Calculating Average Hypothesis

[ripande]

I guess this question has been asked before, but I am not very clear on it.

I understand calculating the average hypothesis of a constant i.e. taking average of the constant values of differnet hypothesis.

But what does calculating average hypothesis of a hypothesis set which consists of lines in y= mx+c mean? How is it calculated ?

[ripande]

I read the earlier post on this question once again. The concept is clear to me now

[ripande]

I wanted to validate that what I have understood is correct.

1. I calculated the value of "a" for which the which minimizes the least square for two points ( x, sin(pi*x) ), x being between -1 and 1.

2. Repeated the above for 100 times and hence got 100 values of "a"

3. Then I chose a fresh point x3 between [-1, 1] and calculated the value of y3 = a*x3 for all 100 points

4. Calculated average value of y3 for 100 points, say y_avg.

5. Calculated "a" for avg hypothesis as : y_avg/x3

Is my method of calculating avg hypothesis correct ? Iam not very confident of the answer I am getting

[yaser]

Originally Posted by ripande

1. I calculated the value of "a" for which the which minimizes the least square for two points ( x, sin(pi*x) ), x being between -1 and 1.

2. Repeated the above for 100 times and hence got 100 values of "a"

These steps are correct (with instead of in step 1) in calculating the final hypothesis for 100 different sets .

3. Then I chose a fresh point x3 between [-1, 1] and calculated the value of y3 = a*x3 for all 100 points

This step evaluates for each in the 100 runs. If is fixed for all 100 runs, this step can be used to evaluate the bias and variance at the point (namely and ).

4. Calculated average value of y3 for 100 points, say y_avg.

If the 100 points are the same with different , then the average approximates . If the points are different, I am not sure about the utility of this quantity for the calculation of bias and variance.

5. Calculated "a" for avg hypothesis as : y_avg/x3

You already have the different values of for different data sets (these are the values of that you used to calculated from ). Because the formula for the hypothesis is linear in , you can directly calculate of the average hypothesis by averaging all the 's. What you are suggesting is equivalent in this case.

[ripande]

Thanks Prof Yaser for the clarification.

So, if understand correctly,since the hypothesis is linear in "a", I can directly take the average of the value "a" in all hypothesis to calculate the average hypothesis; and what I was doing was just equivalent to it.

What should be done in a hypothetical case where the value had not been linear in the hypothesis set?

[yaser]

Originally Posted by ripande

So, if understand correctly,since the hypothesis is linear in "a", I can directly take the average of the value "a" in all hypothesis to calculate the average hypothesis; and what I was doing was just equivalent to it.

Exactly.

What should be done in a hypothetical case where the value had not been linear in the hypothesis set?

You average the hypothesis value at each point , rather than average the parameters.

[gah44]

To help make sure I understood this, I tried to determine the red line for H1 in Example 2.8. That is, the average hypotheses for h(x)=ax+b.

But I get a=0 and b=0, unlike the figure.

I average over a matrix of points, ignoring x1=x2 points.

Anything I should do differently?

I have:

a=(y1-y2)/(x1-x2)
b=(x1*y2-x2*y1)/(x1-x2)

but a averages to zero.

[gah44]

OK, I was summing the wrong variable.

It works now.

a=0.82 b=0

as it looks on the graph.

Now see if I can get the bias and variance for the graph.