;;; PLT Scheme Science Collection
;;; random-distributions/t-distribution.ss
;;; Copyright (c) 2004-2006 M. Douglas Williams
;;;
;;; This library is free software; you can redistribute it and/or
;;; modify it under the terms of the GNU Lesser General Public
;;; License as published by the Free Software Foundation; either
;;; version 2.1 of the License, or (at your option) any later version.
;;;
;;; This library is distributed in the hope that it will be useful,
;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
;;; Lesser General Public License for more details.
;;;
;;; You should have received a copy of the GNU Lesser General Public
;;; License along with this library; if not, write to the Free
;;; Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
;;; 02111-1307 USA.
;;;
;;; -------------------------------------------------------------------
;;;
;;; This module implements the t-distribution.  It is based on the
;;; Random Number Distributions in the GNU Scientific Library.
;;;
;;; Version  Date      Description
;;; 1.0.0    09/28/04  Marked as ready for Release 1.0.  Added
;;;                    contracts for functions.  (Doug Williams)
;;; 1.0.1    02/08/06  Added cdf.  (Doug Williams)

(module t-distribution mzscheme
  
  (require (lib "contract.ss"))
  
  (provide/contract
   (random-t-distribution
    (case-> (-> random-source? real? real?)
            (-> real? real?)))
   (t-distribution-pdf
    (-> real? real? (>=/c 0.0)))
   (t-distribution-cdf
    (-> real? real? (real-in 0.0 1.0))))
  
  (require "../math.ss")
  (require "../random-source.ss")
  (require "../special-functions/gamma.ss")
  (require "cdf-beta-inc.ss")
  (require "gaussian.ss")
  (require "chi-squared.ss")
  (require "exponential.ss")
  
  ;; random-t-distribution: random-source x real -> real
  ;; random-t-distribution: real -> real
  ;; This function returns a random-variate from the t-distribution
  ;;; with nu degrees of freedom.
  (define random-t-distribution
    (case-lambda
      ((r nu)
       (if (<= nu 2)
           (let ((y1 (random-unit-gaussian r))
                 (y2 (random-chi-squared r nu)))
             (/ y1 (sqrt (/ y2 nu))))
           (let ((y1 0)
                 (y2 0)
                 (z 0))
             (let loop ()
               (set! y1 (random-unit-gaussian r))
               (set! y2 (random-exponential r (/ 1 (- (/ nu 2) 1))))
               (set! z (/ (* y1 y1) (- nu 2)))
               (if (or (< (- 1 z) 0)
                       (> (exp (- (- y2) z)) (- 1 z)))
                   (loop)))
             (* (/ y2 (sqrt (- 1 (/ 2 nu)))) (- 1 z)))))
      ((nu)
       (random-t-distribution (current-random-source) nu))))
  
  ;; t-distribution-pdf: real x real -> real
  ;; This function computes the probability density p(x) at x for a
  ;; t-distribution with nu degrees of freedom.
  (define (t-distribution-pdf x nu)
    (let ((lg1 (lngamma (/ nu 2.0)))
          (lg2 (lngamma (/ (+ nu 1.0) 2.0))))
      (* (/ (exp (- lg2 lg1)) (sqrt (* pi nu)))
         (expt (+ 1.0 (/ (* x x) nu)) (/ (- (+ nu 1.0)) 2.0)))))
  
  ;; cdf implementation

  (define (poly-eval c n x)
    (let ((y (* (vector-ref c 0) x)))
      (do ((i 1 (+ i 1)))
          ((= i n)(+ y (vector-ref c n)))
        (set! y (* x (+ y (vector-ref c i)))))))
  
  (define (cornish-fisher t n)
    (define coeffs6 '#(0.265974025974025974026
                       5.449696969696969696970
                       122.20294372294372294372
                       2354.7298701298701298701
                       37625.00902597402597403
                       486996.1392857142857143
                       4960870.65
                       37978595.55
                       201505390.875
                       622437908.625))
    (define coeffs5 '#(0.2742857142857142857142
                       4.499047619047619047619
                       78.45142857142857142857
                       1118.710714285714285714
                       12387.6
                       101024.55
                       559494.0
                       1764959.625))
    (define coeffs4 '#(0.3047619047619047619048
                       3.752380952380952380952
                       46.67142857142857142857
                       427.5
                       2587.5
                       8518.5))
    (define coeffs3 '#(0.4
                       3.3
                       24.0
                       85.5))
    (let* ((a (- n 0.5))
           (b (* 48.0 a a))
           (z2 (* a (log1p (/ (* t t) n))))
           (z (sqrt z2))
           (p5 (* z (poly-eval coeffs6 9 z2)))
           (p4 (* (- z) (poly-eval coeffs5 7 z2)))
           (p3 (* z (poly-eval coeffs4 5 z2)))
           (p2 (* (- z) (poly-eval coeffs3 3 z2)))
           (p1 (* z (+ z2 3.0)))
           (p0 z)
           (y p5))
      (set! y (+ (/ y b) p4))
      (set! y (+ (/ y b) p3))
      (set! y (+ (/ y b) p2))
      (set! y (+ (/ y b) p1))
      (set! y (+ (/ y b) p0))
      (if (< t 0.0)
          (* y -1.0)
          y)))
  
  ;;; t-distribution-cdf: real x real -> real
  (define (t-distribution-cdf x nu)
    (let ((x2 (* x x)))
      (cond ((and (> nu 30.0)
                  (< x2 (* 10.0 nu)))
             (let ((u (cornish-fisher x nu)))
               (unit-gaussian-cdf  u)))
            ((< x2 nu)
             (let* ((u (/ x2 nu))
                    (eps (/ u (+ 1.0 u))))
               (if (>= x 0.0)
                   (beta-inc-axpy 0.5 0.5 0.5 (/ nu 2.0) eps)
                   (beta-inc-axpy -0.5 0.5 0.5 (/ nu 2.0) eps))))
            (else
             (let* ((v (/ nu (* x x)))
                    (eps (/ v (+ 1.0 v))))
               (if (>= x 0.0)
                   (beta-inc-axpy -0.5 1.0 (/ nu 2.0) 0.5 eps)
                   (beta-inc-axpy 0.5 0.0 (/ nu 2.0) 0.5 eps)))))))
  
)