diff --git a/build.sh b/build.sh index 8d4f1b86..dc013c46 100644 --- a/build.sh +++ b/build.sh @@ -1,5 +1,5 @@ #!/bin/bash if [ ! -f packages/FAKE/tools/Fake.exe ]; then - mono .NuGet/NuGet.exe install FAKE -OutputDirectory packages -ExcludeVersion + mono .nuget/nuget.exe install FAKE -OutputDirectory packages -ExcludeVersion fi mono packages/FAKE/tools/FAKE.exe build.fsx $@ diff --git a/docs/content/Distance.fsx b/docs/content/Distance.fsx index 7fbd8549..02ebf190 100644 --- a/docs/content/Distance.fsx +++ b/docs/content/Distance.fsx @@ -23,6 +23,8 @@ Math.NET Numerics provides the following distance functions on vectors and array Sum of Absolute Difference (SAD) -------------------------------- + + The sum of absolute difference is equivalent to the $L_1$-norm of the difference, also known as Manhattan- or Taxicab-norm. The `abs` function makes this metric a bit complicated to deal with analytically, but it is more robust than SSD. @@ -36,7 +38,10 @@ d_{\mathbf{SAD}} : (x, y) \mapsto \|x-y\|_1 = \sum_{i=1}^{n} |x_i-y_i| Sum of Squared Difference (SSD) ------------------------------- + + The sum of squared difference is equivalent to the squared $L_2$-norm, also known as Euclidean norm. +It is therefore also known as Squared Euclidean distance. This is the fundamental metric in least squares problems and linear algebra. The absence of the `abs` function makes this metric convenient to deal with analytically, but the squares cause it to be very sensitive to large outliers. @@ -51,7 +56,9 @@ d_{\mathbf{SSD}} : (x, y) \mapsto \|x-y\|_2^2 = \langle x-y, x-y\rangle = \sum_{ Mean-Absolute Error (MAE) ------------------------- -The mean absolute error is a normalized version of the sum of absolute difference: + + +The mean absolute error is a normalized version of the sum of absolute difference. $$$ d_{\mathbf{MAE}} : (x, y) \mapsto \frac{d_{\mathbf{SAD}}}{n} = \frac{\|x-y\|_1}{n} = \frac{1}{n}\sum_{i=1}^{n} |x_i-y_i| @@ -63,7 +70,9 @@ d_{\mathbf{MAE}} : (x, y) \mapsto \frac{d_{\mathbf{SAD}}}{n} = \frac{\|x-y\|_1}{ Mean-Squared Error (MSE) ------------------------ -The mean squared error is a normalized version of the sum of squared difference: + + +The mean squared error is a normalized version of the sum of squared difference. $$$ d_{\mathbf{MSE}} : (x, y) \mapsto \frac{d_{\mathbf{SSD}}}{n} = \frac{\|x-y\|_2^2}{n} = \frac{1}{n}\sum_{i=1}^{n} (x_i-y_i)^2 @@ -75,7 +84,10 @@ d_{\mathbf{MSE}} : (x, y) \mapsto \frac{d_{\mathbf{SSD}}}{n} = \frac{\|x-y\|_2^2 Euclidean Distance ------------------ -The euclidean distance is the $L_2$-norm of the difference: + + +The euclidean distance is the $L_2$-norm of the difference, a special case of the Minkowski distance with p=2. +It is the natural distance in a geometric interpretation. $$$ d_{\mathbf{2}} : (x, y) \mapsto \|x-y\|_2 = \sqrt{d_{\mathbf{SSD}}} = \sqrt{\sum_{i=1}^{n} (x_i-y_i)^2} @@ -87,7 +99,10 @@ d_{\mathbf{2}} : (x, y) \mapsto \|x-y\|_2 = \sqrt{d_{\mathbf{SSD}}} = \sqrt{\sum Manhattan Distance ------------------ -The manhattan distance is the $L_1$-norm of the difference and equivalent to the sum of absolute difference: + + +The manhattan distance is the $L_1$-norm of the difference, a special case of the Minkowski distance with p=1 +and equivalent to the sum of absolute difference. $$$ d_{\mathbf{1}} \equiv d_{\mathbf{SAD}} : (x, y) \mapsto \|x-y\|_1 = \sum_{i=1}^{n} |x_i-y_i| @@ -99,10 +114,13 @@ d_{\mathbf{1}} \equiv d_{\mathbf{SAD}} : (x, y) \mapsto \|x-y\|_1 = \sum_{i=1}^{ Chebyshev Distance ------------------ -The chebyshev distance is the $L_\infty$-norm of the difference: + + +The chebyshev distance is the $L_\infty$-norm of the difference, a special case of the Minkowski distance +where p goes to infinity. It is also known as Chessboard distance. $$$ -d_{\mathbf{\infty}} : (x, y) \mapsto \|x-y\|_\infty = \lim_{k \rightarrow \infty}\bigg(\sum_{i=1}^{n} |x_i-y_i|^k\bigg)^\frac{1}{k} = \max_{i} |x_i-y_i| +d_{\mathbf{\infty}} : (x, y) \mapsto \|x-y\|_\infty = \lim_{p \rightarrow \infty}\bigg(\sum_{i=1}^{n} |x_i-y_i|^p\bigg)^\frac{1}{p} = \max_{i} |x_i-y_i| [lang=csharp] double d = Distance.Chebyshev(x, y); @@ -111,7 +129,10 @@ d_{\mathbf{\infty}} : (x, y) \mapsto \|x-y\|_\infty = \lim_{k \rightarrow \infty Minkowski Distance ------------------ -The minkovski distance is the generalized $L_p$-norm of the difference: + + +The minkovski distance is the generalized $L_p$-norm of the difference. +The contour plot on the left demonstrates the case of p=3. $$$ d_{\mathbf{p}} : (x, y) \mapsto \|x-y\|_p = \bigg(\sum_{i=1}^{n} |x_i-y_i|^p\bigg)^\frac{1}{p} @@ -123,19 +144,42 @@ d_{\mathbf{p}} : (x, y) \mapsto \|x-y\|_p = \bigg(\sum_{i=1}^{n} |x_i-y_i|^p\big Canberra Distance ----------------- -The canberra distance is a weighted version of the manhattan distance: + + +The Canberra distance is a weighted version of the manhattan distance, introduced and refined 1967 by Lance, Williams and Adkins. +It is often used for data scattered around an origin, as it is biased for measures around the origin and very sensitive for values close to zero. $$$ -d_{\mathbf{Canberra}} : (x, y) \mapsto \sum_{i=1}^{n} \frac{|x_i-y_i|}{|x_i|+|y_i|} +d_{\mathbf{CAD}} : (x, y) \mapsto \sum_{i=1}^{n} \frac{|x_i-y_i|}{|x_i|+|y_i|} [lang=csharp] double d = Distance.Canberra(x, y); +Cosine Distance (planned) +------------------------- + + + +The cosine distance contains the dot product scaled by the product of the Euclidean distances from the origin. +It represents the angular distance of two vectors while ignoring their scale. + +$$$ +d_{\mathbf{cos}} : (x, y) \mapsto 1-\frac{\langle x, y\rangle}{\|x\|_2\|y\|_2} = 1-\frac{\sum_{i=1}^{n} x_i y_i}{\sqrt{\sum_{i=1}^{n} x_i^2}\sqrt{\sum_{i=1}^{n} y_i^2}} + + [lang=csharp] + // Planned (not implemented yet): + double d = Distance.Cosine(x, y); + + Pearson's Distance ------------------ -The pearson's distance is based on pearson's product-momentum correlation coefficient of the two sample vectors: + + +The Pearson distance is a correlation distance based on Pearson's product-momentum correlation coefficient +of the two sample vectors. Since the correlation coefficient falls between [-1, 1], the Pearson distance +lies in [0, 2] and measures the linear relationship between the two vectors. $$$ d_{\mathbf{Pearson}} : (x, y) \mapsto 1 - \mathbf{Corr}(x, y) @@ -148,6 +192,7 @@ Hamming Distance ---------------- The hamming distance represents the number of entries in the two sample vectors which are different. +It is a fundamental distance measure in information theory but less relevant in non-integer numerical problems. 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