III. KERNEL-BASED REGRESSION APPROACH

According to

the Moore-Aronszajn theorem in 17, there exists a unique reproducing kernel

Hilbert space (RKHS) for every positive definite kernel on ?Nt × ?Nt and vice versa. We

consider a positive definite reproducing kernel k: ?Nt × ?Nt ? ? and its corresponding RKHS

with inner product ?? , ??K, where H denotes the RKHS. Our ultimate goal is to

find the relationship, denoted by the model ƒd(?) with d ?{1, ? , D}, between TOA

measurements and its corresponding dth spatial

coordinate. Based on the representer theorem 20-21 and the formulation in

15, the function ƒd(?) can be obtained through minimizing the following mean

square error cost function plus a regularization term (which is included to

prevent over-fitting) with respect to ƒd(?):

1 /N ? (Ri,d ?

ƒd(ri))2 + 5?ƒd(?)?2 (1)

where Ri,d is

the dth spatial coordinate of Ri. In the regularization term, 5 is a

positive parameter and the norm of ƒd(?) is defined in RKHS and is expressed as:

ƒd(?) = ?N 1

?j,dk(?, rj) (2)

Specifically,

we consider the Gaussian kernel:

k(ri, rj) =

exp( 2o2 ) (3)

where o is the

bandwidth of the Gaussian kernel. Here, ?j,d is the coefficient or weight for

the dth spatial coordinate of the jth reference node; N is chosen to be less

than Nr such that N reference nodes are used for training to find ƒd(?) and the

remaining (Nr ? N) reference nodes are used for cross validation. From (3), one

can see that a constant bias on {ri} has no effect on the localization results.

In order to

solve the minimization problem in (1), we need to use the reproducing property:

k(ri, rj)

=< k(?, ri), k(?, rj) >K (4)

So the second

term in equation (1) can be written as

(5)

At last, we

can rewrite our objective function as its dual optimization problem with

respect to a in a matrix format,(6)

where Rd is a N

× 1 matrix whose I entry is Ri,d , i ?{1,? , N} ; K is a N × N

matrix whose (i, j)t? entry is the kernel k(ri,

rj) , i, j ? {1, ? , N} ; ad is the N × 1 matrix whose ith entry is ?i,d

, i ? {1, ? , N}. So the problem is simplified and becomes a

well-known quadratic regression problem for finding a finite dimensional

coefficient vector ad.

Taking

gradient of the cost function in

(6)

with respect

to ad, and making it equals to zero, we have

?KRd + K2ad +

5NKad = 0 (7)

So we can

easily get the solution which is

ad = (K +

5NI)–1Rd (8)

The above derivation

is for the dth spatial dimension. Define

†(?) = ƒ ƒ . .. ƒ and a^ = ?? … ? . Then the mapping in (2) is extended

to all D dimensions:

†(?) = ?N 1 a^jk(?, rj). (9)