Deep Probabilistic Koopman (DPK): Long-term time-series forecasting under periodic uncertainties

Stable, long-term, probabilistic forecasts with calibrated uncertainty measures. Our Koopman-theoretic approach enables powerful forecasts of a variety of quasi-periodic phenomena from electricity demand to neural activity.

For a more user-friendly DPK framework, see this repository.

Dependencies

The following are dependencies of the DPK framework:

• torch
• numpy
• scipy

While these are used for the experiments in the paper:

• pandas
• sklearn
• matplotlib
• jupyter
• allensdk (optional—only install this if you wish to work with the neuroscience data)

All of these are available through pip.

Results

All results can be found in figures.ipynb with instructions on how to replicate them. In a retrospective comparison, our model outperforms all 177 competing teams in Global Energy Forecasting Competition 2017 at the qualifying task. We also show significant improvements over a NASA air quality forecast model.

Training a DPK model

1. Load your time-series data into memory as a time-by-n numpy array.
   (If the samples are not uniform over time, you will also need to provide a 1D array of these times).
2. Input the frequencies your data exhibits. This is usually as easy as 24 hours, 1 week, and 1 year, but sometimes the frequencies must be found by DPK by solving a global optimization problem or via the FFT.
3. Choose a model object from model_obs.py or write your own. We recommend starting out with SkewNLLwithTime, which assumes your data is drawn from a time-varying skew-normal distribution at every point in time. "withTime" indicates that this model object allows for non-periodic trends.
4. Call fit!

x = np.sin(np.linspace(0, 1000 * np.pi, 10000)).reshape(-1, 1)  # for example
periods = [20,]  # 20 idxs is a period
model_obj = model_objs.SkewNLLwithTime(x_dim=x.shape[1], num_freqs=[len(periods),] * 3)

k = koopman_probabilistic.KoopmanProb(model_obj, device='cpu', num_fourier_modes=len(periods))
k.find_fourier_omegas(x, hard_code=periods)
k.fit(x, iterations=25, verbose=True)

Forecasting

1. Call predict! This returns the time-varying parameters of the distribution defined by your model object. To obtain a point forecast with uncertainty, simply call model_obj.mean(params) and model_obj.std(params). This step should be instantaneous since DPK does not require time-stepping for predictions.

params = k.predict(T=15000)
loc_hat, scale_hat, alpha_hat = params  # time-varying parameters of skew-normal distribution
x_hat = model_obj.mean(params)
std_hat = model_obj.std(params)  # uncertainty over time

A similar but slightly more sophisticated example can be found in example.py.