Fix the two piecewise-linear regression calculation

#66

The current implementation of the regression calculation has these flaws:

When processing (x[0], y[0]), L1 must be any line through (x[0], y[0]) which meets
L2 at a point (x’, y’) where x[0] < x' < x[1]. L1 has no error.

When processing (x[n - 2], y[n - 2]), L2 must be any line through (x[n - 1], y[n - 1])
which meets L1 at a point (x’, y’) where x[n - 2] < x' < x[n - 1]. L2 has no error.

The lambda calculation is incorrect. It includes a term called H which is equal
to C - I. Looking at the algorithm of Kundu/Ubhaya, this should be just C.

lambda should to be used with calculating L1 and (1 - lambda) should to be used
with calculating L2. Currently (1 - lambda) is used in calculating L1 and L2.

The current calculation has this condition if (t1 != t2) continue; This condition
is almost always true even if t1 and t2 are essentiallyEqual.

Author: shallawa

MotionMark/resources/statistics.js

@@ -244,19 +244,19 @@ Regression = Utilities.createClass(
     },
 
     _areEssentiallyEqual: function(n1, n2) {
-        const epsilon = 0.001;
+        const epsilon = 0.0001;
         return Math.abs(n1 - n2) < epsilon;
     },
 
-    _setOptimal: function(segment1, segment2, xp, x, options) {
+    _setOptimal: function(segment1, segment2, x, xn, options) {
         if (segment1.e + segment2.e > this.segment1.e + this.segment2.e)
             return false;
 
         let complexity = this._complexity();
         if (!this._areEssentiallyEqual(this.segment1.t, this.segment2.t)) {
             // If segment1 and segment2 are not parallel, then they have to meet
             // at complexity such that xp < complexity < x.
-            if (!(complexity >= xp && complexity <= x))
+            if (!(complexity >= x && complexity <= xn))
                 return false;
         } else {
             // If segment1 and segment2 are parallel, then they have to form one
@@ -395,7 +395,7 @@ Regression = Utilities.createClass(
                 };
             }
 
-            if (this._setOptimal(segment1, segment2, xp, x, options))
+            if (this._setOptimal(segment1, segment2, x, sortedSamples[j + 1][complexityIndex], options))
                 continue
 
             // These values remove the influence of this sample
@@ -426,7 +426,7 @@ Regression = Utilities.createClass(
                 e: (k2 + a2 * s2 * s2 + c2 * t2 * t2 - 2 * d2 * s2 - 2 * h2 * t2 + 2 * b2 * s2 * t2) + lambda1 * Math.pow(y - (s2 + t2 * x), 2)
             };
 
-            this._setOptimal(segment1, segment2, xp, x, options);
+            this._setOptimal(segment1, segment2, x, sortedSamples[j + 1][complexityIndex], options);
         }
     }
 });