;+ ; NAME: ; MPCHITEST ; ; AUTHOR: ; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770 ; craigm@lheamail.gsfc.nasa.gov ; UPDATED VERSIONs can be found on my WEB PAGE: ; http://cow.physics.wisc.edu/~craigm/idl/idl.html ; ; PURPOSE: ; Compute the probability of a given chi-squared value ; ; MAJOR TOPICS: ; Curve and Surface Fitting, Statistics ; ; CALLING SEQUENCE: ; PROB = MPCHITEST(CHI, DOF, [/SIGMA, /CLEVEL, /SLEVEL ]) ; ; DESCRIPTION: ; ; The function MPCHITEST() computes the probability for a value drawn ; from the chi-square distribution to equal or exceed the given value ; CHI. This can be used for confidence testing of a measured value ; obeying the chi-squared distribution. ; ; P_CHI(X > CHI; DOF) = PROB ; ; In specifying the returned probability level the user has three ; choices: ; ; * return the confidence level when the /CLEVEL keyword is passed; ; OR ; ; * return the significance level (i.e., 1 - confidence level) when ; the /SLEVEL keyword is passed (default); OR ; ; * return the "sigma" of the probability (i.e., compute the ; probability based on the normal distribution) when the /SIGMA ; keyword is passed. ; ; Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive. ; ; INPUTS: ; ; CHI - chi-squared value to be tested. ; ; DOF - scalar or vector number, giving the number of degrees of ; freedom in the chi-square distribution. ; ; RETURNS: ; ; Returns a scalar or vector of probabilities, as described above, ; and according to the /SLEVEL, /CLEVEL and /SIGMA keywords. ; ; KEYWORD PARAMETERS: ; ; SLEVEL - if set, then PROB describes the significance level ; (default). ; ; CLEVEL - if set, then PROB describes the confidence level. ; ; SIGMA - if set, then PROB is the number of "sigma" away from the ; mean in the normal distribution. ; ; EXAMPLES: ; ; print, mpchitest(1300d,1252d) ; ; Print the probability for a chi-squared value with 1252 degrees of ; freedom to exceed a value of 1300, as a confidence level. ; ; REFERENCES: ; ; Algorithms taken from CEPHES special function library, by Stephen ; Moshier. (http://www.netlib.org/cephes/) ; ; MODIFICATION HISTORY: ; Completed, 1999, CM ; Documented, 16 Nov 2001, CM ; Reduced obtrusiveness of common block and math error handling, 18 ; Nov 2001, CM ; Convert to IDL 5 array syntax (!), 16 Jul 2006, CM ; Move STRICTARR compile option inside each function/procedure, 9 Oct 2006 ; Add usage message, 24 Nov 2006, CM ; Really add usage message, with /CONTINUE, 23 Sep 2009, CM ; ; \$Id: mpchitest.pro,v 1.10 2009/10/05 16:22:44 craigm Exp \$ ;- ; Copyright (C) 1997-2001, 2006, 2009, Craig Markwardt ; This software is provided as is without any warranty whatsoever. ; Permission to use, copy, modify, and distribute modified or ; unmodified copies is granted, provided this copyright and disclaimer ; are included unchanged. ;- forward_function cephes_igamc, cephes_igam ;; Set machine constants, once for this session. Double precision ;; only. pro cephes_setmachar COMPILE_OPT strictarr common cephes_machar, cephes_machar_vals if n_elements(cephes_machar_vals) GT 0 then return if (!version.release) LT 5 then dummy = check_math(1, 1) mch = machar(/double) machep = mch.eps maxnum = mch.xmax minnum = mch.xmin maxlog = alog(mch.xmax) minlog = alog(mch.xmin) maxgam = 171.624376956302725D cephes_machar_vals = {machep: machep, maxnum: maxnum, minnum: minnum, \$ maxlog: maxlog, minlog: minlog, maxgam: maxgam} if (!version.release) LT 5 then dummy = check_math(0, 0) return end function cephes_igam, a, x ; ; Incomplete gamma integral ; ; ; ; SYNOPSIS: ; ; double a, x, y, igam(); ; ; y = igam( a, x ); ; ; DESCRIPTION: ; ; The function is defined by ; ; x ; - ; 1 | | -t a-1 ; igam(a,x) = ----- | e t dt. ; - | | ; | (a) - ; 0 ; ; ; In this implementation both arguments must be positive. ; The integral is evaluated by either a power series or ; continued fraction expansion, depending on the relative ; values of a and x. ; ; ACCURACY: ; ; Relative error: ; arithmetic domain # trials peak rms ; IEEE 0,30 200000 3.6e-14 2.9e-15 ; IEEE 0,100 300000 9.9e-14 1.5e-14 COMPILE_OPT strictarr common cephes_machar, machvals MAXLOG = machvals.maxlog MACHEP = machvals.machep if x LE 0 OR a LE 0 then return, 0.D if x GT 1. AND x GT a then return, 1.D - cephes_igamc(a, x) ax = a * alog(x) - x - lngamma(a) if ax LT -MAXLOG then begin ; message, 'WARNING: underflow', /info return, 0.D endif ax = exp(ax) r = a c = 1.D ans = 1.D repeat begin r = r + 1 c = c * x/r ans = ans + c endrep until (c/ans LE MACHEP) return, ans*ax/a end function cephes_igamc, a, x ; ; Complemented incomplete gamma integral ; ; ; ; SYNOPSIS: ; ; double a, x, y, igamc(); ; ; y = igamc( a, x ); ; ; DESCRIPTION: ; ; The function is defined by ; ; ; igamc(a,x) = 1 - igam(a,x) ; ; inf. ; - ; 1 | | -t a-1 ; = ----- | e t dt. ; - | | ; | (a) - ; x ; ; ; In this implementation both arguments must be positive. ; The integral is evaluated by either a power series or ; continued fraction expansion, depending on the relative ; values of a and x. ; ; ACCURACY: ; ; Tested at random a, x. ; a x Relative error: ; arithmetic domain domain # trials peak rms ; IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 ; IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 COMPILE_OPT strictarr if n_params() EQ 0 then begin message, 'USAGE: PROB = MPCHITEST(CHI, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])', /info return, !values.d_nan endif common cephes_machar, machvals MAXLOG = machvals.maxlog MACHEP = machvals.machep big = 4.503599627370496D15 biginv = 2.22044604925031308085D-16 if x LE 0 OR a LE 0 then return, 1.D if x LT 1. OR x LT a then return, 1.D - cephes_igam(a, x) ax = a * alog(x) - x - lngamma(a) if ax LT -MAXLOG then begin ; message, 'ERROR: underflow', /info return, 0.D endif ax = exp(ax) y = 1.D - a z = x + y + 1.D c = 0.D pkm2 = 1.D qkm2 = x pkm1 = x + 1.D qkm1 = z * x ans = pkm1 / qkm1 repeat begin c = c + 1.D y = y + 1.D z = z + 2.D yc = y * c pk = pkm1 * z - pkm2 * yc qk = qkm1 * z - qkm2 * yc if qk NE 0 then begin r = pk/qk t = abs( (ans-r)/r ) ans = r endif else begin t = 1.D endelse pkm2 = pkm1 pkm1 = pk qkm2 = qkm1 qkm1 = qk if abs(pk) GT big then begin pkm2 = pkm2 * biginv pkm1 = pkm1 * biginv qkm2 = qkm2 * biginv qkm1 = qkm1 * biginv endif endrep until t LE MACHEP return, ans * ax end ; MPCHITEST ; compute the probability for a chi-squared value to exceed x give ; the number of degrees of freedom dof. function mpchitest, x, dof, slevel=slevel, clevel=clevel, sigma=sigma COMPILE_OPT strictarr if n_params() LT 2 then begin message, 'USAGE: PROB = MPCHITEST(CHI, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])', /cont return, !values.d_nan endif cephes_setmachar ;; Set machine constants p = double(x) * 0 for i = 0, n_elements(x)-1 do begin p[i] = cephes_igamc(0.5D * dof, 0.5D * double(x[i])) endfor if keyword_set(clevel) then return, 1D - double(p) if keyword_set(sigma) then return, mpnormlim(p, /slevel) return, p end