Time Series Forecasting With a Learning Algorithm: An Approximate - PowerPoint PPT Presentation

Time Series Forecasting With a Learning Algorithm: An Approximate Dynamic Programming Approach Ricardo Collado 1 an Creamer 1 Germ´ 1 School of Business Stevens Institute of Technology Hoboken, New Jersey R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 1 / 34

Introduction Time Series Drivers: • Historical data: • Incorporated in classical time series forecast methods • Works best when the underlying model is fix • Exogenous processes: • Not included in historical observations • Difficult to incorporate via classical methods • Could indicate changes in the underlying model R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 2 / 34

Introduction Techniques to handle changes due to external forces: • Jump Diffusion Models • Regime Switching Methods • System of Weighted Experts • Others . . . These methods do not directly integrate alternative data sources available to us R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 3 / 34

Introduction Alternative data sources: • Text & News Analysis • Social Networks Data • Sentiment Analysis R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 4 / 34

Introduction Alternative data sources: • Text & News Analysis • Social Networks Data • Sentiment Analysis R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 4 / 34

Introduction Alternative data sources: • Text & News Analysis • Social Networks Data • Sentiment Analysis R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 4 / 34

Introduction We study time series forecast methods that are: • Dynamic • Context-Based • Capable of Integrating Social, Text, and Sentiment Data In this presentation we develop: • Stochastic dynamic programming model for time series forecast • Rely on an “external forecast” for future values • External forecast allows to incorporate alternative data sources R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 5 / 34

Traditional Approach Time Series Fitting Process R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 6 / 34

Traditional Approach s 0 = x 0 a 0 = φ ∗ 1 X 1 = φ ∗ 1 x 0 + � 1 A = { φ ∈ R } s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Approach s 0 = x 0 a 0 = φ ∗ 1 X 1 = φ ∗ 1 x 0 + � 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Approach s 1 = x 1 a 1 = φ ∗ 2 X 2 = φ ∗ 2 x 1 + � 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Approach s 2 = x 2 a 2 = φ ∗ 3 X 3 = φ ∗ 3 x 2 + � 3 s 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Approach s 3 = x 3 a 3 = φ ∗ 4 X 4 = φ ∗ 4 x 3 + � 4 s 3 s 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Approach s 4 = x 4 a 4 = φ ∗ 5 X 5 = φ ∗ 5 x 4 + � 5 s 3 s 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 7 / 34

Traditional Time Series Fitting Main Problem: � T � � min c t ( s t , a t ) , π ∈ Π E t =1 where a t = π t ( x 1 , . . . , x t ) is an admissible fitting policy. • The time series model is parametrized by Θ ⊆ R d • A ( s ) = Θ for all states s • c t ( s , a ) is the result of a goodness of fit test for the observations s = ( x 0 , . . . , x t ) and model selection a = θ • Solution via Bellman’s optimality equations R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 8 / 34

Traditional Time Series Fitting Conditions to guarantee optimality: • The set of actions A ( s ) is compact • The cost functions c t ( s , · ) are lower semicontinuous • For every measurable selection a t ( · ) ∈ A t ( · ), the functions s �→ c t ( s , a t ( s )) and c T ( · ) are elements of L 1 ( S , B S , P 0 ) • The DP stochastic kernel function Q t ( s , · ) is continuous R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 9 / 34

Traditional Time Series Fitting Value Functions: Value functions v t : S → R , t = 1 , . . . , T , given recursively by: v T ( s ) = c T ( s ) v t ( s ) = min a ∈ A t ( s ) { c t ( s , a ) + E [ v t +1 | s , a ] } , for all s ∈ S and t = T − 1 , . . . , 0. Bellman’s Optimality Equations: Then an optimal Markov policy π ∗ = { π ∗ 0 , . . . , π ∗ T − 1 } exists and satisfies the equations: π ∗ t ( s ) ∈ arg min { c t ( s , a ) + E [ v t +1 | s , a ] } , s ∈ S , t = T − 1 , . . . , 0 . a ∈ A t ( s ) Conversely, any measurable solution of these is an optimal Markov policy π ∗ . R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 10 / 34

Traditional Time Series Fitting Notice that: • Our choice of model does not affect future observations and cost. • So, E [ v | s , a ] = E [ v | s , a ′ ], for any ( s , a ) , ( s , a ′ ) ∈ graph( A ). • Therefore we can rewrite the optimal policy as: π ∗ t ( s ) ∈ arg min { c t ( s , a ) } , s ∈ S , t = T − 1 , . . . , 0 . a ∈ A t ( s ) R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 11 / 34

Traditional Time Series Fitting Notice that: • Our choice of model does not affect future observations and cost. • So, E [ v | s , a ] = E [ v | s , a ′ ], for any ( s , a ) , ( s , a ′ ) ∈ graph( A ). • Therefore we can rewrite the optimal policy as: π ∗ t ( s ) ∈ arg min { c t ( s , a ) } , s ∈ S , t = T − 1 , . . . , 0 . a ∈ A t ( s ) • The optimal policy π ∗ is purely myopic R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 11 / 34

Traditional Time Series Fitting Q: How to break with the myopic policy? R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 12 / 34

Traditional Time Series Fitting A: Introduce a new Markov model R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 13 / 34

New Markov Model New Markov Model: • Given stochastic process { X t | t = 0 , . . . , T } , s.t. X 0 = { φ 0 } • Time series model parameterized by Θ ⊆ R d • State space:  �  �  x t observation from X t ,   �  � S t = ( x t , h t − 1 , θ t − 1 ) h t = x 0 , . . . , x t − 1 sample sequence , �  �    � θ t − 1 = ( φ 1 , . . . , φ p ) ∈ Θ • Action space: A ( s ) = Θ for all states s ∈ S t R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 14 / 34

New Markov Model Cost function: c t ( s , θ t ) = γ ( s t , θ t ) + r δ ( s t , θ t − 1 , θ t ) • γ : Goodness of fit test • δ : Penalty on changes from previous model selection • r ≥ 0: Scaling factor used to balance fit and penalty R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 15 / 34

New Markov Model Example: � � �� � � � ��� χ 2 ( θ t | h t − 1 , x t ) + r � E [ P θ t | h t − 1 , x t ] − E � h t − 1 , x t 1 − exp − λ P θ t − 1 , where r , λ ≥ 0. • γ ( s t , θ t ) := χ 2 ( θ t | h t − 1 , x t ) � � � � �� �� � E [ P θ t | h t − 1 , x t ] − E � h t − 1 , x t • δ ( s t , θ t − 1 , θ t ) := 1 − exp − λ P θ t − 1 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 16 / 34

New Markov Model Dynamic Optimization Model: s 0 = x 0 a 0 = φ ∗ 1 X 1 = φ ∗ 1 x 0 + � 1 A = { φ ∈ R } s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

New Markov Model Dynamic Optimization Model: s 0 = x 0 a 0 = φ ∗ 1 γ enforces good fit X 1 = φ ∗ 1 x 0 + � 1 s 0 downstream δ adds cost for changing model R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

New Markov Model Dynamic Optimization Model: s 0 = x 0 a 0 = φ ∗ 1 γ enforces good fit X 1 = φ ∗ 1 x 0 + � 1 s 0 downstream δ adds cost for changing model R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

New Markov Model Dynamic Optimization Model: s 1 = x 1 a 1 = φ ∗ 2 X 2 = φ ∗ 2 x 1 + � 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

New Markov Model Dynamic Optimization Model: s 2 = x 2 a 2 = φ ∗ 3 X 3 = φ ∗ 3 x 2 + � 3 s 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

New Markov Model Dynamic Optimization Model: s 3 = x 3 a 3 = φ ∗ 4 X 4 = φ ∗ 4 x 3 + � 4 s 3 s 2 s 1 s 0 R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 17 / 34

Lookahead Methods Introducing Exogenous Information Via Lookahead Methods R. Collado (Stevens) Learning Time Series and Dynamic Programming September 10, 2019 18 / 34