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Thanks.
Im not sure I understand the transition between last 3 lines of the derivation. The n.T^2, term, for example, what does it stand for?
Also, in the last and second last step: how did you obtain the sigma^2 and xbar^2 terms? The min calculations should involve derivatives, I guess.
Given a set of random real numbers G = (a, b, c, d, e, f, ...), Im supposed to calculate the number T such that the sum of squared errors between that
number and all the elements in G is mimimum. Intuitively, this should be the average of the numbers in G, but Im not sure how to proceed with the proof (perhaps Im missing something important here).
Would the same answer hold for the question: compute a number T such that the deviation of all the elements in G with respect to T is mimimum. (The standard deviation is with respect to the mean value, but is there some other value T such that the deviation with respecto to it is mimimum (smaller that standard deviation and all others))
In addition, how would the conclusions relate to the median value. Again, intuitively, this would match the number T with the closest value in G.
Your opinion on this is highly appreciated. Thanks
Given a function
f(X)= Tr(X'AX) - 2Tr(X'BC), with X' denoting matrix transpose, I'm supposed to find the expression used to miminize the function with respect to X. The derivatives should be used, but I'm not sure how to proceed.
Any help is appreciated.
I dont understand your comment (how constructive can it be?).
Assume you're given a circle with the line AB containing its center O, such that A and B are on the circle (OA=OB=radius). A tangent t is drawn on the point A, and
I should calculate the mapping of certain points (a,b,c,d...) of the circle to the points on the tangent (at, bt, ct, dt, ...) such that the distance Aa (the distance along the circle) is the same as the distance Aat (the distance along the tangent) (and the same for the distances Ab, Ac, Ad). But, here, certain constraint should be considered: those points of the circle (among (a, b, c, d)) that are from one side of the circle from A to B should be placed on one side of the tangent (the nearer), and those from the other side of the circle form A to B should be placed on the other side. Basically, the circle should be split at B, and then mapped to the tangent. I hope this explanation is sufficient enough.
It should be noted that I have information about coordinates of A, B, O, a, b, c, d. I supposed to calculate (at, bt, ct, dt).
For solving this problem, I have two approaches, but I'm not sure how I could make sure they always work correctly.
1) I calculate the equation of the tangent at point A. Then for each point (a, b, c, d) I calculate the distance from A (along the circle), and use these distances for calculating (at, bt, ct, dt...) along the tangent. What I dont know here is how to calculate the distances
from A to (a, b, c, d). The problem is the 'proper side' determination, meaning how should I determine whether the point should be mapped on one side of the tangent or the other. What would be the way to determine this.
2) I calculate the equation of the tangent at point A. Then for each point (a, b, c, d) I calculate the distance from A (along the circle), and use these distances for calculating (at, bt, ct, dt...) along the tangent. To determine the 'proper side' of a given point, I might use the projection of that point to the tangent. But, even with this, how I know 'which side is which'? Perhaps there are much simpler ways to do this.
Any suggestion on how to do this is welcome. In case I was not precise enough, I'll elaborate.
Thanks.
Exactly. The point is to have approximately the same difference values for first k input values (say, for 1,2,3,4,5) satisfying y(k)<y(k+1), and then the difference should decrease gradually still maintaining y(n)<y(n+1).
Right now I'm trying to combine certain functions (sqrt, log, ...) with some linear to obtain approximately wanted behaviour. Any suggestions on what to try are welcome. Also, if there is a software for which I *might* draw a wanted line, and get the approximate function, please let me know.
That is for this specific example. I chose y(x)=x for x<=5
In simple words:
we are trying to obtain a function whose behaviour should be like this (example):
y(1)=1
y(2)=2
y(3)=3
y(4)=4
y(5)=5
...............up to here the difference between two consecutive y(n) y(n+1) is equal (always equal to 1), and the y() sequence itself is increasing
y(6)=5.9 (difference is 5.9-5=0.9 and 0.9<1, where 1 corresponds to previous difference)
y(7)=6.7 (difference is 6.7-5.9=0.8 and 0.8<0.9, where 0.9 corresponds to previous difference (see above))
y(8)=7.4 (difference is 7.4- 6.7=0.7 and 0.7<0.8, where 0.8 corresponds to previous difference (see above))
y(9)=8 (same....)
Check that the sequence y() is increasing, and that, starting with n=6, the difference is decreasing.
This was an example. I hope I clarified what I look for.
I really appreciate your help.
Ok, I agree. I wrote that I made a mistake(typing). My last post should explain what I want.
Thanks for the functions. I should test their behaviour.
Perhaps a 'single' function might not work for the extension I'm asking for.
Let
y(n)=n for n<=5;
So, now, if I specify the following values for y(5),...y(10)
y(6)=5.9
y(7)=6.7
y(8)=7.4
y(9)=8
y(10)=8.5
y(11)=8.9
....
so, the sequence is increasing in y(n)|n>=5, and the sequence of
differences is decreasing. Could we construct such a function for specific k and n (ending values).
I made a mistake above denoting all the functions with y().
Perhaps I'm missing something important here.
Thanks again
Note that
For the extension;
let y(1)=1; y(2)=2; y(3)=3; y(4)=4; I wonder what would be the function
y(x) for x>=5 that satisfy the above.
Thanks. I put 'derivatives' with '' emphasis, so it does not correspond to the real derivative of a function ('derivative of sqrt(x) might be sqrt(2x), for example, but this is a language obstacle').
In addition, I would like to extend the case for the following:
[tex]
$y(1)<y(2)<y(3)<y(4)<...<y(n-1)<y(n)$ \\
but satisfy
$y(1)=[y(2)-y(1)]=[y(3)-y(2)]=...=[y(k+1)-y(k)]>[y(k+2)-y(k+1)]>[y(k+3)-y(k+2)]>>...>[y(n)-y(n-1)]>$ \\
[/tex]
How could I extend the given functions to obtain this (for given k (and n)).
Thanks
For determination of the distance from a certain point, a function should be used. Namely, the input values 'x' are from N (1,2,3,4,5...), and the corresponding outputs 'y' from R, but such that the sequence 'y' (relative to 'x') is increasing;
[tex]
$y(1)<y(2)<y(3)<y(4)<...$ \\
but to satisfy \\
$y(1)>[y(2)-y(1)]>[y(3)-y(2)]>[y(4)-y(3)]>...$
[/tex]
The function that satisfies this is sqrt(x), but I'm interested in other possible functions that satisfy the above. The 'derivatives' of sqrt(x) should also work.
Thanks
Actually, I should use the checking prior to invoking the Cholesky decomposition. This means the checking is preparation for Cholesky (further used as a preconditioner for Conjugate Gradient). I could, however, start Cholesky and wait until its eventual completion.
Good choice, but expensive in terms of the computational time/power. Completion of the first criterion would be better.
The problem is that the hard way is computationally expensive. I would need simple strategy to check positive definiteness of symmetric matrices.
The above criteria seems incomplete. Any suggestions on how to check this relatively easily?
Anyone?
Note the following:
A symmetric matrix is positive definite if:
1. all the diagonal entries are positive, and
2. each diagonal entry is greater than the sum of the absolute values of all other entries in the same row.
...........................................................................
However, the following matrix seems to have Cholesky decomposition (therefore, it is positive definite):
5.000 2.000 3.000 2.000
2.000 3.000 1.000 1.000
3.000 1.000 4.000 -1.000
2.000 1.000 -1.000 7.000
Here, the second condition is not satisfied. Any suggestions on the positive definite matrix criteria?
I would need a simple positive definiteness test. (The above seems simple, but might not be correct/complete).
Thanks