My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX D. RECONSTRUCTING A PROBABILITY DENSITY FUNCTION
fact that x i and x j are independent when i ≠ j:
〈 ( 〉 2
g(x) − f(x))
= N − 1 〈
w(xi , x) 〉 2
N
x i
+ 1 〈
w(xi , x) 2〉 N
x i
{x i }
− 2f(x) 〈 w(x i , x) 〉 x i
+ f(x) 2
) 2
( 〈w(xi
= , x) 〉 x i
− f(x)
+ 1 ( 〈w(xi
, x) 2〉 N
x i
− 〈 w(x i , x) 〉 )
(D.6)
2
x i
(∫
) 2
= w(y, x)f(y) dy − f(x)
+ 1 ∫ ( ∫
) 2f(y)
w(y, x) − w(z, x)f(z) dz dy.
N
When w is made too narrow and delta-function-like, the second term (the variance
of w for a particular value of x) becomes large. The factor of 1/N means that with
more sample points, w may be made narrower; the benefit of this is that more of
the fine detail in f is then recovered.
The optimum w is therefore a compromise between a broad, slowly-decaying
function which ensures that g is smooth but misses some of the features of f, and
a narrow, quickly-decaying function which captures all the details but makes g very
noisy.
Using an inhomogeneous weighting function, it is possible to incorporate various
boundary conditions. For example, if it is known that f(x) = 0 for x < a, the weight
may be adjusted so that w(y, x < a) = 0, while still satisfying the normalisation
condition expressed in equation (D.2).
It is also possible to incorporate symmetry into this formalism: if f is known to
be symmetric about x = a then letting w(y, x) = w(y, a − x) ensures that g has the
same symmetry.
As an example, consider the homogeneous Gaussian weight function
w(y, x) =
1
√
2πσ
2 e−(y−x)2 /2σ 2 ,
(D.7)
which satisfies the normalisation condition (D.2). The expected value of the recon-
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