@sgratzl/science/src/stats/loess.js

Scientific and statistical computing in JavaScript.

import medianF from './median';

// Based on org.apache.commons.math.analysis.interpolation.LoessInterpolator
// from http://commons.apache.org/math/
export default function loess() {
  var bandwidth = .3,
      robustnessIters = 2,
      accuracy = 1e-12;

  function smooth(xval, yval, weights) {
    var n = xval.length,
        i;

    if (n !== yval.length) throw {error: "Mismatched array lengths"};
    if (n == 0) throw {error: "At least one point required."};

    if (arguments.length < 3) {
      weights = [];
      i = -1; while (++i < n) weights[i] = 1;
    }

    science_stats_loessFiniteReal(xval);
    science_stats_loessFiniteReal(yval);
    science_stats_loessFiniteReal(weights);
    science_stats_loessStrictlyIncreasing(xval);

    if (n == 1) return [yval[0]];
    if (n == 2) return [yval[0], yval[1]];

    var bandwidthInPoints = Math.floor(bandwidth * n);

    if (bandwidthInPoints < 2) throw {error: "Bandwidth too small."};

    var res = [],
        residuals = [],
        robustnessWeights = [];

    // Do an initial fit and 'robustnessIters' robustness iterations.
    // This is equivalent to doing 'robustnessIters+1' robustness iterations
    // starting with all robustness weights set to 1.
    i = -1; while (++i < n) {
      res[i] = 0;
      residuals[i] = 0;
      robustnessWeights[i] = 1;
    }

    var iter = -1;
    while (++iter <= robustnessIters) {
      var bandwidthInterval = [0, bandwidthInPoints - 1];
      // At each x, compute a local weighted linear regression
      var x;
      i = -1; while (++i < n) {
        x = xval[i];

        // Find out the interval of source points on which
        // a regression is to be made.
        if (i > 0) {
          science_stats_loessUpdateBandwidthInterval(xval, weights, i, bandwidthInterval);
        }

        var ileft = bandwidthInterval[0],
            iright = bandwidthInterval[1];

        // Compute the point of the bandwidth interval that is
        // farthest from x
        var edge = (xval[i] - xval[ileft]) > (xval[iright] - xval[i]) ? ileft : iright;

        // Compute a least-squares linear fit weighted by
        // the product of robustness weights and the tricube
        // weight function.
        // See http://en.wikipedia.org/wiki/Linear_regression
        // (section "Univariate linear case")
        // and http://en.wikipedia.org/wiki/Weighted_least_squares
        // (section "Weighted least squares")
        var sumWeights = 0,
            sumX = 0,
            sumXSquared = 0,
            sumY = 0,
            sumXY = 0,
            denom = Math.abs(1 / (xval[edge] - x));

        for (var k = ileft; k <= iright; ++k) {
          var xk   = xval[k],
              yk   = yval[k],
              dist = k < i ? x - xk : xk - x,
              w    = science_stats_loessTricube(dist * denom) * robustnessWeights[k] * weights[k],
              xkw  = xk * w;
          sumWeights += w;
          sumX += xkw;
          sumXSquared += xk * xkw;
          sumY += yk * w;
          sumXY += yk * xkw;
        }

        var meanX = sumX / sumWeights,
            meanY = sumY / sumWeights,
            meanXY = sumXY / sumWeights,
            meanXSquared = sumXSquared / sumWeights;

        var beta = (Math.sqrt(Math.abs(meanXSquared - meanX * meanX)) < accuracy)
            ? 0 : ((meanXY - meanX * meanY) / (meanXSquared - meanX * meanX));

        var alpha = meanY - beta * meanX;

        res[i] = beta * x + alpha;
        residuals[i] = Math.abs(yval[i] - res[i]);
      }

      // No need to recompute the robustness weights at the last
      // iteration, they won't be needed anymore
      if (iter === robustnessIters) {
        break;
      }

      // Recompute the robustness weights.

      // Find the median residual.
      var medianResidual = medianF(residuals);

      if (Math.abs(medianResidual) < accuracy)
        break;

      var arg,
          w;
      i = -1; while (++i < n) {
        arg = residuals[i] / (6 * medianResidual);
        robustnessWeights[i] = (arg >= 1) ? 0 : ((w = 1 - arg * arg) * w);
      }
    }

    return res;
  }

  smooth.bandwidth = function(x) {
    if (!arguments.length) return x;
    bandwidth = x;
    return smooth;
  };

  smooth.robustnessIterations = function(x) {
    if (!arguments.length) return x;
    robustnessIters = x;
    return smooth;
  };

  smooth.accuracy = function(x) {
    if (!arguments.length) return x;
    accuracy = x;
    return smooth;
  };

  return smooth;
};

function science_stats_loessFiniteReal(values) {
  var n = values.length,
      i = -1;

  while (++i < n) if (!isFinite(values[i])) return false;

  return true;
}

function science_stats_loessStrictlyIncreasing(xval) {
  var n = xval.length,
      i = 0;

  while (++i < n) if (xval[i - 1] >= xval[i]) return false;

  return true;
}

// Compute the tricube weight function.
// http://en.wikipedia.org/wiki/Local_regression#Weight_function
function science_stats_loessTricube(x) {
  return (x = 1 - x * x * x) * x * x;
}

// Given an index interval into xval that embraces a certain number of
// points closest to xval[i-1], update the interval so that it embraces
// the same number of points closest to xval[i], ignoring zero weights.
function science_stats_loessUpdateBandwidthInterval(
  xval, weights, i, bandwidthInterval) {

  var left = bandwidthInterval[0],
      right = bandwidthInterval[1];

  // The right edge should be adjusted if the next point to the right
  // is closer to xval[i] than the leftmost point of the current interval
  var nextRight = science_stats_loessNextNonzero(weights, right);
  if ((nextRight < xval.length) && (xval[nextRight] - xval[i]) < (xval[i] - xval[left])) {
    var nextLeft = science_stats_loessNextNonzero(weights, left);
    bandwidthInterval[0] = nextLeft;
    bandwidthInterval[1] = nextRight;
  }
}

function science_stats_loessNextNonzero(weights, i) {
  var j = i + 1;
  while (j < weights.length && weights[j] === 0) j++;
  return j;
}