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Data and its Distribution

Data and its Distribution. The popular table. Table (relation) propositional, attribute-value Example record, row, instance, case Table represents a sample from a larger population independent, identically distributed Attribute variable, column, feature, item Target attribute, class

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Data and its Distribution

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  1. Data and its Distribution

  2. The popular table • Table (relation) • propositional, attribute-value • Example • record, row, instance, case • Table represents a sample from a larger population • independent, identically distributed • Attribute • variable, column, feature, item • Target attribute, class • Sometimes rows and columns are swapped • bioinformatics

  3. Example: play tennis data attributes examples

  4. Example: play tennis data attributes examples target attribute

  5. Example: play tennis data three examples covered, 100% correct if Outlook = sunny and Humidity = high then play = no

  6. Numeric tennis data numeric attributes

  7. Numeric tennis data numeric attributes

  8. Numeric tennis data if Outlook = sunny and Humidity > 83 then play = no if Temperature < Humidity then play = no

  9. Types • Nominal, categorical, symbolic, discrete • only equality (=) • no distance measure • Numeric • inequalities (<, >, ≤, ≥) • arithmetic • distance measure • Ordinal • inequalities • no arithmetic or distance measure • Binary • like nominal, but only two values, and True (1, yes, y) plays special role.

  10. Distributions

  11. Univariate (probability) distribution • What values occur for an attribute and how often? • count occurrences • Counts are complete information about sample • actual data can be ignored from here on • Data is a sample of a population • counts are probability estimates

  12. Attribute information: entropy • How informative is an attribute? • (How informative is an attribute about the value of another attribute?) • if an attribute is not informative, it cannot be informative about another • Entropy • a measure for the amount of information/chaos usefulness 1 bit entropy do you own a Mercedes? social security nr. gender highest degree

  13. Distribution of a Binary Attribute • Only two values • probabilities pand 1-p • Entropy:H(A) = – plg(p) – (1–p)lg(1–p) • lg(p) is the 2-log of p • H(A)is maximal when p = ½ = 1/m(mis the number of values) • uniform distribution • e.g., gender

  14. Entropy, Binary case gender, coin flip, … do you own a Mercedes? do you own a car? are you an alien? Entropy: H(A) = – plg(p) – (1–p)lg(1–p)

  15. Distribution of nominal attribute • Multiple values (m) • each with probability pi • Entropy:H(A) = Σ–pilg(pi) • notice binary as special case • H is maximal when p = 1/m • uniform distribution • Hmax = –m1/m lg(1/m) = –lg(1/m) = lg m • e.g. season of booking date • m = 4 • at most lg(m) = lg(4) = 2bits • Q: what if only summer and winter? bar chart

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