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Scheme
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;; -*- coding: utf-8 -*-
 
#!c-expr
 
;; I always use Scheme as a quick calculators, and a few such expressions
;; happened to be in my scratch buffer so I turned it to C-exprs.
 
(use srfi-42)
 
(define ^^ expt)
 
(define (S0 λ N)
(sum-ec (: n 1 N)
* (exp (- {n * λ})) * (- {2 ^^ (ceiling (log n 2))}) * n}))
 
(define (S1 λ K)
(sum-ec (: k K)
{{2 ^^ k} * (exp (- {{2 ^^ k} * λ}))}))
 
(define (St λ K)
{(- {(/ λ) * (exp (- λ))})
+ (sum-ec (: k 1 K)
{{2 ^^ {k - 1}} * (exp (- {{2 ^^ k} * λ}))})})
 
 
;; Jeremy Gibbons's Unbounded Spigot algorithm.
;; http://sourceforge.net/projects/readable/files/
;; Original code is in Haskell. Here's my port to Gauche:
;; http://blog.practical-scheme.net/shiro/20120713-spigot
;; Since it was one of the times that I envied Haskell's infix syntax,
;; let's try to use C-expr.
 
#!c-expr
 
(use gauche.lazy)
(use util.match)
 
(define-syntax define*
(syntax-rules ()
[(_ (fn . pats) . body) (define fn (match-lambda* [pats . body]))]))
 
(define (stream next safe prod kons seed xs)
(^[] (let loop ([y (next seed)])
(cond [(safe seed y) (set! seed (prod seed y)) y]
[else (set! seed (kons seed (car xs)))
(set! xs (cdr xs))
(loop (next seed))]))))
 
(define (lft q r s t)
(let1 f (gcd q r s t)
(vector {q / f} {r / f} {s / f} {t / f})))
 
(define *unit-lft* '#(1 0 0 1))
 
(define* (lft*v #(q r s t) x y)
(floor {{{q * x} + {r * y}} / {{s * x} + {t * y}}}))
 
(define* (lft*lft #(q r s t) #(u v w x))
(lft {{q * u} + {r * w}} {{q * v} + {r * x}}
{{s * u} + {t * w}} {{s * v} + {t * x}}))
 
(define (pi)
(define lfts (lmap (^k (lft k {{4 * k} + 2} 0 {{2 * k} + 1})) (lrange 1)))
(define (next z) (lft*v z 3 1))
(define (safe z n) {n = (lft*v z 4 1)}
(define (prod z n) (lft*lft (lft 10 {-10 * n} 0 1) z))
(stream next safe prod lft*lft *unit-lft* lfts))
 
(define (piL)
(define lfts (lmap (^i (lft {{2 * i} - 1} {i * i} 1 0)) (lrange 1)))
(define* (next (z . i)) (lft*v z {{2 * i} - 1} 1))
(define* (safe (z . i) n) {n = (lft*v z {{5 * i} - 2} 2)})
(define* (prod (z . i) n) (cons (lft*lft (lft 10 {-10 * n} 0 1) z) i))
(define* (kons (z . i) z_) (cons (lft*lft z z_) {i + 1}))
(stream next safe prod kons `(,(lft 0 4 1 0) . 1) lfts))
 
;; Lazy infinite sequence of prime numbers.
;; http://blog.practical-scheme.net/shiro/20110929-primes
 
(define (prime? n)
(not (any zero? (lmap (pa$ mod n) (take-while (^k {{k * k} <= n}) *primes*)))))
 
(define (primes-from k)
(if (prime? k)
(lcons k (primes-from {k + 2}))
(primes-from {k + 2})))
 
(define *primes* (list* 2 3 (lcons 5 (primes-from 7))))
 
;; Some math-heavy code excerpts from https://github.com/shirok/mpm-experiment
 
(define-constant 2π (sqrt {pi * 2}))
 
(define (φ x) {(exp {-.5 * x * x}) / 2π})
 
(define (Φ x)
(define (Φ+ x)
(let* ([t {/ {1 + {0.2316419 * x}}}]
[t2 {t * t}]
[t3 {t * t2}]
[t4 {t * t3}]
[t5 {t * t4}])
{1 - {(φ x) * {{0.319381530 * t} + {-0.356563782 * t2}
+ {1.781477937 * t3} + {-1.821255978 * t4}
+ {1.330274429 * t5}}}}))
(cond [{x = 0} .5]
[{x > 0} (Φ+ x)]
[else {1 - (Φ+ (- x))}]))
 
(define (cpick μ σ)
(let ([min - {4 * σ}}]
[max + {4 * σ}}]
[u (random-real)])
(let loop ([x0 μ])
(let1 Fx {(Φ {{x0 - μ} / σ}) - u}
(if {(abs Fx) < 10e-6}
x0
(let* ([fx {(φ {{x0 - μ} / σ}) / σ}]
[x1 {x0 - {Fx / fx}}])
(cond [{x1 <= min} min]
[{x1 >= max} max]
[else (loop x1)])))))))
 
(define (derive! tree X)
(define (derive-1 Xs)
(if (null? Xs) (finalize!) (grow! (car Xs) (cdr Xs))))
(define (finalize!)
(let ([λs '()] [αs '()] [βs '()] [Xs '()])
(define (rec tree)
(match tree
[('F <λ>) (push! λs <λ>)]
[('X _ subtree) (push! Xs tree) (rec subtree)]
[((or 'L 'R) <α> <β>) (push! αs <α>) (push! βs <β>)]
[('Tree nodes ...) (for-each rec nodes)]
[_ #f]))
(rec (Tree-ω tree))
(Tree-λs-set! tree (list->vector λs))
(Tree-αs-set! tree (list->vector αs))
(Tree-βs-set! tree (list->vector βs))
(Tree-num-params-set! tree {(vector-length (Tree-λs tree))
+ {(vector-length (Tree-αs tree)) * 2}})
(Tree-Xs-set! tree Xs)
(recalculate-weights! tree)))
(define (grow! X Xs)
(let* ([l (cadr X)] [u (random-real)])
(if {u < {l / M}}
(let1 <λ.> (list (cpick μ_λ σ_λ))
(set! (caddr X) `(Tree (F ,<λ.>) Z))
(derive-1 Xs))
(let ([<λ.> (list (cpick μ_λ σ_λ))]
[<α.> (list (cpick μ_α σ_α))]
[<β.> (list (cpick μ_β σ_β))]
[XL (list 'X {l + 1} #f)]
[XR (list 'X {l + 1} #f)])
(set! (caddr X) `(Tree (F ,<λ.>)
(Tree (L ,<α.> ,<β.>) ,XL)
(Tree (R ,<α.> ,<β.>) ,XR)))
(derive-1 (list* XL XR Xs))))))
(derive-1 `(,X)))
 
(define (recalculate-weights! tree)
(let1 maxdepth (apply max (map cadr (Tree-Xs tree)))
(receive (ws sum)
(map-accum (^[Xi sum]
(let1 wi {2 ^^ {maxdepth - (cadr Xi)}}
(values wi {wi + sum})))
0 (Tree-Xs tree))
(set! (Tree-ws tree) ws)
(set! (Tree-Σw tree) sum))))
 
(define (walk-node node twig leaf)
(define (rec node level x y θ)
(match node
[('F (λ))
(let ([x. {x + * (sin θ)}}]
[y. {y + * (cos θ)}}])
(when twig (twig x y x. y. level))
(values x. y. θ))]
['Z (when leaf (leaf x y level)) (values x y θ)]
[('X level subtree) (rec subtree level x y θ)]
[('Tree nodes ...)
(let loop ([ns nodes] [x. x] [y. y] [θ. θ])
(match ns
[() (values x y θ)]
[(n . ns) (receive (x. y. θ.) (rec n level x. y. θ.)
(loop ns x. y. θ.))]))]
[((and (or 'L 'R) z) (α) (β))
(values x y + (if {z eq? 'L} + α} - α})})]))
(rec node 0 0 0 0))
 
(define kernel-center {2 * (round->exact σ_z)})
(define kernel-size {{2 * kernel-center} - 1})
(define I0 (make-f32vector {I_size * I_size} 1.0))
 
(define kernel
(rlet1 v (make-f32vector {kernel-size * kernel-size})
(do-ec (: y kernel-size)
(: x kernel-size)
(f32vector-set! v {x + {y * kernel-size}}
(exp (- {{{{y - kernel-center - 1} ^^ 2}
+ {{x - kernel-center - 1} ^^ 2}}
/ σ_z}))))))
 
(define (likelihood tree I) ;I :: <f32vector>
(define buf (likelihood-buf tree))
(define penalty 0) ; if we're outside of canvas, we penalize it a lot.
(define (roundI a min max)
(round->exact (clamp {{I_size - 1} * {{a - min} /. {max - min}}}
0 {I_size - 1})))
(define (add-Z x y)
(if {{x < canvas-x0} or {canvas-x1 < x}
or {y < canvas-y0} or {canvas-y1 < y}}
(inc! penalty 5.0)
(let ([ix (roundI x canvas-x0 canvas-x1)]
[iy (roundI y canvas-y0 canvas-y1)])
(do-ec (: y kernel-size)
(: x kernel-size)
(let ([jy {iy + y + (- kernel-center)}]
[jx {ix + x + (- kernel-center)}])
(when {{-1 < jy < I_size} and {-1 < jx < I_size}}
(f32vector-set! buf {jx + {jy * I_size}}
{(f32vector-ref kernel
{x + {y * kernel-size}})
+
(f32vector-ref buf
{jx + {jy * I_size}})}))))
)))
 
(f32vector-fill! buf 0)
(walk-node (Tree-ω tree) #f (^[x y level] (add-Z x y)))
(f32vector-clamp! buf 0 1)
(f32vector-sub! buf I)
;; Just to calculate likelihood, we can do (exp (- (f32vector-dot buf buf)))
;; but we want the squared image for visualization.
(f32vector-mul! buf buf)
(exp (- {penalty + (f32vector-dot buf I0)})))

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