vega-statistics/build/vega-statistics.js

Statistical routines and probability distributions.

import { ascending, quantileSorted, deviation, quantile, median } from 'd3-array';

function* numbers (values, valueof) {
  if (valueof == null) {
    for (let value of values) {
      if (value != null && value !== '' && (value = +value) >= value) {
        yield value;
      }
    }
  } else {
    let index = -1;
    for (let value of values) {
      value = valueof(value, ++index, values);
      if (value != null && value !== '' && (value = +value) >= value) {
        yield value;
      }
    }
  }
}

function quantiles (array, p, f) {
  const values = Float64Array.from(numbers(array, f));

  // don't depend on return value from typed array sort call
  // protects against undefined sort results in Safari (vega/vega-lite#4964)
  values.sort(ascending);
  return p.map(_ => quantileSorted(values, _));
}

function quartiles (array, f) {
  return quantiles(array, [0.25, 0.50, 0.75], f);
}

// Scott, D. W. (1992) Multivariate Density Estimation:
// Theory, Practice, and Visualization. Wiley.
function estimateBandwidth (array, f) {
  const n = array.length,
    d = deviation(array, f),
    q = quartiles(array, f),
    h = (q[2] - q[0]) / 1.34,
    v = Math.min(d, h) || d || Math.abs(q[0]) || 1;
  return 1.06 * v * Math.pow(n, -0.2);
}

function bin (_) {
  // determine range
  const maxb = _.maxbins || 20,
    base = _.base || 10,
    logb = Math.log(base),
    div = _.divide || [5, 2];
  let min = _.extent[0],
    max = _.extent[1],
    step,
    level,
    minstep,
    v,
    i,
    n;
  const span = _.span || max - min || Math.abs(min) || 1;
  if (_.step) {
    // if step size is explicitly given, use that
    step = _.step;
  } else if (_.steps) {
    // if provided, limit choice to acceptable step sizes
    v = span / maxb;
    for (i = 0, n = _.steps.length; i < n && _.steps[i] < v; ++i);
    step = _.steps[Math.max(0, i - 1)];
  } else {
    // else use span to determine step size
    level = Math.ceil(Math.log(maxb) / logb);
    minstep = _.minstep || 0;
    step = Math.max(minstep, Math.pow(base, Math.round(Math.log(span) / logb) - level));

    // increase step size if too many bins
    while (Math.ceil(span / step) > maxb) {
      step *= base;
    }

    // decrease step size if allowed
    for (i = 0, n = div.length; i < n; ++i) {
      v = step / div[i];
      if (v >= minstep && span / v <= maxb) step = v;
    }
  }

  // update precision, min and max
  v = Math.log(step);
  const precision = v >= 0 ? 0 : ~~(-v / logb) + 1,
    eps = Math.pow(base, -precision - 1);
  if (_.nice || _.nice === undefined) {
    v = Math.floor(min / step + eps) * step;
    min = min < v ? v - step : v;
    max = Math.ceil(max / step) * step;
  }
  return {
    start: min,
    stop: max === min ? min + step : max,
    step: step
  };
}

var random = Math.random;
function setRandom(r) {
  random = r;
}

function bootstrapCI (array, samples, alpha, f) {
  if (!array.length) return [undefined, undefined];
  const values = Float64Array.from(numbers(array, f)),
    n = values.length,
    m = samples;
  let a, i, j, mu;
  for (j = 0, mu = Array(m); j < m; ++j) {
    for (a = 0, i = 0; i < n; ++i) {
      a += values[~~(random() * n)];
    }
    mu[j] = a / n;
  }
  mu.sort(ascending);
  return [quantile(mu, alpha / 2), quantile(mu, 1 - alpha / 2)];
}

// Dot density binning for dot plot construction.
// Based on Leland Wilkinson, Dot Plots, The American Statistician, 1999.
// https://www.cs.uic.edu/~wilkinson/Publications/dotplots.pdf
function dotbin (array, step, smooth, f) {
  f = f || (_ => _);
  const n = array.length,
    v = new Float64Array(n);
  let i = 0,
    j = 1,
    a = f(array[0]),
    b = a,
    w = a + step,
    x;
  for (; j < n; ++j) {
    x = f(array[j]);
    if (x >= w) {
      b = (a + b) / 2;
      for (; i < j; ++i) v[i] = b;
      w = x + step;
      a = x;
    }
    b = x;
  }
  b = (a + b) / 2;
  for (; i < j; ++i) v[i] = b;
  return smooth ? smoothing(v, step + step / 4) : v;
}

// perform smoothing to reduce variance
// swap points between "adjacent" stacks
// Wilkinson defines adjacent as within step/4 units
function smoothing(v, thresh) {
  const n = v.length;
  let a = 0,
    b = 1,
    c,
    d;

  // get left stack
  while (v[a] === v[b]) ++b;
  while (b < n) {
    // get right stack
    c = b + 1;
    while (v[b] === v[c]) ++c;

    // are stacks adjacent?
    // if so, compare sizes and swap as needed
    if (v[b] - v[b - 1] < thresh) {
      d = b + (a + c - b - b >> 1);
      while (d < b) v[d++] = v[b];
      while (d > b) v[d--] = v[a];
    }

    // update left stack indices
    a = b;
    b = c;
  }
  return v;
}

function lcg (seed) {
  // Random numbers using a Linear Congruential Generator with seed value
  // Uses glibc values from https://en.wikipedia.org/wiki/Linear_congruential_generator
  return function () {
    seed = (1103515245 * seed + 12345) % 2147483647;
    return seed / 2147483647;
  };
}

function integer (min, max) {
  if (max == null) {
    max = min;
    min = 0;
  }
  let a, b, d;
  const dist = {
    min(_) {
      if (arguments.length) {
        a = _ || 0;
        d = b - a;
        return dist;
      } else {
        return a;
      }
    },
    max(_) {
      if (arguments.length) {
        b = _ || 0;
        d = b - a;
        return dist;
      } else {
        return b;
      }
    },
    sample() {
      return a + Math.floor(d * random());
    },
    pdf(x) {
      return x === Math.floor(x) && x >= a && x < b ? 1 / d : 0;
    },
    cdf(x) {
      const v = Math.floor(x);
      return v < a ? 0 : v >= b ? 1 : (v - a + 1) / d;
    },
    icdf(p) {
      return p >= 0 && p <= 1 ? a - 1 + Math.floor(p * d) : NaN;
    }
  };
  return dist.min(min).max(max);
}

const SQRT2PI = Math.sqrt(2 * Math.PI);
const SQRT2 = Math.SQRT2;

let nextSample = NaN;
function sampleNormal(mean, stdev) {
  mean = mean || 0;
  stdev = stdev == null ? 1 : stdev;
  let x = 0,
    y = 0,
    rds,
    c;
  if (nextSample === nextSample) {
    x = nextSample;
    nextSample = NaN;
  } else {
    do {
      x = random() * 2 - 1;
      y = random() * 2 - 1;
      rds = x * x + y * y;
    } while (rds === 0 || rds > 1);
    c = Math.sqrt(-2 * Math.log(rds) / rds); // Box-Muller transform
    x *= c;
    nextSample = y * c;
  }
  return mean + x * stdev;
}
function densityNormal(value, mean, stdev) {
  stdev = stdev == null ? 1 : stdev;
  const z = (value - (mean || 0)) / stdev;
  return Math.exp(-0.5 * z * z) / (stdev * SQRT2PI);
}

// Approximation from West (2009)
// Better Approximations to Cumulative Normal Functions
function cumulativeNormal(value, mean, stdev) {
  mean = mean || 0;
  stdev = stdev == null ? 1 : stdev;
  const z = (value - mean) / stdev,
    Z = Math.abs(z);
  let cd;
  if (Z > 37) {
    cd = 0;
  } else {
    const exp = Math.exp(-Z * Z / 2);
    let sum;
    if (Z < 7.07106781186547) {
      sum = 3.52624965998911e-02 * Z + 0.700383064443688;
      sum = sum * Z + 6.37396220353165;
      sum = sum * Z + 33.912866078383;
      sum = sum * Z + 112.079291497871;
      sum = sum * Z + 221.213596169931;
      sum = sum * Z + 220.206867912376;
      cd = exp * sum;
      sum = 8.83883476483184e-02 * Z + 1.75566716318264;
      sum = sum * Z + 16.064177579207;
      sum = sum * Z + 86.7807322029461;
      sum = sum * Z + 296.564248779674;
      sum = sum * Z + 637.333633378831;
      sum = sum * Z + 793.826512519948;
      sum = sum * Z + 440.413735824752;
      cd = cd / sum;
    } else {
      sum = Z + 0.65;
      sum = Z + 4 / sum;
      sum = Z + 3 / sum;
      sum = Z + 2 / sum;
      sum = Z + 1 / sum;
      cd = exp / sum / 2.506628274631;
    }
  }
  return z > 0 ? 1 - cd : cd;
}

// Approximation of Probit function using inverse error function.
function quantileNormal(p, mean, stdev) {
  if (p < 0 || p > 1) return NaN;
  return (mean || 0) + (stdev == null ? 1 : stdev) * SQRT2 * erfinv(2 * p - 1);
}

// Approximate inverse error function. Implementation from "Approximating
// the erfinv function" by Mike Giles, GPU Computing Gems, volume 2, 2010.
// Ported from Apache Commons Math, http://www.apache.org/licenses/LICENSE-2.0
function erfinv(x) {
  // beware that the logarithm argument must be
  // commputed as (1.0 - x) * (1.0 + x),
  // it must NOT be simplified as 1.0 - x * x as this
  // would induce rounding errors near the boundaries +/-1
  let w = -Math.log((1 - x) * (1 + x)),
    p;
  if (w < 6.25) {
    w -= 3.125;
    p = -364441206401782e-35;
    p = -16850591381820166e-35 + p * w;
    p = 1.2858480715256400167e-18 + p * w;
    p = 1.115787767802518096e-17 + p * w;
    p = -1333171662854621e-31 + p * w;
    p = 2.0972767875968561637e-17 + p * w;
    p = 6.6376381343583238325e-15 + p * w;
    p = -4054566272975207e-29 + p * w;
    p = -8151934197605472e-29 + p * w;
    p = 2.6335093153082322977e-12 + p * w;
    p = -12975133253453532e-27 + p * w;
    p = -5415412054294628e-26 + p * w;
    p = 1.051212273321532285e-09 + p * w;
    p = -4.112633980346984e-9 + p * w;
    p = -2.9070369957882005e-8 + p * w;
    p = 4.2347877827932403518e-07 + p * w;
    p = -13654692000834679e-22 + p * w;
    p = -13882523362786469e-21 + p * w;
    p = 0.0001867342080340571352 + p * w;
    p = -740702534166267e-18 + p * w;
    p = -0.006033670871430149 + p * w;
    p = 0.24015818242558961693 + p * w;
    p = 1.6536545626831027356 + p * w;
  } else if (w < 16.0) {
    w = Math.sqrt(w) - 3.25;
    p = 2.2137376921775787049e-09;
    p = 9.0756561938885390979e-08 + p * w;
    p = -2.7517406297064545e-7 + p * w;
    p = 1.8239629214389227755e-08 + p * w;
    p = 1.5027403968909827627e-06 + p * w;
    p = -4013867526981546e-21 + p * w;
    p = 2.9234449089955446044e-06 + p * w;
    p = 1.2475304481671778723e-05 + p * w;
    p = -47318229009055734e-21 + p * w;
    p = 6.8284851459573175448e-05 + p * w;
    p = 2.4031110387097893999e-05 + p * w;
    p = -3550375203628475e-19 + p * w;
    p = 0.00095328937973738049703 + p * w;
    p = -0.0016882755560235047 + p * w;
    p = 0.0024914420961078508066 + p * w;
    p = -0.003751208507569241 + p * w;
    p = 0.005370914553590063617 + p * w;
    p = 1.0052589676941592334 + p * w;
    p = 3.0838856104922207635 + p * w;
  } else if (Number.isFinite(w)) {
    w = Math.sqrt(w) - 5.0;
    p = -27109920616438573e-27;
    p = -2555641816996525e-25 + p * w;
    p = 1.5076572693500548083e-09 + p * w;
    p = -3.789465440126737e-9 + p * w;
    p = 7.6157012080783393804e-09 + p * w;
    p = -1.496002662714924e-8 + p * w;
    p = 2.9147953450901080826e-08 + p * w;
    p = -6.771199775845234e-8 + p * w;
    p = 2.2900482228026654717e-07 + p * w;
    p = -9.9298272942317e-7 + p * w;
    p = 4.5260625972231537039e-06 + p * w;
    p = -1968177810553167e-20 + p * w;
    p = 7.5995277030017761139e-05 + p * w;
    p = -21503011930044477e-20 + p * w;
    p = -13871931833623122e-20 + p * w;
    p = 1.0103004648645343977 + p * w;
    p = 4.8499064014085844221 + p * w;
  } else {
    p = Infinity;
  }
  return p * x;
}
function gaussian (mean, stdev) {
  let mu, sigma;
  const dist = {
    mean(_) {
      if (arguments.length) {
        mu = _ || 0;
        return dist;
      } else {
        return mu;
      }
    },
    stdev(_) {
      if (arguments.length) {
        sigma = _ == null ? 1 : _;
        return dist;
      } else {
        return sigma;
      }
    },
    sample: () => sampleNormal(mu, sigma),
    pdf: value => densityNormal(value, mu, sigma),
    cdf: value => cumulativeNormal(value, mu, sigma),
    icdf: p => quantileNormal(p, mu, sigma)
  };
  return dist.mean(mean).stdev(stdev);
}

function kde (support, bandwidth) {
  const kernel = gaussian();
  let n = 0;
  const dist = {
    data(_) {
      if (arguments.length) {
        support = _;
        n = _ ? _.length : 0;
        return dist.bandwidth(bandwidth);
      } else {
        return support;
      }
    },
    bandwidth(_) {
      if (!arguments.length) return bandwidth;
      bandwidth = _;
      if (!bandwidth && support) bandwidth = estimateBandwidth(support);
      return dist;
    },
    sample() {
      return support[~~(random() * n)] + bandwidth * kernel.sample();
    },
    pdf(x) {
      let y = 0,
        i = 0;
      for (; i < n; ++i) {
        y += kernel.pdf((x - support[i]) / bandwidth);
      }
      return y / bandwidth / n;
    },
    cdf(x) {
      let y = 0,
        i = 0;
      for (; i < n; ++i) {
        y += kernel.cdf((x - support[i]) / bandwidth);
      }
      return y / n;
    },
    icdf() {
      throw Error('KDE icdf not supported.');
    }
  };
  return dist.data(support);
}

function sampleLogNormal(mean, stdev) {
  mean = mean || 0;
  stdev = stdev == null ? 1 : stdev;
  return Math.exp(mean + sampleNormal() * stdev);
}
function densityLogNormal(value, mean, stdev) {
  if (value <= 0) return 0;
  mean = mean || 0;
  stdev = stdev == null ? 1 : stdev;
  const z = (Math.log(value) - mean) / stdev;
  return Math.exp(-0.5 * z * z) / (stdev * SQRT2PI * value);
}
function cumulativeLogNormal(value, mean, stdev) {
  return cumulativeNormal(Math.log(value), mean, stdev);
}
function quantileLogNormal(p, mean, stdev) {
  return Math.exp(quantileNormal(p, mean, stdev));
}
function lognormal (mean, stdev) {
  let mu, sigma;
  const dist = {
    mean(_) {
      if (arguments.length) {
        mu = _ || 0;
        return dist;
      } else {
        return mu;
      }
    },
    stdev(_) {
      if (arguments.length) {
        sigma = _ == null ? 1 : _;
        return dist;
      } else {
        return sigma;
      }
    },
    sample: () => sampleLogNormal(mu, sigma),
    pdf: value => densityLogNormal(value, mu, sigma),
    cdf: value => cumulativeLogNormal(value, mu, sigma),
    icdf: p => quantileLogNormal(p, mu, sigma)
  };
  return dist.mean(mean).stdev(stdev);
}

function mixture (dists, weights) {
  let m = 0,
    w;
  function normalize(x) {
    const w = [];
    let sum = 0,
      i;
    for (i = 0; i < m; ++i) {
      sum += w[i] = x[i] == null ? 1 : +x[i];
    }
    for (i = 0; i < m; ++i) {
      w[i] /= sum;
    }
    return w;
  }
  const dist = {
    weights(_) {
      if (arguments.length) {
        w = normalize(weights = _ || []);
        return dist;
      }
      return weights;
    },
    distributions(_) {
      if (arguments.length) {
        if (_) {
          m = _.length;
          dists = _;
        } else {
          m = 0;
          dists = [];
        }
        return dist.weights(weights);
      }
      return dists;
    },
    sample() {
      const r = random();
      let d = dists[m - 1],
        v = w[0],
        i = 0;

      // first select distribution
      for (; i < m - 1; v += w[++i]) {
        if (r < v) {
          d = dists[i];
          break;
        }
      }
      // then sample from it
      return d.sample();
    },
    pdf(x) {
      let p = 0,
        i = 0;
      for (; i < m; ++i) {
        p += w[i] * dists[i].pdf(x);
      }
      return p;
    },
    cdf(x) {
      let p = 0,
        i = 0;
      for (; i < m; ++i) {
        p += w[i] * dists[i].cdf(x);
      }
      return p;
    },
    icdf() {
      throw Error('Mixture icdf not supported.');
    }
  };
  return dist.distributions(dists).weights(weights);
}

function sampleUniform(min, max) {
  if (max == null) {
    max = min == null ? 1 : min;
    min = 0;
  }
  return min + (max - min) * random();
}
function densityUniform(value, min, max) {
  if (max == null) {
    max = min == null ? 1 : min;
    min = 0;
  }
  return value >= min && value <= max ? 1 / (max - min) : 0;
}
function cumulativeUniform(value, min, max) {
  if (max == null) {
    max = min == null ? 1 : min;
    min = 0;
  }
  return value < min ? 0 : value > max ? 1 : (value - min) / (max - min);
}
function quantileUniform(p, min, max) {
  if (max == null) {
    max = min == null ? 1 : min;
    min = 0;
  }
  return p >= 0 && p <= 1 ? min + p * (max - min) : NaN;
}
function uniform (min, max) {
  let a, b;
  const dist = {
    min(_) {
      if (arguments.length) {
        a = _ || 0;
        return dist;
      } else {
        return a;
      }
    },
    max(_) {
      if (arguments.length) {
        b = _ == null ? 1 : _;
        return dist;
      } else {
        return b;
      }
    },
    sample: () => sampleUniform(a, b),
    pdf: value => densityUniform(value, a, b),
    cdf: value => cumulativeUniform(value, a, b),
    icdf: p => quantileUniform(p, a, b)
  };
  if (max == null) {
    max = min == null ? 1 : min;
    min = 0;
  }
  return dist.min(min).max(max);
}

function constant (data, x, y) {
  let mean = 0,
    n = 0;
  for (const d of data) {
    const val = y(d);
    if (x(d) == null || val == null || isNaN(val)) continue;
    mean += (val - mean) / ++n;
  }
  return {
    coef: [mean],
    predict: () => mean,
    rSquared: 0
  };
}

// Ordinary Least Squares
function ols (uX, uY, uXY, uX2) {
  const delta = uX2 - uX * uX,
    slope = Math.abs(delta) < 1e-24 ? 0 : (uXY - uX * uY) / delta,
    intercept = uY - slope * uX;
  return [intercept, slope];
}

function points(data, x, y, sort) {
  data = data.filter(d => {
    let u = x(d),
      v = y(d);
    return u != null && (u = +u) >= u && v != null && (v = +v) >= v;
  });
  if (sort) {
    data.sort((a, b) => x(a) - x(b));
  }
  const n = data.length,
    X = new Float64Array(n),
    Y = new Float64Array(n);

  // extract values, calculate means
  let i = 0,
    ux = 0,
    uy = 0,
    xv,
    yv,
    d;
  for (d of data) {
    X[i] = xv = +x(d);
    Y[i] = yv = +y(d);
    ++i;
    ux += (xv - ux) / i;
    uy += (yv - uy) / i;
  }

  // mean center the data
  for (i = 0; i < n; ++i) {
    X[i] -= ux;
    Y[i] -= uy;
  }
  return [X, Y, ux, uy];
}
function visitPoints(data, x, y, callback) {
  let i = -1,
    u,
    v;
  for (const d of data) {
    u = x(d);
    v = y(d);
    if (u != null && (u = +u) >= u && v != null && (v = +v) >= v) {
      callback(u, v, ++i);
    }
  }
}

// Adapted from d3-regression by Harry Stevens
// License: https://github.com/HarryStevens/d3-regression/blob/master/LICENSE
function rSquared (data, x, y, uY, predict) {
  let SSE = 0,
    SST = 0;
  visitPoints(data, x, y, (dx, dy) => {
    const sse = dy - predict(dx),
      sst = dy - uY;
    SSE += sse * sse;
    SST += sst * sst;
  });
  return 1 - SSE / SST;
}

// Adapted from d3-regression by Harry Stevens
// License: https://github.com/HarryStevens/d3-regression/blob/master/LICENSE
function linear (data, x, y) {
  let X = 0,
    Y = 0,
    XY = 0,
    X2 = 0,
    n = 0;
  visitPoints(data, x, y, (dx, dy) => {
    ++n;
    X += (dx - X) / n;
    Y += (dy - Y) / n;
    XY += (dx * dy - XY) / n;
    X2 += (dx * dx - X2) / n;
  });
  const coef = ols(X, Y, XY, X2),
    predict = x => coef[0] + coef[1] * x;
  return {
    coef: coef,
    predict: predict,
    rSquared: rSquared(data, x, y, Y, predict)
  };
}

// Adapted from d3-regression by Harry Stevens
// License: https://github.com/HarryStevens/d3-regression/blob/master/LICENSE
function log (data, x, y) {
  let X = 0,
    Y = 0,
    XY = 0,
    X2 = 0,
    n = 0;
  visitPoints(data, x, y, (dx, dy) => {
    ++n;
    dx = Math.log(dx);
    X += (dx - X) / n;
    Y += (dy - Y) / n;
    XY += (dx * dy - XY) / n;
    X2 += (dx * dx - X2) / n;
  });
  const coef = ols(X, Y, XY, X2),
    predict = x => coef[0] + coef[1] * Math.log(x);
  return {
    coef: coef,
    predict: predict,
    rSquared: rSquared(data, x, y, Y, predict)
  };
}

function exp (data, x, y) {
  const [xv, yv, ux, uy] = points(data, x, y);
  let YL = 0,
    XY = 0,
    XYL = 0,
    X2Y = 0,
    n = 0,
    dx,
    ly,
    xy;
  visitPoints(data, x, y, (_, dy) => {
    dx = xv[n++];
    ly = Math.log(dy);
    xy = dx * dy;
    YL += (dy * ly - YL) / n;
    XY += (xy - XY) / n;
    XYL += (xy * ly - XYL) / n;
    X2Y += (dx * xy - X2Y) / n;
  });
  const [c0, c1] = ols(XY / uy, YL / uy, XYL / uy, X2Y / uy),
    predict = x => Math.exp(c0 + c1 * (x - ux));
  return {
    coef: [Math.exp(c0 - c1 * ux), c1],
    predict: predict,
    rSquared: rSquared(data, x, y, uy, predict)
  };
}

// Adapted from d3-regression by Harry Stevens
// License: https://github.com/HarryStevens/d3-regression/blob/master/LICENSE
function pow (data, x, y) {
  let X = 0,
    Y = 0,
    XY = 0,
    X2 = 0,
    YS = 0,
    n = 0;
  visitPoints(data, x, y, (dx, dy) => {
    const lx = Math.log(dx),
      ly = Math.log(dy);
    ++n;
    X += (lx - X) / n;
    Y += (ly - Y) / n;
    XY += (lx * ly - XY) / n;
    X2 += (lx * lx - X2) / n;
    YS += (dy - YS) / n;
  });
  const coef = ols(X, Y, XY, X2),
    predict = x => coef[0] * Math.pow(x, coef[1]);
  coef[0] = Math.exp(coef[0]);
  return {
    coef: coef,
    predict: predict,
    rSquared: rSquared(data, x, y, YS, predict)
  };
}

function quad (data, x, y) {
  const [xv, yv, ux, uy] = points(data, x, y),
    n = xv.length;
  let X2 = 0,
    X3 = 0,
    X4 = 0,
    XY = 0,
    X2Y = 0,
    i,
    dx,
    dy,
    x2;
  for (i = 0; i < n;) {
    dx = xv[i];
    dy = yv[i++];
    x2 = dx * dx;
    X2 += (x2 - X2) / i;
    X3 += (x2 * dx - X3) / i;
    X4 += (x2 * x2 - X4) / i;
    XY += (dx * dy - XY) / i;
    X2Y += (x2 * dy - X2Y) / i;
  }
  const X2X2 = X4 - X2 * X2,
    d = X2 * X2X2 - X3 * X3,
    a = (X2Y * X2 - XY * X3) / d,
    b = (XY * X2X2 - X2Y * X3) / d,
    c = -a * X2,
    predict = x => {
      x = x - ux;
      return a * x * x + b * x + c + uy;
    };

  // transform coefficients back from mean-centered space
  return {
    coef: [c - b * ux + a * ux * ux + uy, b - 2 * a * ux, a],
    predict: predict,
    rSquared: rSquared(data, x, y, uy, predict)
  };
}

// Adapted from d3-regression by Harry Stevens
// License: https://github.com/HarryStevens/d3-regression/blob/master/LICENSE
// ... which was adapted from regression-js by Tom Alexander
// Source: https://github.com/Tom-Alexander/regression-js/blob/master/src/regression.js#L246
// License: https://github.com/Tom-Alexander/regression-js/blob/master/LICENSE
function poly (data, x, y, order) {
  // use more efficient methods for lower orders
  if (order === 0) return constant(data, x, y);
  if (order === 1) return linear(data, x, y);
  if (order === 2) return quad(data, x, y);
  const [xv, yv, ux, uy] = points(data, x, y),
    n = xv.length,
    lhs = [],
    rhs = [],
    k = order + 1;
  let i, j, l, v, c;
  for (i = 0; i < k; ++i) {
    for (l = 0, v = 0; l < n; ++l) {
      v += Math.pow(xv[l], i) * yv[l];
    }
    lhs.push(v);
    c = new Float64Array(k);
    for (j = 0; j < k; ++j) {
      for (l = 0, v = 0; l < n; ++l) {
        v += Math.pow(xv[l], i + j);
      }
      c[j] = v;
    }
    rhs.push(c);
  }
  rhs.push(lhs);
  const coef = gaussianElimination(rhs),
    predict = x => {
      x -= ux;
      let y = uy + coef[0] + coef[1] * x + coef[2] * x * x;
      for (i = 3; i < k; ++i) y += coef[i] * Math.pow(x, i);
      return y;
    };
  return {
    coef: uncenter(k, coef, -ux, uy),
    predict: predict,
    rSquared: rSquared(data, x, y, uy, predict)
  };
}
function uncenter(k, a, x, y) {
  const z = Array(k);
  let i, j, v, c;

  // initialize to zero
  for (i = 0; i < k; ++i) z[i] = 0;

  // polynomial expansion
  for (i = k - 1; i >= 0; --i) {
    v = a[i];
    c = 1;
    z[i] += v;
    for (j = 1; j <= i; ++j) {
      c *= (i + 1 - j) / j; // binomial coefficent
      z[i - j] += v * Math.pow(x, j) * c;
    }
  }

  // bias term
  z[0] += y;
  return z;
}

// Given an array for a two-dimensional matrix and the polynomial order,
// solve A * x = b using Gaussian elimination.
function gaussianElimination(matrix) {
  const n = matrix.length - 1,
    coef = [];
  let i, j, k, r, t;
  for (i = 0; i < n; ++i) {
    r = i; // max row
    for (j = i + 1; j < n; ++j) {
      if (Math.abs(matrix[i][j]) > Math.abs(matrix[i][r])) {
        r = j;
      }
    }
    for (k = i; k < n + 1; ++k) {
      t = matrix[k][i];
      matrix[k][i] = matrix[k][r];
      matrix[k][r] = t;
    }
    for (j = i + 1; j < n; ++j) {
      for (k = n; k >= i; k--) {
        matrix[k][j] -= matrix[k][i] * matrix[i][j] / matrix[i][i];
      }
    }
  }
  for (j = n - 1; j >= 0; --j) {
    t = 0;
    for (k = j + 1; k < n; ++k) {
      t += matrix[k][j] * coef[k];
    }
    coef[j] = (matrix[n][j] - t) / matrix[j][j];
  }
  return coef;
}

const maxiters = 2,
  epsilon = 1e-12;

// Adapted from science.js by Jason Davies
// Source: https://github.com/jasondavies/science.js/blob/master/src/stats/loess.js
// License: https://github.com/jasondavies/science.js/blob/master/LICENSE
function loess (data, x, y, bandwidth) {
  const [xv, yv, ux, uy] = points(data, x, y, true),
    n = xv.length,
    bw = Math.max(2, ~~(bandwidth * n)),
    // # nearest neighbors
    yhat = new Float64Array(n),
    residuals = new Float64Array(n),
    robustWeights = new Float64Array(n).fill(1);
  for (let iter = -1; ++iter <= maxiters;) {
    const interval = [0, bw - 1];
    for (let i = 0; i < n; ++i) {
      const dx = xv[i],
        i0 = interval[0],
        i1 = interval[1],
        edge = dx - xv[i0] > xv[i1] - dx ? i0 : i1;
      let W = 0,
        X = 0,
        Y = 0,
        XY = 0,
        X2 = 0;
      const denom = 1 / Math.abs(xv[edge] - dx || 1); // avoid singularity!

      for (let k = i0; k <= i1; ++k) {
        const xk = xv[k],
          yk = yv[k],
          w = tricube(Math.abs(dx - xk) * denom) * robustWeights[k],
          xkw = xk * w;
        W += w;
        X += xkw;
        Y += yk * w;
        XY += yk * xkw;
        X2 += xk * xkw;
      }

      // linear regression fit
      const [a, b] = ols(X / W, Y / W, XY / W, X2 / W);
      yhat[i] = a + b * dx;
      residuals[i] = Math.abs(yv[i] - yhat[i]);
      updateInterval(xv, i + 1, interval);
    }
    if (iter === maxiters) {
      break;
    }
    const medianResidual = median(residuals);
    if (Math.abs(medianResidual) < epsilon) break;
    for (let i = 0, arg, w; i < n; ++i) {
      arg = residuals[i] / (6 * medianResidual);
      // default to epsilon (rather than zero) for large deviations
      // keeping weights tiny but non-zero prevents singularites
      robustWeights[i] = arg >= 1 ? epsilon : (w = 1 - arg * arg) * w;
    }
  }
  return output(xv, yhat, ux, uy);
}

// weighting kernel for local regression
function tricube(x) {
  return (x = 1 - x * x * x) * x * x;
}

// advance sliding window interval of nearest neighbors
function updateInterval(xv, i, interval) {
  const val = xv[i];
  let left = interval[0],
    right = interval[1] + 1;
  if (right >= xv.length) return;

  // step right if distance to new right edge is <= distance to old left edge
  // step when distance is equal to ensure movement over duplicate x values
  while (i > left && xv[right] - val <= val - xv[left]) {
    interval[0] = ++left;
    interval[1] = right;
    ++right;
  }
}

// generate smoothed output points
// average points with repeated x values
function output(xv, yhat, ux, uy) {
  const n = xv.length,
    out = [];
  let i = 0,
    cnt = 0,
    prev = [],
    v;
  for (; i < n; ++i) {
    v = xv[i] + ux;
    if (prev[0] === v) {
      // average output values via online update
      prev[1] += (yhat[i] - prev[1]) / ++cnt;
    } else {
      // add new output point
      cnt = 0;
      prev[1] += uy;
      prev = [v, yhat[i]];
      out.push(prev);
    }
  }
  prev[1] += uy;
  return out;
}

// subdivide up to accuracy of 0.5 degrees
const MIN_RADIANS = 0.5 * Math.PI / 180;

// Adaptively sample an interpolated function over a domain extent
function sampleCurve (f, extent, minSteps, maxSteps) {
  minSteps = minSteps || 25;
  maxSteps = Math.max(minSteps, maxSteps || 200);
  const point = x => [x, f(x)],
    minX = extent[0],
    maxX = extent[1],
    span = maxX - minX,
    stop = span / maxSteps,
    prev = [point(minX)],
    next = [];
  if (minSteps === maxSteps) {
    // no adaptation, sample uniform grid directly and return
    for (let i = 1; i < maxSteps; ++i) {
      prev.push(point(minX + i / minSteps * span));
    }
    prev.push(point(maxX));
    return prev;
  } else {
    // sample minimum points on uniform grid
    // then move on to perform adaptive refinement
    next.push(point(maxX));
    for (let i = minSteps; --i > 0;) {
      next.push(point(minX + i / minSteps * span));
    }
  }
  let p0 = prev[0];
  let p1 = next[next.length - 1];
  const sx = 1 / span;
  const sy = scaleY(p0[1], next);
  while (p1) {
    // midpoint for potential curve subdivision
    const pm = point((p0[0] + p1[0]) / 2);
    const dx = pm[0] - p0[0] >= stop;
    if (dx && angleDelta(p0, pm, p1, sx, sy) > MIN_RADIANS) {
      // maximum resolution has not yet been met, and
      // subdivision midpoint is sufficiently different from endpoint
      // save subdivision, push midpoint onto the visitation stack
      next.push(pm);
    } else {
      // subdivision midpoint sufficiently similar to endpoint
      // skip subdivision, store endpoint, move to next point on the stack
      p0 = p1;
      prev.push(p1);
      next.pop();
    }
    p1 = next[next.length - 1];
  }
  return prev;
}
function scaleY(init, points) {
  let ymin = init;
  let ymax = init;
  const n = points.length;
  for (let i = 0; i < n; ++i) {
    const y = points[i][1];
    if (y < ymin) ymin = y;
    if (y > ymax) ymax = y;
  }
  return 1 / (ymax - ymin);
}
function angleDelta(p, q, r, sx, sy) {
  const a0 = Math.atan2(sy * (r[1] - p[1]), sx * (r[0] - p[0])),
    a1 = Math.atan2(sy * (q[1] - p[1]), sx * (q[0] - p[0]));
  return Math.abs(a0 - a1);
}

export { estimateBandwidth as bandwidthNRD, bin, bootstrapCI, cumulativeLogNormal, cumulativeNormal, cumulativeUniform, densityLogNormal, densityNormal, densityUniform, dotbin, quantileLogNormal, quantileNormal, quantileUniform, quantiles, quartiles, random, integer as randomInteger, kde as randomKDE, lcg as randomLCG, lognormal as randomLogNormal, mixture as randomMixture, gaussian as randomNormal, uniform as randomUniform, constant as regressionConstant, exp as regressionExp, linear as regressionLinear, loess as regressionLoess, log as regressionLog, poly as regressionPoly, pow as regressionPow, quad as regressionQuad, sampleCurve, sampleLogNormal, sampleNormal, sampleUniform, setRandom };
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