Basic time series modeling with Stan and Pystan
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Basic time series modeling with Stan and Pystan.

This is a small set of code to make it easy to do basic time series modeling with stan, and particularly with the pystan interface. In brief:

from pystan_time_series import TimeSeriesModel
Y = [your data here]
model = TimeSeriesModel(Y=Y)

What makes pystan_time_series useful is that you can turn on options to modify to the model to do everything and the kitchen sink. The Jupyter notebook Examples_and_Tests shows the different types of models, how to use the Python interface to call them, and how the model correctly recovers the parameters of simulated data. Below is an overview of the tricks that pystan_time_series can do.

Things that pystan_time_series can model easily

  • Many different time series
T = 100 #Number of time points
K = 10 #Number of time series
Y = [data of shape (T,K)]

model = TimeSeriesModel(Y=Y) 
  • missing data
Y[5:10] = nan #Missing data are passed as nans
model = TimeSeriesModel(Y=Y)
model.sampling()['Y_latent'] #Includes all the observed points in Y, along with sampled estimates for what the missing observations are
  • AR(p) models
model = TimeSeriesModel(Y=Y, p=2)
  • MA(q) models
model = TimeSeriesModel(Y=Y, q=2)
  • ARMA(p,q) models
model = TimeSeriesModel(Y=Y, p=1, q=2)
  • Setting p and q to multiple, non-sequential lags
model = TimeSeriesModel(Y=Y, p=[1,5,10], q=[2,7])
  • Multi-dimensional time series, which can be modeled as affecting each other as a VAR model
D = 3 #Number of dimensions for each time series
Y = [data of shape (K,T,D)]
model = TimeSeriesModel(Y=Y, p=1, q=1) #All 3 dimensions affect each other at a lag of 1 (p). The moving average (q) element is separate for each dimension.
  • Setting one or more dimensions to be difference (An I(1) model)
D = 3
Y = [data of shape (K,T,D)]
model = TimeSeriesModel(Y=Y, difference=[0,1,0]) #Says the 2nd dimension should be differenced before modeling, making it an I(1) model
  • Setting one or more dimensions to be monotonic
D = 3
Y = [data of shape (K,T,D)]
model = TimeSeriesModel(Y=Y, monotonic=[0,0,1]) #Says the 3rd dimension is monotonically increasing
  • Partial pooling of parameter estimates across the K time series
model = TimeSeriesModel(Y=Y, use_partial_pooling=True)
  • Using noise that is distributed not as a normal, but as a student's t distrbution
model = TimeSeriesModel(Y=Y, use_student=True)
  • Do something crazy
K = 100
T = 200
D = 5
Y = [your data of shape (K,T,D)]
Y[:5] = nan
Y[100:105] = nan
Y[150:] = nan

model = TimeSeriesModel(Y=Y, p=[1,5,10], q=[1,5], partial_pooling=True, use_student=True, difference=[1,0,0,0,0], monotonic=[1,0,0,0,0])


Each parameter is modeled as a prior distribution with a scale and location:

mu ~ normal(mu_prior_scale, mu_prior_location);

There are default values for these prior scales/locations (all rather uninformative). Upon creating the model the prior values, along with everything else that will go into stan, is in model.stan_data

model = TimeSeriesModel(Y=Y)
> 0

Changes to these priors can be made when creating the model:

model = TimeSeriesModel(Y=Y, mu_prior_location=5)
> 5

It's possible to sample from the priors directly, without updating to any data.

model = TimeSeriesModel(Y=Y, return_priors=True)

The results are the distribution of the priors, which can be visualized, compared to the distribution of the posteriors after updating to data, etc.

Under the hood with the Stan models

The directory stan_models/ has a collection of time series models as individual .stan files, each paired with a .py file that TimeSeriesModel calls for easier interfacing with the model. However, only one of these models actually matters: VAR.stan. This model is fairly complex, but it implements all the capabilities of all the other models, which can then be turned on or off by passing options to the model as data. This is the stan model that TimeSeriesModel actually uses by default. The other .stan files are included for people that want to peruse the code of a simpler implementation, so they may better learn a model or stan.

This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.