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Release calc version 2.11.5t1.1
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Landon Curt Noll
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cal/chi.cal
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251
cal/chi.cal
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/*
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* chi - chi^2 probabilities with degrees of freedom for null hypothesis
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*
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* Copyright (C) 2001 Landon Curt Noll
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*
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* Calc is open software; you can redistribute it and/or modify it under
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* the terms of the version 2.1 of the GNU Lesser General Public License
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* as published by the Free Software Foundation.
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*
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* Calc is distributed in the hope that it will be useful, but WITHOUT
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* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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* Public License for more details.
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*
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* A copy of version 2.1 of the GNU Lesser General Public License is
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* distributed with calc under the filename COPYING-LGPL. You should have
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* received a copy with calc; if not, write to Free Software Foundation, Inc.
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* 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
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*
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* @(#) $Revision: 29.2 $
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* @(#) $Id: chi.cal,v 29.2 2001/04/08 10:21:23 chongo Exp $
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* @(#) $Source: /usr/local/src/cmd/calc/cal/RCS/chi.cal,v $
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*
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* Under source code control: 2001/03/27 14:10:11
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* File existed as early as: 2001
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*
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* chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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* Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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*/
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/*
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* Z(x)
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*
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* From Handbook of Mathematical Functions
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* 10th printing, Dec 1972 with corrections
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* National Bureau of Standards
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*
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* Section 26.2.1, p931.
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*/
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define Z(x, eps_term)
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{
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local eps; /* error term */
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/* obtain the error term */
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if (isnull(eps_term)) {
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eps = epsilon();
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} else {
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eps = eps_term;
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}
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/* compute Z(x) value */
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return exp(-x*x/2, eps) / sqrt(2*pi(eps), eps);
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}
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/*
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* P(x[, eps]) asymtotic P(x) expansion for x>0 to an given epsilon error term
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*
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* NOTE: If eps is omitted, the stored epsilon value is used.
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*
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* From Handbook of Mathematical Functions
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* 10th printing, Dec 1972 with corrections
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* National Bureau of Standards
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*
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* 26.2.11, p932:
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*
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* P(x) = 1/2 + Z(x) * sum(n=0; n < infinity){x^(2*n+1)/(1*3*5*...(2*n+1)};
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*
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* We continue the fraction until it is less than epsilon error term.
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*
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* Also note 26.2.5:
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*
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* P(x) + Q(x) = 1
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*/
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define P(x, eps_term)
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{
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local eps; /* error term */
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local s; /* sum */
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local x2; /* x^2 */
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local x_term; /* x^(2*r+1) */
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local odd_prod; /* 1*3*5* ... */
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local odd_term; /* next odd value to multiply into odd_prod */
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local term; /* the recent term added to the sum */
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/* obtain the error term */
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if (isnull(eps_term)) {
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eps = epsilon();
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} else {
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eps = eps_term;
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}
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/* firewall */
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if (x <= 0) {
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if (x == 0) {
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return 0; /* hack */
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} else {
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quit "Q(x[,eps]) 1st argument must be >= 0";
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}
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}
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if (eps <= 0) {
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quit "Q(x[,eps]) 2nd argument must be > 0";
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}
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/*
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* aproximate sum(n=0; n < infinity){x^(2*n+1)/(1*3*5*...(2*n+1)}
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*/
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x2 = x*x;
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x_term = x;
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s = x_term; /* 1st term */
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odd_term = 1;
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odd_prod = 1;
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do {
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/* compute the term */
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odd_term += 2;
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odd_prod *= odd_term;
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x_term *= x2;
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term = x_term / odd_prod;
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s += term;
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} while (term >= eps);
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/* apply term and factor */
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return 0.5 + Z(x,eps)*s;
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}
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/*
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* chi_prob(chi_sq, v[, eps]) - Prob of >= chi^2 with v degrees of freedom
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*
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* Computes the Probability, given the Null Hypothesis, that a given
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* Chi squared values >= chi_sq with v degrees of freedom.
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*
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* The chi_prob() function does not work well with odd degrees of freedom.
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* It is reasonable with even degrees of freedom, although one must give
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* a sifficently small error term as the degress gets large (>100).
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*
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* NOTE: This function does not work well with odd degrees of freedom.
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* Can somebody help / find a bug / provide a better method of
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* this odd degrees of freedom case?
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*
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* NOTE: This function works well with even degrees of freedom. However
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* when the even degrees gets large (say, as you approach 100), you
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* need to increase your error term.
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*
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* From Handbook of Mathematical Functions
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* 10th printing, Dec 1972 with corrections
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* National Bureau of Standards
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*
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* Section 26.4.4, p941:
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*
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* For odd v:
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*
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* Q(chi_sq, v) = 2*Q(chi) + 2*Z(chi) * (
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* sum(r=1, r<=(r-1)/2) {(chi_sq^r/chi) / (1*3*5*...(2*r-1)});
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*
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* chi = sqrt(chi_sq)
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*
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* NOTE: Q(x) = 1-P(x)
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*
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* Section 26.4.5, p941.
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*
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* For even v:
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*
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* Q(chi_sq, v) = sqrt(2*pi()) * Z(chi) * ( 1 +
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* sum(r=1, r=((v-2)/2)) { chi_sq^r / (2*4*...*(2r)) } );
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*
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* chi = sqrt(chi_sq)
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*
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* Observe that:
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*
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* Z(x) = exp(-x*x/2) / sqrt(2*pi()); (Section 26.2.1, p931)
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*
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* and thus:
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*
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* sqrt(2*pi()) * Z(chi) =
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* sqrt(2*pi()) * Z(sqrt(chi_sq)) =
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* sqrt(2*pi()) * exp(-sqrt(chi_sq)*sqrt(chi_sq)/2) / sqrt(2*pi()) =
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* exp(-sqrt(chi_sq)*sqrt(chi_sq)/2) =
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* exp(-sqrt(-chi_sq/2)
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*
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* So:
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*
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* Q(chi_sq, v) = exp(-sqrt(-chi_sq/2) * ( 1 + sum(....){...} );
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*/
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define chi_prob(chi_sq, v, eps_term)
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{
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local eps; /* error term */
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local r; /* index in finite sum */
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local r_lim; /* limit value for r */
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local s; /* sum */
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local d; /* demoninator (2*4*6*... or 1*3*5...) */
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local chi_term; /* chi_sq^r */
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local ret; /* return value */
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/* obtain the error term */
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if (isnull(eps_term)) {
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eps = epsilon();
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} else {
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eps = eps_term;
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}
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/*
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* odd degrees of freedom
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*/
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if (isodd(v)) {
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local chi; /* sqrt(chi_sq) */
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/* setup for sum */
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s = 1;
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d = 1;
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chi = sqrt(abs(chi_sq), eps);
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chi_term = chi;
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r_lim = (v-1)/2;
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/* compute sum(r=1, r=((v-1)/2)) {(chi_sq^r/chi) / (1*3*5...*(2r-1))} */
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for (r=2; r <= r_lim; ++r) {
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chi_term *= chi_sq;
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d *= (2*r)-1;
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s += chi_term/d;
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}
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/* apply term and factor, Q(x) = 1-P(x) */
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ret = 2*(1-P(chi)) + 2*Z(chi)*s;
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/*
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* even degrees of freedom
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*/
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} else {
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/* setup for sum */
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s =1;
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d = 1;
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chi_term = 1;
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r_lim = (v-2)/2;
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/* compute sum(r=1, r=((v-2)/2)) { chi_sq^r / (2*4*...*(2r)) } */
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for (r=1; r <= r_lim; ++r) {
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chi_term *= chi_sq;
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d *= r*2;
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s += chi_term/d;
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}
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/* apply factor - see observation in the main comment above */
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ret = exp(-chi_sq/2, eps) * s;
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}
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return ret;
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}
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