4.6. linear models

  • Material from “Statistical Models Theory and Practice” - David Freedman

4.6.1. introduction

  • \(Y = X \beta + \epsilon\)

    type of linear regression

    X

    Y

    simple

    univariate

    univariate

    multiple

    multivariate

    univariate

    multivariate

    either

    multivariate

    generalized

    either

    error not normal

  • minimize: \(L(\theta) = \|\|Y-X\beta\|\|_2^2 \implies \hat{\theta} = (X^TX)^{-1} X^TY\)

  • 2 proofs

    1. set deriv and solve

    2. use projection matrix H to show HY is proj of Y onto R(X)

      1. the projection matrix maps the responses to the predictions: \(\hat{y} = Hy\)

    • define projection (hat) matrix \(H = X(X^TX)^{-1} X^T\)

      • show \(\|\|Y-X \theta\|\|^2 \geq \|\|Y - HY\|\|^2\)

      • key idea: subtract and add HY

  • interpretation

    • if feature correlated, weights aren’t stable / can’t be interpreted

    • curvature inverse \((X^TX)^{-1}\) - dictates stability

    • importance: weight * feature value

  • LS doesn’t work when p >> n because of colinearity of X columns

  • assumptions

    • \(\epsilon \sim N(X\beta,\sigma^2)\)

    • homoscedasticity: \(var(Y_i\|X)\) is the same for all i

      • opposite of heteroscedasticity

  • multicollinearity - predictors highly correlated

    • variance inflation factor (VIF) - measure how much the variances of the estimated regression coefficients are inflated as compared to when the predictors are not linearly related

  • normal linear regression

    • variance MLE \(\hat{\sigma}^2 = \sum (y_i - \hat{\theta}^T x_i)^2 / n\)

      • in unbiased estimator, we divide by n-p

    • LS has a distr. \(N(\theta, \sigma^2(X^TX)^{-1})\)

  • linear regression model

    • when n is large, LS estimator ahs approx normal distr provided that X^TX /n is approx. PSD

  • weighted LS: minimize \(\sum [w_i (y_i - x_i^T \theta)]^2\)

    • \(\hat{\theta} = (X^TWX)^{-1} X^T W Y\)

    • heteroscedastic normal lin reg model: erorrs ~ N(0, 1/w_i)

  • leverage scores - measure how much each \(x_i\) influences the LS fit

    • for data point i, \(H_{ii}\) is the leverage score

  • LAD (least absolute deviation) fit

    • MLE estimator when error is Laplacian

4.6.2. recent notes

4.6.2.1. regularization

  • when \((X^T X)\) isn’t invertible can’t use normal equations and gradient descent is likely unstable

    • X is nxp, usually n >> p and X almost always has rank p

    • problems when n < p

  • intuitive way to fix this problem is to reduce p by getting rid of features

  • a lot of papers assume your data is already zero-centered

    • conventionally don’t regularize the intercept term

  1. ridge regression (L2)

  • if (X^T X) not invertible, add a small element to diagonal

  • then it becomes invertible

  • small lambda -> numerical solution is unstable

  • proof of why it’s invertible is difficult

  • argmin \(\sum_i (y_i - \hat{y_i})^2+ \lambda \vert \vert \beta\vert \vert _2^2 \)

  • equivalent to minimizing \(\sum_i (y_i - \hat{y_i})^2\) s.t. \(\sum_j \beta_j^2 \leq t\)

  • solution is \(\hat{\beta_\lambda} = (X^TX+\lambda I)^{-1} X^T y\)

  • for small \(\lambda\) numerical solution is unstable

  • When \(X^TX=I\), \(\beta _{Ridge} = \frac{1}{1+\lambda} \beta_{Least Squares}\)

  1. lasso regression (L1)

  • \(\sum_i (y_i - \hat{y_i})^2+\lambda \vert \vert \beta\vert \vert _1 \)

  • equivalent to minimizing \(\sum_i (y_i - \hat{y_i})^2\) s.t. \(\sum_j \vert \beta_j\vert \leq t\)

  • “least absolute shrinkage and selection operator”

  • lasso - least absolute shrinkage and selection operator - L1

  • acts in a nonlinear manner on the outcome y

  • keep the same SSE loss function, but add constraint of L1 norm

  • doesn’t have closed form for Beta

    • because of the absolute value, gradient doesn’t exist

    • can use directional derivatives

    • best solver is LARS - least angle regression

  • if tuning parameter is chose well, will set lots of coordinates to 0

  • convex functions / convex sets (like circle) are easier to solve

  • disadvantages

    • if p>n, lasso selects at most n variables

    • if pairwise correlations are very high, lasso only selects one variable

  1. elastic net - hybrid of the other two

  • \(\beta_{Naive ENet} = \sum_i (y_i - \hat{y_i})^2+\lambda_1 \vert \vert \beta\vert \vert _1 + \lambda_2 \vert \vert \beta\vert \vert _2^2\)

  • l1 part generates sparse model

  • l2 part encourages grouping effect, stabilizes l1 regularization path

    • grouping effect - group of highly correlated features should all be selected

  • naive elastic net has too much shrinkage so we scale \(\beta_{ENet} = (1+\lambda_2) \beta_{NaiveENet}\)

  • to solve, fix l2 and solve lasso

4.6.2.2. the regression line (freedman ch 2)

  • regression line

    • goes through \((\bar{x}, \bar{y})\)

    • slope: \(r s_y / s_x\)

    • intercept: \(\bar{y} - slope \cdot \bar{x}\)

    • basically fits graph of averages (minimizes MSE)

  • SD line

    • same except slope: \(sign(r) s_y / s_x\)

    • intercept changes accordingly

  • for regression, MSE = \((1-r^2) Var(Y)\)

4.6.2.3. multiple regression (freedman ch 4)

  • assumptions

    1. assume \(n > p\) and X has full rank (rank p - columns are linearly independent)

    2. \(\epsilon_i\) are iid, mean 0, variance \(\sigma^2\)

    3. \(\epsilon\) independent of \(X\)

    • \(e_i\) still orthogonal to \(X\)

  • OLS is conditionally unbiased

    • \(E[\hat{\theta} \| X] = \theta\)

  • \(Cov(\hat{\theta}\|X) = \sigma^2 (X^TX)^{-1}\)

    • \(\hat{\sigma^2} = \frac{1}{n-p} \sum_i e_i^2\)

      • this is unbiased - just dividing by n is too small since we have minimized \(e_i\) so their variance is lower than var of \(\epsilon_i\)

  • random errors \(\epsilon\)

  • residuals \(e\)

  • \(H = X(X^TX)^{-1} X^T\)

    1. e = (I-H)Y = \((I-H) \epsilon\)

    2. H is symmetric

    3. \(H^2 = H, (I-H)^2 = I-H\)

    4. HX = X

    5. \(e \perp X\)

  • basically H projects Y int R(X)

  • \(E[\hat{\sigma^2}\|X] = \sigma^2\)

  • random errs don’t need to be normal

  • variance

    • \(var(Y) = var(X \hat{\theta}) + var(e)\)

      • \(var(X \hat{\theta})\) is the explained variance

      • fraction of variance explained: \(R^2 = var(X \hat{\theta}) / var(Y)\)

      • like summing squares by projecting

    • if there is no intercept in a regression eq, \(R^2 = \|\|\hat{Y}\|\|^2 / \|\|Y\|\|^2\)

4.6.3. advanced topics

4.6.3.1. BLUE

  • drop assumption: \(\epsilon\) independent of \(X\)

    • instead: \(E[\epsilon\|X]=0, cov[\epsilon\|X] = \sigma^2 I\)

    • can rewrite: \(E[\epsilon]=0, cov[\epsilon] = \sigma^2 I\) fixing X

  • Gauss-markov thm - assume linear model and assumption above: when X is fixed, OLS estimator is BLUE = best linear unbiased estimator

    • has smallest variance.

    • prove this

4.6.3.2. GLS = GLM

  • GLMs roughly solve the problem where outcomes are non-Gaussian

    • mean is related to \(w^tx\) through a link function (ex. logistic reg assumes sigmoid)

    • also assume different prob distr on Y (ex. logistic reg assumes Bernoulli)

  • generalized least squares regression model: instead of above assumption, use \(E[\epsilon\|X]=0, cov[\epsilon\|X] = G, \: G \in S^K_{++}\)

    • covariance formula changes: \(cov(\hat{\theta}_{OLS}\|X) = (X^TX)^{-1} X^TGX(X^TX)^{-1}\)

    • estimator is the same, but is no longer BLUE - can correct for this: \((G^{-1/2}Y) = (G^{-1/2}X)\theta + (G^{-1/2}\epsilon)\)

  • feasible GLS=Aitken estimator - use \(\hat{G}\)

  • examples

    • simple

    • iteratively reweighted

  • 3 assumptions can break down:

    1. if \(E[\epsilon\|X] \neq 0\) - GLS estimator is biased

    2. else if \(cov(\epsilon\|X) \neq G\) - GLS unbiased, but covariance formula breaks down

    3. if G from data, but violates estimation procedure, estimator will be misealding estimate of cov

4.6.3.3. path models

  • path model - graphical way to represent a regression equation

  • making causal inferences by regression requires a response schedule

4.6.3.4. simultaneous equations

  • simultaneous-equation models - use instrumental variables / two-stage least squares

    • these techniques avoid simultaneity bias = endogeneity bias

4.6.3.5. binary variables

  • indicator variables take on the value 0 or 1

    • dummy coding - matrix is singular so we drop the last indicator variable - called reference class / baseline class

    • effect coding

      • one vector is all -1s

      • B_0 should be weighted average of the class averages

    • orthogonal coding

  • additive effects assume that each predictor’s effect on the response does not depend on the value of the other predictor (as long as the other one was fixed

    • assume they have the same slope

  • interaction effects allow the effect of one predictor on the response to depend on the values of other predictors.

    • \(y_i = β_0 + β_1x_{i1} + β_2x_{i2} + β_3x_{i1}x_{i2} + ε_i\)

4.6.3.6. LR with non-linear basis functions

  • can have nonlinear basis functions (ex. polynomial regression)

  • radial basis function - ex. kernel function (Gaussian RBF)

    • \(exp(-(x-r)^2 / (2 \lambda ^2))\)

  • non-parametric algorithm - don’t get any parameters theta; must keep data

4.6.3.7. locally weighted LR (lowess)

  • recompute model for each target point

  • instead of minimizing SSE, we minimize SSE weighted by each observation’s closeness to the sample we want to query

4.6.3.8. kernel regression

  • nonparametric method

  • \(\operatorname{E}(Y | X=x) = \int y f(y|x) dy = \int y \frac{f(x,y)}{f(x)} dy\)

    Using the [[kernel density estimation]] for the joint distribution ‘’f(x,y)’’ and ‘’f(x)’’ with a kernel ‘’’’’K’’’’’,

    \(\hat{f}(x,y) = \frac{1}{n}\sum_{i=1}^{n} K_h\left(x-x_i\right) K_h\left(y-y_i\right)\) \(\hat{f}(x) = \frac{1}{n} \sum_{i=1}^{n} K_h\left(x-x_i\right)\)

    we get

    \(\begin{align} \operatorname{\hat E}(Y | X=x) &= \int \frac{y \sum_{i=1}^{n} K_h\left(x-x_i\right) K_h\left(y-y_i\right)}{\sum_{j=1}^{n} K_h\left(x-x_j\right)} dy,\\ &= \frac{\sum_{i=1}^{n} K_h\left(x-x_i\right) \int y \, K_h\left(y-y_i\right) dy}{\sum_{j=1}^{n} K_h\left(x-x_j\right)},\\ &= \frac{\sum_{i=1}^{n} K_h\left(x-x_i\right) y_i}{\sum_{j=1}^{n} K_h\left(x-x_j\right)},\end{align}\)

4.6.3.9. GAM = generalized additive model

  • generalized additive models: assume mean output is sum of functions of individual variables (no interactions)

    • learn individual functions using splines

  • \(g(\mu) = b + f(x_0) + f(x_1) + f(x_2) + ...\)

  • can also add some interaction terms (e.g. \(f(x_0, x_1)\))

4.6.4. sums interpretation

  • SST - total sum of squares - measure of total variation in response variable

    • \(\sum(y_i-\bar{y})^2\)

  • SSR - regression sum of squares - measure of variation explained by predictors

    • \(\sum(\hat{y_i}-\bar{y})^2\)

  • SSE - measure of variation not explained by predictors

    • \(\sum(y_i-\hat{y_i})^2\)

  • SST = SSR + SSE

  • \(R^2 = \frac{SSR}{SST}\) - coefficient of determination

    • measures the proportion of variation in Y that is explained by the predictor

4.6.5. geometry - J. 6

  • LMS = least mean squares (p-dimensional geometries)

    • \(y_n = \theta^T x_n + \epsilon_n\)

    • \(\theta^{(t+1)}=\theta^{(t)} + \alpha (y_n - \theta^{(t)T} x_n) x_n\)

      • converges if \(0 < \alpha < 2/||x_n||^2\)

    • if N=p and all \(x^{(i)}\) are lin. indepedent, then there exists exact solution \(\theta\)

  • solving requires finding orthogonal projection of y on column space of X (n-dimensional geometries)

    1. 3 Pfs

      1. geometry - \(y-X\theta^*\) must be orthogonal to columns of X: \(X^T(y-X\theta)=0\)

      2. minimize least square cost function and differentiate

      3. show HY projects Y onto col(X)

    • either of these approaches yield the normal eqns: \(X^TX \theta^* = X^Ty\)

  • SGD

    • SGD converges to normal eqn

  • convergence analysis: requires \(0 < \rho < 2/\lambda_{max} [X^TX]\)

    • algebraic analysis: expand \(\theta^{(t+1)}\) and take \(t \to \infty\)

    • geometric convergence analysis: consider contours of loss function

  • weighted least squares: \(J(\theta)=\frac{1}{2}\sum_n w_n (y_n - \theta^T x_n)^2\)

    • yields \(X^T WX \theta^* = X^T Wy\)

  • probabilistic interpretation

    • \(p(y\|x, \theta) = \frac{1}{(2\pi\sigma^2)^{N/2}} exp \left( \frac{-1}{2\sigma^2} \sum_{n=1}^N (y_n - \theta^T x_n)^2 \right)\)

    • \(l(\theta; x,y) = - \frac{1}{2\sigma^2} \sum_{n=1}^N (y_n - \theta^T x_n)^2\)

      • log-likelihood is equivalent to least-squares cost function

4.6.6. likelihood calcs

4.6.6.1. normal equation

  • \(L(\theta) = \frac{1}{2} \sum_{i=1}^n (\hat{y}_i-y_i)^2\)

  • \(L(\theta) = 1/2 (X \theta - y)^T (X \theta -y)\)

  • \(L(\theta) = 1/2 (\theta^T X^T - y^T) (X \theta -y)\)

  • \(L(\theta) = 1/2 (\theta^T X^T X \theta - 2 \theta^T X^T y +y^T y)\)

  • \(0=\frac{\partial L}{\partial \theta} = 2X^TX\theta - 2X^T y\)

  • \(\theta = (X^TX)^{-1} X^Ty\)

4.6.6.2. ridge regression

  • \(L(\theta) = \sum_{i=1}^n (\hat{y}_i-y_i)^2+ \lambda \vert \vert \theta\vert \vert _2^2\)

  • \(L(\theta) = (X \theta - y)^T (X \theta -y)+ \lambda \theta^T \theta\)

  • \(L(\theta) = \theta^T X^T X \theta - 2 \theta^T X^T y +y^T y + \lambda \theta^T \theta\)

  • \(0=\frac{\partial L}{\partial \theta} = 2X^TX\theta - 2X^T y+2\lambda \theta\)

  • \(\theta = (X^TX+\lambda I)^{-1} X^T y\)