Kalman filter and simulating Gaussian noise

[Dustinsfl]

Water level assumed constant. Static state model \(P_0 = 1000\), system noise \(Q = 0.0001\), measurement noise \(R = 0.1\).

I want to use Kalman filtering to solve this problem. I know how to do this but I need to generate \(\mathbf{y} = y\) using Gaussian random noise. \(P_0\) is also call the variance which is high do to uncertanity in \(x_0\).

How do I generate 10 y values based on Gaussian random noise with these conditions?

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These values come from here.

Then I have a copy the same presentation but page 7-9 are additionally hand written pages with more info and everything before and after these pages are identical to the first link.

 

[Dustinsfl]

I found a presentation on the topic by Stanford; unfortunately, I don't see how to apply it to the problem. If someone can help with this, that would be great.

On page 14, they go over linear measurements of the form \(y = Ax + v\).

Stanford lecture

This is just a beamer style presentation so it isn't dense.

 

[zzephod]

Water level assumed constant. Static state model \(P_0 = 1000\), system noise \(Q = 0.0001\), measurement noise \(R = 0.1\).

I want to use Kalman filtering to solve this problem. I know how to do this but I need to generate \(\mathbf{y} = y\) using Gaussian random noise. \(P_0\) is also call the variance which is high do to uncertanity in \(x_0\).

How do I generate 10 y values based on Gaussian random noise with these conditions?

---

These values come from here.

Then I have a copy the same presentation but page 7-9 are additionally hand written pages with more info and everything before and after these pages are identical to the first link.

The \(\displaystyle y\)'s are \(\displaystyle \sim N(y_{true},R)\) with \(\displaystyle y_{true}=1\) and \(\displaystyle R=1/10\), so in Matlab code a sample of 10 \(\displaystyle y\)'s would be generated thusly:

--> ysamp=1+randn(1,10)/sqrt(10)

ysamp =

 Columns 1 to 6
    1.0032    1.0506    0.5366    0.9875    1.3739    0.7645
 Columns 7 to 10
    1.2047    0.8725    0.6744    1.2186

.

 

[Dustinsfl]

The \(\displaystyle y\)'s are \(\displaystyle \sim N(y,R)\) with \(\displaystyle y=1\) and \(\displaystyle R=1/10\), so in Matlab code a sample of 10 \(\displaystyle y\)'s would be generated thusly:

--> ysamp=1+randn(1,10)/sqrt(10)

ysamp =

 Columns 1 to 6
    1.0032    1.0506    0.5366    0.9875    1.3739    0.7645
 Columns 7 to 10
    1.2047    0.8725    0.6744    1.2186

.

Can you explain the plus one and divided by sqrt(10). I understand the randn.

 

[Dustinsfl]

If y was represented as \(y = 0.1k + \) noise, \(r = 0.1\), and we want 6 values, would it be

for i = (1:1:6)
     sampley = 1 + 0.1*i + randn(1,6)/sqrt(10)
end

I just realized this doesn't make sense because we will have 6 1x6 column vectors. I am not sure how to use this definition of \(y\).

Or would it be

for i = (1:1:6)
     sampley = randn(1,6)/sqrt(10)
end

and then where I use my sample ys (which is in a loop), I can augment them as

for i = (2:1:8)
     some stuff
     x(i) = ax(i) + K(i)*((y(i - 1) + (i - 1)*.1) - ax(i))
     some stuff
end

 

[zzephod]

Can you explain the plus one and divided by sqrt(10). I understand the randn.

randn(nrows,ncols) generates a matrix of nrows rows and ncols columns of samples from a zero mean unit variance normal distribution.

-->Sample_Mean=1.0;

-->Sample_Variance=1/10.0;

-->ysamp=Sample_Mean+randn(1,10)*sqrt(Sample_Variance)

ysamp = 

Columns 1 to 6    1.0032    1.0506    0.5366    0.9875    1.3739    0.7645
 
Columns 7 to 10    1.2047    0.8725    0.6744    1.2186