A Comparison of McCullagh’s Proportional Odds Model to Modern Ordinal Regression Algorithms
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چکیده
We introduce McCullagh’s Proportional Odds as the foundation for modern Ordinal Regression approaches. Proportional Odds introduced the ideas of (1) mapping examples to the real number line, and (2) segmenting the real number line using a set of thresholds. We compare against two modern approaches to Ordinal Regression which use the framework established by Proportional Odds and find some surprising similarities. 1 Proportional Odds McCullagh’s Proportional Odds model (1980) assumes that: • Examples are represented as d-dimensional real-valued feature vectors, x ∈ R, • Each example has an underlying score, defined by the dot-product between the feature vector and a weight vector, w ∈ R, • Each example is associated with a discrete, ordinal label, y ∈ {1, . . . , l}, • The real number line is segmented via a set of threshold values, −∞ ≡ θ0 < θ1 < θ2 < · · · < θl−1 < θl ≡ +∞, and • Each label, y, is associated with a segment of the real number line, (θy−1, θy). Define g(z) ≡ 1 1+e−z , the sigmoid function. Proportional Odds defines the cumulative likelihood of an example being associated with a label less-than-orequal-to j (for j ≤ l − 1) as the sigmoid function, PPO(y ≤ j|x) = g(θj − w x) = 1 1 + exp(wTx− θj) . (1) Updated October 30, 2006.
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تاریخ انتشار 2006