ΣΤΑΤΙΣΤΙΚΗ ΓΝΩΣΗΣ Εγγυτης και Αποσταση - PDF

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ΣΤΑΤΙΣΤΙΚΗ ΓΝΩΣΗΣ Εγγυτης και Αποσταση Ioannis Antoniou Charalambos Bratsas Mathematics Department Aristotle University 54124,Thessaloniki,Greece Εγγύηηρ = Proximity

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ΣΤΑΤΙΣΤΙΚΗ ΓΝΩΣΗΣ Εγγυτης και Αποσταση Ioannis Antoniou Charalambos Bratsas Mathematics Department Aristotle University 54124,Thessaloniki,Greece Εγγύηηρ = Proximity = Affinity = Similarity a real-valued function that quantifies how close two Entities are. Εγγςηηρ από Αποζηαζη H Εγγστης είναι μεγαλη οσο η αποσταση είναι μικρη Αποσταση (Distance) D1 Positivity D2 Identical beings are Indiscernible: d(x, x) =min = 0, x D3 Identity of Indiscernibles: d(x,y) = 0, implies x =y d(x, B) d(x, x) for all x y D4 Triangle Inequality: d(x, y) + d(y, z) d(x, z), x,y,z D5 Symmetry: d(x, y) = d(y, x), for all x and y Εγγστης (Proximity, Affinty, Similarity) Positive Identical beings have max Affinity: (x, x) =max, x max Affinity beings are Identical: (x, y) (x, x) for all x y if x and y are similar and y and z are similar, then x and z must also be similar (x, y) = (y, x) for all x and y Σσνηθεις Αποστασεις Αποζηαζειρ ζε Διανςζμαηικοςρ Χωποςρ The two most popular distance are Euclidean distance Let the coordinates of x, y in an N-dimensional space are The Euclidean distance between x, y: The Angle distance between x, y The city-block distance between x, y: The r-norm distance between x, y: the Hamming Distance between two strings of equal length, { the number of positions at which the corresponding symbols are different. the Mahalanobis Distance between two strings of equal length = ο Πιναξ Σςνδιαζποπαρ ηων Μεηαβληηων x,y, αν ο Πιναξ Σςνδιαζποπαρ είναι Διαγωνιορ (Μεηαζσημαηιζμορ ζηιρ Κςπιερ Σςνιζηωζερ) The Frobenius Distance between Μatrices, - The Jaccard Distance between Sets in a Measure Space for a given measure μ, for the cardinality measure, A,B finite Sets the Symmetric Difference of the Sets A, B Applications in Learning: Cuckier F., Smale S. 2001, On the Mathematical Foundations of Learning, Bull. Am. Math. Soc. 39, 1-49 Shepard Similarity distance and perceived Similarity are related via an exponential function Shepard s Universal law Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237, Shepard further proposed that this exponential function describes the probability that two stimuli fall in a region of stimulus space associated with the same response, which he called a consequential region. Shepard noted some failures of the exponential function in the case of confusable objects, although it was later shown that these problems can be resolved by treating percepts as probabilistic and applying Shepard s similarity function at the moment of decision-making (Ennis, 1988). If Similarity is increasing as Distance is Increasing, then perceived similarity must also obey the corresponding axioms. Test the validity of the distance axioms. Evidence has now been collected raising questions about the validity of these axioms. Evidence against Triangle Inequality: William James (1890) described an apparent counterexample to the triangle inequality. A flame is similar to the moon because they are both luminous, and the moon is similar to a ball because they are both round, but in contradiction to the triangle inequality, a flame is not similar to a ball. Evidence against symmetry Asymmetric Distance = Quasimetric Tversky (1977) reported that most people judge the similarity of North Korea to China to be greater than the similarity of China to North Korea. In response to examples, theories were proposed relaxing the distance axioms. Distance D1 Positivity D2 Identical beings are Indiscernible: d(x, x) =min = 0, x D3 Identity of Indiscernibles: d(x,y) = 0, implies x =y d(x, y) d(x, x) for all x y Affinity = Similarity Positive Identical beings have max Affinity: (x, x) =max, x max Affinity beings are Identical: (x, y) (x, x) for all x y Deza, M., Deza, E. 2013, Encyclopedia of Distances, 2nd revised edition, Springer-Verlag, Definition Renyi Proximity (Dependence Assessment) (X;Y) is defined for any pair of Random Variables X and Y. A1. Symmetry: (X;Y ) = (Y;X) A2. (X; Y ) = (f(x); g(y )), for bijective Borel-measurable functions A3. Normalization: 0 (X; Y ) 1 A4. Limit Values: (X; Y ) = 0 X and Y are statistically independent. (X; Y ) = 1 X and Y are deterministically dependent, ie. Y = f(x) Borel-measurable function or X = g(y), Borel-measurable function A8. (X; Y) =, r is the Pearson coefficient, if X, Y are Normal Distributions with the same mean m and the same variance Renyi A. 1959, On Measures of Dependence, Acta Mathematica X/3-4, Typical Renyi Proximity Normalized Mutual Information as Renyi Proximity Between Random Variables, - (, -, -), -, -, -, - H Αμοιβαια Πληποθοπια ηων Μεηαβληηων Χ, Υ, - η Ενηποπια ηηρ Μεηαβληηηρ Χ, - η Ενηποπια ηηρ Μεηαβληηηρ Y, - η Κοινη Ενηποπια ηων Μεηαβληηων Χ,Υ Information Distance Between Random Variables, - (, -, -) Ralski C. 1961, A Metric Space of Discrete Probability Distributions Information and Control 4, Rokhlin V. 1967, Lectures on the Entropy Theory of Measure Preserving Transformations, Russ. Math. Surv. 22, No 5,1-52, p19 Horibe Y A Note on Entropy Metrics, Information and Control 22, (provided a simple Proof of Distance) Google Search (Statistical) Distance between the terms x, y N the Number of Searches * + * + f(x) and f(y) are the number of hits for search terms x and y, f(x, y) is the number of web pages on which both x and y occur. Cilibrasi R. and Vitanyi P. 2007, The Google Similarity Distance IEEE Transactions on Knowledge and Data Engineering, Vol. 19, NO 3, arxiv.org/pdf/ v3
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