Patent Document ID: 8775082
Application ID: 12789497
Patent Flag: 1

Claim One:
1. A computer-implemented method for estimating a set of regression coefficients for use in estimating a model forecast variable from a small data set, the method including the steps of: (a) receiving, using a computer, data of an n-vector a MOS and an n-vector b MOS the elements of a MOS and b MOS comprising a set of modeled output statistics (MOS) estimates of the true values of forecast correction coefficients a and b; (b) receiving, using the computer, data of an n-vector a prior and an n-vector b prior , the elements of a prior and b prior comprising a set of prior estimates of forecast correction coefficients a and b; (c) estimating a spatial covariance matrix G a for a prior and a spatial covariance matrix G b for b prior ; (i) wherein the spatial covariance matrix G a is estimated as G a =E(diag(g))E T , where E is an n×n matrix listing a real set of discrete orthonormal basis functions for domain of interest, E T is the transpose of E, and diag(g) is a diagonal matrix whose non-zero elements are given by 
 g k,l =B exp(− b 2 ( k 2 +l 2 )) 
 and 
 g T =[g k,j ,k= 1 ,. .. ,n k ,l= 1 ,. .. ,n l ] in which k and l are sinusoidal basis functions with a wavenumber k in the x-direction and a wavenumber l in the y-direction, and B and b minimize the function J = 1 2 ⁢ ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ( ( g k , l ) - ( d k , l ) ) 2 by ⁢ ⁢ setting ∂ J ∂ B = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ∂ g k , l ∂ B ⁢ ( ( g k , l ) - ( d k , l ) ) = 0 and ∂ J ∂ ( b 2 ) = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ∂ g k , l ∂ ( b 2 ) ⁢ ( ( g k , l ) - ( d k , l ) ) = 0 , where d k,l is the element of the vector d=[E T (a prior −a MOS )]⊙[E T (a prior −a MOS )] corresponding to the wavenumber k and the wavenumber l, where the symbol ⊙ indicates the elementwise product, and where the derivatives of J are of the form ⁢ ∂ J ∂ B = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] ⁢ ( B ⁢ ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l ) ) = 0 ⁢ and ∂ J ∂ ( b 2 ) = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ - ( k 2 + l 2 ) ⁢ b ⁢ ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] ⁢ ( B ⁢ ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l ) ) = 0 ⇒ ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ - ( k 2 + l 2 ) ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] ⁢ ( B ⁢ ⁢ exp ⁡ [ - b 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l ) ) = 0 ; and (ii) wherein the spatial covariance matrix G b is estimated as G b =E(diag(g b ))E T , where diag(g b ) is a diagonal matrix whose non-zero elements are given by 
 g b k,l =A exp(− a 2 ( k 2 +l 2 )) 
 and 
 g bT =[g b k,l k= 1 ,. .. ,n k ,l= 1 ,. .. ,n l ] where k and l are the sinusoidal basis functions with the wavenumber k and the wavenumber l, and where A and a minimize the function J b = 1 2 ⁢ ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ( ( g k , l b ) - ( d k , l b ) ) 2 by ⁢ ⁢ setting ∂ J b ∂ A = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ∂ g k , l b ∂ A ⁢ ( ( g k , l b ) - ( d k , l b ) ) = 0 and ∂ J b ∂ ( a 2 ) = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ ∂ g k , l b ∂ ( a 2 ) ⁢ ( ( g k , l b ) - ( d k , l b ) ) = 0 , where d b k,l is an element of the vector d b =[E T (b prior −b MOS )]⊙[E T (b prior −b MOS )] corresponding to the wavenumber k and the wavenumber l and the derivatives of J b take the form ⁢ ∂ J b ∂ A = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ exp ⁡ [ - a 2 ⁡ ( k 2 + l 2 ) ] ⁢ ( A b ⁢ exp ⁡ [ - a 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l b ) ) = 0 ⁢ and ∂ J b ∂ ( a 2 ) = ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ - ( k 2 + l 2 ) ⁢ A ⁢ ⁢ exp ⁡ [ - a 2 ⁡ ( k 2 + l 2 ) ] ⁢ ( A ⁢ ⁢ exp ⁡ [ - a 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l b ) ) = 0 ⇒ ∑ k = 1 n k ⁢ ∑ l = 1 n l ⁢ - ( k 2 + l 2 ) ⁢ ⁢ exp [ ⁢ - ⁢ ⁢ a 2 ⁡ ( k 2 + l 2 ) ] ⁢ ⁢ ( A ⁢ ⁢ exp [ ⁢ - a 2 ⁡ ( k 2 + l 2 ) ] - ( d k , l b ) ) = 0 ; (d) estimating, using the computer, data of a spatial error covariance matrix R a for a MOS and a spatial error covariance matrix R b for b MOS , each element {R a } ij of R a and each element {R b } ij of R b being estimated over a set of m=1 to M events, M being smaller than a number of points of interest in a spatial grid, (i) wherein R b is estimated using the relation R b ≈ E ⁢ { diag ⁢ { E T ⁢ { 1 M - 1 ⁢ ∑ m = 1 M ⁢ ( ζ _ m - ζ _ _ ) ⁢ ( ζ _ m - ζ _ _ ) M } ⁢ E } } ⁢ E T ξ m ≈f m −a MOS ⊙y m −b MOS gives a vector of differences between a forecast [(f m ) T =[f 1 m , f 2 m ,. .. , f n m ]] and a MOS prediction f m MOS =a MOS ⊙y m +b MOS of a forecast from an mth event and ζ _ _ = 1 M ⁢ ∑ m = 1 M ⁢ ζ _ m , and wherein {R b } ij denotes an element of the matrix R b lying on its ith row and jth column; and (ii) wherein R a is estimated as R a ≈D(Y⊙R b )D, where { Y } ij = { ∑ m = 1 M ⁢ ( y i m - y _ i ) ⁢ ( y j m - y _ j ) M - 1 } , and ⁢ ⁢ D = { diag ⁡ ( Y ) } - 1 , and wherein each element {R a } ij of error covariance matrix R a is estimated as { R a } ij = 〈 ( y i m - y _ i ) ⁢ ( y j m - y _ j ) 〉 ⁢ { R b } ij [ 1 M ⁢ ∑ m = 1 M ⁢ ( y i m - y _ i ) 2 ] ⁡ [ 1 M ⁢ ∑ m = 1 M ⁢ ( y j M - y _ j ) 2 ] = { ∑ m = 1 M ⁢ ( y i m - y _ i ) ⁢ ( y j m - y _ j ) M - 1 } ⁢ { R b } ij [ 1 M - 1 ⁢ ∑ m = 1 M ⁢ ( y i m - y _ i ) 2 ] ⁡ [ 1 M - 1 ⁢ ∑ m = 1 M ⁢ ( y j M - y _ j ) 2 ] , where y i is a sample mean of a verifying variable y i m over a plurality of m events; and (e) transforming the data of a MOS , b MOS , a prior , b prior , G a , G b , R a , and R b into a set of estimated regression coefficients a FMOS and b FMOS , wherein 
 a FMOS =a prior +G a ( G a +R a ) −1 ( a MOS −a prior ); and 
 b FMOS =b prior +G b ( G b +R b ) −1 ( b MOS −b prior ).