Merge pull request #30 from NunoSempere/meanAndVariance

Mean and variance

__tests__/Distributions__Test.re

@@ -12,6 +12,15 @@ let makeTest = (~only=false, str, item1, item2) =>
         expect(item1) |> toEqual(item2)
       );
 
+let makeTestCloseEquality = (~only=false, str, item1, item2, ~digits) =>
+  only
+    ? Only.test(str, () =>
+        expect(item1) |> toBeSoCloseTo(item2, ~digits)
+      )
+    : test(str, () =>
+        expect(item1) |> toBeSoCloseTo(item2, ~digits)
+      );
+
 describe("Shape", () => {
   describe("Continuous", () => {
     open Distributions.Continuous;
@@ -185,6 +194,13 @@ describe("Shape", () => {
       0.9,
     );
     makeTest("integralEndY", T.Integral.sum(~cache=None, discrete), 1.0);
+    makeTest("mean", T.getMean(discrete), 3.9);
+    makeTestCloseEquality(
+      "variance",
+      T.getVariance(discrete),
+      5.89,
+      ~digits=7,
+    );
   });
 
   describe("Mixed", () => {
@@ -291,7 +307,7 @@ describe("Shape", () => {
     );
   });
 
-  describe("Mixed", () => {
+  describe("Distplus", () => {
     open Distributions.DistPlus;
     let discrete: DistTypes.xyShape = {
       xs: [|1., 4., 8.|],
@@ -362,4 +378,42 @@ describe("Shape", () => {
       ),
     );
   });
+
+  describe("Shape", () => {
+    let mean = 10.0;
+    let stdev = 4.0;
+    let variance = stdev ** 2.0;
+    let numSamples = 10000;
+    open Distributions.Shape;
+    let normal: SymbolicDist.dist = `Normal({mean, stdev});
+    let normalShape = SymbolicDist.GenericSimple.toShape(normal, numSamples);
+    let lognormal = SymbolicDist.Lognormal.fromMeanAndStdev(mean, stdev);
+    let lognormalShape =
+      SymbolicDist.GenericSimple.toShape(lognormal, numSamples);
+
+    makeTestCloseEquality(
+      "Mean of a normal",
+      T.getMean(normalShape),
+      mean,
+      ~digits=2,
+    );
+    makeTestCloseEquality(
+      "Variance of a normal",
+      T.getVariance(normalShape),
+      variance,
+      ~digits=1,
+    );
+    makeTestCloseEquality(
+      "Mean of a lognormal",
+      T.getMean(lognormalShape),
+      mean,
+      ~digits=2,
+    );
+    makeTestCloseEquality(
+      "Variance of a lognormal",
+      T.getVariance(lognormalShape),
+      variance,
+      ~digits=0,
+    );
+  });
 });

src/distPlus/distribution/Distributions.re

@@ -17,6 +17,9 @@ module type dist = {
   let integralEndY: (~cache: option(integral), t) => float;
   let integralXtoY: (~cache: option(integral), float, t) => float;
   let integralYtoX: (~cache: option(integral), float, t) => float;
+
+  let getMean: t => float;
+  let getVariance: t => float;
 };
 
 module Dist = (T: dist) => {
@@ -35,6 +38,8 @@ module Dist = (T: dist) => {
   let toDiscrete = T.toDiscrete;
   let toScaledContinuous = T.toScaledContinuous;
   let toScaledDiscrete = T.toScaledDiscrete;
+  let getMean = T.getMean;
+  let getVariance = T.getVariance;
 
   // TODO: Move this to each class, have use integral to produce integral in DistPlus class.
   let scaleBy = (~scale=1.0, t: t) => t |> mapY((r: float) => r *. scale);
@@ -135,6 +140,23 @@ module Continuous = {
       let toDiscrete = _ => None;
       let toScaledContinuous = t => Some(t);
       let toScaledDiscrete = _ => None;
+
+      let getMean = (t: t) => {
+        let indefiniteIntegralStepwise = (p, h1) => h1 *. p ** 2.0 /. 2.0;
+        let indefiniteIntegralLinear = (p, a, b) =>
+          a *. p ** 2.0 /. 2.0 +. b *. p ** 3.0 /. 3.0;
+        XYShape.Analysis.integrateContinuousShape(
+          ~indefiniteIntegralStepwise,
+          ~indefiniteIntegralLinear,
+          t,
+        );
+      };
+      let getVariance = (t: t): float =>
+        XYShape.Analysis.getVarianceDangerously(
+          t,
+          getMean,
+          XYShape.Analysis.getMeanOfSquaresContinuousShape,
+        );
     });
 };
 
@@ -144,11 +166,22 @@ module Discrete = {
   let sortedByX = (t: DistTypes.discreteShape) =>
     t |> XYShape.T.zip |> XYShape.Zipped.sortByX;
   let empty = XYShape.T.empty;
-  let combine = (fn, t1: DistTypes.discreteShape, t2: DistTypes.discreteShape): DistTypes.discreteShape => {
-    XYShape.Combine.combine(~xsSelection=ALL_XS, ~xToYSelection=XYShape.XtoY.stepwiseIfAtX, ~fn, t1, t2) 
-  }
-  let _default0 = ((fn, a,b) => fn(E.O.default(0.0, a), E.O.default(0.0, b)));
-  let reduce = (fn, items) => items |> E.A.fold_left(combine(_default0((fn))), empty);
+  let combine =
+      (fn, t1: DistTypes.discreteShape, t2: DistTypes.discreteShape)
+      : DistTypes.discreteShape => {
+    XYShape.Combine.combine(
+      ~xsSelection=ALL_XS,
+      ~xToYSelection=XYShape.XtoY.stepwiseIfAtX,
+      ~fn,
+      t1,
+      t2,
+    );
+  };
+  let _default0 = (fn, a, b) =>
+    fn(E.O.default(0.0, a), E.O.default(0.0, b));
+  let reduce = (fn, items) =>
+    items |> E.A.fold_left(combine(_default0(fn)), empty);
+
   module T =
     Dist({
       type t = DistTypes.discreteShape;
@@ -195,6 +228,14 @@ module Discrete = {
         |> integral(~cache)
         |> Continuous.getShape
         |> XYShape.YtoX.linear(f);
+
+      let getMean = (t: t): float =>
+        E.A.reducei(t.xs, 0.0, (acc, x, i) => acc +. x *. t.ys[i]);
+      let getVariance = (t: t): float => {
+        let getMeanOfSquares = t =>
+          getMean(XYShape.Analysis.squareXYShape(t));
+        XYShape.Analysis.getVarianceDangerously(t, getMean, getMeanOfSquares);
+      };
     });
 };
 
@@ -366,6 +407,41 @@ module Mixed = {
           discreteProbabilityMassFraction,
         };
       };
+
+      let getMean = (t: t): float => {
+        let discreteProbabilityMassFraction =
+          t.discreteProbabilityMassFraction;
+        switch (discreteProbabilityMassFraction) {
+        | 1.0 => Discrete.T.getMean(t.discrete)
+        | 0.0 => Continuous.T.getMean(t.continuous)
+        | _ =>
+          Discrete.T.getMean(t.discrete)
+          *. discreteProbabilityMassFraction
+          +. Continuous.T.getMean(t.continuous)
+          *. (1.0 -. discreteProbabilityMassFraction)
+        };
+      };
+
+      let getVariance = (t: t): float => {
+        let discreteProbabilityMassFraction =
+          t.discreteProbabilityMassFraction;
+        let getMeanOfSquares = (t: t) => {
+          Discrete.T.getMean(XYShape.Analysis.squareXYShape(t.discrete))
+          *. t.discreteProbabilityMassFraction
+          +. XYShape.Analysis.getMeanOfSquaresContinuousShape(t.continuous)
+          *. (1.0 -. t.discreteProbabilityMassFraction);
+        };
+        switch (discreteProbabilityMassFraction) {
+        | 1.0 => Discrete.T.getVariance(t.discrete)
+        | 0.0 => Continuous.T.getVariance(t.continuous)
+        | _ =>
+          XYShape.Analysis.getVarianceDangerously(
+            t,
+            getMean,
+            getMeanOfSquares,
+          )
+        };
+      };
     });
 };
 
@@ -470,6 +546,20 @@ module Shape = {
           Discrete.T.mapY(fn),
           Continuous.T.mapY(fn),
         ));
+
+      let getMean = (t: t): float =>
+        switch (t) {
+        | Mixed(m) => Mixed.T.getMean(m)
+        | Discrete(m) => Discrete.T.getMean(m)
+        | Continuous(m) => Continuous.T.getMean(m)
+        };
+
+      let getVariance = (t: t): float =>
+        switch (t) {
+        | Mixed(m) => Mixed.T.getVariance(m)
+        | Discrete(m) => Discrete.T.getVariance(m)
+        | Continuous(m) => Continuous.T.getVariance(m)
+        };
     });
 };
 
@@ -584,6 +674,8 @@ module DistPlus = {
       let integralYtoX = (~cache as _, f, t: t) => {
         Shape.T.Integral.yToX(~cache=Some(t.integralCache), f, toShape(t));
       };
+      let getMean = (t: t) => Shape.T.getMean(t.shape);
+      let getVariance = (t: t) => Shape.T.getVariance(t.shape);
     });
 };
 

src/distPlus/distribution/XYShape.re

@@ -17,7 +17,7 @@ module T = {
   type ts = array(xyShape);
   let xs = (t: t) => t.xs;
   let ys = (t: t) => t.ys;
-  let empty = ({xs: [||], ys: [||]});
+  let empty = {xs: [||], ys: [||]};
   let minX = (t: t) => t |> xs |> E.A.Sorted.min |> extImp;
   let maxX = (t: t) => t |> xs |> E.A.Sorted.max |> extImp;
   let firstY = (t: t) => t |> ys |> E.A.first |> extImp;
@@ -297,4 +297,64 @@ let logScorePoint = (sampleCount, t1, t2) =>
   |> Range.integrateWithTriangles
   |> E.O.fmap(T.accumulateYs((+.)))
   |> E.O.fmap(Pairs.last)
-  |> E.O.fmap(Pairs.y);
+  |> E.O.fmap(Pairs.y);
+
+module Analysis = {
+  let integrateContinuousShape =
+      (
+        ~indefiniteIntegralStepwise=(p, h1) => h1 *. p,
+        ~indefiniteIntegralLinear=(p, a, b) => a *. p +. b *. p ** 2.0 /. 2.0,
+        t: DistTypes.continuousShape,
+      )
+      : float => {
+    let xs = t.xyShape.xs;
+    let ys = t.xyShape.ys;
+
+    E.A.reducei(
+      xs,
+      0.0,
+      (acc, _x, i) => {
+        let areaUnderIntegral =
+          switch (t.interpolation, i) {
+          | (_, 0) => 0.0
+          | (`Stepwise, _) =>
+            indefiniteIntegralStepwise(xs[i], ys[i - 1])
+            -. indefiniteIntegralStepwise(xs[i - 1], ys[i - 1])
+          | (`Linear, _) =>
+            let x1 = xs[i - 1];
+            let x2 = xs[i];
+            let h1 = ys[i - 1];
+            let h2 = ys[i];
+            let b = (h1 -. h2) /. (x1 -. x2);
+            let a = h1 -. b *. x1;
+            indefiniteIntegralLinear(x2, a, b)
+            -. indefiniteIntegralLinear(x1, a, b);
+          };
+        acc +. areaUnderIntegral;
+      },
+    );
+  };
+
+  let getMeanOfSquaresContinuousShape = (t: DistTypes.continuousShape) => {
+    let indefiniteIntegralLinear = (p, a, b) =>
+      a *. p ** 3.0 /. 3.0 +. b *. p ** 4.0 /. 4.0;
+    let indefiniteIntegralStepwise = (p, h1) => h1 *. p ** 3.0 /. 3.0;
+    integrateContinuousShape(
+      ~indefiniteIntegralStepwise,
+      ~indefiniteIntegralLinear,
+      t,
+    );
+  };
+
+  let getVarianceDangerously =
+      (t: 't, getMean: 't => float, getMeanOfSquares: 't => float): float => {
+    let meanSquared = getMean(t) ** 2.0;
+    let meanOfSquares = getMeanOfSquares(t);
+    meanOfSquares -. meanSquared;
+  };
+
+  let squareXYShape = (t): DistTypes.xyShape => {
+    ...t,
+    xs: E.A.fmap(x => x ** 2.0, t.xs),
+  };
+};

src/distPlus/utility/E.re

@@ -259,6 +259,9 @@ module A = {
   let fold_right = Array.fold_right;
   let concatMany = Belt.Array.concatMany;
   let keepMap = Belt.Array.keepMap;
+  let init = Array.init;
+  let reduce = Belt.Array.reduce;
+  let reducei = Belt.Array.reduceWithIndex;
   let min = a =>
     get(a, 0)
     |> O.fmap(first => Belt.Array.reduce(a, first, (i, j) => i < j ? i : j));