Time series deep learning
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This document summarizes different methods for time series analysis and prediction in the deep learning era. It discusses classical autoregressive and Bayesian models, general machine learning approaches, and various deep learning techniques including DeepAR, Deep Ensembles, Deep State Space models, and combinations of deep neural networks with Gaussian processes. The document compares the pros and cons of each approach in terms of scalability, ability to share information across time series, handling cold starts with limited data, estimating predictive uncertainty, and dealing with unevenly spaced time series data.
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Time series deep learning
- 1. Time series analysis and prediction in the deep learning era Alberto Arrigoni, PhD February 2019
- 2. Time series: analysis and prediction What will the future hold? FuturePast Now
- 3. Time series applications + context Time series prediction: e.g. demand/sales forecasting... Use prediction for anomaly detection: e.g. manufacturing settings... Counterfactual prediction: e.g. marketing campaigns... Show ads Counterfactual
- 4. Time series applications + context Time series prediction: e.g. demand/sales forecasting... Use prediction for anomaly detection: e.g. manufacturing settings... Counterfactual prediction: e.g. marketing campaigns... Show ads Counterfactual
- 5. Time series prediction methods (non-comprehensive list) Classical autoregressive models Bayesian AR models General machine learning approaches Deep learning t+3
- 6. Number of time series (~ thousands) [the SCALE problem] Time series are often highly erratic, intermittent or bursty (...and on highly different scales) ~ 10 items 2 items Product A Product B ... (1) (2) Time series prediction and sales forecasting: issues E.g. retail businesses
- 7. Time series belong to a hierarchy of products/categories E.g. online retailer selling clothes Time series prediction and sales forecasting: issues Now Nike t-shirts Clothes (total sales) T-shirts total sales ~ 100 ~ 1000(3) For new products historical data is missing (the cold-start problem) (4) Adidas t-shirts
- 8. Classical autoregressive models Estimate model order (AIC, BIC) Fit model parameters (maximum likelihood) Autoregressive component Moving average component Test residuals for randomness De-trending by differencing Variance stabilization by log or Box-Cox transformation Workflow
- 9. Classical autoregressive models THE PROS: - Good explainability - Solid theoretical background - Very explicit model - A lot of control as it is a manual process THE CONS: - Data is seldom stationary: trend, seasonality, cycles need to modeled as well - Computationally intensive (one model for each time series) - No information sharing across time series (apart from Hyndman’s hts approach) * - Historical data are essential for forecasting, (no cold-start) * https://robjhyndman.com/publications/hierarchical/ Tech stack and packages - Rob Hyndman’s online text: https://otexts.com/fpp2/ - Infamous auto.arima package, ets, tbats, garch, stl... - Python’s Pyramid
- 10. - Aggregate histograms over time scales - Transform into Fourier space - Add low/high pass filters as variables General machine learning approach for ts prediction Past Yt t Autoregressive component - Can use any number of methods (linear, trees, neural networks...) - Turn the time series prediction problem into a supervised learning problem - Easily extendable to support multiple input variables - Covariates can be easily handled and transformed through feature engineering Covariates E.g. feature engineering
- 11. THE PROS: - Can model non-linear relationships - Can model the “hierarchical structure” of the time series through categorical variables - Support for covariates (predictors) + feature engineering - One model is shared among multiple time series - Cold-start predictions are possible by iteratively feeding the predictions back to the feature space THE CONS: - Feature engineering takes time - Long-term relationships between data points need to be explicitly modeled (autoregressive features) General machine learning approach for ts prediction Tech stack and packages - Sklearn, PySpark for feature engineering, data reduction
- 12. Bayesian AR models (Facebook Prophet) Prophet is a Bayesian GAM (Generalized Additive Model) Linear trend with changepoints Seasonal component Holiday-specific componentt Sales 1) Detect changepoints in the time series 2) Fit linear trend parameters (k and delta) (piecewise) linear trends Growth rate Growth rate adjustment ** ** An additional ‘offset’ term has been omitted from the formula * Implemented using STAN *
- 13. Bayesian AR models (Facebook Prophet) E.g. P = 365 for yearly data Need to estimate 2N parameters (an and bn ) using MCMC! Prophet is a Bayesian GAM (Generalized Additive Model) Linear trend with changepoints Seasonal component Holiday-specific componentt Sales
- 14. THE PROS: - Uncertainty estimation - Bayesian changepoint detection - User-in-the-loop paradigm (Prophet) - Black-box variational inference is revolutionizing Bayesian inference THE CONS: - Bayesian inference takes time (the “scale” issue) - One model for each time series - No information sharing among series (unless you specify a hierarchical bayesian model with shared parameters, but still...) - Historical data are needed for prediction! - Performance is often on par* with autoregressive models Tech stack and packages - Python/R clients for Prophet * - R package for structural bayesian time series models: Bsts Bayesian AR models * Taylor et al., Forecasting at scale* This may open endless discussions. Bottom line: depends on your data :)
- 15. Interlude: uncertainty estimation with deep learning - Uncertainty estimation is a prerogative of Bayesian methods. - Black box variational inference (ADVI) has sprung renewed interest towards Bayesian neural networks, but we are not there yet in terms of performance - A DeepMind paper from NIPS 2017 introduces a simple yet effective way to estimate predictive uncertainty using Deep Ensembles For a TensorFlow implementation of this paper: https://arrigonialberto86.github.io/funtime/deep_ensembles.html “Engineering Uncertainty Estimation in Neural Networks for Time Series Prediction at Uber” https://eng.uber.com/neural-network s-uncertainty-estimation/ 1) 2)
- 16. Interlude: Deep Ensembles Train a deep learning model using a custom final layer which parametrizes a Gaussian distribution Sample x from the Gaussian distribution using fitted parameters Calculate loss to backpropagate the error (using Gaussian likelihood) (1) (3) (2) Network output
- 17. What the network is learning: different regions of the x space have different variances Generate a synthetic dataset with different variances Interlude: Deep Ensembles PREDICTION ON TRAINING DATASET SYNTHETIC TRAINING DATASET Use the network from previous slide to predict on the training set to see if it actually detects variance reduction
- 18. Interlude: Deep Ensembles The authors suggest to train different NNs on the same data (the whole training set) with random initialization Ensemble networks (improve generalization power) Uniformly weighted mixture model Predictions for regions outside of the training dataset show increasing variance (due to ensembling) In addition to ‘distribution’ modeling and ensembling the authors suggest to use the fast gradient sign method * to produce adversarial training example (Not shown here) * Goodfellow et al., 2014
- 19. Interlude: Deep Ensembles Custom GaussianLayer Let’s just do some extra work and define a custom layer For a TensorFlow implementation of this paper: https://arrigonialberto86.github.io/funtime/deep_ensembles.html
- 20. Interlude: Deep Ensembles Custom layer returns both mu and sigma Build 2 weight matrices + 2 biase terms
- 21. DeepAR (Amazon) Instead of fitting separate models for each time series we create a global model from related time series to handle widely-varying scales through rescaling and velocity-based sampling. Differentscales Probabilities ~1000 time series Past Future Covariates Flunkert et al., 2017
- 22. DeepAR (Amazon) ht-1 ht ht+1 - Use LSTM interactions in the time series - As seen with the Deep Ensemble architecture, we can predict parameters of distributions at each time point (theta vector) - Time series need to be scaled for the network to learn time-varying dynamics
- 23. DeepAR (Amazon) * Likelihood/loss is customizable: Gaussian/negative binomial for count data + overdispersion Training Prediction *
- 24. For a commentary + code review: https://arrigonialberto86.github.io/funtime/deepar.html DeepAR (Amazon) The mandatory ‘AirPassengers’ prediction example (results shown on training set) It is given that this is not the use case Amazon had in mind...
- 25. DeepAR (Amazon) - Long-term relationships are handled by design using LSTMs - One model is fitted for all the time series - The hierarchical ts structure and inter-dependencies are captured by using covariates (even holidays, recurrent events etc...) - The model can be used for cold-start predictions (using categorical covariates with ‘descriptive’ product information) - Hassle-free uncertainty estimation DeepAR and the AWS ecosystem AWS SageMaker
- 26. Deep State Space (NIPS 2018)* A state space model or SSM is just like an Hidden Markov Model, except the hidden states are continuous Observation (zt ) update Latent state (lt ) update In normal settings we would need to fit these parameters for each time series zt-1 zt zt+1 ??? * Rangapuram et al, 2018, Deep State Space Models for Time Series Forecasting
- 27. Deep State Space (NIPS 2018) Training Prediction Compute the negative likelihood, derive the time-varying SS parameters using backpropagation Use Kalman filtering to estimate lt , then recursively apply the transition equation and the observation model to generate prediction samples
- 28. - Long-term relationships are handled by design using LSTMs - One model is fitted for all the time series - The hierarchical ts structure and inter-dependencies are captured by ad-hoc design and components of the SS model (even holidays, recurrent events etc...) - The model can be used for cold-start predictions (using categorical covariates with ‘descriptive’ product information) Deep State Space (NIPS 2018)
- 29. Going forward: Deep factors with GPs * * Maddix et al., “Deep Factors with Gaussian Processes for Forecasting”, NIPS 2018 The combination of probabilistic graphical models with deep neural networks has been an active research area recently Global DNN backbone and local Gaussian Process (GP). The main idea is to represent each time series as a combination of a global time series and a corresponding local model. gt gt gt gt RNN zit + covariates Backpropagation to find RNN parameters to produce global factors (gt ) + GP hyperparameters
- 30. M4 forecasting competition winner algo (Uber, 2018) The winning idea is often the simplest! Hybrid Exponential Smoothing-Recurrent Neural Networks (ES-RNN) method. It mixes hand-coded parts like ES formulas with a black-box recurrent neural network (RNN) forecasting engine. yt-1 yt yt+1 Deseasonalized and normalized vector of covariates + previous state RNN results are now part of a parametric model
- 31. Classical autoregressive models Bayesian models (GAM/structural) Classical machine learning Deep learning approaches Scalability Info sharing across ts Cold-start predictions Uncertainty estimation Unevenly spaced time series * Summary of performance * DeepAR Deep Factors * Chen et al., Neural ordinary differential equations, 2018 / Futoma et al., 2017, Multitask GP + RNN
- 32. BACKUP SLIDES
- 33. Deep State Space (Amazon) Level-trend model parametrization:
- 34. DeepAR (Amazon) Step 1 Step 2 Step 3 Training procedure: - Predict parameters (e.g. mu, sigma) - Compute likelihood of the prediction (can be Gaussian as we have seen with Deep Ensembles) * - Sample next point * Likelihood/loss is customizable: Gaussian/negative binomial for count data + overdispersion Training Prediction (~ Monte Carlo)