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.
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@ -12,7 +12,17 @@ let makeTest = (~only=false, str, item1, item2) =>
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expect(item1) |> toEqual(item2)
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);
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let makeTestCloseEquality = (~only=false, str, item1, item2, ~digits) =>
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only
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? Only.test(str, () =>
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expect(item1) |> toBeSoCloseTo(item2, ~digits)
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)
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: test(str, () =>
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expect(item1) |> toBeSoCloseTo(item2, ~digits)
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);
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describe("Shape", () => {
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describe("Continuous", () => {
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open Distributions.Continuous;
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let continuous = make(`Linear, shape);
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@ -119,7 +129,7 @@ describe("Shape", () => {
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1.0,
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);
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});
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describe("Discrete", () => {
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open Distributions.Discrete;
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let shape: DistTypes.xyShape = {
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@ -185,6 +195,7 @@ describe("Shape", () => {
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0.9,
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);
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makeTest("integralEndY", T.Integral.sum(~cache=None, discrete), 1.0);
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});
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describe("Mixed", () => {
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@ -289,9 +300,10 @@ describe("Shape", () => {
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},
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),
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);
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});
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describe("Mixed", () => {
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describe("Distplus", () => {
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open Distributions.DistPlus;
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let discrete: DistTypes.xyShape = {
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xs: [|1., 4., 8.|],
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@ -362,4 +374,39 @@ describe("Shape", () => {
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),
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);
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});
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});
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describe("Shape", () => {
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let mean = 10.0;
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let stdev = 4.0;
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let variance = stdev ** 2.0;
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let numSamples = 10000;
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open Distributions.Shape;
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let normal: SymbolicDist.dist = `Normal({ mean, stdev});
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let normalShape = SymbolicDist.GenericSimple.toShape(normal, numSamples);
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let lognormal = SymbolicDist.Lognormal.fromMeanAndStdev(mean, stdev);
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let lognormalShape = SymbolicDist.GenericSimple.toShape(lognormal, numSamples);
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makeTestCloseEquality(
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"Mean of a normal",
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T.getMean(normalShape),
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mean,
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~digits=2);
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makeTestCloseEquality(
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"Variance of a normal",
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T.getVariance(normalShape),
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variance,
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~digits=1);
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makeTestCloseEquality(
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"Mean of a lognormal",
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T.getMean(lognormalShape),
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mean,
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~digits=2);
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makeTestCloseEquality(
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"Variance of a lognormal",
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T.getVariance(lognormalShape),
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variance,
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~digits=0);
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});
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});
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@ -17,6 +17,9 @@ module type dist = {
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let integralEndY: (~cache: option(integral), t) => float;
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let integralXtoY: (~cache: option(integral), float, t) => float;
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let integralYtoX: (~cache: option(integral), float, t) => float;
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let getMean: t => float;
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let getVariance: t => float;
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};
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module Dist = (T: dist) => {
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@ -35,6 +38,8 @@ module Dist = (T: dist) => {
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let toDiscrete = T.toDiscrete;
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let toScaledContinuous = T.toScaledContinuous;
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let toScaledDiscrete = T.toScaledDiscrete;
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let getMean = T.getMean;
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let getVariance = T.getVariance;
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// TODO: Move this to each class, have use integral to produce integral in DistPlus class.
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let scaleBy = (~scale=1.0, t: t) => t |> mapY((r: float) => r *. scale);
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@ -99,7 +104,7 @@ module Continuous = {
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)
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|> DistTypes.MixedPoint.makeContinuous;
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};
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// let combineWithFn = (t1: t, t2: t, fn: (float, float) => float) => {
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// switch(t1, t2){
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// | ({interpolation: `Stepwise}, {interpolation: `Stepwise}) => 3.0
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@ -135,6 +140,9 @@ module Continuous = {
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let toDiscrete = _ => None;
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let toScaledContinuous = t => Some(t);
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let toScaledDiscrete = _ => None;
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let getMean = (t: t) => XYShape.Analysis.integrateContinuousShape(t);
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let getVariance = (t: t): float => XYShape.Analysis.getVarianceDangerously(t, getMean, XYShape.Analysis.getMeanOfSquaresContinuousShape);
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});
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};
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@ -144,11 +152,22 @@ module Discrete = {
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let sortedByX = (t: DistTypes.discreteShape) =>
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t |> XYShape.T.zip |> XYShape.Zipped.sortByX;
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let empty = XYShape.T.empty;
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let combine = (fn, t1: DistTypes.discreteShape, t2: DistTypes.discreteShape): DistTypes.discreteShape => {
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XYShape.Combine.combine(~xsSelection=ALL_XS, ~xToYSelection=XYShape.XtoY.stepwiseIfAtX, ~fn, t1, t2)
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}
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let _default0 = ((fn, a,b) => fn(E.O.default(0.0, a), E.O.default(0.0, b)));
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let reduce = (fn, items) => items |> E.A.fold_left(combine(_default0((fn))), empty);
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let combine =
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(fn, t1: DistTypes.discreteShape, t2: DistTypes.discreteShape)
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: DistTypes.discreteShape => {
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XYShape.Combine.combine(
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~xsSelection=ALL_XS,
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~xToYSelection=XYShape.XtoY.stepwiseIfAtX,
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~fn,
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t1,
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t2,
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);
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};
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let _default0 = (fn, a, b) =>
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fn(E.O.default(0.0, a), E.O.default(0.0, b));
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let reduce = (fn, items) =>
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items |> E.A.fold_left(combine(_default0(fn)), empty);
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module T =
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Dist({
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type t = DistTypes.discreteShape;
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@ -195,7 +214,14 @@ module Discrete = {
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|> integral(~cache)
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|> Continuous.getShape
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|> XYShape.YtoX.linear(f);
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let getMean = (t: t): float => E.A.reducei(t.xs, 0.0, (acc, x, i) => acc +. x*. t.ys[i]);
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let getVariance = (t: t): float => {
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let getMeanOfSquares = t => getMean(XYShape.Analysis.squareXYShape(t));
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XYShape.Analysis.getVarianceDangerously(t, getMean, getMeanOfSquares);
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};
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});
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};
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// TODO: I think this shouldn't assume continuous/discrete are normalized to 1.0, and thus should not need the discreteProbabilityMassFraction being separate.
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@ -366,6 +392,30 @@ module Mixed = {
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discreteProbabilityMassFraction,
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};
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};
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let getMean = (t: t) : float => {
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let discreteProbabilityMassFraction = t.discreteProbabilityMassFraction;
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let mean = switch(discreteProbabilityMassFraction){
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| 1.0 => Discrete.T.getMean(t.discrete);
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| 0.0 => Continuous.T.getMean(t.continuous);
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| _ => (Discrete.T.getMean(t.discrete) *. discreteProbabilityMassFraction)
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+. (Continuous.T.getMean(t.continuous) *. (1.0 -. discreteProbabilityMassFraction))
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};
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mean;
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};
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let getVariance = (t: t) : float => {
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let discreteProbabilityMassFraction = t.discreteProbabilityMassFraction;
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let getMeanOfSquares = (t: t) => {
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Discrete.T.getMean(XYShape.Analysis.squareXYShape(t.discrete))*.t.discreteProbabilityMassFraction
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+. XYShape.Analysis.getMeanOfSquaresContinuousShape(t.continuous)*.(1.0 -. t.discreteProbabilityMassFraction)
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};
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switch(discreteProbabilityMassFraction){
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| 1.0 => Discrete.T.getVariance(t.discrete);
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| 0.0 => Continuous.T.getVariance(t.continuous);
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| _ => XYShape.Analysis.getVarianceDangerously(t, getMean, getMeanOfSquares);
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};
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};
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});
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};
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Discrete.T.mapY(fn),
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Continuous.T.mapY(fn),
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));
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let getMean = (t: t): float => switch (t) {
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| Mixed(m) => Mixed.T.getMean(m);
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| Discrete(m) => Discrete.T.getMean(m);
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| Continuous(m) => Continuous.T.getMean(m);
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};
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let getVariance = (t: t): float => switch (t) {
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| Mixed(m) => Mixed.T.getVariance(m);
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| Discrete(m) => Discrete.T.getVariance(m);
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| Continuous(m) => Continuous.T.getVariance(m);
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};
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});
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};
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let integralYtoX = (~cache as _, f, t: t) => {
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Shape.T.Integral.yToX(~cache=Some(t.integralCache), f, toShape(t));
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};
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let getMean = (t: t) => Shape.T.getMean(t.shape);
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let getVariance = (t: t) => Shape.T.getVariance(t.shape);
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});
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};
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@ -297,4 +297,57 @@ let logScorePoint = (sampleCount, t1, t2) =>
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|> Range.integrateWithTriangles
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|> E.O.fmap(T.accumulateYs((+.)))
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|> E.O.fmap(Pairs.last)
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|> E.O.fmap(Pairs.y);
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|> E.O.fmap(Pairs.y);
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module Analysis = {
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let integrateContinuousShape = (
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~indefiniteIntegralStepwise = (p,h1) => (h1*.(p**2.0)/. 2.0),
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~indefiniteIntegralLinear = (p, a, b) => (a *. (p ** 2.0) /.2.0) +. (b *. (p**3.0) /. 3.0),
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t: DistTypes.continuousShape
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): float => {
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let xs = t.xyShape.xs;
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let ys = t.xyShape.ys;
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E.A.reducei(xs, 0.0, (acc, _x, i) => {
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let areaUnderIntegral = switch(t.interpolation, i){
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| (_, 0) => 0.0;
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| (`Stepwise, _) => indefiniteIntegralStepwise(xs[i],ys[i-1])
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-. indefiniteIntegralStepwise(xs[i-1],ys[i-1]);
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| (`Linear, _) => {
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let x1 = xs[i-1];
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let x2 = xs[i];
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let h1 = ys[i-1];
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let h2 = ys[i];
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let b = (h1 -. h2 ) /. (x1 -.x2)
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let a = h1 -. b *.x1;
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indefiniteIntegralLinear(x2, a, b) -. indefiniteIntegralLinear(x1, a, b);
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};
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};
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acc +. areaUnderIntegral;
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});
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};
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let getVarianceDangerously = (
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t: 't,
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getMean: ('t => float),
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getMeanOfSquares: ('t => float),
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): float => {
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let meanSquared = getMean(t)**2.0;
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let meanOfSquares = getMeanOfSquares(t);
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meanOfSquares -. meanSquared;
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};
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let squareXYShape = t: DistTypes.xyShape => {...t, xs: E.A.fmap(x => x**2.0, t.xs)};
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let getMeanOfSquaresContinuousShape = (t: DistTypes.continuousShape) => {
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let indefiniteIntegralLinear = (p, a, b) => (a *. (p ** 3.0) /.3.0) +. (b *. (p**4.0) /. 4.0);
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let indefiniteIntegralStepwise = (p,h1) => h1*.(p**3.0)/. 3.0;
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integrateContinuousShape(
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~indefiniteIntegralStepwise,
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~indefiniteIntegralLinear,
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t
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);
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}
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};
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@ -259,6 +259,9 @@ module A = {
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let fold_right = Array.fold_right;
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let concatMany = Belt.Array.concatMany;
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let keepMap = Belt.Array.keepMap;
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let init = Array.init;
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let reduce = Belt.Array.reduce;
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let reducei = Belt.Array.reduceWithIndex;
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let min = a =>
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get(a, 0)
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|> O.fmap(first => Belt.Array.reduce(a, first, (i, j) => i < j ? i : j));
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