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The first thing we want to talk about today is the introduction and a little reconsideration

what pattern recognition was about. So the first chapter is a brief introduction. I should

take Arabic numbers. A brief introduction. And we will talk about two problems about

regression and classification. So the first thing we look at is regression. Is there anybody

in the audience who has an idea what regression actually is or what regression means? You have

to solve this question. So the first thing we look at is regression. So the first thing

we look at is regression. Is there anybody in the audience who has an idea what regression

actually is or what regression means? You have to solve a regression problem. A regression

problem. The term should be known from mathematics. I mean there are a few things that you might

be interested in engineering mathematics. And if you look at the following situation.

So let us look at this. If you have points and you have let us say this is bad here.

I should do it this way. You have measurements. So you measure two dimensional points. And

if you do interpolation. What does it mean interpolation? Let us do interpolation here.

What is interpolation? Interpolation means you look for a function that exactly goes

through your sample points and tells you how the function behaves in between the points.

So that is interpolation. Interpolation means you exactly hit with the function the points

you measure. So if you evaluate the function exactly at these points where your measurements

are or at these positions where your measurements are you end up with the proper y coordinate

that you have measured. And if you want to know what is the function value in between

two points you evaluate the interpolation function at this point and you get a function

value here. That is interpolation. What we quite often do and do we have color chalk

here? No. Is we approximate a function for instance with linear line segments and do

a polygon approximation of the function and interpolate. Interpolation means hit the sample

points. Then if we have interpolation here. Let me do it this way. And you want to have

the function value outside of the interval you have measurements. For instance if you

want to have a function value here outside of the interval we call it extrapolation.

So if we want to measure the function in between two samples we do interpolation. Here we do

extrapolation. And regression means that we fit a function through the points such that

in average we fit the function and that is regression. So in regression it is not necessarily

required to hit the points. So points are not required to sit on the function.

So if we talk about interpolation we mean we hit the samples. Regression means we fit

a line or a function through a set of sample points. And I will tell you why this is important

in a few minutes. And these sample points just for notation the set of sample points

let me write complete sentences. The set of sample points is denoted by S is x1, x2 up

to xn where xi is a d-dimensional vector. So we call these points xi and we have N capital

N sample points and these are our samples and measures and we call the set S in the

future. So whenever I write S this is the set of sample points we have observed. Let's

look at an example how to do regression. Regression requires the estimation of a continuous vector

y and this vector can have a certain dimension let's call this dimension d. So that's a

continuous variable. Now let's look at an example. If I lost you already please let

me know. Example. Given a set S x1, x2 up to xn where we have two dimensional feature

vectors that means xi is element rd, d is equal to 2. So we have a set of sample points

and we can draw these sample points like this. We have here our x1 coordinate and our x2

coordinate and here our sample points and we want to compute a regression function.

So our task is compute a degree two polynomial regression function.

That means we want to compute a degree two function that goes through these points. How

would we do that? What's your name? Sir? Eid. How would you compute that? A regression function

I give you points let's say 1, 2, 2, 3, 4, 3, 3, 3, whatever. How would you compute the

regression function? So the polynomial representation is so our y is in this case y1, y2, y3. So