My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX D. RECONSTRUCTING A PROBABILITY DENSITY FUNCTION
structed function is
∫
〈
g(x)
〉{x = 1
√
i } 2πσ
2
e −(y−x)2 /2σ 2 f(y) dy.
(D.8)
The integrand is only significant in the region close to y = x; the size of this region
is determined by σ. Expanding f(y) about this point gives
∫
〈 〉 e
−(y−x) 2 /2σ 2
g(x) = √
{x i
(f(x) + (y − x)f ′ (x) +
} 2πσ
2
= f(x) + 1 2 σ2 f ′′ (x) + O[σ 4 ].
)
(y − x)2
f ′′ (x) + · · · dy
2
(D.9)
Thus the expected value of g(x) differs from f(x) by a term of order σ 2 ; this suggests
that σ should be as small as possible.
In order to calculate the expected square deviation from f(x), it is first necessary
to evaluate the mean square weight:
Expanding f(y) about y = x as before gives
〈
w(xi , x) 2〉 ∫ e
−(y−x) 2 /σ 2
x i
=
f(y) dy. (D.10)
2πσ 2
〈
w(xi , x) 2〉 x i
= 1
2σ √ π f(x) + σ
8 √ π f ′′ (x) + O[σ 3 ]. (D.11)
This, together with equation (D.9), can now be substituted into equation (D.6):
〈 (
g(x) − f(x)) 2
〉
{x i }
( ) 2
1
=
2 σ2 f ′′ (x) + O[σ 4 ]
{ (
+ 1 1
N 2σ √ π f(x) + σ
)
8 √ π f ′′ (x) + O[σ 3 ]
(
− f(x) + 1 ) } 2
2 σ2 f ′′ (x) + O[σ 4 ] .
(D.12)
When the number of samples N is finite, this function diverges as σ → 0 and has
a minimum at some non-zero value of σ. The optimum value of σ decreases as N
increases.
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