Distance Measure (𝑑) - is a function that takes two elements in some set (𝑋) and outputs a non-negative real number
- used to measure the distance between two elements
Distance Metric (𝑑) - is a distance measure that satisfies 4 properties
- when paired with its elements (𝑋) it forms a metric space (𝑋,𝑑)
- a norm (||·||) induces a distance metric (𝑑_||·||) defined as: 𝑑_||·||(𝑥,𝑦) = ||𝑥-𝑦||
- a distance metric (𝑑) induces a topology (𝜏)Topologywhere the set { 𝐵_𝑑(𝑥,𝜖) | 𝑥∈𝑋 and 𝜖>0 } forms the basis of a metric topology (𝜏𝑑), where 𝐵_𝑑(𝑥,𝜖) is an open-set defined as 𝐵_𝑑(𝑥,𝜖) = { 𝑦∈𝑋 : 𝑑(𝑥,𝑦) < 𝜖 }
Distance Semi-Metric (𝑑) - is a distance measure that satisfies 3 properties

Distance Measure - Categories

Category	Properties
Category	positivity 𝑑(𝑥,𝑦) ≥ 0 ∀𝑥,𝑦∊𝑋	identity-discerning 𝑑(𝑥,𝑦) = 0 if and only if 𝑥=𝑦	symmetry 𝑑(𝑥,𝑦) = 𝑑(𝑦,𝑥) ∀𝑥,𝑦∊𝑋	triangle-inequality 𝑑(𝑥,𝑧) ≤ 𝑑(𝑥,𝑦) + 𝑑(𝑦,𝑧) ∀𝑥,𝑦,𝑧∊𝑋
Distance Measure	REQUIRED	NOT-REQUIRED	NOT-REQUIRED	NOT-REQUIRED
Distance Metric	REQUIRED	REQUIRED	REQUIRED	REQUIRED
Distance Semi-Metric	REQUIRED	REQUIRED	REQUIRED	NOT REQUIRED

Distance Measure - Types

taxonomy-of-principal-distances-and-divergences.pdf

Type	Category	Input Type	Description
Norm Distance Metric	METRIC	depends	an norm (\|\|·\|\|)induces a distance metric (𝑑_\|\|·\|\|) defined as: 𝑑_\|\|·\|\|(𝑥,𝑦) = \|\|𝑥-𝑦\|\|
Rectilinear Distance Metric	METRIC	vectors	𝑑(𝑢̅,𝑣̅) = 𝐿₁ = \|\|𝑢̅-𝑣̅\|\|₁ = 𝛴_{1≤𝑖≤𝑛}\|𝑢̅_𝑖-𝑣̅_𝑖\|
Euclidean Distance Metric	METRIC	vectors	𝑑(𝑢̅,𝑣̅) = 𝐿₂ = \|\|𝑢̅-𝑣̅\|\|₂ = \|\|𝑢̅-𝑣̅\|\| = [ 𝛴_{1≤𝑖≤𝑛}(𝑢̅_𝑖-𝑣̅_𝑖)² ]^(1/2)
Minkowski Distance Metric	METRIC	vectors	𝑑(𝑢̅,𝑣̅) = 𝐿_𝑝 = \|\|𝑢̅-𝑣̅\|\|_𝑝 = [ 𝛴_{1≤𝑖≤𝑛}\|𝑢̅_𝑖-𝑣̅_𝑖\|^𝑝 ]^(1/𝑝)
Tchebychev Distance Metric	METRIC	vectors	𝑑(𝑢̅,𝑣̅) = 𝐿_∞ = \|\|𝑢̅-𝑣̅\|\|_∞ = [ 𝛴_{1≤𝑖≤𝑛}\|𝑢̅_𝑖-𝑣̅_𝑖\|^∞ ]^(1/∞) = 𝑚𝑎𝑥(\|𝑢̅_𝑖-𝑣̅_𝑖\|)
Squared Euclidean Distance	MEASURE	vectors	𝑑(𝑢̅,𝑣̅) = 𝛴_{1≤𝑖≤𝑛}(𝑢̅_𝑖-𝑣̅_𝑖)²
Cosine Distance Measure	MEASURE	vectors	𝑑(𝑢̅,𝑣̅) = 𝑐𝑜𝑠-𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑢̅,𝑣̅) = 1 - 𝑐𝑜𝑠-𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑦(𝑢̅,𝑣̅) # 𝑐𝑜𝑠-𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑢̅,𝑣̅) = [𝑢̅·𝑣̅] / [\|\|𝑢̅\|\|*\|\|𝑣̅\|\|] used for measuring distance when the magnitude of the vectors does not matter
Jaccard Distance Metric	METRIC	sets	𝑑(𝐴,𝐵) = 𝐽_{𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒}(𝐴,𝐵) = 1 - 𝐽_{𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦}(𝐴,𝐵) # 𝐽(𝐴,𝐵) = \|𝐴⋂𝐵\| / \|𝐴⋃𝐵\|
KL-Divergence Relative Entropy	MEASURE	random variables/vectors	measures the difference between 2 probability distributions
Hamming Distance Metric	METRIC	strings	measures the similarity between 2 strings of equal length
Discrete Distance Metric Discrete Metric	METRIC	anything	$d (x, y) = {0, 1, if x=y otherwise$

／var／log marcus chiu

Explorer

Distance Measures／Metrics／Semi-Metrics／Functions

Distance Measure - Categories

Distance Measure - Types

Distance Measure - Other

／var／logmarcus chiu

Explorer

Distance Measures／Metrics／Semi-Metrics／Functions

Distance Measure - Categories

Distance Measure - Types

Distance Measure - Other

／var／log marcus chiu