Merge pull request #2 from mcgirjau/master

Add normal distribution, calculus, and basic math

src/tools/calculus.cljc

@@ -0,0 +1,33 @@
+(ns tools.calculus)
+
+(defonce ^:const dx 0.0001)
+
+(defn deriv
+  "Returns the derivative of f evaluated at c. If called with only one argument,
+  it returns the derivative function."
+  ([f c]
+   ((deriv f) c))
+  ([f]
+   (fn [x]
+     (/ (- (f (+ x dx)) (f x)) dx))))
+
+(defn integrate
+  "Returns the definite integral of f over [a, b] using Simpson's method.
+  If called with only one argument (the function), returns the indefinite
+  integral, which takes as input a value x and (optionally) a constant c."
+  ([f]
+   (fn this
+     ([x] (this x 0))
+     ([x c] (+ (integrate f 0 x) c))))
+  ([f a b]
+   (let [n (/ 1 dx)
+         h (/ (- b a) n)]
+     (loop [i 1
+            sum1 (f (+ a (/ h 2)))
+            sum2 0]
+       (if (< i n)
+         (recur (inc i)
+                (+ sum1 (f (+ a (* h i) (/ h 2))))
+                (+ sum2 (f (+ a (* h i)))))
+         (* (/ h 6) (+ (f a) (f b) (* 4 sum1) (* 2 sum2))))))))
+

src/tools/distributions.cljc

@@ -0,0 +1,59 @@
+(ns tools.distributions
+  (:require [tools.math :as math])
+  (:require [tools.calculus :as calculus]))
+
+;; -----------------------------------------------------------------------------
+;; NORMAL
+;; -----------------------------------------------------------------------------
+
+(defn- box-muller
+  "Given two uniformly distributed random variables (from 0 to 1), returns a
+  Standard Normal variable computed using the Box-Muller Transform."
+  [u1 u2]
+  (* (math/sqrt (* -2 (math/log u1)))
+     (math/cos (* 2 math/PI u2))))
+
+(defn- normal-pdf
+  "Given a mean and standard deviation, returns the corresponding Normal
+  Probability Distribution Function."
+  [mu sigma]
+  (fn [x]
+    (* (/ 1 (* sigma (math/sqrt (* 2 math/PI))))
+       (math/exp (- (/ (math/pow (/ (- x mu) sigma) 2) 2))))))
+
+(defn rand-norm
+  "Generates n Normally-distributed random variables with given mean and
+  standard deviation. If no parameters are provided, defaults to a
+  single random observation from a Standard Normal distribution.
+  Accepts an argument map with optional keys :n, :mu, and :sigma."
+  [{:keys [n mu sigma]
+    :or   {n 1, mu 0, sigma 1}}]
+  (repeatedly n #(box-muller (rand 1) (rand 1))))
+
+(defn pdf-norm
+  "Returns the value of the Normal Probability Distribution Function at a
+  particular value x. If no distributional parameters are provided, defaults to
+  the Standard Normal PDF.
+  Accepts an argument map with keys :x, and optionally :mu and :sigma."
+  [{:keys [x mu sigma]
+    :or   {mu 0, sigma 1}}]
+  ((normal-pdf mu sigma) x))
+
+(defn cdf-norm
+  "Parameters: {:keys [x mu sigma]}
+  Returns the value of the Normal Cumulative Distribution Function at a
+  particular value x. If no distributional parameters are provided, defaults to
+  the Standard Normal CDF.
+  Accepts an argument map with keys :x, and optionally :mu and :sigma."
+  [{:keys [x mu sigma]
+    :or   {mu 0, sigma 1}}]
+  (calculus/integrate (normal-pdf mu sigma) (- mu (* 6 sigma)) x))
+
+(defn quant-norm
+  "For a given probability p, returns the corresponding value of the quantile
+  function (i.e. the inverse Cumulative Distribution Function). If no
+  distributional parameters are provided, defaults to Standard Normal quantiles.
+  Accepts an argument map with keys :p, and optionally :mu and :sigma."
+  [{:keys [p mu sigma]
+    :or   {mu 0, sigma 1}}]
+  ())                                                       ; unfinished...

src/tools/math.cljc

@@ -0,0 +1,99 @@
+(ns tools.math)
+
+(defonce PI #?(:clj  Math/PI
+               :cljs js/Math.PI))
+
+(defonce E #?(:clj  Math/E
+              :cljs js/Math.PI))
+
+(defn abs
+  "Returns the absolute value of a number."
+  [x]
+  (if (neg? x) (- x) x))
+
+(defn approx= [x y epsilon]
+  "Returns true if the absolute difference between x and y is less than or
+  equal to some specified error level, epsilon."
+  (<= (abs (- y x)) epsilon))
+
+(defn ceil
+  "Returns the smallest integer greater than or equal to x."
+  [x]
+  #?(:clj  (Math/ceil x)
+     :cljs (js/Math.ceil x)))
+
+(defn cos
+  "Returns the cosine of an angle (specified in radians)."
+  [x]
+  #?(:clj  (Math/cos x)
+     :cljs (js/Math.cos x)))
+
+(defn exp
+  "Returns Euler's number (approx. 2.71) raised to the given power."
+  [x]
+  #?(:clj  (Math/exp x)
+     :cljs (js/Math.exp x)))
+
+(defn floor
+  "Returns the largest integer less than or equal to x."
+  [x]
+  #?(:clj  (Math/floor x)
+     :cljs (js/Math.floor x)))
+
+(defn log
+  "Returns the logarithm of x with the given base. If called with only one
+  argument, returns the natural logarithm (base e) of the given value."
+  ([x base]
+   (/ (log x) (log base)))
+  ([x]
+   #?(:clj  (Math/log x)
+      :cljs (js/Math.log x))))
+
+(defn pow
+  "Returns the value obtained by raising the first argument to the power of
+  the second argument."
+  [x n]
+  #?(:clj  (Math/pow x n)
+     :cljs (js/Math.pow x n)))
+
+(defn root
+  "Returns the root of x with base n."
+  [x n]
+  (pow x (/ 1 n)))
+
+(defn round
+  "Returns the value of x rounded to the nearest integer."
+  [x]
+  #?(:clj  (Math/round x)
+     :cljs (js/Math.round x)))
+
+(defn sign
+  "Returns the 1 if the argument is positive, -1 if the argument is negative,
+  and 0 if the argument is zero."
+  [x]
+  (cond (< x 0) -1
+        (> x 0) 1
+        :else 0))
+
+(defn sin
+  "Returns the sine of an angle (specified in radians)."
+  [x]
+  #?(:clj  (Math/sin x)
+     :cljs (js/Math.sin x)))
+
+(defn sqrt
+  "Returns the square root of the given value."
+  [x]
+  #?(:clj  (Math/sqrt x)
+     :cljs (js/Math.sqrt x)))
+
+(defn square
+  "Returns the square of the given value."
+  [x]
+  (* x x))
+
+(defn tan
+  "Returns the tangent of an angle (specified in radians)."
+  [x]
+  #?(:clj  (Math/tan x)
+     :cljs (js/Math.tan x)))