Selecting a time series forecasting model is just the beginning.
Using the chosen model in practice can pose challenges, including data transformations and storing the model parameters on disk.
In this tutorial, you will discover how to finalize a time series forecasting model and use it to make predictions in Python.
After completing this tutorial, you will know:
 How to finalize a model and save it and required data to file.
 How to load a finalized model from file and use it to make a prediction.
 How to update data associated with a finalized model in order to make subsequent predictions.
Let’s get started.
Process for Making a Prediction
A lot is written about how to tune specific time series forecasting models, but little help is given to how to use a model to make predictions.
Once you can build and tune forecast models for your data, the process of making a prediction involves the following steps:
 Model Selection. This is where you choose a model and gather evidence and support to defend the decision.
 Model Finalization. The chosen model is trained on all available data and saved to file for later use.
 Forecasting. The saved model is loaded and used to make a forecast.
 Model Update. Elements of the model are updated in the presence of new observations.
We will take a look at each of these elements in this tutorial, with a focus on saving and loading the model to and from file and using a loaded model to make predictions.
Before we dive in, let’s first look at a standard univariate dataset that we can use as the context for this tutorial.
Daily Female Births Dataset
This dataset describes the number of daily female births in California in 1959.
The units are a count and there are 365 observations. The source of the dataset is credited to Newton (1988).
Learn more and download the dataset from Data Market.
Download the dataset and place it in your current working directory with the filename “dailytotalfemalebirths.csv“.
We can load the dataset as a Pandas series. The snippet below loads and plots the dataset.

from pandas import Series from matplotlib import pyplot series = Series.from_csv(‘dailytotalfemalebirths.csv’, header=0) print(series.head()) series.plot() pyplot.show() 
Running the example prints the first 5 rows of the dataset.

Date 19590101 35 19590102 32 19590103 30 19590104 31 19590105 44 
The series is then graphed as a line plot.
1. Select Time Series Forecast Model
You must select a model.
This is where the bulk of the effort will be in preparing the data, performing analysis, and ultimately selecting a model and model hyperparameters that best capture the relationships in the data.
In this case, we can arbitrarily select an autoregression model (AR) with a lag of 6 on the differenced dataset.
We can demonstrate this model below.
First, the data is transformed by differencing, with each observation transformed as:

value(t) = obs(t) – obs(t – 1) 
Next, the AR(6) model is trained on 66% of the historical data. The regression coefficients learned by the model are extracted and used to make predictions in a rolling manner across the test dataset.
As each time step in the test dataset is executed, the prediction is made using the coefficients and stored. The actual observation for the time step is then made available and stored to be used as a lag variable for future predictions.
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from pandas import Series from matplotlib import pyplot from statsmodels.tsa.ar_model import AR from sklearn.metrics import mean_squared_error import numpy
# create a difference transform of the dataset def difference(dataset): diff = list() for i in range(1, len(dataset)): value = dataset[i] – dataset[i – 1] diff.append(value) return numpy.array(diff)
# Make a prediction give regression coefficients and lag obs def predict(coef, history): yhat = coef[0] for i in range(1, len(coef)): yhat += coef[i] * history[–i] return yhat
series = Series.from_csv(‘dailytotalfemalebirths.csv’, header=0) # split dataset X = difference(series.values) size = int(len(X) * 0.66) train, test = X[0:size], X[size:] # train autoregression model = AR(train) model_fit = model.fit(maxlag=6, disp=False) window = model_fit.k_ar coef = model_fit.params # walk forward over time steps in test history = [train[i] for i in range(len(train))] predictions = list() for t in range(len(test)): yhat = predict(coef, history) obs = test[t] predictions.append(yhat) history.append(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 first prints the Mean Squared Error (MSE) of the predictions, which is 52 (or about 7 births on average, if we take the square root to return the error score to the original units).
This is how well we expect the model to perform on average when making forecasts on new data.
Finally, a graph is created showing the actual observations in the test dataset (blue) compared to the predictions (red).
This may not be the very best possible model we could develop on this problem, but it is reasonable and skillful.
2. Finalize and Save Time Series Forecast Model
Once the model is selected, we must finalize it.
This means save the salient information learned by the model so that we do not have to recreate it every time a prediction is needed.
This involves first training the model on all available data and then saving the model to file.
The statsmodels implementations of time series models do provide builtin capability to save and load models by calling save() and load() on the fit ARResults object.
For example, the code below will train an AR(6) model on the entire Female Births dataset and save it using the builtin save() function, which will essentially pickle the ARResults object.
The differenced training data must also be saved, both the for the lag variables needed to make a prediction, and for knowledge of the number of observations seen, required by the predict() function of the ARResults object.
Finally, we need to be able to transform the differenced dataset back into the original form. To do this, we must keep track of the last actual observation. This is so that the predicted differenced value can be added to it.
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from pandas import Series from statsmodels.tsa.ar_model import AR from statsmodels.tsa.ar_model import ARResults import numpy
# create a difference transform of the dataset def difference(dataset): diff = list() for i in range(1, len(dataset)): value = dataset[i] – dataset[i – 1] diff.append(value) return numpy.array(diff)
# load dataset series = Series.from_csv(‘dailytotalfemalebirths.csv’, header=0) X = difference(series.values) # fit model model = AR(X) model_fit = model.fit(maxlag=6, disp=False) # save model to file model_fit.save(‘model.pkl’) # save the differenced dataset numpy.save(‘data_differenced.npy’, X) # save the last ob numpy.save(‘last_obs.npy’, [series.values[–1]]) 
This code will create a file model.pkl that you can load later and use to make predictions.
The entire training dataset is saved as data_differenced.npy and the last observation is saved in the file last_obs.npy as an array with one item.
The NumPy save() function is used to save the differenced training data and the observation. The load() function can then be used to load these arrays later.
The snippet below will load the model, differenced data, and last observation.

# load model from statsmodels.tsa.ar_model import ARResults import numpy loaded = ARResults.load(‘model.pkl’) print(loaded.params) data = numpy.load(‘data_differenced.npy’) last_ob = numpy.load(‘last_obs.npy’) print(last_ob) 
Running the example prints the coefficients and the last observation.

[ 0.12129822 0.75275857 0.612367 0.51097172 0.4176669 0.32116469 0.23412997] [50] 
I think this is good for most cases, but is also pretty heavy. You are subject to changes to the statsmodels API.
My preference is to work with the coefficients of the model directly, as in the case above, evaluating the model using a rolling forecast.
In this case, you could simply store the model coefficients and later load them and make predictions.
The example below saves just the coefficients from the model, as well as the minimum differenced lag values required to make the next prediction and the last observation needed to transform the next prediction made.
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from pandas import Series from statsmodels.tsa.ar_model import AR import numpy
# create a difference transform of the dataset def difference(dataset): diff = list() for i in range(1, len(dataset)): value = dataset[i] – dataset[i – 1] diff.append(value) return numpy.array(diff)
# load dataset series = Series.from_csv(‘dailytotalfemalebirths.csv’, header=0) X = difference(series.values) # fit model window_size = 6 model = AR(X) model_fit = model.fit(maxlag=window_size, disp=False) # save coefficients coef = model_fit.params numpy.save(‘model_coef.npy’, coef) # save lag lag = X[–window_size:] numpy.save(‘model_history.npy’, lag) # save the last ob numpy.save(‘last_obs.npy’, [series.values[–1]]) 
The coefficients are saved in the local file model_coef.npy, the lag history is saved in the file model_history.npy, and the last observation is saved in the file last_obs.npy.
These values can then be loaded again as follows:

# load saved model import numpy coef = numpy.load(‘model_coef.npy’) print(coef) lag = numpy.load(‘model_history.npy’) print(lag) last_ob = numpy.load(‘last_obs.npy’) print(last_ob) 
Running this example prints the loaded data for review. We can see the coefficients and last observation match the output from the previous example.

[ 0.12129822 0.75275857 0.612367 0.51097172 0.4176669 0.32116469 0.23412997] [10 3 15 4 7 5] [50] 
Now that we know how to save a finalized model, we can use it to make forecasts.
3. Make a Time Series Forecast
Making a forecast involves loading the saved model and estimating the observation at the next time step.
If the ARResults object was serialized, we can use the predict() function to predict the next time period.
The example below shows how the next time period can be predicted.
The model, training data, and last observation are loaded from file.
The period is specified to the predict() function as the next time index after the end of the training data set. This index may be stored directly in a file instead of storing the entire training data, which may be an efficiency.
The prediction is made, which is in the context of the differenced dataset. To turn the prediction back into the original units, it must be added to the last known observation.

from statsmodels.tsa.ar_model import ARResults import numpy # load model model = ARResults.load(‘model.pkl’) data = numpy.load(‘data_differenced.npy’) last_ob = numpy.load(‘last_obs.npy’) # make prediction predictions = model.predict(start=len(data), end=len(data)) # transform prediction yhat = predictions[0] + last_ob[0] print(‘Prediction: %f’ % yhat) 
Running the example prints the prediction.
4. Update Forecast Model
Our work is not done.
Once the next real observation is made available, we must update the data associated with the model.
Specifically, we must update:
 The differenced training dataset used as inputs to make the subsequent prediction.
 The last observation, providing a context for the predicted differenced value.
Let’s assume the next actual observation in the series was 48. We could then update the data as follows:

import numpy # get real observation observation = 48 # update and save differenced observation diffed = observation – last_ob[0] data = numpy.append(data, [diffed], axis=0) numpy.save(‘data_differenced.npy’, data) # update and save real observation last_ob[0] = observation numpy.save(‘last_obs.npy’, last_ob) 
We can also use a similar trick to load the raw coefficients and make a manual prediction.
The complete example is listed below.
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import numpy
def predict(coef, history): yhat = coef[0] for i in range(1, len(coef)): yhat += coef[i] * history[–i] return yhat
# load model coef = numpy.load(‘model_coef.npy’) lag = numpy.load(‘model_history.npy’) last_ob = numpy.load(‘last_obs.npy’) # make prediction prediction = predict(coef, lag) # transform prediction yhat = prediction + last_ob[0] print(‘Prediction: %f’ % yhat) 
Running the example, we achieve the same prediction as we would expect, given the underlying model and method for making the prediction are the same.
Finally, we can update the lag differenced observations and the last observed value as follows.

import numpy # get real observation observation = 48 # update and save differenced observation lag = numpy.load(‘model_history.npy’) last_ob = numpy.load(‘last_obs.npy’) diffed = observation – last_ob[0] lag = numpy.append(lag[1:], [diffed], axis=0) numpy.save(‘model_history.npy’, lag) # update and save real observation last_ob[0] = observation numpy.save(‘last_obs.npy’, last_ob) 
We have focused on onestep forecasts.
These methods would work just as easily for multistep forecasts, by using the model repetitively and using forecasts of previous time steps as input lag values to predict observations for subsequent time steps.
Consider Storing All Observations
Generally, it is a good idea to keep track of all the observations.
This will allow you to:
 Provide a context for further time series analysis to understand new changes in the data.
 Train a new model in the future on the most recent data.
 Backtest new and different models to see if performance can be improved.
For small applications, perhaps you could store the raw observations in a file alongside your model.
It may also be desirable to store the model coefficients and required lag data and last observation in plain text for easy review.
For larger applications, perhaps a database system could be used to store the observations.
Summary
In this tutorial, you discovered how to finalize a time series model and use it to make predictions with Python.
Specifically, you learned:
 How to save a time series forecast model to file.
 How to load a saved time series forecast from file and make a prediction.
 How to update a time series forecast model with new observations.
Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.