I need help understanding and finding MMSE for these two, I researched on my own but i found many solutions that dealt with matrices(from what I understand those dealt with multiple variables) I don't want those answers I want to get some answers using this approach:
Number1: $$ y= \sin(wx)$$ Number2: $$ x^2+y^2=r^2$$
For understanding problem see these two pictures
Here consider that both x and y are known we are to minimize the remaining variable in these questions.
What I did regarding Number1:
For $x^2+y^2=r^2$ , I made the loss function as the sum of differences of distance between the true fit(the one that has least error) and a random fit, using distance formula for a point and circle.
$$ Loss=\sum_{j=1}^{n}(d_{j} -(\sqrt{x^2_{j}+y^2_{j}}-r))^2 $$ By solving using MMSE for r i got: $$r=\frac{1}{n}\sum_{j=1}^{n}(d_{j}-\sqrt{x^2_{j}+y^2_{j}})$$ Question-1: Is the approach and the answer correct?
What I did regarding Number2:
For $y= \sin(wx)$ , I made the loss function as the sum of difference between two sine curves one best(minimized) other random.
$$Loss=\sum_{j=1}^{n}(y_{j}-sin(w*x_{j}))^2$$ By solving using MMSE for w I got:
$$\sum_{j=1}^{n}(y_{j}-sin(w*x_{j}))*(cos(w*x_{j})*x_{j})=0$$
Another approach that might be right is to find the distance between data points and the sine curve.If the points on the sine curve are of the form (x,sin(wx)). The distance of a point (e,f) to the points on sine curve is:
$$D = \sqrt{(x - e)^2 + (\sin (wx) - f)^2}$$ And then minimize Loss function for $$D^2$$ as the minimization will be same for both.
$$ Loss=\sum_{j=1}^{n}(D^2_{j}-((x_{j} - e)^2 + (\sin (wx_{j}) - f)^2))^2 $$
Question-2: Which approach is correct here and in both cases i am unable to seperate $$w$$ what am I doing wrong?
Thanking you in advance.
In addition to my above comment, a numerical example of circular regression :