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Chapter 4.2 Regression Topics

Chapter 4.2 Regression Topics. Credits Hastie, Tibshirani, Friedman Chapter 3 Padhraic Smyth Lecture Notes Wolfgang Jank Lecture Notes. Regression Review. Linear Regression models a numeric outcome as a linear function of several predictors.

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Chapter 4.2 Regression Topics

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  1. Chapter 4.2 Regression Topics Credits Hastie, Tibshirani, Friedman Chapter 3 Padhraic Smyth Lecture Notes Wolfgang Jank Lecture Notes Data Mining - 2011 - Volinsky - Columbia University

  2. Regression Review • Linear Regression models a numeric outcome as a linear function of several predictors. • It is the king of all statistical and data mining models • ease of interpretation • mathematically concise • tends to perform well for prediction, even under violations of assumptions • Characteristics • numeric response - ideally real valued • numeric predictors- but not necessarily Data Mining - 2011 - Volinsky - Columbia University

  3. Linar Regression Model • Basic model: • you are not modelling y, but you are modelling the mean of y for a given x! • Simple Regression - one x. • easy to describe, good for mathematics, but not used often in data mining • Multiple regression - many x - • response surface is a plane…harder to conceptualize • Useful as a baseline model Data Mining - 2011 - Volinsky - Columbia University

  4. Linear Regression Model • Assumptions: • linearity • constant variance • normality of errors • residuals ~ Normal(mu,sigma^2) • Assumptions must be checked, • but if inference is not the goal, you can accept some deviation from assumptions (don’t’ tell the statisticians I said that!) • Multicollinearity also an issue • creates unstable estimates Data Mining - 2011 - Volinsky - Columbia University

  5. Fitting the Model • We can look at regression as a matrix problem • We want a score function which minimizes “a”: = which is minimized by Data Mining - 2011 - Volinsky - Columbia University

  6. Fitting models: in-sample Minimize the sum of the squared errors: • S = S e2 = e’ e = (y – X a)’ (y – X a) • = y’ y – a’ X’ y – y’ X a + a’ X’ X a • = y’ y – 2 a’ X’ y + a’ X’ X a Take derivative of S with respect to a: • dS/da = -2X’y + 2 X’ X a Set this to 0 to find the (minimum) of S as a function of a… • - 2X’y + 2 X’ X a = 0 • X’Xa = X’ y • a = ( X’ X )-1 X’ y • Prediction follows easily: Data Mining - 2011 - Volinsky - Columbia University

  7. Fitting regression: out-of-sample • Can also optimize “a” based on a hold-out sample and a search over all “a”s • But how to search over all values of all a’s? • This will minimize MSE – might give a different answer • MSE=Bias + Variance • Because of the nice algebraic form, typically in-sample is used • But different loss function may change things • R2 measures a ratio between • regression sum of squares - how much of the variance does the regression explain, and • the total sum of squares - how much variation is there altogether • If it is close to 1, your fit is good. But be careful. Data Mining - 2011 - Volinsky - Columbia University

  8. Limitations of Linear Regression • True relationship of X and Y might be non-linear • Suggests generalizations to non-linear models • Correlation/Collinearity among the X variables • Can cause numerical instability • Problems in interpretability (identifiability) • Includes all variables in the model… • But what if p=100 and only 3 variables are related to Y? Data Mining - 2011 - Volinsky - Columbia University

  9. Checking assumptions • linearity • look to see if transformations make relationships ‘more’ linear • normality of errors • Histograms and qqplots • Non-constant variance • Beware of ‘fanning’ residuals • Time effects • Can be revealed in an ordering plot • Influence • Use hat matrix Data Mining - 2011 - Volinsky - Columbia University

  10. Checking influence • Influence ^ • H is called the hat matrix (why?): • The element of H for a given observation is its influence • The leverage hi quantifies the influence that the observed response yi has on its predicted value y • It measures the distance between the X values for the ith case and the means of the X values for all n cases. • influence hi is a number between 0 and 1 inclusive. Data Mining - 2011 - Volinsky - Columbia University

  11. Influence Measures for Linear Model • There are a few quite influential (and extreme) points… • What to do? Data Mining - 2011 - Volinsky - Columbia University

  12. Diagnostic Plots Data Mining - 2011 - Volinsky - Columbia University

  13. Data Mining - 2011 - Volinsky - Columbia University

  14. Model selection: finding the best k variables • If noisy variables are included in the model, it can effect the overall performance. • Best to remove an predictors which have no effect, lest random patterns look significant. • Searching all possible models • How many are there? • Heuristic search is used to search over model space: • Forward or backward stepwise search • Leaps and bound techniques do exhaustive search • In-sample: penalize for complexity (AIC, BIC, Mallow’s Cp) • Out-of-sample: use cross validation Data Mining - 2011 - Volinsky - Columbia University

  15. R ‘step’: uses AIC Data Mining - 2011 - Volinsky - Columbia University

  16. Leaps output R ‘leaps’ : uses Cp Data Mining - 2011 - Volinsky - Columbia University

  17. Generalizing Linear Regression Data Mining - 2011 - Volinsky - Columbia University

  18. Complexity versus Goodness of Fit Training data y x Data Mining - 2011 - Volinsky - Columbia University

  19. Complexity versus Goodness of Fit Too simple? Training data y y x x Data Mining - 2011 - Volinsky - Columbia University

  20. Complexity versus Goodness of Fit Too simple? Training data y y x x Too complex ? y x Data Mining - 2011 - Volinsky - Columbia University

  21. Complexity versus Goodness of Fit Too simple? Training data y y x x Too complex ? About right ? y y x x Data Mining - 2011 - Volinsky - Columbia University

  22. Complexity and Generalization Score Function e.g., squared error Stest(q) Strain(q) Complexity = degrees of freedom in the model (e.g., number of variables) Optimal model complexity Data Mining - 2011 - Volinsky - Columbia University

  23. Non-linear models, linear in parameters • We can add additional polynomial terms in our equations, • non-linear functional form, but linear in the parameters (so still referred to as “linear regression”) • We can just treat the xi xj termsas additional fixed inputs • In fact we can add in any non-linear input functions!, e.g. Comments: • Number of parameters can explode => greater chance of overfitting • Adding complexity: must use penalties! Data Mining - 2011 - Volinsky - Columbia University

  24. Non-linear (both model and parameters) • We can generalize further to models that are nonlinear in all aspects where the g’s are non-linear functions (k of them) This is called a Neural Network (we’ll talk about it later) Closed form (analytical) solutions are rare. This is a a multivariate non-linear optimization problem (which may be quite difficult!) Data Mining - 2011 - Volinsky - Columbia University

  25. Generalizing Regression • Generalized Linear Models (GLM) linear combination of the predictors independent RV with distribution based on the error term function which connects the two GLMs are defined by error structure (Gaussian, Poisson, Binomial) linear predictor (single variables, interactions, polynomials) link function (identity, log, reciprocal) Data Mining - 2011 - Volinsky - Columbia University

  26. Logistic Regression • Logistic regression is the most common GLM. • response in this case is binary (0,1). (Y follows a bernoulli or Binomial distribution) • we model the probability of a 1 (p) occurring. • for mathematical convenience, we model the odds: • p/(1-p) • log odds are even better - logit function • scales on the real line, rather than [0,1] • Deviance: -2 x (difference in log-likelihood from saturated model) Data Mining - 2011 - Volinsky - Columbia University

  27. Logistic Regression • Interpretation of coefficients changes! Data Mining - 2011 - Volinsky - Columbia University

  28. Logistic example • womensrole data (R handbook) • Survey in 1975: “Women should take care of running their homes and leave running the coutnry up to men” education sex agree disagree 1 0 Male 4 2 2 1 Male 2 0 3 2 Male 4 0 4 3 Male 6 3 5 4 Male 5 5 6 5 Male 13 7 7 6 Male 25 9 8 7 Male 27 15 9 8 Male 75 49 10 9 Male 29 29 11 10 Male 32 45 • … Data Mining - 2011 - Volinsky - Columbia University

  29. Womensrole Logistic fit Data Mining - 2011 - Volinsky - Columbia University

  30. Other GLMs • Another useful GLM is for count data • model Y ~ Poisson(lambda) • link is log(Y) • Also called ‘log-linear’ models • Typically used for counts: • People at a store • Calls at a help center • Spams in an hour Data Mining - 2011 - Volinsky - Columbia University

  31. Shrinkage Models: Ridge Regression • Variable selection is a binary process • That makes it high variance: small changes can effect final model • Can we have a more continuous process, where each variable is ‘partly’ included? • Ridge regression “shrinks” coefficients on by imposing a penalty for the model “size” • Minimize the penalized sum of squares: L is a complexity parameter which controls the amount of shrinkage - the larger l is, the more the coefficients are shrunk towards 0. Data Mining - 2011 - Volinsky - Columbia University

  32. Ridge Regression • Model is imposing a penalty on the coefficient size • Since a’s depend on the units, care must be taken to standardize inputs. • Also, you can show that the ridge estimates are a linear function of y: • this adds a positive constant to the diagonal and allows inverision even if the matrix is not full rank • So, can be used in cases where p > n! • In general: increasing bias, decreasing variance • Often decreases MSE Data Mining - 2011 - Volinsky - Columbia University

  33. Ridge coefficients • df(l) is a one-to-one monotone function of l such that df(l) ranges from 0 to p. • l= 0; s=p : least squares solution; p degrees of freedom • l= inf; s=0; heaviest shrinkage; all parameter estimates = 0; zero degrees of freedom • Look at plot as a function of degrees of freedom df(l) Data Mining - 2011 - Volinsky - Columbia University

  34. Lasso • Very similar to ridge with one important difference: • L2 penalty replaced by L1 • has an interesting effect on the profile plot: • if lambda is large then estimates go to zero • continuous variable selection • s=1 is least squares answer • s=0 all estimates are 0 • s=0.5 was the value chosen by cross validation Data Mining - 2011 - Volinsky - Columbia University

  35. lasso coefficients Note how parameters shrink to zero! This is the appeal of lasso (in addition to good performance) s = df(l) / p Data Mining - 2011 - Volinsky - Columbia University

  36. Principal Components Regression • Create PC from the original data vectors and use them in any of the above regression schemes • Removes the ‘less important’ parts of the data space, while creating a reduced data set • Since each PC is a linear combination of the original variables, we can express the solution in terms of the initial coefficients. Data Mining - 2011 - Volinsky - Columbia University

  37. Comparison of results (prostate data) Cross validation allows all of these different methods to be comparable to each other Data Mining - 2011 - Volinsky - Columbia University

  38. Nonparametric Modeling • A nonparametric model does not assume any parameters to be estimated (thus the name nonparametric) • Its general form is Y = f(X) + ε • Typically, we only assume that f() is some smooth, continuous function • Also, we typically assume independent and identically distributed errors, ε~N(0,σ^2), but that’s not necessary. • 1-D nonparametric regression = density estimation Data Mining - 2011 - Volinsky - Columbia University

  39. Advantages & Disadvantages • Advantage • More flexibility leads to better data-fit, often also to better predictive capabilities • Smoothness can also lead to entirely new concepts, such as dynamics (via derivatives) and thus to flexible differential equation models, etc • Disadvantage • Much more complexity, hard to explain Data Mining - 2011 - Volinsky - Columbia University

  40. Fitting Nonparametric models • How do we estimate the function f()? • Restrictions on f: smoothness, continuity, existence of the first and second derivatives • options for estimating f include scatterplot smoothers, regression splines, smoothing splines, B-splines, thin-plate splines, wavelets, and many, many more… • one particularly popular option, the smoothing spline Data Mining - 2011 - Volinsky - Columbia University

  41. Splines • Splines are piecewise polynomials smoothly connected together. The joining points of the polynomial pieces are called knots. • Smoothing splines are splines that are penalized against too much local variability (and thus appear smoother) • Must be differentiable at the knots • linear spline: 0-times differentiable • cubic spline: twice differentiable Data Mining - 2011 - Volinsky - Columbia University

  42. Piecewise Polynomial cont. • Piecewise constant and piecewise linear “Knots” Data Mining - 2011 - Volinsky - Columbia University

  43. Spline cont. (Linear Spline) Data Mining - 2011 - Volinsky - Columbia University

  44. Spline cont. (Cubic Spline) Cubic spline Data Mining - 2011 - Volinsky - Columbia University

  45. Definition of Smoothing Splines • Smoothing Splines arise as the solution to the following simple regression problem • Find a piecewise polynomial f(x) with smooth breakpoints • f(x) minimizes the penalized sum-of-squares fit curvature Data Mining - 2011 - Volinsky - Columbia University

  46. Example of Smoothing Splines • Two Smoothing Splines fit to the Prestige Data • Little smoothing, λ small (red line) • Heavy smoothing, λ large (blue line) Data Mining - 2011 - Volinsky - Columbia University

  47. The smoothing parameter • The magnitude of λ affects the quality of the smoother; many ad-hoc approaches to find a “good” smoothing parameter • Visual trial and error • Minimize mean-squared error of the fit • Cross-validation, optimization on hold-out sample, etc Data Mining - 2011 - Volinsky - Columbia University

  48. Prestige Data Revisited • Education (X1) and Income (X2) influence the perceived Prestige (Y) of a profession • Is there a linear relationship between the X’s and Y? • If we’re not sure of the type of relationship between X and Y, nonparametric regression can be a very useful exploratory tool. Data Mining - 2011 - Volinsky - Columbia University

  49. Additive Model Estimates Parametric coefficients: Estimate std. err. t ratio Pr(>|t|) constant 46.833 0.6889 67.98 <2e-16 Approximate significance of smooth terms: edf chi.sq p-value s(income) 3.118 58.12 8.39e-10 s(education) 3.177 152.79 <2e-16 R-sq.(adj) = 0.836 Deviance explained = 84.7% GCV score = 52.143 Intercept! Inference for Income and Education, similar to F-test Measures of model fit Data Mining - 2011 - Volinsky - Columbia University

  50. Compare to Classical Regression Parametric coefficients: Estimate std. err. t ratio Pr(>|t|) (Intercept) -6.8478 3.219 -2.127 0.0359 income 0.0013612 0.000224 6.071 2.36e-08 education 4.1374 0.3489 11.86 <2e-16 R-sq.(adj) = 0.794 Deviance explained = 79.8% GCV score = 62.847 Better model fit for the nonparametric model!! Data Mining - 2011 - Volinsky - Columbia University

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