Data Mining - 2011 - Volinsky - Columbia University 1 Chapter 4.2 Regression Topics Credits Hastie,...

Data Mining - 2011 - Volinsky - Columbia University 1

Chapter 4.2 Regression Topics

CreditsHastie, Tibshirani, Friedman Chapter 3

Padhraic Smyth Lecture NotesWolfgang Jank Lecture Notes


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


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


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


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 = 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

€

ˆ y k =Xka

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– 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.



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?


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

0.120.100.080.060.040.020.00

0.03

0.02

0.01

0.00

-0.01

-0.02

-0.03

-0.04

Fitted Value

Residual

Residuals Versus the Fitted Values(response is alpha)

5 10 15 20 25 30 35 40 45 50

-10

-5

0

5

Observation Order

Residual

Residuals Versus the Order of the Data

(response is strength)


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.

^

Influence Measures for Linear Model

• There are a few quite influential (and extreme) points…

• What to do?

11Data Mining - 2011 - Volinsky - Columbia University


Diagnostic Plots


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


R ‘step’: uses AIC

Leaps output


R ‘leaps’ : uses Cp


Generalizing Linear Regression


Complexity versus Goodness of Fit

x

yTraining data



x

y

x

yToo simple?Training data



x

y

x

y

x

y

Too simple?

Too complex ?

Training data



x

y

x

y

x

y

x

y

Too simple?

Too complex ? About right ?

Training data


Complexity and Generalization

Strain()

Stest()

Complexity = degreesof freedom in the model(e.g., number of variables)

Score Functione.g., squarederror

Optimal modelcomplexity


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 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!


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


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 byerror structure (Gaussian, Poisson, Binomial)linear predictor (single variables, interactions,

polynomials)link function (identity, log, reciprocal)


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!



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 disagree1 0 Male 4 22 1 Male 2 03 2 Male 4 04 3 Male 6 35 4 Male 5 56 5 Male 13 77 6 Male 25 98 7 Male 27 159 8 Male 75 4910 9 Male 29 2911 10 Male 32 45• …


Womensrole Logistic fit


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


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.


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


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()


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

lasso coefficients

Note how parameters shrink to zero!

This is the appeal of lasso (in addition to good performance)


35s = 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.


Comparison of results (prostate data)

Term LS Best Subset

Ridge Lasso PCR

Intercept 2.465 2.477 2.452 2.468 2.497

Lcavol 0.680 0.740 0.420 0.533 0.543

Lwight 0.236 0.316 0.238 0.169 0.289

Age -0.141 -0.046 -0.152

Lbph 0.210 0.162 0.002 0.214

Svi 0.305 0.227 0.094 0.315

Lcp -0.288 0.000 -0.051

Gleason -0.021 0.040 0.232

Pgg45 0.267 0.133 -0.056

Test Error

0.521 0.492 0.492 0.479 0.449

Std Error 0.179 0.143 0.165 0.164 0.105


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


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


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


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


Piecewise Polynomial cont.

• Piecewise constant and piecewise linear

“Knots”


Spline cont. (Linear Spline)


Spline cont. (Cubic Spline)

Cubic spline


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

€

RSS( f ,λ ) = {y i − f (x)}2

i=1

n

∑ + λ { ′ ′ f (t)}2∫ dt

fit curvature


Example of Smoothing Splines

• Two Smoothing Splines fit to the Prestige Data– Little smoothing,

λ small (red line)– Heavy

smoothing, λ large (blue line)


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


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.


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-10s(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-testMeasures of

model fit


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!!


Function Estimates from Additive Regression Model

• What is the nature of the relationship of the individual predictor variables and prestige?


Data Mining - 2011 - Volinsky - Columbia University 1 Chapter 4.2 Regression Topics Credits Hastie,...

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