Added mean and variance functions, and tests for those.

- A getMean and getVariance in each module of src/distPlus/distribution/Distributions.re
    - They get the exact answer for the functions in Distributions.re, according to the approximation used.
    - There is now an XYShape.Analysis.integrateContinuousShape function.
- Tests in the __tests__/Distributions__Test.re function.
    - Calculation of the mean and variance for the normal and lognnormal distributions, at the end.
- I also added some reduce array functions to the E.A. module.

__tests__/Distributions__Test.re

@@ -12,7 +12,17 @@ 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;
     let continuous = make(`Linear, shape);
@@ -185,6 +195,7 @@ describe("Shape", () => {
       0.9,
     );
     makeTest("integralEndY", T.Integral.sum(~cache=None, discrete), 1.0);
+
   });
 
   describe("Mixed", () => {
@@ -289,9 +300,10 @@ describe("Shape", () => {
         },
       ),
     );
+  
   });
 
-  describe("Mixed", () => {
+  describe("Distplus", () => {
     open Distributions.DistPlus;
     let discrete: DistTypes.xyShape = {
       xs: [|1., 4., 8.|],
@@ -362,4 +374,39 @@ 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 +140,9 @@ module Continuous = {
       let toDiscrete = _ => None;
       let toScaledContinuous = t => Some(t);
       let toScaledDiscrete = _ => None;
+
+      let getMean = (t: t) => XYShape.Analysis.integrateContinuousShape(t);
+      let getVariance = (t: t): float => XYShape.Analysis.getVarianceDangerously(t, getMean, XYShape.Analysis.getMeanOfSquaresContinuousShape);
     });
 };
 
@@ -144,11 +152,22 @@ 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;
@@ -195,7 +214,14 @@ 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);
+      };
     });
+
 };
 
 // TODO: I think this shouldn't assume continuous/discrete are normalized to 1.0, and thus should not need the discreteProbabilityMassFraction being separate.
@@ -366,6 +392,30 @@ module Mixed = {
           discreteProbabilityMassFraction,
         };
       };
+
+      let getMean = (t: t) : float => {
+        let discreteProbabilityMassFraction = t.discreteProbabilityMassFraction;
+        let mean = 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))
+        };
+        mean;
+      };
+
+      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 +520,18 @@ 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 +646,8 @@ 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

@@ -298,3 +298,56 @@ let logScorePoint = (sampleCount, t1, t2) =>
   |> E.O.fmap(T.accumulateYs((+.)))
   |> E.O.fmap(Pairs.last)
   |> E.O.fmap(Pairs.y);
+
+
+module Analysis = { 
+  let integrateContinuousShape = (
+    ~indefiniteIntegralStepwise = (p,h1) => (h1*.(p**2.0)/. 2.0), 
+    ~indefiniteIntegralLinear = (p, a, b) => (a *. (p ** 2.0) /.2.0) +. (b *. (p**3.0) /. 3.0),
+    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 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)};
+
+  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
+    );
+  }
+};

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));