Renamed make-flexp/base' to make-flexpt'
Renamed `dist' struct type to `distribution' ("dist" is too common)
This commit is contained in:
Neil Toronto
parent
6ca52be0ae
commit
055512b4e8
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@ -14,8 +14,8 @@
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discrete-dist-probs)
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(begin-encourage-inline
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(struct: (In Out) discrete-dist-struct dist ([values : (Listof Out)]
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[probs : (Listof Positive-Flonum)])
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(struct: (In Out) discrete-dist-struct distribution ([values : (Listof Out)]
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[probs : (Listof Positive-Flonum)])
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#:property prop:custom-print-quotable 'never
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#:property prop:custom-write
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(λ (v port mode)
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@ -1,166 +1,143 @@
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#lang racket/base
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#lang typed/racket/base
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(require racket/performance-hint
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racket/promise
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"../../flonum.rkt")
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(provide
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(all-from-out (submod "." typed-defs))
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;; Rename transformers
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dist-cdf
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dist-inv-cdf
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dist-min
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dist-max)
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PDF Sample CDF Inverse-CDF
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(struct-out distribution)
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(struct-out ordered-dist)
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Real-Dist
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pdf sample cdf inv-cdf real-dist-prob)
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(module typed-defs typed/racket/base
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(require racket/performance-hint
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racket/promise
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"../../flonum.rkt")
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(define-type (PDF In)
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(case-> (In -> Flonum)
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(In Any -> Flonum)))
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(define-type (Sample Out)
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(case-> (-> Out)
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(Integer -> (Listof Out))))
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(define-type (CDF In)
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(case-> (In -> Flonum)
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(In Any -> Flonum)
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(In Any Any -> Flonum)))
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(define-type (Inverse-CDF Out)
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(case-> (Real -> Out)
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(Real Any -> Out)
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(Real Any Any -> Out)))
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(struct: (In Out) distribution ([pdf : (PDF In)]
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[sample : (Sample Out)])
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#:transparent)
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(struct: (In Out) ordered-dist distribution
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([cdf : (CDF In)]
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[inv-cdf : (Inverse-CDF Out)]
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[min : Out]
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[max : Out]
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[median : (Promise Out)])
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#:transparent)
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(define-type Real-Dist (ordered-dist Real Flonum))
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;; =================================================================================================
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(begin-encourage-inline
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(provide
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PDF Sample CDF Inverse-CDF
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(struct-out dist)
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(struct-out ordered-dist)
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Real-Dist
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dist-median
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pdf sample cdf inv-cdf real-dist-prob)
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(: pdf (All (In Out) (case-> ((distribution In Out) In -> Flonum)
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((distribution In Out) In Any -> Flonum))))
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(define (pdf d v [log? #f])
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((distribution-pdf d) v log?))
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(define-type (PDF In)
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(case-> (In -> Flonum)
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(In Any -> Flonum)))
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(: sample (All (In Out) (case-> ((distribution In Out) -> Out)
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((distribution In Out) Integer -> (Listof Out)))))
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(define sample
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(case-lambda
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[(d) ((distribution-sample d))]
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[(d n) ((distribution-sample d) n)]))
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(define-type (Sample Out)
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(case-> (-> Out)
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(Integer -> (Listof Out))))
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(: cdf (All (In Out) (case-> ((ordered-dist In Out) In -> Flonum)
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((ordered-dist In Out) In Any -> Flonum)
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((ordered-dist In Out) In Any Any -> Flonum))))
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(define (cdf d v [log? #f] [1-p? #f])
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((ordered-dist-cdf d) v log? 1-p?))
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(define-type (CDF In)
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(case-> (In -> Flonum)
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(In Any -> Flonum)
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(In Any Any -> Flonum)))
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(define-type (Inverse-CDF Out)
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(case-> (Real -> Out)
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(Real Any -> Out)
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(Real Any Any -> Out)))
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(struct: (In Out) dist ([pdf : (PDF In)]
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[sample : (Sample Out)])
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#:transparent)
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(struct: (In Out) ordered-dist dist
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([cdf : (CDF In)]
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[inv-cdf : (Inverse-CDF Out)]
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[min : Out]
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[max : Out]
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[median : (Promise Out)])
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#:transparent)
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(define-type Real-Dist (ordered-dist Real Flonum))
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;; =================================================================================================
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(begin-encourage-inline
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(: dist-median (All (In Out) ((ordered-dist In Out) -> Out)))
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(define (dist-median d) (force (ordered-dist-median d)))
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(: pdf (All (In Out) (case-> ((dist In Out) In -> Flonum)
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((dist In Out) In Any -> Flonum))))
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(define (pdf d v [log? #f])
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((dist-pdf d) v log?))
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(: sample (All (In Out) (case-> ((dist In Out) -> Out)
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((dist In Out) Integer -> (Listof Out)))))
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(define sample
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(case-lambda
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[(d) ((dist-sample d))]
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[(d n) ((dist-sample d) n)]))
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(: cdf (All (In Out) (case-> ((ordered-dist In Out) In -> Flonum)
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((ordered-dist In Out) In Any -> Flonum)
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((ordered-dist In Out) In Any Any -> Flonum))))
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(define (cdf d v [log? #f] [1-p? #f])
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((ordered-dist-cdf d) v log? 1-p?))
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(: inv-cdf (All (In Out) (case-> ((ordered-dist In Out) Real -> Out)
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((ordered-dist In Out) Real Any -> Out)
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((ordered-dist In Out) Real Any Any -> Out))))
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(define (inv-cdf d p [log? #f] [1-p? #f])
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((ordered-dist-inv-cdf d) p log? 1-p?))
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)
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(: real-dist-prob* (Real-Dist Flonum Flonum Any -> Flonum))
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;; Assumes a <= b
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(define (real-dist-prob* d a b 1-p?)
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(define c (dist-median d))
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(define cdf (ordered-dist-cdf d))
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(define p
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(cond [(a . fl= . b) (if 1-p? 1.0 0.0)]
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[1-p? (fl+ (cdf a #f #f) (cdf b #f #t))]
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[(b . fl<= . c)
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;; Both less than the median; use lower tail only
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(fl- (cdf b #f #f) (cdf a #f #f))]
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[(a . fl>= . c)
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;; Both greater than the median; use upper tail only
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(fl- (cdf a #f #t) (cdf b #f #t))]
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[else
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;; Median between a and b; use lower for (a,c] and upper for (c,b]
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(define P_x<=a (cdf a #f #f))
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(define P_x>b (cdf b #f #t))
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(fl+ (fl- 0.5 P_x<=a) (fl- 0.5 P_x>b))]))
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(max 0.0 (min 1.0 p)))
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(: real-dist-log-prob* (Real-Dist Flonum Flonum Any -> Flonum))
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;; Assumes a <= b
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(define (real-dist-log-prob* d a b 1-p?)
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(define c (dist-median d))
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(define cdf (ordered-dist-cdf d))
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(define log-p
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(cond [(a . fl= . b) (if 1-p? 0.0 -inf.0)]
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[1-p? (lg+ (cdf a #t #f) (cdf b #t #t))]
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[(b . fl<= . c)
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;; Both less than the median; use lower tail only
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(define log-P_x<=a (cdf a #t #f))
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(define log-P_x<=b (cdf b #t #f))
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(cond [(log-P_x<=b . fl< . log-P_x<=a) -inf.0]
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[else (lg- log-P_x<=b log-P_x<=a)])]
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[(a . fl>= . c)
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;; Both greater than the median; use upper tail only
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(define log-P_x>a (cdf a #t #t))
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(define log-P_x>b (cdf b #t #t))
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(cond [(log-P_x>a . fl< . log-P_x>b) -inf.0]
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[else (lg- log-P_x>a log-P_x>b)])]
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[else
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;; Median between a and b; try 1-upper first
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(define log-P_x<=a (cdf a #t #f))
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(define log-P_x>b (cdf b #t #t))
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(define log-p (lg1- (lg+ log-P_x<=a log-P_x>b)))
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(cond [(log-p . fl> . (log 0.1)) log-p]
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[else
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;; Subtracting from 1.0 (in log space) lost bits; split and add instead
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(define log-P_a<x<=c
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(cond [((fllog 0.5) . fl< . log-P_x<=a) -inf.0]
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[else (lg- (fllog 0.5) log-P_x<=a)]))
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(define log-P_c<x<=b
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(cond [((fllog 0.5) . fl< . log-P_x>b) -inf.0]
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[else (lg- (fllog 0.5) log-P_x>b)]))
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(lg+ log-P_a<x<=c log-P_c<x<=b)])]))
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(min 0.0 log-p))
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(: real-dist-prob (case-> (Real-Dist Real Real -> Flonum)
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(Real-Dist Real Real Any -> Flonum)
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(Real-Dist Real Real Any Any -> Flonum)))
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(define (real-dist-prob d a b [log? #f] [1-p? #f])
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(let ([a (fl a)] [b (fl b)])
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(let ([a (flmin a b)] [b (flmax a b)])
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(cond [log? (define p (real-dist-prob* d a b 1-p?))
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(cond [(and (p . fl> . +max-subnormal.0) (p . fl< . 0.9)) (fllog p)]
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[else (real-dist-log-prob* d a b 1-p?)])]
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[else (real-dist-prob* d a b 1-p?)]))))
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(: inv-cdf (All (In Out) (case-> ((ordered-dist In Out) Real -> Out)
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((ordered-dist In Out) Real Any -> Out)
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((ordered-dist In Out) Real Any Any -> Out))))
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(define (inv-cdf d p [log? #f] [1-p? #f])
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((ordered-dist-inv-cdf d) p log? 1-p?))
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)
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(require (submod "." typed-defs)
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(for-syntax racket/base))
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(: real-dist-prob* (Real-Dist Flonum Flonum Any -> Flonum))
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;; Assumes a <= b
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(define (real-dist-prob* d a b 1-p?)
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(define c (force (ordered-dist-median d)))
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(define cdf (ordered-dist-cdf d))
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(define p
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(cond [(a . fl= . b) (if 1-p? 1.0 0.0)]
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[1-p? (fl+ (cdf a #f #f) (cdf b #f #t))]
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[(b . fl<= . c)
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;; Both less than the median; use lower tail only
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(fl- (cdf b #f #f) (cdf a #f #f))]
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[(a . fl>= . c)
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;; Both greater than the median; use upper tail only
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(fl- (cdf a #f #t) (cdf b #f #t))]
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[else
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;; Median between a and b; use lower for (a,c] and upper for (c,b]
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(define P_x<=a (cdf a #f #f))
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(define P_x>b (cdf b #f #t))
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(fl+ (fl- 0.5 P_x<=a) (fl- 0.5 P_x>b))]))
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(max 0.0 (min 1.0 p)))
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(define-syntax dist-cdf (make-rename-transformer #'ordered-dist-cdf))
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(define-syntax dist-inv-cdf (make-rename-transformer #'ordered-dist-inv-cdf))
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(define-syntax dist-min (make-rename-transformer #'ordered-dist-min))
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(define-syntax dist-max (make-rename-transformer #'ordered-dist-max))
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(: real-dist-log-prob* (Real-Dist Flonum Flonum Any -> Flonum))
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;; Assumes a <= b
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(define (real-dist-log-prob* d a b 1-p?)
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(define c (force (ordered-dist-median d)))
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(define cdf (ordered-dist-cdf d))
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(define log-p
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(cond [(a . fl= . b) (if 1-p? 0.0 -inf.0)]
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[1-p? (lg+ (cdf a #t #f) (cdf b #t #t))]
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[(b . fl<= . c)
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;; Both less than the median; use lower tail only
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(define log-P_x<=a (cdf a #t #f))
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(define log-P_x<=b (cdf b #t #f))
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(cond [(log-P_x<=b . fl< . log-P_x<=a) -inf.0]
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[else (lg- log-P_x<=b log-P_x<=a)])]
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[(a . fl>= . c)
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;; Both greater than the median; use upper tail only
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(define log-P_x>a (cdf a #t #t))
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(define log-P_x>b (cdf b #t #t))
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(cond [(log-P_x>a . fl< . log-P_x>b) -inf.0]
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[else (lg- log-P_x>a log-P_x>b)])]
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[else
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;; Median between a and b; try 1-upper first
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(define log-P_x<=a (cdf a #t #f))
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(define log-P_x>b (cdf b #t #t))
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(define log-p (lg1- (lg+ log-P_x<=a log-P_x>b)))
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(cond [(log-p . fl> . (log 0.1)) log-p]
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[else
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;; Subtracting from 1.0 (in log space) lost bits; split and add instead
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(define log-P_a<x<=c
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(cond [((fllog 0.5) . fl< . log-P_x<=a) -inf.0]
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[else (lg- (fllog 0.5) log-P_x<=a)]))
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(define log-P_c<x<=b
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(cond [((fllog 0.5) . fl< . log-P_x>b) -inf.0]
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[else (lg- (fllog 0.5) log-P_x>b)]))
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(lg+ log-P_a<x<=c log-P_c<x<=b)])]))
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(min 0.0 log-p))
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(: real-dist-prob (case-> (Real-Dist Real Real -> Flonum)
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(Real-Dist Real Real Any -> Flonum)
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(Real-Dist Real Real Any Any -> Flonum)))
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(define (real-dist-prob d a b [log? #f] [1-p? #f])
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(let ([a (fl a)] [b (fl b)])
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(let ([a (flmin a b)] [b (flmax a b)])
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(cond [log? (define p (real-dist-prob* d a b 1-p?))
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(cond [(and (p . fl> . +max-subnormal.0) (p . fl< . 0.9)) (fllog p)]
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[else (real-dist-log-prob* d a b 1-p?)])]
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[else (real-dist-prob* d a b 1-p?)]))))
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@ -24,16 +24,16 @@
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[(d a) (truncated-dist d -inf.0 a)]
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[(d a b)
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(let*-values ([(a b) (values (fl a) (fl b))]
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[(a b) (values (max (dist-min d) (min a b))
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(min (dist-max d) (max a b)))])
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[(a b) (values (max (ordered-dist-min d) (min a b))
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(min (ordered-dist-max d) (max a b)))])
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(unsafe-truncated-dist d a b))]))
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(: unsafe-truncated-dist (Real-Dist Float Float -> Truncated-Dist))
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(define (unsafe-truncated-dist d a b)
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(define orig-pdf (dist-pdf d))
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(define orig-cdf (dist-cdf d))
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(define orig-inv-cdf (dist-inv-cdf d))
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(define orig-sample (dist-sample d))
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(define orig-pdf (distribution-pdf d))
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(define orig-cdf (ordered-dist-cdf d))
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(define orig-inv-cdf (ordered-dist-inv-cdf d))
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(define orig-sample (distribution-sample d))
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(define log-P_a<x<=b (real-dist-prob d a b #t #f))
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(define log-P_x<=a (delay (orig-cdf a #t #f)))
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(define log-P_x>b (delay (orig-cdf b #t #t)))
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@ -53,45 +53,6 @@
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(: proc-names (type-name -> arg-types)) ...
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(define (proc-names v) (struct-proc-names v)) ...))))]))
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(define-syntax (define-distribution-type: stx)
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(syntax-case stx (:)
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[(_ (type-name T ...) (parent-type-name In Out)
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(A B) name ([arg-names : arg-type] ...))
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(let ([arg-name-lst (syntax->list #'(arg-names ...))])
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(with-syntax* ([struct-type-name (format-id #'type-name "~a-struct" #'type-name)]
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[(struct-proc-names ...) (map (λ (arg-name)
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(format-id #'type-name
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"~a-~a"
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#'struct-type-name
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arg-name))
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arg-name-lst)]
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[struct-pred-name (format-id #'type-name "~a?" #'struct-type-name)]
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[make-name (format-id #'name "make-~a" #'name)]
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[(proc-names ...) (map (λ (arg-name)
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(format-id #'name "~a-~a" #'name arg-name))
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arg-name-lst)]
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[pred-name (format-id #'name "~a?" #'name)]
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[format-str
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(string-append "(~a "
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(string-join (build-list (length arg-name-lst) (λ _ "~v")))
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")")])
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(syntax/loc stx
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(begin
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(struct: (A B) struct-type-name parent-type-name ([arg-names : arg-types] ...)
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#:property prop:custom-print-quotable 'never
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#:property prop:custom-write
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(λ (v port write?)
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(fprintf port format-str 'name (proc-names v) ...)))
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(define-type (type-name T ...) (struct-type-name In Out))
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(: proc-names ((type-name T ...) -> arg-types)) ...
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(define (proc-names d) (struct-proc-names d)) ...
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(define make-name struct-type-name)
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(define pred-name struct-pred-name)))))]
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[(_ type-name (parent-type-name In Out) name ([arg-names arg-opts ...] ...))
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(syntax/loc stx
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(define-distribution-type: (type-name) (parent-type-name In Out)
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(A B) name ([arg-names arg-opts ...] ...)))]))
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;; ===================================================================================================
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;; One-sided scale family distributions (e.g. exponential)
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@ -11,7 +11,7 @@
|
|||
(provide flsqrt1pm1
|
||||
flsinh flcosh fltanh
|
||||
flasinh flacosh flatanh
|
||||
make-flexp/base flexpt+ flexpt1p)
|
||||
make-flexpt flexpt+ flexpt1p)
|
||||
|
||||
;; ---------------------------------------------------------------------------------------------------
|
||||
;; sqrt(1+x)-1
|
||||
|
|
@ -163,8 +163,8 @@
|
|||
|
||||
(begin-encourage-inline
|
||||
|
||||
(: make-flexp/base (Positive-Exact-Rational -> (Flonum -> Flonum)))
|
||||
(define (make-flexp/base b)
|
||||
(: make-flexpt (Positive-Exact-Rational -> (Flonum -> Flonum)))
|
||||
(define (make-flexpt b)
|
||||
(define b-hi (fl b))
|
||||
(define b-lo (fl (- (/ (inexact->exact b-hi) b) 1)))
|
||||
(cond [(fl= b-lo 0.0) (λ: ([x : Flonum]) (flexpt b-hi x))]
|
||||
|
|
|
|||
|
|
@ -211,7 +211,7 @@
|
|||
(define pi.128 267257146016241686964920093290467695825/85070591730234615865843651857942052864)
|
||||
|
||||
(: flexppi (Flonum -> Flonum))
|
||||
(define flexppi (make-flexp/base pi.128))
|
||||
(define flexppi (make-flexpt pi.128))
|
||||
|
||||
(: flpsi (Integer Flonum -> Flonum))
|
||||
(define (flpsi m x)
|
||||
|
|
|
|||
|
|
@ -107,7 +107,7 @@ P. Borwein. An Efficient Algorithm for the Riemann Zeta Function.
|
|||
(define pi.128 267257146016241686964920093290467695825/85070591730234615865843651857942052864)
|
||||
|
||||
(: flexppi (Flonum -> Flonum))
|
||||
(define flexppi (make-flexp/base pi.128))
|
||||
(define flexppi (make-flexpt pi.128))
|
||||
|
||||
(require math/bigfloat)
|
||||
(: fleta (Flonum -> Flonum))
|
||||
|
|
@ -124,7 +124,7 @@ P. Borwein. An Efficient Algorithm for the Riemann Zeta Function.
|
|||
[(fleven? (fltruncate (* 0.5 s))) +inf.0]
|
||||
[else -inf.0])]
|
||||
[(s . fl< . -171.0)
|
||||
(define f (make-flexp/base (/ (* euler.64 pi.64) (inexact->exact (- s)))))
|
||||
(define f (make-flexpt (/ (* euler.64 pi.64) (inexact->exact (- s)))))
|
||||
(* (/ 4.0 pi)
|
||||
(flexp-stirling (- s))
|
||||
(flsqrt (/ pi (* -0.5 s)))
|
||||
|
|
@ -160,7 +160,7 @@ P. Borwein. An Efficient Algorithm for the Riemann Zeta Function.
|
|||
[else (- (flexpm1 (fl* (fl- 1.0 s) (fllog 2.0))))]))
|
||||
(fl/ (fleta s) c)]
|
||||
[(and (s . fl> . -266.5) (s . fl< . 0.0)) ; s < 0 keeps TR happy
|
||||
(define f (make-flexp/base (/ (* euler.64 pi.64) (inexact->exact (- s)))))
|
||||
(define f (make-flexpt (/ (* euler.64 pi.64) (inexact->exact (- s)))))
|
||||
(* (/ 4.0 pi)
|
||||
(flexp-stirling (- s))
|
||||
(flsqrt (/ pi (* -0.5 s)))
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@
|
|||
@(define untyped-eval (make-untyped-math-eval))
|
||||
@interaction-eval[#:eval untyped-eval (require racket/list)]
|
||||
|
||||
@title[#:tag "dist"]{Probability Distributions}
|
||||
@title[#:tag "dist" #:style 'toc]{Probability Distributions}
|
||||
@(author-neil)
|
||||
|
||||
@defmodule[math/distributions]
|
||||
|
|
@ -57,7 +57,7 @@ half-open interval (1/2,1], and computes another approximation from random sampl
|
|||
|
||||
This plots the pdf and a kernel density estimate of the pdf from random samples:
|
||||
@interaction[#:eval untyped-eval
|
||||
(plot (list (function (dist-pdf d) #:color 0 #:style 'dot)
|
||||
(plot (list (function (distribution-pdf d) #:color 0 #:style 'dot)
|
||||
(density xs))
|
||||
#:x-label "x" #:y-label "density of N(2,5)")]
|
||||
|
||||
|
|
@ -69,7 +69,7 @@ the probabilities within the interval:
|
|||
(define d-trunc (truncated-dist d -inf.0 5))
|
||||
(real-dist-prob d-trunc 5 6)
|
||||
(real-dist-prob d-trunc 0.5 1.0)
|
||||
(plot (list (function (dist-pdf d-trunc) #:color 0 #:style 'dot)
|
||||
(plot (list (function (distribution-pdf d-trunc) #:color 0 #:style 'dot)
|
||||
(density (sample d-trunc 1000)))
|
||||
#:x-label "x" #:y-label "density of T(N(2,5),-∞,5)")]
|
||||
|
||||
|
|
@ -132,7 +132,7 @@ When distribution object constructors receive parameters outside their domains,
|
|||
(cdf (beta-dist 0 0) 1/2)
|
||||
(inv-cdf (geometric-dist 1.1) 0.2)]
|
||||
|
||||
@section{Basic Distribution Types and Operations}
|
||||
@section{Distribution Types and Operations}
|
||||
|
||||
@defform[(PDF In)]{
|
||||
The type of probability density functions, or @tech{pdfs}, defined as
|
||||
|
|
@ -150,38 +150,6 @@ When given a nonnegative integer @racket[n] as an argument, a sampling procedure
|
|||
a length-@racket[n] list of independent, random samples.
|
||||
}
|
||||
|
||||
@defstruct*[dist ([pdf (PDF In)] [sample (Sample Out)])]{
|
||||
The parent type of @tech{distribution objects}. The @racket[In] type parameter is the data type a
|
||||
distribution accepts as arguments to its @tech{pdf}. The @racket[Out] type parameter is the data
|
||||
type a distribution returns as random samples.
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(dist? (discrete-dist '(a b c)))
|
||||
(dist? (normal-dist))
|
||||
((dist-pdf (normal-dist)) 0)
|
||||
((dist-sample (normal-dist)))]
|
||||
See @racket[pdf] and @racket[sample] for uncurried forms of @racket[dist-pdf] and
|
||||
@racket[dist-sample].
|
||||
}
|
||||
|
||||
@defproc[(pdf [d (dist In Out)] [v In] [log? Any #f]) Flonum]{
|
||||
An uncurried form of @racket[dist-pdf]. When @racket[log?] is not @racket[#f], returns a
|
||||
log density.
|
||||
@examples[#:eval untyped-eval
|
||||
(pdf (discrete-dist '(a b c) '(1 2 3)) 'a)
|
||||
(pdf (discrete-dist '(a b c) '(1 2 3)) 'a #t)]
|
||||
}
|
||||
|
||||
@defproc*[([(sample [d (dist In Out)]) Out]
|
||||
[(sample [d (dist In Out)] [n Integer]) (Listof Out)])]{
|
||||
An uncurried form of @racket[dist-sample].
|
||||
@examples[#:eval untyped-eval
|
||||
(sample (exponential-dist))
|
||||
(sample (exponential-dist) 3)]
|
||||
}
|
||||
|
||||
@section{Ordered Distribution Types and Operations}
|
||||
|
||||
@defform[(CDF In)]{
|
||||
The type of cumulative distribution functions, or @tech{cdfs}, defined as
|
||||
@racketblock[(case-> (In -> Flonum)
|
||||
|
|
@ -200,17 +168,32 @@ For any function of this type, both optional arguments should default to @racket
|
|||
interpreted as specified in the description of @racket[inv-cdf].
|
||||
}
|
||||
|
||||
@defstruct*[(ordered-dist dist) ([cdf (CDF In)]
|
||||
[inv-cdf (Inverse-CDF Out)]
|
||||
[min Out]
|
||||
[max Out]
|
||||
[median (Promise Out)])]{
|
||||
@defstruct*[distribution ([pdf (PDF In)] [sample (Sample Out)])]{
|
||||
The parent type of @tech{distribution objects}. The @racket[In] type parameter is the data type a
|
||||
distribution accepts as arguments to its @tech{pdf}. The @racket[Out] type parameter is the data
|
||||
type a distribution returns as random samples.
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(distribution? (discrete-dist '(a b c)))
|
||||
(distribution? (normal-dist))
|
||||
((distribution-pdf (normal-dist)) 0)
|
||||
((distribution-sample (normal-dist)))]
|
||||
See @racket[pdf] and @racket[sample] for uncurried forms of @racket[distribution-pdf] and
|
||||
@racket[distribution-sample].
|
||||
}
|
||||
|
||||
@defstruct*[(ordered-dist distribution) ([cdf (CDF In)]
|
||||
[inv-cdf (Inverse-CDF Out)]
|
||||
[min Out]
|
||||
[max Out]
|
||||
[median (Promise Out)])]{
|
||||
The parent type of @italic{ordered} @tech{distribution objects}.
|
||||
|
||||
Similarly to @racket[dist], the @racket[In] type parameter is the data type an ordered distribution
|
||||
accepts as arguments to its @tech{pdf}, and the @racket[Out] type parameter is the data type an
|
||||
ordered distribution returns as random samples. Additionally, its @tech{cdf} accepts values of type
|
||||
@racket[In], and its @tech{inverse cdf} returns values of type @racket[Out].
|
||||
Similarly to @racket[distribution], the @racket[In] type parameter is the data type an ordered
|
||||
distribution accepts as arguments to its @tech{pdf}, and the @racket[Out] type parameter is the
|
||||
data type an ordered distribution returns as random samples. Additionally, its @tech{cdf}
|
||||
accepts values of type @racket[In], and its @tech{inverse cdf} returns values of type
|
||||
@racket[Out].
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(ordered-dist? (discrete-dist '(a b c)))
|
||||
|
|
@ -227,8 +210,24 @@ The parent type of real-valued distributions, such as any distribution returned
|
|||
@racket[normal-dist]. Equivalent to the type @racket[(ordered-dist Real Flonum)].
|
||||
}
|
||||
|
||||
@defproc[(pdf [d (dist In Out)] [v In] [log? Any #f]) Flonum]{
|
||||
An uncurried form of @racket[distribution-pdf]. When @racket[log?] is not @racket[#f], returns a
|
||||
log density.
|
||||
@examples[#:eval untyped-eval
|
||||
(pdf (discrete-dist '(a b c) '(1 2 3)) 'a)
|
||||
(pdf (discrete-dist '(a b c) '(1 2 3)) 'a #t)]
|
||||
}
|
||||
|
||||
@defproc*[([(sample [d (dist In Out)]) Out]
|
||||
[(sample [d (dist In Out)] [n Integer]) (Listof Out)])]{
|
||||
An uncurried form of @racket[distribution-sample].
|
||||
@examples[#:eval untyped-eval
|
||||
(sample (exponential-dist))
|
||||
(sample (exponential-dist) 3)]
|
||||
}
|
||||
|
||||
@defproc[(cdf [d (ordered-dist In Out)] [v In] [log? Any #f] [1-p? Any #f]) Flonum]{
|
||||
An uncurried form of @racket[dist-cdf].
|
||||
An uncurried form of @racket[ordered-dist-cdf].
|
||||
|
||||
When @racket[log?] is @racket[#f], @racket[cdf] returns a probability; otherwise, it returns a log
|
||||
probability.
|
||||
|
|
@ -238,7 +237,7 @@ When @racket[1-p?] is @racket[#f], @racket[cdf] returns a lower-tail probability
|
|||
}
|
||||
|
||||
@defproc[(inv-cdf [d (ordered-dist In Out)] [p Real] [log? Any #f] [1-p? Any #f]) Out]{
|
||||
An uncurried form of @racket[dist-inv-cdf].
|
||||
An uncurried form of @racket[ordered-dist-inv-cdf].
|
||||
|
||||
When @racket[log?] is @racket[#f], @racket[inv-cdf] interprets @racket[p] as a probability; otherwise,
|
||||
it interprets @racket[p] as a log probability.
|
||||
|
|
@ -254,22 +253,6 @@ the two endpoints are swapped first.) The @racket[log?] and @racket[1-p?] argume
|
|||
meaning of the return value in the same way as the corresponding arguments to @racket[cdf].
|
||||
}
|
||||
|
||||
@deftogether[(@defproc[(dist-cdf [d (ordered-dist In Out)]) (CDF In)]
|
||||
@defproc[(dist-inv-cdf [d (ordered-dist In Out)]) (Inverse-CDF Out)]
|
||||
@defproc[(dist-min [d (ordered-dist In Out)]) Out]
|
||||
@defproc[(dist-max [d (ordered-dist In Out)]) Out]
|
||||
@defproc[(dist-median [d (ordered-dist In Out)]) Out])]{
|
||||
The first four are synonyms for @racket[ordered-dist-cdf], @racket[ordered-dist-inv-cdf],
|
||||
@racket[ordered-dist-min] and @racket[ordered-dist-max]. The last is equivalent to
|
||||
@racket[(force (ordered-dist-median d))].
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(dist-min (normal-dist))
|
||||
(dist-min (geometric-dist 1/3))
|
||||
(dist-max (uniform-dist 1 2))
|
||||
(dist-median (gamma-dist 4 2))]
|
||||
}
|
||||
|
||||
@section{Finite Distribution Families}
|
||||
|
||||
@subsection{Unordered Discrete Distributions}
|
||||
|
|
@ -299,7 +282,7 @@ random samples.
|
|||
(define n 500)
|
||||
(define h (samples->hash (sample d n)))
|
||||
(plot (list (discrete-histogram
|
||||
(map vector xs (map (dist-pdf d) xs))
|
||||
(map vector xs (map (distribution-pdf d) xs))
|
||||
#:x-min 0 #:skip 2 #:label "P[x]")
|
||||
(discrete-histogram
|
||||
(map vector xs (map (λ (x) (/ (hash-ref h x) n)) xs))
|
||||
|
|
@ -354,11 +337,11 @@ on it are faster.
|
|||
|
||||
@examples[#:eval untyped-eval
|
||||
(define d (bernoulli-dist 0.75))
|
||||
(map (dist-pdf d) '(0 1))
|
||||
(map (dist-cdf d) '(0 1))
|
||||
(map (distribution-pdf d) '(0 1))
|
||||
(map (ordered-dist-cdf d) '(0 1))
|
||||
(define d (binomial-dist 1 0.75))
|
||||
(map (dist-pdf d) '(0 1))
|
||||
(map (dist-cdf d) '(0 1))]
|
||||
(map (distribution-pdf d) '(0 1))
|
||||
(map (ordered-dist-cdf d) '(0 1))]
|
||||
}
|
||||
|
||||
@subsection{Binomial Distributions}
|
||||
|
|
@ -375,9 +358,9 @@ probability of success.
|
|||
@examples[#:eval untyped-eval
|
||||
(define d (binomial-dist 15 0.6))
|
||||
(plot (discrete-histogram
|
||||
(map vector (build-list 16 values) (build-list 16 (dist-pdf d))))
|
||||
(map vector (build-list 16 values) (build-list 16 (distribution-pdf d))))
|
||||
#:x-label "number of successes" #:y-label "probability")
|
||||
(plot (function-interval (λ (x) 0) (dist-cdf d) -0.5 15.5)
|
||||
(plot (function-interval (λ (x) 0) (ordered-dist-cdf d) -0.5 15.5)
|
||||
#:x-label "at-most number of successes" #:y-label "probability")]
|
||||
}
|
||||
|
||||
|
|
@ -395,9 +378,9 @@ first success starting from zero.
|
|||
@examples[#:eval untyped-eval
|
||||
(define d (geometric-dist 0.25))
|
||||
(plot (discrete-histogram
|
||||
(map vector (build-list 16 values) (build-list 16 (dist-pdf d))))
|
||||
(map vector (build-list 16 values) (build-list 16 (distribution-pdf d))))
|
||||
#:x-label "first success index" #:y-label "probability")
|
||||
(plot (function-interval (λ (x) 0) (dist-cdf d) -0.5 15.5)
|
||||
(plot (function-interval (λ (x) 0) (ordered-dist-cdf d) -0.5 15.5)
|
||||
#:x-label "at-most first success index" #:y-label "probability"
|
||||
#:y-max 1)]
|
||||
}
|
||||
|
|
@ -415,9 +398,9 @@ independent events.
|
|||
@examples[#:eval untyped-eval
|
||||
(define d (poisson-dist 6.2))
|
||||
(plot (discrete-histogram
|
||||
(map vector (build-list 16 values) (build-list 16 (dist-pdf d))))
|
||||
(map vector (build-list 16 values) (build-list 16 (distribution-pdf d))))
|
||||
#:x-label "number of events" #:y-label "probability")
|
||||
(plot (function-interval (λ (x) 0) (dist-cdf d) -0.5 15.5)
|
||||
(plot (function-interval (λ (x) 0) (ordered-dist-cdf d) -0.5 15.5)
|
||||
#:x-label "at-most number of events" #:y-label "probability"
|
||||
#:y-max 1)]
|
||||
}
|
||||
|
|
@ -468,14 +451,14 @@ which must both be nonnegative.
|
|||
(plot (for/list ([α (in-list '(1 2 3 1/2))]
|
||||
[β (in-list '(1 3 1 1/2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (beta-dist α β))
|
||||
(function (distribution-pdf (beta-dist α β))
|
||||
#:color i #:label (format "Beta(~a,~a)" α β)))
|
||||
#:x-min 0 #:x-max 1 #:y-max 4 #:y-label "density")
|
||||
|
||||
(plot (for/list ([α (in-list '(1 2 3 1/2))]
|
||||
[β (in-list '(1 3 1 1/2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (beta-dist α β))
|
||||
(function (ordered-dist-cdf (beta-dist α β))
|
||||
#:color i #:label (format "Beta(~a,~a)" α β)))
|
||||
#:x-min 0 #:x-max 1 #:y-label "probability")]
|
||||
|
||||
|
|
@ -502,7 +485,7 @@ Represents the Cauchy distribution family parameterized by mode and scale.
|
|||
(plot (for/list ([m (in-list '(0 -1 0 2))]
|
||||
[s (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (cauchy-dist m s))
|
||||
(function (distribution-pdf (cauchy-dist m s))
|
||||
#:color i #:label (format "Cauchy(~a,~a)" m s)))
|
||||
#:x-min -8 #:x-max 8 #:y-label "density"
|
||||
#:legend-anchor 'top-right)
|
||||
|
|
@ -510,7 +493,7 @@ Represents the Cauchy distribution family parameterized by mode and scale.
|
|||
(plot (for/list ([m (in-list '(0 -1 0 2))]
|
||||
[s (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (cauchy-dist m s))
|
||||
(function (ordered-dist-cdf (cauchy-dist m s))
|
||||
#:color i #:label (format "Cauchy(~a,~a)" m s)))
|
||||
#:x-min -8 #:x-max 8 #:y-label "probability")]
|
||||
}
|
||||
|
|
@ -529,7 +512,7 @@ Represents the family of distributions whose densities are Dirac delta functions
|
|||
(pdf (delta-dist) 1)
|
||||
(plot (for/list ([μ (in-list '(-1 0 1))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (delta-dist μ))
|
||||
(function (ordered-dist-cdf (delta-dist μ))
|
||||
#:color i #:style i #:label (format "δ(~a)" μ)))
|
||||
#:x-min -2 #:x-max 2 #:y-label "probability")]
|
||||
}
|
||||
|
|
@ -551,14 +534,14 @@ is the reciprocal of mean or scale. Construct exponential distributions from rat
|
|||
@examples[#:eval untyped-eval
|
||||
(plot (for/list ([μ (in-list '(2/3 1 2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (exponential-dist μ))
|
||||
(function (distribution-pdf (exponential-dist μ))
|
||||
#:color i #:label (format "Exponential(~a)" μ)))
|
||||
#:x-min 0 #:x-max 5 #:y-label "density"
|
||||
#:legend-anchor 'top-right)
|
||||
|
||||
(plot (for/list ([μ (in-list '(2/3 1 2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (exponential-dist μ))
|
||||
(function (ordered-dist-cdf (exponential-dist μ))
|
||||
#:color i #:label (format "Exponential(~a)" μ)))
|
||||
#:x-min 0 #:x-max 5 #:y-label "probability"
|
||||
#:legend-anchor 'bottom-right)]
|
||||
|
|
@ -583,7 +566,7 @@ which is the reciprocal of scale. Construct gamma distributions from rates using
|
|||
(plot (for/list ([k (in-list '(1 2 3 9))]
|
||||
[s (in-list '(2 2 3 1/2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (gamma-dist k s))
|
||||
(function (distribution-pdf (gamma-dist k s))
|
||||
#:color i #:label (format "Gamma(~a,~a)" k s)))
|
||||
#:x-min 0 #:x-max 15 #:y-label "density"
|
||||
#:legend-anchor 'top-right)
|
||||
|
|
@ -591,7 +574,7 @@ which is the reciprocal of scale. Construct gamma distributions from rates using
|
|||
(plot (for/list ([k (in-list '(1 2 3 9))]
|
||||
[s (in-list '(2 2 3 1/2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (gamma-dist k s))
|
||||
(function (ordered-dist-cdf (gamma-dist k s))
|
||||
#:color i #:label (format "Gamma(~a,~a)" k s)))
|
||||
#:x-min 0 #:x-max 15 #:y-label "probability"
|
||||
#:legend-anchor 'bottom-right)]
|
||||
|
|
@ -621,7 +604,7 @@ and scale. In this parameterization, the variance is @racket[(* 1/3 (sqr (* pi s
|
|||
(plot (for/list ([μ (in-list '(0 -1 0 2))]
|
||||
[s (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (logistic-dist μ s))
|
||||
(function (distribution-pdf (logistic-dist μ s))
|
||||
#:color i #:label (format "Logistic(~a,~a)" μ s)))
|
||||
#:x-min -8 #:x-max 8 #:y-label "density"
|
||||
#:legend-anchor 'top-right)
|
||||
|
|
@ -629,7 +612,7 @@ and scale. In this parameterization, the variance is @racket[(* 1/3 (sqr (* pi s
|
|||
(plot (for/list ([μ (in-list '(0 -1 0 2))]
|
||||
[s (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (logistic-dist μ s))
|
||||
(function (ordered-dist-cdf (logistic-dist μ s))
|
||||
#:color i #:label (format "Logistic(~a,~a)" μ s)))
|
||||
#:x-min -8 #:x-max 8 #:y-label "probability")]
|
||||
}
|
||||
|
|
@ -652,14 +635,14 @@ which is the square of standard deviation. Construct normal distributions from v
|
|||
(plot (for/list ([μ (in-list '(0 -1 0 2))]
|
||||
[σ (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (normal-dist μ σ))
|
||||
(function (distribution-pdf (normal-dist μ σ))
|
||||
#:color i #:label (format "N(~a,~a)" μ σ)))
|
||||
#:x-min -5 #:x-max 5 #:y-label "density")
|
||||
|
||||
(plot (for/list ([μ (in-list '(0 -1 0 2))]
|
||||
[σ (in-list '(1 1/2 2.25 0.7))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (normal-dist μ σ))
|
||||
(function (ordered-dist-cdf (normal-dist μ σ))
|
||||
#:color i #:label (format "N(~a,~a)" μ σ)))
|
||||
#:x-min -5 #:x-max 5 #:y-label "probability")]
|
||||
}
|
||||
|
|
@ -685,7 +668,7 @@ before constructing the distribution object.
|
|||
[b (in-list '(0 1 3))]
|
||||
[m (in-list '(-2 0 2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (triangle-dist a b m)) #:color i
|
||||
(function (distribution-pdf (triangle-dist a b m)) #:color i
|
||||
#:label (format "Triangle(~a,~a,~a)" a b m)))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "density")
|
||||
|
||||
|
|
@ -693,7 +676,7 @@ before constructing the distribution object.
|
|||
[b (in-list '(0 1 3))]
|
||||
[m (in-list '(-2 0 2))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (triangle-dist a b m)) #:color i
|
||||
(function (ordered-dist-cdf (triangle-dist a b m)) #:color i
|
||||
#:label (format "Triangle(~a,~a,~a)" a b m)))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "probability")]
|
||||
|
||||
|
|
@ -724,11 +707,11 @@ Samples are taken by applying the truncated distribution's @tech{inverse cdf} to
|
|||
(define d (normal-dist))
|
||||
(define t (truncated-dist d -2 1))
|
||||
t
|
||||
(plot (list (function (dist-pdf d) #:label "N(0,1)" #:color 0)
|
||||
(function (dist-pdf t) #:label "T(N(0,1),-2,1)"))
|
||||
(plot (list (function (distribution-pdf d) #:label "N(0,1)" #:color 0)
|
||||
(function (distribution-pdf t) #:label "T(N(0,1),-2,1)"))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "density")
|
||||
(plot (list (function (dist-cdf d) #:label "N(0,1)" #:color 0)
|
||||
(function (dist-cdf t) #:label "T(N(0,1),-2,1)"))
|
||||
(plot (list (function (ordered-dist-cdf d) #:label "N(0,1)" #:color 0)
|
||||
(function (ordered-dist-cdf t) #:label "T(N(0,1),-2,1)"))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "probability")]
|
||||
}
|
||||
|
||||
|
|
@ -754,14 +737,14 @@ they are swapped before constructing the distribution object.
|
|||
(plot (for/list ([a (in-list '(-3 -1 -2))]
|
||||
[b (in-list '(0 1 3))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-pdf (uniform-dist a b)) #:color i
|
||||
(function (distribution-pdf (uniform-dist a b)) #:color i
|
||||
#:label (format "Uniform(~a,~a)" a b)))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "density")
|
||||
|
||||
(plot (for/list ([a (in-list '(-3 -1 -2))]
|
||||
[b (in-list '(0 1 3))]
|
||||
[i (in-naturals)])
|
||||
(function (dist-cdf (uniform-dist a b)) #:color i
|
||||
(function (ordered-dist-cdf (uniform-dist a b)) #:color i
|
||||
#:label (format "Uniform(~a,~a)" a b)))
|
||||
#:x-min -3.5 #:x-max 3.5 #:y-label "probability")]
|
||||
|
||||
|
|
|
|||
|
|
@ -164,7 +164,7 @@ But see @racket[flexpt1p], which is more accurate still.
|
|||
Like @racket[(flexpt (+ 1.0 x) y)], but accurate for any @racket[x] and @racket[y].
|
||||
}
|
||||
|
||||
@defproc[(make-flexp/base [x Real]) (Flonum -> Flonum)]{
|
||||
@defproc[(make-flexpt [x Real]) (Flonum -> Flonum)]{
|
||||
Equivalent to @racket[(λ (y) (flexpt x y))] when @racket[x] is a flonum, but much more
|
||||
accurate for large @racket[y] when @racket[x] cannot be exactly represented
|
||||
by a flonum.
|
||||
|
|
@ -181,9 +181,9 @@ the error is low near zero, but grows with distance from the origin:
|
|||
(eval:result ""))
|
||||
(eval:alts (flulp-error (flexpt pi y) pi^y)
|
||||
(eval:result @racketresultfont{43.12619934359266}))]
|
||||
Using @racket[make-flexp/base], the error is near rounding error everywhere:
|
||||
Using @racket[make-flexpt], the error is near rounding error everywhere:
|
||||
@interaction[#:eval untyped-eval
|
||||
(eval:alts (define flexppi (make-flexp/base (bigfloat->rational pi.bf)))
|
||||
(eval:alts (define flexppi (make-flexpt (bigfloat->rational pi.bf)))
|
||||
(eval:result ""))
|
||||
(eval:alts (flulp-error (flexppi y) pi^y)
|
||||
(eval:result @racketresultfont{0.8738006564073412}))]
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user