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 xt, and a noisy observation yt. Follows a Markov process: xt depends only upon xt−1, yt depends

• nly upon xt.

xt-1 xt xt+1 yt-1 yt yt+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 yt = Gtxt + vt, vt ∼ N(0, Vt), xt = Htxt−1 + wt, wt ∼ N(0, Wt) where Gt and Ht are known matrices, and vt and wt are the system and

• bservation 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(xt−1|y1:t−1) = N(µt−1, Σt−1), by calculating µt and Σt for t = 1 : T. Estimating the filtered density is split into two parts: Prediction: Predict the data xt given the observations up to time t − 1. Update: Update the predictive density to include the newest

• bservation yt.

Sarah Oscroft Mixture Kalman Filter 4th September 2015 5 / 13

Kalman Filter

Consider the DLM yt = xt + N(0, Vt), xt = xt−1 + N(0, Wt). The Kalman filter estimates the state as shown below:

Time Data 20 40 60 80 100 120 −400 −200 200 400

−

• −

Data Observations Filtered Estimates

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 yt = Gλtxt + vt, vt ∼ N(0, Vλt), xt = Hλtxt−1 + wt, wt ∼ N(0, Wλt) if Λt = λ 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 yt = xt + N(0, Θ), xt = xt−1 + N(0, (1 − λt)Π), λt ∼ Bern(1 − β). The Kalman filter assumes that all the values for Π, Θ and β are known.

100 200 300 400 500 −15 −10 −5 5 Time Data

−

• −

Data Observations Filtered Estimates

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.

100 200 300 400 500 −15 −10 −5 5 Time Data

−

• −

Data Observations Filtered Estimates

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 ˆ θ2

t = t − 1

t ˆ θ2

t−1 + 1

t (yt − µt)2.

100 200 300 400 500 −15 −10 −5 5 Time Data

−

• −

Data Observations Filtered Estimates

Time Theta 100 200 300 400 500 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Sarah Oscroft Mixture Kalman Filter 4th September 2015 11 / 13

Multivariate Mixture Kalman Filter

We now assume that: xt and yt are vectors. V and W are covariance matrices. Off diagonal elements of covariance matrices are zero.

100 200 300 400 500 5 10 15 20 Time Data

−

• −

Data Observations Observations Filtered Estimates

Time Theta 100 200 300 400 500 0.4 0.6 0.8 1.0

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