thanks in advance.

Perhaps model each series separately?

How many minimum data points do we require for creating accurate prediction using ARIMA model. We are predicting future cut-off values of colleges using previous records, how many years of records would we need to predict just the cutoff value of next year.

I recommend testing with different amounts of history on your specific dataset and discover the right amount of data for modeling.

If I am not wrong, ACF plot is used to get MA value for ARIMA. But here, you have taken AR value as 5 using ACF plot?

Hi Jason Brownlee!

I have been following your blog since some time and the concepts and code snippets here often come handy.

I’m totally new to time series analysis and have read some posts (mostly yours), a few lectures and of course questions from stackoverflow.

What confuses me is, to make a series stationary we difference it, double differencing in case seasonality and trend both are present in the series. Now while performing ARIMA, the parameter ‘I’ depicts what? Number of times we have performed differencing or lag value we chose for differencing (for the removal of seasonality).

For example, let say there is a dataset of monthly average temperatures of a place (possibly affected by global warming). Now there is seasonality (lag value of 12) and a global upward trend too.

before performing ARIMA I need to make the series stationary, right?

To do that I Difference twice like this:

differenced = series – series.shift(1) # to remove trend

double_differenced = differenced – differenced.shift(12) # to remove seasonality.

Now what should be passed as ‘I’ to ARIMA?

2? As we did double(2) differencing

or

1 or 12 as that’s the value we used for shifting.

Also if you’re kind enough, can you elaborate more how *exactly* did you decide the value of ‘p’ and ‘q’ from acf and pacf plots.

Or link me to some post if you have already explained that in layman terms somewhere else!

Extremely thankful for your time and effort!

It might be better to let the ARIMA model perform the differencing rather than do it manually.

And, if you have seasonality, you can use SARIMA to difference the trend and seasonality for you.

If you difference manually, you don’t need the model to do it again.

The computed initial MA coefficients are not invertible

You should induce invertibility, choose a different model order, or you can

pass your own start_params.

How do I fix this error? Best ARIMA params are (4,1,3)

Perhaps try a different configuration or try to prepare the data before modeling.

Do we have a similar function in python like we have auto.arima in R?

I wrote one here:

https://machinelearningmastery.com/how-to-grid-search-sarima-model-hyperparameters-for-time-series-forecasting-in-python/

And another here:

https://machinelearningmastery.com/grid-search-arima-hyperparameters-with-python/

Thank you very much, your blogs really come in handy for a beginner in python. when I run the ARIMA forecasting using above codes, getting some format error. I have tried to use Shampoo sales data too. below is the error note,

File “”, line 1, in

runfile(‘C:/Users/43819008/untitled2.py’, wdir=’C:/Users/43819008′)

File “C:\ProgramData\Anaconda3\lib\site-packages\spyder\utils\site\sitecustomize.py”, line 880, in runfile

execfile(filename, namespace)

ValueError: time data ‘19019-01-2019’ does not match format ‘%Y-%m’

I have tried all the format in excel and saved as CSV. but nothing helped me. hope you can help me.

Looks like an issue loading the data.

You could try removing the date column and changing the load function call to not use the custom function?

Hello Everyone , I want to implement ARIMA model but this error is not leaving me.

from . import kalman_loglike

ImportError: cannot import name ‘kalman_loglike’

Looks like you’re trying to import a module that does not exist or is not installed.

I got that.

Thank you very very much ,

Hi Jason, I recently came accross your blog and really like the things I have learned in a short period of time. Machine learning and AI are still relatively new to me, but I try to catch up with your information. As the ARIMA Model comes from the statistics field and predicts from past data, could it be used as the basis of a machine learning algorithm? For example: if you would create a system that would update the predictions as soon as the data of a new month arrives, can it be called a machine learning algorithm? Or are there better standarized machine learning solutions to make sales predictions?

Sure.

Yes, ARIMA is a great place to start.

Hello AI,

>>the last line of the data set, at least in the current version that you can download, is the text line “Sales of shampoo over a three year period”. The parser barfs on this because it is not in the specified format for the data lines. Try using the “nrows” parameter in read_csv.

series = read_csv(‘~/Downloads/shampoo-sales.csv’, header=0, nrows=36, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

worked for me.

Thank you for posting this. I was having the same issue. This solved it.

Thanks Jason for another great tutorial.

Thanks, I’m glad it helped.

Jason,

thank you it was very helpful in many different ways. I just want to know how you predict and how far you can predict in the future.

Thanks, good question. This post will show you how to predict:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Hi Jason,

Thanks for your write-up. I’ve tried all the suggestions here but still getting these two errors.

in parser(x)

5 def parser(x):

—-> 6 return datetime.strptime(‘190’+x, ‘%Y-%m’)

7

TypeError: strptime() argument 1 must be str, not numpy.ndarray

ValueError: time data ‘1901-Jan’ does not match format ‘%Y-%m

I removed the footer, tried with your csv file , tried with nrows but nothing worked. Please give me your valuable feedback.Thanks.

Perhaps confirm that you downloaded the dataset in the correct format?

i use R to get the p,q but it does work in the statsmodel’s arima model which always raise SVD did not converge even i set the p,q very small

Hmm, maybe the R version is preparing the data automatically before modelling in some way?

how can I get future forecast value with arima?

See this tutorial:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Hi Jason,

Your materials on Time Series have been extremely useful. I want to clarify a basic question on Model results. For an ARMA(3,0) , the statsmodel prints the output as

coef P>Z

const c 0.00

ar.L1 x1 0.003

ar.L2 x2 0.10

ar.L3 x3 0.0001

And the Data is:

Actual Daily Traffic Predicted Traffic

Jan7 100

Jan8 95

Jan9 85

Jan10 105

If I want to convert the output to a linear equation will the Predicted Traffic for Jan10 be :Pred= c+ x1*85 + 0*x2 + x3*100 ?? Appreciate your thoughts

Great question, I have an example of making a manual prediction here:

https://machinelearningmastery.com/make-manual-predictions-arima-models-python/

Thank you very much Jason. That post was very helpful. Putting the options no constant has given me the exact result for prediction. i.e., dot product of coefficients and the lagged values.

Glad it hear it.

i am newbie and trying to learn time series. getting following error, please help.

series = read_csv(‘sales.csv’, delimiter=’,’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

Traceback (most recent call last):

File “”, line 1, in

series = read_csv(‘sales.csv’, delimiter=’,’, header=0, parse_dates=[0], index_col=0, squeeze=True, date_parser=parser)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 678, in parser_f

return _read(filepath_or_buffer, kwds)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 446, in _read

data = parser.read(nrows)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1036, in read

ret = self._engine.read(nrows)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1922, in read

index, names = self._make_index(data, alldata, names)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1426, in _make_index

index = self._agg_index(index)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 1504, in _agg_index

arr = self._date_conv(arr)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\parsers.py”, line 3033, in converter

return generic_parser(date_parser, *date_cols)

File “E:\ProgramData\Anaconda3\lib\site-packages\pandas\io\date_converters.py”, line 39, in generic_parser

results[i] = parse_func(*args)

File “”, line 2, in parser

return datetime.strptime(‘190’+x, ‘%Y-%m’)

File “E:\ProgramData\Anaconda3\lib\_strptime.py”, line 565, in _strptime_datetime

tt, fraction = _strptime(data_string, format)

File “E:\ProgramData\Anaconda3\lib\_strptime.py”, line 362, in _strptime

(data_string, format))

ValueError: time data ‘1901-Jan’ does not match format ‘%Y-%m’

Looks like you need to download the data with numeric date format, or change the data parsing string.

Thanks, it is resolved, i have to download another file.

Glad to hear that.

Hi Jason!

I have a compile error: insufficient degree of freedom to estimate, when finishing my program on ARIMA in Python. Could you tell me what leads to this error? Cuz I found little answer in other solution website like stack overflow.

Hoping to hear from you!

Thank you, Jason!

Perhaps your data requires further preparation – it can happen if you have lots of zero values or observations with the same value.

Hi, Jason.

Thanks for the writeup. When running your code with a small dataset (60-ish values) it runs without a hitch, but when I run it with an identically-formatted, much larger database (~1200 values) it throws this error:

“TypeError: must be str, not list”

Any idea why this is? Thanks in advance.

Perhaps confirm that you have loaded your data correctly, as a floating point values?

Hi Jason,

Do you know how predict from estimated ARIMA model with new data, preserving the parameters just fitted in the previus model?

I’m trying to accomplish in python something similar to R:

# Refit the old model with newData

new_model <- Arima(as.ts(Data), model = old_model)

Yes, you can use the forecast() or predict() functions.

More here:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Jason, great tutorial! I follow your blogs and book regularly and they help me a lot!

However I have some conceptual doubts that I hope you can help me with.

1. If you don’t do a rolling forecast and only use the predict function, it gives us various predicted values (number of predicted values are equal to length of training data). How are the predictions made in this case? Does it use the previous predicted values to make next predictions?

2. When I validate a neural network made of one or more LSTM layers, I pass actual test data to the predict function and hence it uses that data to make predictions, so is walk forward validation/ rolling forecast redundant there?

Good question, ideally you want to fit the ARIMA model on all available data – up to the point of prediction.

So, in a walk-forward validation you might want to re-fit the ARIMA each iteration.

Hi Jason, thank you so much for all your tutorials. They have been of great help to me.

I had a question about the ARIMA model in statsmodels. If I want to select certain lags for the parameter p instead of all lags up until p how would I to do it ? I have not seen functionality for this in statsmodels, I wondered if you knew.

Whenever you find the time. Kind regards Karl

You might have to write a custom implementation I’m afraid.

Yes, I totally understand why we use walk forward validation, but I see a major drawback of it i.e it works great with shorter time series, however when you have a longer time series and multiple variables, it takes a really really long time to re-fit a SARIMAX model and get the predictions.

That’s why what I intended to ask in the second point is, if instead of a SARIMA model, I use an LSTM model, do I still need to do walk forward validation, since it already uses the actual values up to the point of prediction.

Yes. But you may not have to refit the model each step. I often do not.

Hi Jason, thanks for the great post. My time series problem is kind of different. The data lag I have is large and inconsistent. For example, I want to know for the order I received 6 pm today, how many hours we will use to fulfill this order. We might not know the fulfillment time for order received at 5 pm, 4 pm, or not even yesterday since they might not be fulfilled yet. We have no access to the future data in real life, do you have any suggestion on this? Thank you so much.

That sounds like a great problem.

I recommend using this framework to help think about different ways you can frame the problem for prediction:

http://machinelearningmastery.com/how-to-define-your-machine-learning-problem/

Ok, Have you covered it in any of your articles? Can you refer me to it?

Hi Jason. Thank You very much for teach how ti make Forecast. Butaca i have a doubt, in this example only we have 12 prediction for 12 observations (or expected values).

Un this case, i would like yo know. What is the prediction to the short future.

Thank so much.

Atte. Luis

Perhaps this post will help:

https://machinelearningmastery.com/make-sample-forecasts-arima-python/

Hi! Thank You for your teach.I have a problem when I use the ARIMA to build a model for the multivariate data,but appear some error”TypeError:must be str,not list”at”model=ARIMA(history,order=(5,1,0))”.The history data is a list of 500*2.

Sounds like you might not have loaded the dataset correctly.

Perhaps confirm it was loaded as real values, not strings.

Hi can you please show us some plots ,spcific to ARIMA ?

thank you

Like what exactly?

Hi Jason:

Thanks for this tutorial.

Just wondering how was a value of 0 was decided for q? For that don’t you need the PACF plot?

Any help will be much appreciated.

Regards,

Anindya

I may have configured the model in this tutorial based on a trial and error.

Hey man, great tutorial. I just wanted to ask you how does residual error or its graph fit into time series analysis, I mean i am not able to understand the importance of residual error, what does it show. I am still in the learning phase.

Thanks, we expect the residual error to be random – if there is a pattern to it, it means our model is missing something important.

Hi Jason,

I’m considering buying your book. Will the code examples be up to date seeing as it is now 2019? Also, what success have you had forecasting several time series, lets say 30, with the same model. Would you suggest more of an ensemble approach?

Oh, is any other reading material you would suggest? We did not cover time series in my masters program, so I’m a newbie.

Yes, you can get started with the basics here:

https://machinelearningmastery.com/start-here/#timeseries

Advanced topics here:

https://machinelearningmastery.com/start-here/#deep_learning_time_series

Yes, I update the books frequently. After purchasing, you can email me any time to get the latest version.

Hmm, 30 is not a large number, it might be best developing a separate model for each and compare the results to any model that tries to learn across the series.

Hello Jason,

I still don’t understand why the forecast is one step ahead of the actual value. Why is this behavior expected, If for instance my model predicts very well the timeseries but with a lag, does this mean that my model is good or I should go on tuning to take off the lag?

In the case of the lag, the line print(‘predicted=%f, expected=%f’ % (yhat, obs)) isn’t it also lagged and not representative of the actual comparison?

Thanks

I think you are describing a persistence forecast, this might help:

https://machinelearningmastery.com/faq/single-faq/why-is-my-forecasted-time-series-right-behind-the-actual-time-series