Using the Mixture Kalman Filter to Track a Hidden State in - PowerPoint PPT Presentation

Using the Mixture Kalman Filter to Track a Hidden State in Changepoint Models Sarah Oscroft Supervisor: Andrew Wright 4th September 2015 Sarah Oscroft Mixture Kalman Filter 4th September 2015 1 / 13

Outline 1. State Space Models 2. Kalman Filter 2. 1. Dynamic Linear Models 2. 2. Kalman Filter 3. Mixture Kalman Filter 3. 1. Conditional Dynamic Linear Models 3. 2. Mixture Kalman Filter 3. 3. Multivariate Mixture Kalman Filter Sarah Oscroft Mixture Kalman Filter 4th September 2015 2 / 13

State Space Models A state space model: Is an unobservable process x t , and a noisy observation y t . Follows a Markov process: x t depends only upon x t − 1 , y t depends only upon x t . x t-1 x t x t+1 y t-1 y t y t+1 Sarah Oscroft Mixture Kalman Filter 4th September 2015 3 / 13

Dynamic Linear Models A state space model which is linear Gaussian is known as a DLM, and has the form y t = G t x t + v t , v t ∼ N (0 , V t ) , x t = H t x t − 1 + w t , w t ∼ N (0 , W t ) where G t and H t are known matrices, and v t and w t are the system and observation noise. Sarah Oscroft Mixture Kalman Filter 4th September 2015 4 / 13

Kalman Filter The Kalman filter finds the filtered density at time t , assuming that the filtered density at time t − 1 is p ( x t − 1 | y 1: t − 1 ) = N ( µ t − 1 , Σ t − 1 ) , by calculating µ t and Σ t for t = 1 : T . Estimating the filtered density is split into two parts: Prediction: Predict the data x t given the observations up to time t − 1 . Update: Update the predictive density to include the newest observation y t . Sarah Oscroft Mixture Kalman Filter 4th September 2015 5 / 13

Kalman Filter Consider the DLM y t = x t + N (0 , V t ) , x t = x t − 1 + N (0 , W t ) . The Kalman filter estimates the state as shown below: 400 200 Data 0 − 200 − Data − 400 o Observations − Filtered Estimates 0 20 40 60 80 100 120 Time Sarah Oscroft Mixture Kalman Filter 4th September 2015 6 / 13

Conditional Dynamic Linear Models For systems that are only partially linear and conditional Gaussian, a conditional dynamic linear model (CDLM) can be used. A CDLM is defined as y t = G λ t x t + v t , v t ∼ N (0 , V λ t ) , if Λ t = λ x t = H λ t x t − 1 + w t , w t ∼ N (0 , W λ t ) where, given λ , all coefficient matrices are known. Sarah Oscroft Mixture Kalman Filter 4th September 2015 7 / 13

Mixture Kalman Filter For j = 1 , ..., m : Run the MKF for each value of Λ . Calculate the probability of obtaining each value of Λ . Sample a value for Λ based on these probabilities. Let KF ( j ) t +1 be the one for this Λ . Calculate the weight, w ( j ) t +1 . Resample if the weight exceeds a threshold value. The MKF takes a weighted sum of µ for each j to provide the state estimate. Sarah Oscroft Mixture Kalman Filter 4th September 2015 8 / 13

Mixture Kalman Filter Simulation Study Consider the CDLM y t = x t + N (0 , Θ) , x t = x t − 1 + N (0 , (1 − λ t )Π) , λ t ∼ Bern (1 − β ) . The Kalman filter assumes that all the values for Π , Θ and β are known. 5 0 − 5 Data − 10 − 15 − Data o Observations − Filtered Estimates 0 100 200 300 400 500 Time Sarah Oscroft Mixture Kalman Filter 4th September 2015 9 / 13

Mixture Kalman Filter In reality, not all values are known. E.g. If you don’t know the value for Θ and guess incorrectly, it could result in estimates like the ones shown in the plot below. 5 0 − 5 Data − 10 − 15 − Data o Observations − Filtered Estimates 0 100 200 300 400 500 Time Sarah Oscroft Mixture Kalman Filter 4th September 2015 10 / 13

Mixture Kalman Filter Simultaneously estimating the state and Θ Estimate the value for Θ by t = t − 1 t − 1 + 1 ˆ ˆ θ 2 θ 2 t ( y t − µ t ) 2 . t 5 2.2 2.0 0 1.8 − 5 Theta 1.6 Data 1.4 − 10 1.2 − 15 − Data 1.0 o Observations − Filtered Estimates 0 100 200 300 400 500 0 100 200 300 400 500 Time Time Sarah Oscroft Mixture Kalman Filter 4th September 2015 11 / 13

Multivariate Mixture Kalman Filter We now assume that: x t and y t are vectors. V and W are covariance matrices. Off diagonal elements of covariance matrices are zero. − Data o Observations 20 1.0 o Observations − Filtered Estimates 15 0.8 Theta Data 10 0.6 5 0.4 0 0 100 200 300 400 500 0 100 200 300 400 500 Time Time This could be very useful for tracking abruptly changing classes in classification problems. Sarah Oscroft Mixture Kalman Filter 4th September 2015 12 / 13

References Petris, Giovanni and Petrone, Sonia and Campagnoli, Patrizia. (2007). Dynamic linear models with R. Nemeth, Christopher. (2014). Parameter estimation for state space models using sequential Monte Carlo algorithms. Chen, Rong and Liu, Jun S. (1999). Mixture kalman filters. Sarah Oscroft Mixture Kalman Filter 4th September 2015 13 / 13