| Summary: | Neither of these kernel functions are used here, although note that the inverse distance function is equivalent to a triangular kernel if HT <math altimg="urn:x-wiley:00791636:media:trps12222:trps12222-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math> ht . If our variant frequencies at each point are already defined as usage proportions between 0 and 1, then: HT <math altimg="urn:x-wiley:00791636:media:trps12222:trps12222-math-0016" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>f</mi><mo> </mo></mover><mfenced separators=" open="(" close=")"><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msubsup><mo> </mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mi>k</mi><mfenced separators=" open="(" close=")"><mrow><msub><mi>d</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>,</mo><mi>b</mi></mrow></mfenced><msub><mi>X</mi><mrow><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow><mrow><msubsup><mo> </mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mi>k</mi><mfenced separators=" open="(" close=")"><mrow><msub><mi>d</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>,</mo><mi>b</mi></mrow></mfenced></mrow></mfrac><mo>.</mo></mrow></math> ht Having so defined the core of our method, we must determine the kernel function: the importance of points as evidence and therefore the weight assigned to them should fall off with distance, but what should the shape of this decline be? We must first calculate a matrix of distances between all points HT <math altimg="urn:x-wiley:00791636:media:trps12222:trps12222-math-0012" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mrow><mn>1</mn><mi>...</mi><mi>p</mi><mo>,</mo><mn>1</mn><mi>...</mi><mi>p</mi></mrow></msub></math> ht .
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