Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

 Quantlet: kfilter2 Description: Calculates a filtered time serie (uni- or multivariate) using the Kalman filter equations. The state-space model is assumed to be in the following form: y_t = H x_t + v_t x_t = F x_t-1 + w_t x_0 ~ (mu,Sig), v_t ~ (0,Q), w_t ~ (0,R) All parameters are assumed known.

 Usage: {filtX,KG,PreviousPs} = kfilter2(y,mu,Sig,H,F,Q,R) Input: y T x m matrix of observed time series, T is the number of observations, m is the dimension of time series mu n x 1 vector, the mean of the initial state Sig n x n covariance matrix of the initial state H m x n matrix F n x n matrix Q m x m variance-covariance matrix R n x n variance-covariance matrix Output: filtX T x n matrix of (Kalman-)filtered states x_{t|t}, T is the number of observations, n is the dimension of the states KG T x n x m "vector" of Kalman-Gain-matrices for different times PreviousPs T x n x n "vector" of P_t|t's (Filter-Error-Covariances)

Example:
```library("kalman")
library("plot")
y = serie[,2]
mu = 10
Sig = 0
H = 1
F = 1
Q = 9
R = 9
res= kfilter2(y,mu,Sig, H,F,Q,R)
fx=res.filtX
fy =(H*fx')'
fserie = serie[,1]~serie[,2]~fy
data = fserie[,1]~fserie[,2]
data = setmask(data, "line", "red", "thin")
fdata = fserie[,1]~fserie[,3]
fdata = setmask(fdata, "line", "blue", "thin")
disp = createdisplay(1,1)
show(disp,1,1, data, fdata)
setgopt(disp,1,1, "title", "Kalman filter 1")

```
Result:
```Original serie is displayed with red colour,
filtered serie is displayed with blue colour.
(y is a lagged random walk with errors.)
```
Example:
```library("kalman")
library("plot")
y  = serie[,2:3]
mu = #(20,0)
Sig = #(0,0)~#(0,0)
H  = #(0.3,-0.3)~#(1,1)
F  = #(1,0)~#(1,0)
Q  = #(9,0)~#(0,9)
R  = #(0,0)~#(0,9)
res= kfilter2(y,mu,Sig, H,F,Q,R)
fx=res.filtX
fy =(H*fx')'
fserie = serie[,1]~serie[,2]~serie[,3]~fy[,1]~fy[,2]
data1 = fserie[,1]~fserie[,2]
data1 = setmask(data1, "line", "red", "thin")
fdata1 = fserie[,1]~fserie[,4]
fdata1 = setmask(fdata1, "line", "blue", "thin")
data2 = fserie[,1]~fserie[,3]
data2 = setmask(data2, "line", "red", "thin")
fdata2 = fserie[,1]~fserie[,5]
fdata2 = setmask(fdata2, "line", "blue", "thin")
disp = createdisplay(2,1)
show(disp,1,1, data1, fdata1)
setgopt(disp, 1, 1, "title", "Kalman filter 2 - 1st element")
show(disp,2,1, data2, fdata2)
setgopt(disp,2,1, "title", "Kalman filter 2 - 2nd element")

```
Result:
```Original serie is displayed with red colour,
filtered serie is displayed with blue colour.
(y is a lagged bivariate MA process with errors.)
```
Example:
```library("kalman")
library("plot")
y = serie[,2]
mu = #(0,0)
Sig = #(0,0)~#(0,0)
H = #(1,0)'
F = #(0.5,1)~#(-0.3,0)
R = #(1,0)~#(0,0)
Q = 4
res= kfilter2(y,mu,Sig, H,F,Q,R)
fx=res.filtX
fy =(H*fx')'
fserie = serie[,1]~serie[,2]~fy
data1 = fserie[,1]~fserie[,2]
data1 = setmask(data1, "line", "red", "thin")
fdata1 = fserie[,1]~fserie[,3]
fdata1 = setmask(fdata1, "line", "blue", "thin")
disp = createdisplay(1,1)
show(disp,1,1, data1, fdata1)
setgopt(disp,1,1, "title", "Kalman filter 3")

```
Result:
```Original serie is displayed with red colour,
filtered serie is displayed with blue colour.
(y is an AR(2) process with errors.)
```

Author: P. Ruckdeschel, P. Franek, 19 991010
(C) MD*TECH Method and Data Technologies, 05.02.2006