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Higher order linear regression

I have a matrix system:

A x B = C

A Ana n, a B- nna B. And A, and are Bunknown, but I have partial information about C(I have some values, but not all), and it is nchosen so small that it is expected that the system will be more limited. All rows in Aor in columns are not required to Bbe bounded.

I am looking for something like the least squares linear regression to find the best option for this system (Note: I knew that there would be no unique solution, but all I want is one of the best solutions)

Make a concrete example; all a and b are unknown, all c are known, eh? are ignored. I want to find a solution of least squares only taking into account the knowledge c.

[ a11, a12 ]                                     [ c11, c12, c13, c14, ?   ]
[ a21, a22 ]   [ b11, b12, b13, b14, b15]        [ c21, c22, c23, c24, c25 ]
[ a31, a32 ] x [ b21, b22, b23, b24, b25] = C ~= [ c31, c32, c33, ?,   c35 ]
[ a41, a42 ]                                     [ ?,   ?,   c43, c44, c45 ]
[ a51, a52 ]                                     [ c51, c52, c53, c54, c55 ]

Note that if B is trimmed to only b11 and b21, and the unknown row 4 is knocked out, then this is an almost standard problem of linear least squares regression.

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5 answers

I have no idea how to handle your missing values, so I will ignore this problem.

There are no unique solutions. To find the best solution, you need some kind of indicator to judge them. I assume that you want to use the least squares metric, i.e. The best values ​​of Assumptions A and B are those that minimize the sum of the numbers [C_ij- (AB) _ij] ^ 2.

, , - , n. , "" , 1 <= n <= b. , 1 <= ( (C)) <= b. rank (span (C)) = C. , a >= b. 1 <= rank (span (C)) <= min (a, b).

, n , 1 <= n <= b. , A, , span (A) = span ( n C). , A n C. , A, B . B = (A'A) ^ (- 1) A 'C

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, .

A, B C = 5, . A * B = 5 .

, , g,

g (A, B) = || AB-C || ^ 2 = ((AB-C) * (AB-C)) ^ 2

(BFGS).
( ). M * - M, . ( - ... F, )

, .

, .

: , , , " ". , . , " " . , , (. Terry Tao et al., ) , , .

, , . , LM - , . .

, : , V U, 21 ( C).

V - C, , ( MATLAB)

V = [C11; C21; C31; C51; 12;....; 55]

U - , AB, , '?' C. x x = [a11, a21,.. a52, b11, b21..., b25].

f (x) = U ( ).

f (x) = V .

, , . , , . , - , . ( , )

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: ?

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. Levenberg-Marquadt LS. . , , -, .

, . . . . .

P = exp(-(x-x0)/T)

T - , x-x0 - . x T ( ). . T , " ", , . ( > 10 ), , - . 20 . , .

FYI, , . , LM, / .

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, B , :

  • A .
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  • , A, B, 2.
  • 2 , .

, .

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Source: https://habr.com/ru/post/1707615/

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