+
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.
@@ -32,7 +32,7 @@ d_{\mathbf{SAD}} : (x, y) \mapsto \|x-y\|_1 = \sum_{i=1}^{n} |x_i-y_i|
Sum of Squared Difference (SSD)
-------------------------------
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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.
@@ -50,7 +50,7 @@ d_{\mathbf{SSD}} : (x, y) \mapsto \|x-y\|_2^2 = \langle x-y, x-y\rangle = \sum_{
Mean-Absolute Error (MAE)
-------------------------
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The mean absolute error is a normalized version of the sum of absolute difference.
@@ -64,7 +64,7 @@ d_{\mathbf{MAE}} : (x, y) \mapsto \frac{d_{\mathbf{SAD}}}{n} = \frac{\|x-y\|_1}{
Mean-Squared Error (MSE)
------------------------
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The mean squared error is a normalized version of the sum of squared difference.
@@ -78,7 +78,7 @@ d_{\mathbf{MSE}} : (x, y) \mapsto \frac{d_{\mathbf{SSD}}}{n} = \frac{\|x-y\|_2^2
Euclidean Distance
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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.
@@ -93,7 +93,7 @@ d_{\mathbf{2}} : (x, y) \mapsto \|x-y\|_2 = \sqrt{d_{\mathbf{SSD}}} = \sqrt{\sum
Manhattan Distance
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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.
@@ -108,7 +108,7 @@ d_{\mathbf{1}} \equiv d_{\mathbf{SAD}} : (x, y) \mapsto \|x-y\|_1 = \sum_{i=1}^{
Chebyshev Distance
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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.
@@ -123,7 +123,7 @@ d_{\mathbf{\infty}} : (x, y) \mapsto \|x-y\|_\infty = \lim_{p \rightarrow \infty
Minkowski Distance
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The Minkowski distance is the generalized $L_p$-norm of the difference.
The contour plot on the left demonstrates the case of p=3.
@@ -138,7 +138,7 @@ d_{\mathbf{p}} : (x, y) \mapsto \|x-y\|_p = \bigg(\sum_{i=1}^{n} |x_i-y_i|^p\big
Canberra Distance
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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.
@@ -153,7 +153,7 @@ d_{\mathbf{CAD}} : (x, y) \mapsto \sum_{i=1}^{n} \frac{|x_i-y_i|}{|x_i|+|y_i|}
Cosine Distance
---------------
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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.
@@ -168,7 +168,7 @@ d_{\mathbf{cos}} : (x, y) \mapsto 1-\frac{\langle x, y\rangle}{\|x\|_2\|y\|_2} =
Pearson's Distance
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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
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