Data Mining Volinsky - Columbia University 1 Chapter 4.2 Regression Topics Credits Hastie, Tibshirani, Friedman Chapter 3 Padhraic Smyth Lecture Notes Wolfgang Jank Lecture Notes
Data Mining 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 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 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 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
Fitting models: in-sample Minimize the sum of the squared errors: S = e 2 = 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 Volinsky - Columbia University 6
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 –R 2 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 Volinsky - Columbia University 7
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 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 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 h i quantifies the influence that the observed response y i has on its predicted value y It measures the distance between the X values for the i th case and the means of the X values for all n cases. influence h i is a number between 0 and 1 inclusive. ^
Influence Measures for Linear Model There are a few quite influential (and extreme) points… What to do? 11 Data Mining Volinsky - Columbia University
12 Diagnostic Plots
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Data Mining 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 C p ) –Out-of-sample: use cross validation
Data Mining Volinsky - Columbia University 15 R ‘step’: uses AIC
Leaps output Data Mining Volinsky - Columbia University 16 R ‘leaps’ : uses C p
Data Mining Volinsky - Columbia University 17 Generalizing Linear Regression
Data Mining Volinsky - Columbia University 18 Complexity versus Goodness of Fit x y Training data
Data Mining Volinsky - Columbia University 19 Complexity versus Goodness of Fit x y x y Too simple? Training data
Data Mining Volinsky - Columbia University 20 Complexity versus Goodness of Fit x y x y x y Too simple? Too complex ? Training data
Data Mining Volinsky - Columbia University 21 Complexity versus Goodness of Fit x y x y x y x y Too simple? Too complex ?About right ? Training data
Data Mining Volinsky - Columbia University 22 Complexity and Generalization S train ( ) S test ( ) Complexity = degrees of freedom in the model (e.g., number of variables) Score Function e.g., squared error Optimal model complexity
Data Mining 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 x i x j terms as 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 Volinsky - Columbia University 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!) 24
Data Mining Volinsky - Columbia University 25 Generalizing Regression Generalized Linear Models (GLM) independent RV with distribution based on the error term linear combination of the predictors 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 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)
Logistic Regression Interpretation of coefficients changes! Data Mining Volinsky - Columbia University 27
Data Mining 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 Male Male Male Male Male Male Male Male Male Male …
Data Mining Volinsky - Columbia University 29 Womensrole Logistic fit
Data Mining 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 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: is a complexity parameter which controls the amount of shrinkage - the larger is, the more the coefficients are shrunk towards 0.
Data Mining 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 Volinsky - Columbia University 33 Ridge coefficients df( ) is a one-to-one monotone function of such that df( ) ranges from 0 to p. = 0; s=p : least squares solution; p degrees of freedom = inf; s=0; heaviest shrinkage; all parameter estimates = 0; zero degrees of freedom Look at plot as a function of degrees of freedom df( )
Data Mining Volinsky - Columbia University 34 Lasso Very similar to ridge with one important difference: L 2 penalty replaced by L 1 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
lasso coefficients Note how parameters shrink to zero! This is the appeal of lasso (in addition to good performance) Data Mining Volinsky - Columbia University 35 s = df( ) / p
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 Volinsky - Columbia University 36
Comparison of results (prostate data) TermLSBest Subset RidgeLassoPCR Intercept Lcavol Lwight Age Lbph Svi Lcp Gleason Pgg Test Error Std Error Data Mining Volinsky - Columbia University 37 Cross validation allows all of these different methods to be comparable to each other
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 38 Data Mining Volinsky - Columbia University
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 39 Data Mining Volinsky - Columbia University
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 40 Data Mining Volinsky - Columbia University
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 41 Data Mining Volinsky - Columbia University
Piecewise Polynomial cont. Piecewise constant and piecewise linear “Knots” 42 Data Mining Volinsky - Columbia University
Spline cont. (Linear Spline) 43 Data Mining Volinsky - Columbia University
Spline cont. (Cubic Spline) Cubic spline 44 Data Mining Volinsky - Columbia University
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 fitcurvature 45 Data Mining Volinsky - Columbia University
Example of Smoothing Splines Two Smoothing Splines fit to the Prestige Data –Little smoothing, λ small (red line) –Heavy smoothing, λ large (blue line) 46 Data Mining Volinsky - Columbia University
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 47 Data Mining Volinsky - Columbia University
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. 48 Data Mining Volinsky - Columbia University
Additive Model Estimates Parametric coefficients: Estimate std. err. t ratio Pr(>|t|) constant <2e-16 Approximate significance of smooth terms: edf chi.sq p-value s(income) e-10 s(education) <2e-16 R-sq.(adj) = Deviance explained = 84.7% GCV score = Intercept! Inference for Income and Education, similar to F-test Measures of model fit 49 Data Mining Volinsky - Columbia University
Compare to Classical Regression Parametric coefficients: Estimate std. err. t ratio Pr(>|t|) (Intercept) income e-08 education <2e-16 R-sq.(adj) = Deviance explained = 79.8% GCV score = Better model fit for the nonparametric model!! 50 Data Mining Volinsky - Columbia University
Function Estimates from Additive Regression Model What is the nature of the relationship of the individual predictor variables and prestige? 51 Data Mining Volinsky - Columbia University