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## Measurement Update

[Homework Assignment #2] This is homework assignment #2 in CS 373, and it's all about Kalman filters. [Kalman Filters] [Measurement Update] So question #1 is measurement update. Let's start with 2 Gaussians and say they have identical variances. Let's multiple them. We know that the resulting Gaussian's mean will be just at the center between those 2 Gaussians. I wonder if the variance Î£Â² is now smaller, larger, or the same as either of the individual Gaussian. Please check one of the three.

## Measurement Update Solution

And the correct answer is smaller. We are more certain afterwards. You get a more peaky Gaussian whose standard deviation is smaller. This is the correct answer.

## New Variance

Say we have a prior of a Gaussian with a mean mu and covariance sigma squared, and our measurement has exactly the same mean and same covariance. Suppose we multiply both and get a new mean, which is the same as the old mean, and we get a new covarience, sigma squared, which as you all know, corresponds to a more peaked Gaussian. I want you to express nu squared as a multiple of sigma squared. Just put the answer here as a real number.

## New Variance Solution

The answer is a Â½ or 0.5 To see, let us multiply these two gaussians over here-- which the exponent becomes an addition-- we can now rewrite this as follows-- when we bring the factor 2, for these two terms in to the numerator as our 2 sigma square, and from that we see that the new variance is 0.5 as large as the old one when applied to the squares. So, as a result, you would have this equation over here.

## Heavytail Guassian

I have another Gaussian question for you-- this is called a heavytail Gaussian. So, as you go to +/- infinity, the Gaussian levels off at some finite value alpha, as opposed to zero. My question is-- "Is it possible to represent this function as a Gaussian?" This is the formula for the Gaussian again. Particularly, can you find a Mu and a sigma square for which this exact function over here is obtained? Please just check one of these two boxes.

## Heavytail Guassian Solution

The correct answer here is no. Suppose we let "x" go to infinity, then x-Mu2 for any fixed Mu would go to infinity. So if x with a -infinity would go to zero-- because it is a constant-- Therefore, we know that in the limit of x goes to infinity-- this expression must be zero. However, in this graph--it stays at alpha, and doesn't go to zero-- So, therefore, there can't be a valid Mu at sigma square. If a deep improbability, you know that the area in the Gaussian has to integrate into one, and this area diverges, it is actually infinite in size, so it's not even a valid execution.

## How Many Dimensions

My next question pertains to the tracking problem that we talked about in class. In class we addressed a 1-dimensional tracking problem where we estimated the location of the system and its velocity. I'd like now to generalize this to a 2-dimensional problem. We'll be given coordinants x and y, and the object we're tracking moves in 2-dimensional space, and we wish to use a Kalman filter to understand where the object is, what its velocity is, and even be able to predict a future location based on the estimate of the position and its velocity. So the only difference, class, is that our object now moves in a 2-dimensional space, where as in class, it moved in a 1-dimensional space. So my first question is what's the dimension of the state vector in the Kalman filter? [Dimension of the state vector?] In the class, it was this kind of state vector. Now, we have a new one. How many dimensions or how many variables are there?

## How Many Dimensions Solution

And the answer is 4. So rather than X and áº, we have X and Y and áº and áº as our state vectors. And there's 4 variables

## State Transition Matrix

Now comes the tricky question. In the Kalman filter program that we studied, the 2-D Kalman filter, we had a matrix F. And for delta T equals 0.1, our F matrix, the state transition matrix, had a main diagonal of 1, which means in exportation our location stays the same and our velocity stays the same. And we also knew that the velocity affected our state in the following way. And you could now place 0.1 instead of the delta T to get this specific F matrix. Now I want to know from you for this new 2-D case with a 4-dimensional state vector what is the new F? It is a 4 by 4 matrix, so I want you to fill in all of those values. Again please assume that delta T is 0.1, and don't write delta T, write 0.1.

## State Transition Matrix Solution

And the answer is the main diagonal remains one. This expresses the effect that in the absence of any velocity, we expect the x-coordinate and the y-coordinate not to change. We also don't expect the velocities to change. But the x-velocity will impact the x-position through a 0.1 over here. The third coordinate is the velocity, and it affects the first coordinate. The same over here. All the other values should be zero. So this is the 4-dimensionalization of this 2-dimensional state-transition matrix over here.