My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX D. RECONSTRUCTING A PROBABILITY DENSITY FUNCTION
The weighting function w(x i , x) is not necessarily homogeneous (although in many
cases, this is a sensible choice). Each sample point carries the same total weight;
mathematically, this is expressed as
∫
w(x i , x) dx = 1.
(D.2)
The reason for choosing the total weight to be one rather than some other constant
is to ensure that g has the normalisation appropriate to a probability density:
∫
g(x) dx = 1 N
= 1.
N∑
∫
i=1
w(x i , x) dx
(D.3)
The expected value of the reconstructed function is
〈 〉
g(x) = 〈 {x i
w(x
} i , x) 〉 ∫
x i
= w(y, x)f(y) dy.
(D.4)
Evidently, the best approximation from this point of view is achieved when w(y, x) =
δ(y − x). A good approximation is one in which w(y, x) is localised, tending to zero
quickly as |y − x| increases; the extent of w should be smaller than the length scale
on which changes in f occur.
However, this is not the whole story; the expected error in g must also be considered:
〈 (
g(x) − f(x)) 2
〉
{x i }
= 〈 g(x) 2〉 {x i } − 2f(x)〈 g(x) 〉 {x i } + f(x)2
= 1 ∑〈 w(xi , x)w(x
N 2 j , x) 〉 x i ,x j
− 2f(x) 〈 w(x i , x) 〉 x i
i,j
+ f(x) 2 .
(D.5)
To proceed further, it is necessary to separate the terms with i = j, and to use the
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