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Topic: Computing tables for use with Peirce's Criterion - in 1855 and 2008
Replies: 3   Last Post: Nov 21, 2012 9:28 AM

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 ninovanhooff@gmail.com Posts: 2 Registered: 11/21/12
Re: Computing tables for use with Peirce's Criterion - in 1855 and 2008
Posted: Nov 21, 2012 9:28 AM

Op zondag 5 oktober 2008 15:41:37 UTC+2 schreef Knud Thomsen het volgende:
> In 1852 Peirce described his criterion for identifying outliers in a
> sample from an otherwise normal distribution (1).
> In 1855 Gould, having being applying the criterion professionally,
> published a set of tables of critical deviations together with a
> description of the algorithm he had used to calculate them (2). The
> calculations were done manually, aided only by log-tables and some
> special function tables he managed to get his hands on!
>
> Gould's first two tables list the (squares of the) critical deviation
> x as a function of N, the number of values in the sample, and k, the
> number of outliers being tested for. These tables differ with respect
> the value for m, the 'number of unknown quantities', set to 1 and 2,
> respectively.
>
> Below I have tried to be consistent with Gould's notation.
> Here are the exceptions:
>
> Gould Here
> --------------------------------
> Lower case n k
> psi x erfc(x/sqrt(2))
> --------------------------------
>
> Next, two equations are formulated from which the critical deviation
> can be found (x>0).
>
> Gould's equation D can be stated:
>
> R = R1(x) = exp((x^2-1)/2) erfc(x/sqrt(2))
>
> Defining (Gould's equation B):
>
> Q^N = k^k (N-k)^(N-k) / N^N
>
> Gould's equation A' can be combined with the unlabeled one just above
> equation C into:
>
> R = R2(x) = Q^(N/k) [(N-m-k x^2)/(N-m-k)]^(-(N-k)/(2k))
>
> Now a numerical solution for x can be found after equating R1 and R2.
>
> Gould's algorithm was based on the following pseudoconstant strategy
> (3), the initial value for R being my own suggestion:
>
> R = 0.25
> <Find x from: R2(x)=R>
> repeat
> R = R1(x)
> oldx = x
> <Find x from: R2(x)=R>
> until abs(x-oldx)<Eps
>
> Although this works quite well on a computer (with some extra
> termination logistics), much faster convergence can be achieved
> through root-finding by Newton's Method.
> I currently use the algorithm below (suitable for double precision
> floating point). Some loop-invariant code have been kept explicit for
>
> ----------------------------------------------------------------------
> function PeirceDeviation(N,k,m)
> // Peirce's critical deviation for N samples, k potential outliers and
> m
> // 'unknown quantities' (usually 1).
>
> if N-m-k<=0 then return(0) // result undefined
>
> LnQN = k*Ln(k) + (N-k)*Ln(N-k) - N*Ln(N)
>
> x = 1
>
> repeat
> x = min(x, sqrt((N-m)/k) - 1e-10)
>
> // R1(x) and R2(x)
> R1 = exp((x*x-1)/2)*erfc(x/sqrt(2))
> R2 = exp( (LnQN - 0.5*(N-k)*ln((N-m-k*x*x)/(N-m-k)))/k )
>
> // R1'(x) and R2'(x)
> R1d = x*R1 - sqrt(2/pi/exp(1))
> R2d = x*(N-k)/(N-m-k*x*x)*R2
>
> // Newton
> oldx = x
> x = oldx - (R1-R2)/(R1d-R2d)
> until abs(x-oldx) < N * 2E-16
>
> return(x)
> ----------------------------------------------------------------------
>
> Gould did not list x-values below 1, and when glancing down the tables
> along the edge of the filled-in cells, one is tempted to ask if
> something mathematically interesting can be said about the special
> case x=1.
> For that purpose we shall accept non-integer values for N and/or k.
>
> Setting R1(1)=R2(1) we get:
>
> erfc(1/sqrt(2)) = Q^(N/k) = k [(N-k)^(N-k) / N^N]^(1/k)
>
> or, defining the ratio C = k/N:
>
> erfc(1/sqrt(2)) = C (1-C)^(1/C-1)
>
> So, x=1 implies a fixed k/N ratio, which does not depend on m.
> C must be estimated numerically; here it is to 30 decimal places:
>
> C = 0.58974 36745 10713 07248 06181 37574 ...
>
> which just happens to be rather close to the simple fraction:
>
> 23/39 = 0.58974 35897 43589 ...
>
>
> 1. Peirce, B.: Criterion for the rejection of doubtful observations.
> Astronomical Journal, vol. 2, iss. 45, p. 161-163 (1852).
>
> 2. Gould, B.A.: On Peirce's Criterion for the Rejection of Doubtful
> Observations,
> with tables for facilitating its application.
> Astronomical Journal, vol. 4, iss. 83, p. 81-87 (1855).
>
> 3. Acton, F.S.: Real Computing Made Real: Preventing Errors in
> Scientific and
> Engineering Calculations. Princeton University Press (1996).
>
>
> Best regards,
>
> Knud Thomsen
> Geologist, Denmark