Time Series Analysis

Transcription

1 Time Series Analysis Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1

2 Outline of the lecture State space models, 1st part: Model: Sec The Kalman filter: Sec Cursory material: Sec (Empirical-Bayesian description) 2

3 State space models System model; A full description of the dynamical system (i.e. including the parameters) Observations; Noisy measurements on some parts (states) of the system Goal; reconstruct and predict the state of the system Input u t System State: X t Output Y t 3

4 State space models; examples Estimate the temperature inside a solid block of material when we measure the temperature on the surface (with noise) Noisy measurements of the position of a ship; give a better estimate of the current position A model of a car engine: Input; fuel. State; Fuel and temperature in various parts. Observarions: Sensor output PK/PD-modeling: State: Amount of drug in blood, liver, muscules,... Observations: Amount in blood (with noise), Input: Drug. 4

5 Determining the model structure The system model is often based on physical considerations; this often leads to dynamical models consisting of differential equations An m th order differential equation can be formulated as m 1st order differential equations Sampling such a system leads to a linear state space model and there exist a way of coming from the coefficients in continuous time to the coefficients in discrete time 5

6 The linear stochastic state space model System equation: X t = AX t 1 + Bu t 1 + e 1,t Observation equation: Y t = CX t + e 2,t X: State vector Y : Observation vector u: Input vector e 1 : System noise e 2 : Observation noise dim(x t ) = m is called the order of the system {e 1,t } and {e 2,t } mutually independent white noise V [e 1 = Σ 1, V [e 2 = Σ 2 A, B, C, Σ 1, and Σ 2 are known matrices The state vector contains all information available for future evaluation; the state vector is a Markov process 6

7 Example a falling body Height above ground: z(t) Initial conditions: Position z(t 0 ) and velocity z (t 0 ) Physical considerations: d2 z dt 2 = g States: Position x 1 (t) = z(t) and velocity x 2 (t) = z (t) Only the position is measured y(t) = x 1 (t) Continuous time description x(t) = [x 1 (t) x 2 (t) T : x (t) = [ [ x(t) + y(t) = [ 1 0 x(t) 0 1 g 7

8 Example a falling body (cont nd) Solving the equations: x 2 (t) = g(t t 0 ) + x 2 (t 0 ) x 1 (t) = g 2 (t t 0) 2 + (t t 0 )x 2 (t 0 ) + x 1 (t 0 ) Sampling: t = st, t 0 = (s 1)T, and T = 1 [ [ 1 1 1/2 x s = x s g y s = [ 1 0 x s Adding disturbances and measurement noise: [ [ 1 1 1/2 x s = x s 1 + g + e 1,s y s = [ 1 0 x s + e 2,s 8

9 Example a falling body (cont nd) Given measurements of the position at time points 1,2,...,s we could: Predict the future position and velocity x s+k s (k > 0) Reconstruct the current position and velocity from noisy measurements x s s Interpolate to find the best estimate of the position and velocity at a previous time point x s+k s (k < 0) (estimate the path in the state space; vary k so that s + k varied from 1 to s) we will focus on reconstruction and prediction 9

10 Requirement In order to predict, reconstruct or interpolate the m-dimensional state in the system X t = AX t 1 + Bu t 1 + e 1,t Y t = CX t + e 2,t the system must be observable, i.e. rank [ C T.(CA) T.. ( CA m 1) T = m. For the falling body (S-PLUS): > qr( cbind(t(c), t(c %*% A)) )$rank [1 2 Where A and C is taken from the discrete time description of the system. 10

11 The Kalman filter Initialization: X1 0 = E [X 1 = µ 0, 1 0 = V [X 1 = V 0, and thereby Σ yy 1 0 = CΣxx 1 0 CT + Σ 2 For: t = 1,2,3,... Reconstruction: ( ) 1 K t = t t 1 CT Σ yy t t 1 X t t = X ( t t 1 + K t Y t C X ) t t 1 t t = t t 1 K tσ yy t t 1 KT t Prediction: X t+1 t t+1 t Σ yy t+1 t = A X t t + Bu t = A t t AT + Σ 1 = C t+1 t CT + Σ 2 11

12 Multi step predictions Not part of the Kalman filter as stated above Can be calculated recursively for a given t starting with k = 1 for which X t+k t and Σ t+k t are calculated as part of the Kalman filter X t+k+1 t t+k+1 t = A X t+k t + Bu t+k = A t+k t AT + Σ 1 The future input must be decided 12

13 Naming and history The filter is named after Rudolf E. Kalman, though Thorvald Nicolai Thiele and Peter Swerling actually developed a similar algorithm earlier. It was during a visit of Kalman to the NASA Ames Research Center that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program, leading to its incorporation in the Apollo navigation computer. From 13

14 The Foundation of the Kalman filter Theorem 2.6 (Linear projection) The theorem is concerned with the random vectors X and Y for which the means, variances and covariances are used The state is called X t and the observation is called Y t and we could write down the theorem for these We have additional information; Y T t 1 = (Y T 1,...,Y T t 1) We include this information by considering the random vectors X t Y t 1 and Y t Y t 1 instead E[(X t Y t 1 ) (Y t Y t 1 ) = E[X t Y t, Y t 1 = E[X t Y t 1 + C[X t,y t Y t 1 V 1 [Y t Y t 1 (Y t E[Y t Y t 1 ) V [(X t Y t 1 ) (Y t Y t 1 ) = V [X t Y t, Y t 1 = V [X t Y t 1 C[X t,y t Y t 1 V 1 [Y t Y t 1 C T [X t,y t Y t 1 14

15 The Foundation of the Kalman filter (cont nd) E[X t Y t, Y t 1 = E[X t Y t 1 + C[X t,y t Y t 1 V 1 [Y t Y t 1 (Y t E[Y t Y t 1 ) V [X t Y t, Y t 1 = V [X t Y t 1 C[X t,y t Y t 1 V 1 [Y t Y t 1 C T [X t,y t Y t 1 X t t = X t t 1 + Σ xy t t 1 t t = t t 1 Σxy t t 1 K t = Σ xy t t 1 ( ) 1 ( ) Σ yy Y t t 1 t Ŷ t t 1 ( Σ yy t t 1 ( ) 1 Σ yy t t 1 ) 1 ( Σ xy t t 1 K t is called the Kalman gain, because it determine how much the 1-step prediction error influence the update of the state estimate ) T 15

16 The Foundation of the Kalman filter (cont nd) The 1-step predictions are obtained directly from the state space model: Xt+1 t = A X t t + Bu t Ŷ t+1 t = C X t+1 t Which results in the prediction errors: X t+1 t = X t+1 X t+1 t = A X t t + e 1,t+1 Ỹ t+1 t = Y t+1 Ŷ t+1 t = C X t+1 t + e 2,t+1 And in therefore: t+1 t Σ yy t+1 t = A t t AT + Σ 1 = C t+1 t CT + Σ 2 Σ xy t+1 t can also be calculated 16

17 Kalman filter applied to a falling body Description of the system: A = [ B = [ 1/2 1 C = [ 1 0 Σ 1 = [ Σ 2 = [ Initialization: Released m above ground at 0 m/s X 1 0 = [ = [ Σ yy 1 0 = [

18 Kalman filter applied to a falling body (cont nd) 1st observation (t = 1): y 1 = Reconstruction: K 1 = [ 0 0 T X 1 1 = Prediction: [ X 2 1 = 9.82 [ = [ = [ Σ yy 2 1 = [

19 Kalman filter applied to a falling body (cont nd) 2nd observation (t = 2): y 2 = Reconstruction: K 2 = [ T X 2 2 = [ = [ Prediction: [ X 3 2 = = [ Σ yy 3 2 = [

20 Kalman filter applied to a falling body (cont nd) 3rd observation (t = 3): y 3 = Reconstruction: K 3 = [ T X 3 3 = [ = [ Prediction: [ X 4 3 = = [ Σ yy 4 3 = [

21 Falling body the 10 first time points Position (m) Observed position Reconstructed position Reconstructed velocity Velocity (m/s) 21

22 Falling body wrong initial state Position (m) Observed position Reconstructed position Reconstructed velocity Velocity (m/s)