wiener

Wiener estimate

Syntax

[xs,ps,xf,pf]=wiener(y,x0,p0,f,g,h,q,r)

Arguments

f, g, h

system matrices in the interval [t0,tf]

f: =[f0,f1,...,ff], and fk is a nxn matrix
g: =[g0,g1,...,gf], and gk is a nxn matrix
h: =[h0,h1,...,hf], and hk is a mxn matrix

q, r

covariance matrices of dynamics and observation noise

q: =[q0,q1,...,qf], and qk is a nxn matrix
r: =[r0,r1,...,rf], and gk is a mxm matrix

x0, p0

initial state estimate and error variance

y

observations in the interval [t0,tf]. y=[y0,y1,...,yf], and yk is a column m-vector

xs

Smoothed state estimate xs= [xs0,xs1,...,xsf], and xsk is a column n-vector

ps

Error covariance of smoothed estimate ps=[p0,p1,...,pf], and pk is a nxn matrix

xf

Filtered state estimate xf= [xf0,xf1,...,xff], and xfk is a column n-vector

pf

Error covariance of filtered estimate pf=[p0,p1,...,pf], and pk is a nxn matrix

Description

function which gives the Wiener estimate using the forward-backward Kalman filter formulation

Sample

Examples

// initialize state statistics (mean and er. variance)
m0=[10 10]';
p0=[100 0;0 100];
// create system
f2=[1.1 50.1;0 0.8];
g=[1 0;0 1];
h=[1 0;0 1];
[hi,hj]=size(h);
// noise statistics
q=[.01 0;0 0.01];
r=20*eye(2,2);
// initialize system process
rand("seed",66);
rand("normal");
p0c=chol(p0);
x0=m0+p0c'*rand(ones(m0));
y=h*x0+chol(r)'*rand(ones(1:hi))';
yt=y;
// initialize plotted variables
x=x0;
// loop
ft=[f2];
gt=[g];
ht=[h];
qt=[q];
rt=[r];
n=10;
for k=1:n
    // generate the state and observation
    // at time k (i.e. xk and yk)
    [x1,y]=system(x0,f2,g,h,q,r);
    x=[x x1];
    yt=[yt y];
    x0=x1;
    ft=[ft f2];
    gt=[gt g];
    ht=[ht h];
    qt=[qt q];
    rt=[rt r];
end
// get the wiener filter estimate
[xs,ps,xf,pf]=wiener(yt,m0,p0,ft,gt,ht,qt,rt);
// plot result
a=min([x(1,:)-2*sqrt(ps(1,1:2:2*(n+1))),xf(1,:),xs(1,:)]);
b=max([x(1,:)+2*sqrt(ps(1,1:2:2*(n+1))),xf(1,:),xs(1,:)]);
c=min([x(2,:)-2*sqrt(ps(2,2:2:2*(n+1))),xf(2,:),xs(2,:)]);
d=max([x(2,:)+2*sqrt(ps(2,2:2:2*(n+1))),xf(2,:),xs(2,:)]);
xmargin=max([abs(a),abs(b)]);
ymargin=max([abs(c),abs(d)]);
a=-0.1*xmargin+a;
b=.1*xmargin+b;
c=-0.1*ymargin+c;
d=.1*ymargin+d;
// plot frame, real state (x), and estimates (xf, and xs)
scf();
plot([a a b],[d c c]);
plot2d(x(1,:)',x(2,:)',[2],"000")
plot2d(xf(1,:)',xf(2,:)',[2],"000")
plot2d(xs(1,:)',xs(2,:)',[2],"000")
// mark data points (* for real data, o for estimates)
plot2d(xs(1,:)',xs(2,:)',[-2],"000")
plot2d(xf(1,:)',xf(2,:)',[-3],"000")
plot2d(x(1,:)',x(2,:)',[-4],"000")

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Last updated:
Mon Jan 03 14:33:06 CET 2022