Regression-BasisFns

Regression-BasisFns - Machine Learning ! ! ! ! !Srihari...

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Machine Learning Srihari 1 Linear Models for Regression Sargur Srihari srihari@cedar.buffalo.edu
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Machine Learning Srihari 2 Overview • Plan: Discuss supervised learning starting with regression • Goal: predict value of one or more target variables t • Given d -dimensional vector x of input variables • Terminology – Regression • When t is continuous-valued – Classification • if t has a value consisting of labels (non-ordered categories) – Ordinal Regression • Discrete values, ordered categories • Learning to Rank problem t is discrete (eg, 1,2. .6 ) in training set but a continuous value in [1,6] is learnt and used to rank objects
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Machine Learning Srihari 3 Types of Linear Regression Models • Simplest form of linear regression models: – linear function of single input variable y(x ,w ) = w 0 +w 1 x • More useful class of functions: – Polynomial curve fitting y(x ,w ) = w 0 +w 1 x+w 2 x 2 +…= Σ w i x i – linear combination of non-linear functions of input variables φ i ( x ) instead of x i called basis functions • Linear functions of parameters (which gives them simple analytical properties), yet are nonlinear with respect to input variables Task is to learn weights w 0 , w 1 from data D={(y i ,x i )}i=1,,N
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Machine Learning Srihari Regression: Learning To Rank Log frequency of query in anchor text Query word in color on page # of images on page # of (out) links on page PageRank of page URL length URL contains “~” Page length Input ( x i ): ( d Features of Query-URL pair) Output ( y ): Relevance Value In LETOR 4.0 dataset 46 query-document features Maximum of 124 URLs/query ( d >200 ) Yahoo! data set has d=700 Target Variable - Point-wise (0,1,2,3) - Regression returns continuous value -Allows fine-grained ranking of URLs Traditional IR uses TF/IDF
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Machine Learning Srihari NRC Ranking of PhD programs (2006) • S (survey) ranking – Ask faculty to rate how important d = 20 characteristics are to program quality – Randomly draw half of faculty program ratings 500 times • to produce 500 sets of direct weights • R (regression) ranking – Ask faculty to rate quality ( t =1. .6 ) of N specific programs in their field • Values of t i used for regression – Randomly draw half of program ratings 500 times • Obtain 500 sets of regression weights for the d characteristics 5 • Gather raw data ( x i ) from institutions and other sources measures of faculty productivity, student support and outcomes, diversity
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Machine Learning Srihari NRC Weights ( w i ) for CS R-based 0.05 R- 0.95 R-Stdev S-based 0.05 S- 0.95 S-Stdev Publications per Allocated Faculty 0.110 0.135 0.009 0.132 0.138 0.002 Cites per Publication 0.067 0.095 0.011 0.148 0.155 0.002 Grants per Allocated Faculty -0.001 0.047 0.014 0.129 0.135 0.002 Percent Faculty Interdisciplinary 0.044 0.068 0.008 0.044 0.049 0.001 Percent Non-Asian Minority Faculty 0.038 0.070 0.010 0.005 0.007 0.001 Percent Female Faculty 0.038 0.086 0.016 0.008 0.010 0.001 Awards per allocated faculty 0.083 0.125 0.015 0.099 0.106 0.002 Average GRE-Q 0.066 0.119 0.015 0.059 0.063 0.001 Percent 1st yr. students w/ full support -0.011 0.050 0.020 0.066 0.070 0.001 Percent 1st yr students with portable fellowships -0.054 -0.010 0.014 0.042 0.045 0.001 Percent Non-Asian Minority Students -0.047 0.004 0.015 0.011 0.014 0.001
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Regression-BasisFns - Machine Learning ! ! ! ! !Srihari...

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