# Basic

• https://en.wikipedia.org/wiki/Lasso_(statistics). It has a discussion when two covariates are highly correlated. For example if gene $\displaystyle{ i }$ and gene $\displaystyle{ j }$ are identical, then the values of $\displaystyle{ \beta _{j} }$ and $\displaystyle{ \beta _{k} }$ that minimize the lasso objective function are not uniquely determined. Elastic Net has been designed to address this shortcoming.
• Strongly correlated covariates have similar regression coefficients, is referred to as the grouping effect. From the wikipedia page "one would like to find all the associated covariates, rather than selecting only one from each set of strongly correlated covariates, as lasso often does. In addition, selecting only a single covariate from each group will typically result in increased prediction error, since the model is less robust (which is why ridge regression often outperforms lasso)".
• Glmnet Vignette. It tries to minimize $\displaystyle{ RSS(\beta) + \lambda [(1-\alpha)\|\beta\|_2^2/2 + \alpha \|\beta\|_1] }$. The elastic-net penalty is controlled by $\displaystyle{ \alpha }$, and bridge the gap between lasso ($\displaystyle{ \alpha = 1 }$) and ridge ($\displaystyle{ \alpha = 0 }$). Following is a CV curve (adaptive lasso) using the example from glmnet(). Two vertical lines are indicated: left one is lambda.min (that gives minimum mean cross-validated error) and right one is lambda.1se (the most regularized model such that error is within one standard error of the minimum). For the tuning parameter $\displaystyle{ \lambda }$,
• The larger the $\displaystyle{ \lambda }$, more coefficients are becoming zeros (think about coefficient path plots) and thus the simpler (more regularized) the model.
• If $\displaystyle{ \lambda }$ becomes zero, it reduces to the regular regression and if $\displaystyle{ \lambda }$ becomes infinity, the coefficients become zeros.
• In terms of the bias-variance tradeoff, the larger the $\displaystyle{ \lambda }$, the higher the bias and the lower the variance of the coefficient estimators.
• Video by Trevor Hastie
set.seed(1010)
n=1000;p=100
nzc=trunc(p/10)
x=matrix(rnorm(n*p),n,p)
beta=rnorm(nzc)
fx= x[,seq(nzc)] %*% beta
eps=rnorm(n)*5
y=drop(fx+eps)
px=exp(fx)
px=px/(1+px)
ly=rbinom(n=length(px),prob=px,size=1)

## Full lasso
set.seed(999)
cv.full <- cv.glmnet(x, ly, family='binomial', alpha=1, parallel=TRUE)
plot(cv.full)  # cross-validation curve and two lambda's
plot(glmnet(x, ly, family='binomial', alpha=1), xvar="lambda", label=TRUE) # coefficient path plot
plot(glmnet(x, ly, family='binomial', alpha=1))  # L1 norm plot
log(cv.full$lambda.min) # -4.546394 log(cv.full$lambda.1se) # -3.61605
sum(coef(cv.full, s=cv.full$lambda.min) != 0) # 44 ## Ridge Regression to create the Adaptive Weights Vector set.seed(999) cv.ridge <- cv.glmnet(x, ly, family='binomial', alpha=0, parallel=TRUE) wt <- 1/abs(matrix(coef(cv.ridge, s=cv.ridge$lambda.min)
[, 1][2:(ncol(x)+1)] ))^1 ## Using gamma = 1, exclude intercept
## Adaptive Lasso using the 'penalty.factor' argument
set.seed(999)
cv.lasso <- cv.glmnet(x, ly, family='binomial', alpha=1, parallel=TRUE, penalty.factor=wt)
# defautl type.measure="deviance" for logistic regression
plot(cv.lasso)
log(cv.lasso$lambda.min) # -2.995375 log(cv.lasso$lambda.1se) # -0.7625655
sum(coef(cv.lasso, s=cv.lasso$lambda.min) != 0) # 34  ## Lambda $\displaystyle{ \lambda }$ • A list of potential lambdas: see Linear Regression case. The λ sequence is determined by lambda.max and lambda.min.ratio. • lambda.max is computed from the input x and y; it is the maximal value for lambda such that all the coefficients are zero. How does glmnet compute the maximal lambda value?. However see the Random data example. • lambda.min.ratio (default is ifelse(nobs<nvars,0.01,0.0001)) is the ratio of smallest value of the generated λ sequence (say lambda.min) to lambda.max. • The program then generated nlambda values linear on the log scale from lambda.max down to lambda.min. • The sequence of lambda does not change with different data partitions from running cv.glmnet(). • Avoid supplying a single value for lambda (for predictions after CV use predict() instead). Supply instead a decreasing sequence of lambda values. I'll get different coefficients when I use coef(cv.glmnet obj) and coef(glmnet obj). See the complete code on gist. cvfit <- cv.glmnet(x, y, family = "cox", nfolds=10) fit <- glmnet(x, y, family = "cox", lambda = lambda) coef.cv <- coef(cvfit, s = lambda) coef.fit <- coef(fit) length(coef.cv[coef.cv != 0]) # 31 length(coef.fit[coef.fit != 0]) # 30 all.equal(lambda, cvfit$lambda[40])  # TRUE
length(cvfit$lambda) # [1] 100  • Choosing hyper-parameters (α and λ) in penalized regression by Florian Privé • lambda.min vs lambda.1se • The lambda.1se represents the value of λ in the search that was simpler than the best model (lambda.min), but which has error within 1 standard error of the best model ( lambda.min < lambda.1se ). In other words, using the value of lambda.1se as the selected value for λ results in a model that is slightly simpler than the best model but which cannot be distinguished from the best model in terms of error given the uncertainty in the k-fold CV estimate of the error of the best model. • lambda.1se can be checked by looking at the CV plot. The CV plot contains the SE of RMSE/Deviance and cv.glmnet object contains cvm and cvup elements. If we draw a horizontal line at cvup[i] where i is the index of the lambda.min, then the largest lambda satisfying cvm[j] <= cvup[i] will be lambda.1se. This can be seen on the example of Logistic Regression: family = "binomial". • lambda.1se not being in one standard error of the error When we say 1 standard error, we are not talking about the standard error across the lambda's but the standard error across the folds for a given lambda. Suppose A=cvfit$cvsd[which(cv.lasso$lambda == cv.lasso$lambda.min)] is the sd at lambda.min and B=cvfit$cvm[which(cv.lasso$lambda == cv.lasso$lambda.min)] the minimum cross-validated MSE. If we draw a horizontal line at A+B, then the largest lambda less than the cross point (on x-axis) should be lambda.1se. • The lambda.min option refers to value of λ at the lowest CV error. The error at this value of λ is the average of the errors over the k folds and hence this estimate of the error is uncertain. • https://www.rdocumentation.org/packages/glmnet/versions/2.0-10/topics/glmnet • glmnetUtils: quality of life enhancements for elastic net regression with glmnet • Mixing parameter: alpha=1 is the lasso penalty, and alpha=0 the ridge penalty and anything between 0–1 is Elastic net. • RIdge regression uses Euclidean distance/L2-norm as the penalty. It won't remove any variables. • Lasso uses L1-norm as the penalty. Some of the coefficients may be shrunk exactly to zero. • In ridge regression and lasso, what is lambda? • Lambda is a penalty coefficient. Large lambda will shrink the coefficients. • cv.glment()$lambda.1se gives the most regularized model such that error is within one standard error of the minimum
• A deep dive into glmnet: penalty.factor, standardize, offset

### Optimal lambda for Cox model

See Regularization Paths for Cox’s Proportional Hazards Model via Coordinate Descent Simon 2011. We choose the λ value which maximizes $\displaystyle{ \hat{CV}(\lambda) }$.

\displaystyle{ \begin{align} \hat{CV}_{i}(\lambda) = l(\beta_{-i}(\lambda)) - l_{-i}(\beta_{-i}(\lambda)). \end{align} }

Our total goodness of fit estimate, $\displaystyle{ \hat{CV}(\lambda) }$, is the sum of all $\displaystyle{ \hat{CV}_{i}(\lambda). }$ By using the equation above – subtracting the log-partial likelihood evaluated on the non-left out data from that evaluated on the full data – we can make efficient use of the death times of the left out data in relation to the death times of all the data.

### Random data

• For random survival (response) data, glmnet easily returns 0 covariates by using lambda.min.
• For random binary or quantitative trait (response) data, it seems glmnet returns at least 1 covariate at lambda.min which is max(lambda). This seems contradicts the documentation which describes all coefficients are zero with max(lambda).

See the code here.

## Mixing parameter $\displaystyle{ \alpha }$

cva.glmnet() from the glmnetUtils package to choose both the alpha and lambda parameters via cross-validation, following the approach described in the help page for cv.glmnet.

## Plots

library(glmnet)
data(QuickStartExample) # x is 100 x 20 matrix

cvfit = cv.glmnet(x, y)
fit = glmnet(x, y)

oldpar <- par(mfrow=c(2,2))
plot(cvfit) # mse vs log(lambda)
plot(fit) # coef vs L1 norm
plot(fit, xvar = "lambda", label = TRUE) # coef vs log(lambda)
plot(fit, xvar = "dev", label = TRUE) # coef vs Fraction Deviance Explained
par(oldpar)


### print() method

?print.glmnet and ?print.cv.glmnet

### Extract/compute deviance

deviance(fitted_glmnet_object)

coxnet.deviance(pred = NULL, y, x = 0, offset = NULL,
weights = NULL, beta = NULL) # This calls a Fortran function loglike()


According to the source code, coxnet.deviance() returns 2 *(lsat-fitflog). coxnet.deviance() was used in assess.coxnet.R (CV, not intended for use by users) and buildPredmat.coxnetlist.R (CV). See also Survival → glmnet. ## cv.glmnet and deviance Usage cv.glmnet(x, y, weights = NULL, offset = NULL, lambda = NULL, type.measure = c("default", "mse", "deviance", "class", "auc", "mae", "C"), nfolds = 10, foldid = NULL, alignment = c("lambda", "fraction"), grouped = TRUE, keep = FALSE, parallel = FALSE, gamma = c(0, 0.25, 0.5, 0.75, 1), relax = FALSE, trace.it = 0, ...)  type.measure parameter (loss to use for CV): • default • type.measure = deviance which uses squared-error for gaussian models (a.k.a type.measure="mse"), logistic and poisson regression (PS: for binary response data I found that type='class' gives a discontinuous CV curve while 'deviance' give a smooth CV curve), • type.measure = partial-likelihood for the Cox model (note that the y-axis from plot.cv.glmnet() gives deviance but the values are quite different from what deviace() gives from a non-CV modelling). • mse or mae (mean absolute error) can be used by all models except the "cox"; they measure the deviation from the fitted mean to the response • class applies to binomial and multinomial logistic regression only, and gives misclassification error. • auc is for two-class logistic regression only, and gives area under the ROC curve. • C is Harrel's concordance measure, only available for cox models grouped parameter • This is an experimental argument, with default TRUE, and can be ignored by most users. • For the "cox" family, grouped=TRUE obtains the CV partial likelihood for the Kth fold by subtraction; by subtracting the log partial likelihood evaluated on the full dataset from that evaluated on the on the (K-1)/K dataset. This makes more efficient use of risk sets. With grouped=FALSE the log partial likelihood is computed only on the Kth fold • Gradient descent for the elastic net Cox-PH model package deviance CV survival -2*fitloglik[1]
glmnet deviance(fit)
assess.glmnet(fit, newx, newy, family = "cox")
coxnet.deviance(x, y, beta)
cv.glmnet(x, y, lambda, family="cox", nfolds, foldid)$cvm which calls cv.glmnet.raw() (save lasso models for each CV) which calls buildPredmat.coxnetlist(foldid) to create a matrix cvraw (eg 10x100) and then calls cv.coxnet(foldid) (it creates weights of length 10 based on status and do cvraw/=weights) and cvstats(foldid) which will calculate weighted mean of cvraw matrix for each lambda and return the cvm vector; cvm[1]=sum(original cvraw[,1])/sum(weights) Note: the document says the measure (cvm) is partial likelihood for survival data. But cvraw calculation from buildPredmat.coxnetlist() shows the original/unweighted cvraw is CVPL. BhGLM measure.bh(fit, new.x, new.y) cv.bh(fit, nfolds=10, foldid, ncv)$measures[1]
which calls an internal function measure.cox(y.obj, lp) (lp is obtained from CV)
which calls bcoxph() [coxph()] using lp as the covariate and returns
deviance = -2 * ff$loglik[2] (a CV deviance) cindex = summary(ff)$concordance[[1]] (a CV c-index)

An example from the glmnet vignette The deviance value is the same from both survival::deviance() and glmnet::deviance(). But how about cv.glmnet()$cvm (partial-likelihood)? library(glmnet) library(survival) library(tidyverse); library(magrittr) data(CoxExample) dim(x) # 1000 x 30 # I'll focus some lambdas based on one run of cv.glmnet() set.seed(1); cvfit = cv.glmnet(x, y, family = "cox", lambda=c(10,2,1,.237,.016,.003)) rbind(cvfit$lambda, cvfit$cvm, deviance(glmnet(x, y, family = "cox", lambda=c(10, 2, 1, .237, .016, .003))))%>% set_rownames(c("lambda", "cvm", "deviance")) [,1] [,2] [,3] [,4] [,5] [,6] lambda 10.00000 2.00000 1.00000 0.23700 0.01600 0.0030 cvm 13.70484 13.70484 13.70484 13.70316 13.07713 13.1101 deviance 8177.16378 8177.16378 8177.16378 8177.16378 7707.53515 7697.3357 -2* coxph(Surv(y[,1], y[, 2]) ~ x)$loglik[1]
[1] 8177.164
coxph(Surv(y[,1], y[, 2]) ~ x)$loglik[1] [1] -4088.582 # coxnet.deviance: compute the deviance (-2 log partial likelihood) for right-censored survival data fit1 = glmnet(x, y, family = "cox", lambda=.016) coxnet.deviance(x=x, y=y, beta=fit1$coef)
# [1] 8177.164
fit2 = glmnet(x, y, family = "cox", lambda=.003)
coxnet.deviance(x=x, y=y, beta=fit2$coef) # [1] 8177.164 # assess.glmnet assess.glmnet(fit1, newx=x, newy=y) #$deviance
# [1] 7707.444
# attr(,"measure")
# [1] "Partial Likelihood Deviance"
#
# $C # [1] 0.7331241 # attr(,"measure") # [1] "C-index" assess.glmnet(fit2, newx=x, newy=y) #$deviance
# [1] 7697.314
# attr(,"measure")
# [1] "Partial Likelihood Deviance"
#
# $C # [1] 0.7342417 # attr(,"measure") # [1] "C-index"  ### No need for glmnet if we have run cv.glmnet https://stats.stackexchange.com/a/77549 Do not supply a single value for lambda. Supply instead a decreasing sequence of lambda values. glmnet relies on its warms starts for speed, and its often faster to ﬁt a whole path than compute a single ﬁt. ### cv.glmnet in cox model Note: the CV result may changes unless we fix the random seed. Note that the y-axis on the plot depends on the type.measure parameter. It is not the objective function used to find the estimator. For survival data, the y-axis is deviance (-2*loglikelihood) [so the optimal lambda should give a minimal deviance value]. It is not always partial likelihood device has a largest value at a large lambda. In the following two plots, the first one is from the glmnet vignette and the 2nd one is from the coxnet vignette. The survival data are not sparse in both examples. Sparse data library(glmnet); library(survival) n = 100; p <- 1000 beta1 = 2; beta2 = -1; beta3 =1; beta4 = -2 lambdaT = .002 # baseline hazard lambdaC = .004 # hazard of censoring set.seed(1234) x1 = rnorm(n) x2 = rnorm(n) x3 <- rnorm(n) x4 <- rnorm(n) # true event time T = Vectorize(rweibull)(n=1, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2-beta3*x3-beta4*x4)) C = rweibull(n, shape=1, scale=lambdaC) #censoring time time = pmin(T,C) #observed time is min of censored and true event = time==T # set to 1 if event is observed cox <- coxph(Surv(time, event)~ x1 + x2 + x3 + x4); cox -2*cox$loglik[2] # deviance [1] 301.7461
summary(cox)$concordance[1] # 0.9006085 # create a sparse matrix X <- cbind(x1, x2, x3, x4, matrix(rnorm(n*(p-4)), nr=n)) colnames(X) <- paste0("x", 1:p) # X <- data.frame(X) y <- Surv(time, event) set.seed(1234) nfold <- 10 foldid <- sample(rep(seq(nfold), length = n)) cvfit <- cv.glmnet(X, y, family = "cox", foldid = foldid) plot(cvfit) plot(cvfit$lambda, log = "y")
assess.glmnet(cvfit, newx=X, newy = y, family="cox") # return deviance 361.4619 and C 0.897421
# Question: what lambda has been used?
# Ans: assess.glmnet() calls predict.cv.glmnet() which by default uses s = "lambda.1se"

fit <- glmnet(X, y, family = "cox", lambda = cvfit$lambda.min) assess.glmnet(fit, newx=X, newy = y, family="cox") # deviance 308.3646 and C 0.9382788 cvfit$cvm[cvfit$lambda == cvfit$lambda.min]
# [1] 7.959283

fit <- glmnet(X, y, family = "cox", lambda = cvfit$lambda.1se) assess.glmnet(fit, newx=X, newy = y, family="cox") # deviance 361.4786 and C 0.897421 deviance(fit) # [1] 361.4786 fit <- glmnet(X, y, family = "cox", lambda = 1e-3) assess.glmnet(fit, newx=X, newy = y, family="cox") # deviance 13.33405 and C 1 fit <- glmnet(X, y, family = "cox", lambda = 1e-8) assess.glmnet(fit, newx=X, newy = y, family="cox") # deviance 457.3695 and C .5 fit <- glmnet(cbind(x1,x2,x3,x4), y, family = "cox", lambda = 1e-8) assess.glmnet(fit, newx=X, newy = y, family="cox") # Error in h(simpleError(msg, call)) : # error in evaluating the argument 'x' in selecting a method for function 'as.matrix': Cholmod error # 'X and/or Y have wrong dimensions' at file ../MatrixOps/cholmod_sdmult.c, line 90 deviance(fit) # [1] 301.7462 library(BhGLM) X2 <- data.frame(X) f1 = bmlasso(X2, y, family = "cox", ss = c(.04, .5)) measure.bh(f1, X2, y) # deviance Cindex # 303.39 0.90 o <- cv.bh(f1, foldid = foldid) o$measures  # deviance and C
# deviance   Cindex
#  311.743    0.895


### update() function

update() will update and (by default) re-fit a model. It does this by extracting the call stored in the object, updating the call and (by default) evaluating that call. Sometimes it is useful to call update with only one argument, for example if the data frame has been corrected.

It can be used in glmnet() object without a new implementation method.

• Linear regression
lm(y ~ x + z, data=myData)
lm(y ~ x + z, data=subset(myData, sex=="female"))
lm(y ~ x + z, data=subset(myData, age > 30))

• Lasso regression
R> fit <- glmnet(glmnet(X, y, family="cox", lambda=cvfit$lambda.min); fit Call: glmnet(x = X, y = y, family = "cox", lambda = cvfit$lambda.min)

Df   %Dev Lambda
1 21 0.3002 0.1137

R> fit2 <- update(fit, subset = c(rep(T, 50), rep(F, 50)); fit2
Call:  glmnet(x = X[1:50, ], y = y[1:50], family = "cox", lambda = cvfit$lambda.min) Df %Dev Lambda 1 24 0.4449 0.1137 R> fit3 <- update(fit, lambda=cvfit$lambda); fit3

Call:  glmnet(x = X, y = y, family = "cox", lambda = cvfitlambda) Df %Dev Lambda 1 1 0.00000 0.34710 2 2 0.01597 0.33130 ...  ## Relaxed fit and $\displaystyle{ \gamma }$ parameter Relaxed fit: Take a glmnet fitted object, and then for each lambda, refit the variables in the active set without any penalization. Suppose the glmnet fitted linear predictor at $\displaystyle{ \lambda }$ is $\displaystyle{ \hat\eta_\lambda(x) }$ and the relaxed version is $\displaystyle{ \tilde\eta_\lambda(x) }$. We also allow for shrinkage between the two: \displaystyle{ \begin{align} \tilde \eta_{\lambda,\gamma}= \gamma\hat\eta_\lambda(x) + (1-\gamma)\tilde\eta_\lambda(x). \end{align} } $\displaystyle{ \gamma\in[0,1] }$ is an additional tuning parameter which can be selected by cross validation. The debiasing will potentially improve prediction performance, and CV will typically select a model with a smaller number of variables. The default behavior of extractor functions like predict and coef, as well as plot will be to present results from the glmnet fit (not cv.glmnet), unless a value of $\displaystyle{ \gamma }$ is given different from the default value $\displaystyle{ \gamma=1 }$. Question: how does cv.glmnet() select $\displaystyle{ \gamma }$ parameter? Ans: it depends on the parameter type.measure in cv.glmnet.  library(glmnet) data(QuickStartExample) fitr=glmnet(x,y, relax=TRUE) set.seed(1) cfitr=cv.glmnet(x,y,relax=TRUE) c(fitrlambda.min, fitr$lambda.1se) # [1] 0.08307327 0.15932708 str(cfitr$relaxed) plot(cfitr),oldpar <- par(mfrow=c(1,3), mar = c(5,4,6,2)) plot(fitr, main = expression(gamma == 1)) plot(fitr,gamma=0.5, main = expression(gamma == .5)) plot(fitr,gamma=0, main = expression(gamma == 0)) par(oldpar)  Special cases: $\displaystyle{ \gamma=1 }$: only regularized fit, no relaxed fit. $\displaystyle{ \gamma=0 }$: only relaxed fit; a faster version of forward stepwise regression. set.seed(1) cfitr2=cv.glmnet(x,y,gamma=0,relax=TRUE) # default gamma = c(0, 0.25, 0.5, 0.75, 1) plot(cfitr2) c(cfitr2$lambda.min, cfitr2$lambda.1se) # [1] 0.08307327 0.15932708 str(cfitr2$relaxed)  ### Computation time beta1 = 2; beta2 = -1 lambdaT = .002 # baseline hazard lambdaC = .004 # hazard of censoring set.seed(1234) x1 = rnorm(n) x2 = rnorm(n) # true event time T = Vectorize(rweibull)(n=1, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2)) # No censoring event2 <- rep(1, length(T)) system.time(fit <- cv.glmnet(x, Surv(T,event2), family = 'cox')) # user system elapsed # 4.701 0.016 4.721 system.time(fitr <- cv.glmnet(x, Surv(T,event2), family = 'cox', relax= TRUE)) # user system elapsed # 161.002 0.382 161.573  ## predict() and coef() methods ?predict.glmnet OR ?coef.glmnet OR ?coef.relaxed. Similar to other predict methods, this functions predicts fitted values, logits, coefficients and more from a fitted "glmnet" object. ## S3 method for class 'glmnet' predict(object, newx, s = NULL, type = c("link", "response", "coefficients", "nonzero", "class"), exact = FALSE, newoffset, ...) ## S3 method for class 'relaxed' predict(object, newx, s = NULL, gamma = 1, type = c("link", "response", "coefficients", "nonzero", "class"), exact = FALSE, newoffset, ...) ## S3 method for class 'glmnet' coef(object, s = NULL, exact = FALSE, ...)  ?predict.cv.glmnet OR ?coef.cv.glmnet OR ?coef.cv.relaxed. This function makes predictions from a cross-validated glmnet model, using the stored "glmnet.fit" object, and the optimal value chosen for lambda (and gamma for a 'relaxed' fit). ## S3 method for class 'cv.glmnet' predict(object, newx, s = c("lambda.1se", "lambda.min"), ...) ## S3 method for class 'cv.relaxed' predict(object, newx, s = c("lambda.1se", "lambda.min"), gamma = c("gamma.1se", "gamma.min"), ...)  ### Cindex Usage: Cindex(pred, y, weights = rep(1, nrow(y)))  ### assess.glmnet Usage: assess.glmnet(object, newx = NULL, newy, weights = NULL, family = c("gaussian", "binomial", "poisson", "multinomial", "cox", "mgaussian"), ...) confusion.glmnet(object, newx = NULL, newy, family = c("binomial", "multinomial"), ...) roc.glmnet(object, newx = NULL, newy, ...)  ### Variable importance ### R-square/R2 ## ridge regression # example from SGL package set.seed(1) n = 50; p = 100; size.groups = 10 X = matrix(rnorm(n * p), ncol = p, nrow = n) beta = (-2:2) y = X[,1:5] %*% beta + 0.1*rnorm(n) data = list(x = X, y = y) cvfit <- cv.glmnet(X, y, alpha = 0) plot(cvfit) o <- coef(cvfit, lambda = cvfit$lambda.min) %>% drop()

sum(o != 0) # [1] 101.
# Too biased.
o[1:10]
#   (Intercept)            V1            V2            V3            V4
# -3.269401e-01 -2.253226e-36 -8.900799e-37  5.198885e-37  1.311976e-36
#            V5            V6            V7            V8            V9
#  1.873125e-36  1.582532e-37  2.085781e-37  4.732839e-37  2.997614e-37

y_predicted <- predict(cvfit, s = cvfit$lambda.min, newx = X) # Sum of Squares Total and Error sst <- sum((y - mean(y))^2) sse <- sum((y_predicted - y)^2) # R squared rsq <- 1 - sse / sst rsq # 0.46 library(SGL) # sparse group lasso set.seed(1) index <- ceiling(1:p / size.groups) cvFit = cvSGL(data, index, type = "linear", alpha=.95) # this alpha is the default plot(cvFit) cvFit$fit$beta[, 20] # 20th lambda gives smallest negative log likelihood # identify correct predictors # [1] -10.942712 -6.167799 0.000000 6.595406 14.442019 0.000000 ... set.seed(1) cvFit2 = cvSGL(data, index, type = "linear", alpha=0) plot(cvFit2) cvFit2$fit$beta[, 20] # [1] -10.8417371 -6.5251240 0.2476438 6.7223001 14.1605263 0.2149542 # [7] 0.2481450 0.1404282 0.1799818 0.3784596 0.0000000 0.0000000 ...  • Tikhonov regularization (ridge regression). It was used to handle ill-posed/overfitting situation. Ridge regression shrinks the coefficients by a uniform factor of $\displaystyle{ {\displaystyle (1+N\lambda )^{-1}}{\displaystyle (1+N\lambda )^{-1}} }$ and does not set any coefficients to zero. • cvSGL • How and when: ridge regression with glmnet. On training data, ridge regression fits less well than the OLS but the parameter estimate is more stable. So it does better in prediction because it is less sensitive to extreme variance in the data such as outliers. ## Group lasso • pcLasso: Principal Components Lasso package • pclasso paper, slides, Blog • Each feature must be assigned to a group • It allows to assign each feature to groups (including overlapping). library(pcLasso) set.seed(1) n = 50; p = 100; size.groups = 10 index <- ceiling(1:p / size.groups) X = matrix(rnorm(n * p), ncol = p, nrow = n) beta = (-2:2) y = X[,1:5] %*% beta + 0.1*rnorm(n) groups <- vector("list", 3) for (k in 1:2) { groups[[k]] <- 5 * (k-1) + 1:5 } groups[[3]] <- 11:p cvfit <- cv.pcLasso(X, y, ratio = 0.8, groups = groups) plot(cvfit) pred.pclasso <- predict(cvfit, xnew = X, s = "lambda.min") mean((y-pred.pclasso)^2) # [1] 1.956387 library(SGL) index <- ceiling(1:p / size.groups) data = list(x = X, y = y) set.seed(1) cvFit = cvSGL(data, index, type = "linear") Fit = SGL(data, index, type = "linear") # SGL() uses a different set of lambdas than cvSGL() does # After looking at cvFit$lambdas; Fit$lambdas # I should pick the last lambda pred.SGL <- predictSGL(Fit, X, length(Fit$lambdas))
mean((y-pred.SGL)^2)    # [1] 0.146027

library(ggplot2)
library(tidyr)
dat <- tibble(y=y, SGL=pred.SGL, pclasso=pred.pclasso) %>% gather("method", "predict", 2:3)

ggplot(dat, aes(x=y, y=predict, color=method)) + geom_point(shape=1)


## penalty.factor

The is available in glmnet() but not in cv.glmnet().

• Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties, Fan & Li (2001) JASA
• Adaptive Lasso: What it is and how to implement in R. Adaptive lasso weeks to minimize $\displaystyle{ RSS(\beta) + \lambda \sum_1^p \hat{\omega}_j |\beta_j| }$ where $\displaystyle{ \lambda }$ is the tuning parameter, $\displaystyle{ \hat{\omega}_j = \frac{1}{(|\hat{\beta}_j^{ini}|)^\gamma} }$ is the adaptive weights vector and $\displaystyle{ \hat{\beta}_j^{ini} }$ is an initial estimate of the coefficients obtained through ridge regression. Adaptive Lasso ends up penalizing more those coefficients with lower initial estimates. $\displaystyle{ \gamma }$ is a positive constant for adjustment of the adaptive weight vector, and the authors suggest the possible values of 0.5, 1 and 2.
• When n goes to infinity, $\displaystyle{ \hat{\omega}_j |\beta_j| }$ converges to $\displaystyle{ I(\beta_j \neq 0) }$. So the adaptive Lasso procedure can be regarded as an automatic implementation of best-subset selection in some asymptotic sense.
• What is the oracle property of an estimator? An oracle estimator must be consistent in 1) variable selection and 2) consistent parameter estimation.
• Oracle property: Oracle property is a name given to techniques for estimating the regression parameters in the models fitted to high-dimensional data which have the property that they can correctly select the nonzero coefficients with the probability converging to one and that the estimators of nonzero coefficients are asymptotically normal with the identical means and covariances that they would have if the zero coefficients were known in advance that is the estimators are asymptotically as efficient as the ideal estimation assisted by an 'oracle' who knows which coefficients are nonzero.
• (Linear regression) The adaptive lasso and its oracle properties Zou (2006, JASA)
• (Cox model) Adaptive-LASSO for Cox's proportional hazard model by Zhang and Lu (2007, Biometrika)
• When the LASSO fails???. Adaptive lasso is used to demonstrate its usefulness.

## Survival data

data(CoxExample)
dim(x) # 1000 x 30
ind.train <- 1:nrow(x)/2
cv.fit <- cv.glmnet(x[ind.train, ], y[ind.train, ], family="cox")
coef(cv.fit, s = "lambda.min")
# 30 x 1, coef(...) is equivalent to predict(type="coefficients",...)
pr <- predict(cv.fit, x[-ind.train, ], type = "link", s = "lambda.min")
# the default option for type is 'link' which gives the linear predictors for "cox" models.
pr2 <- x[-ind.train, ] %*% coef(cv.fit, s = "lambda.min")
all.equal(pr[, 1], pr2[, 1]) # [1] TRUE
class(pr) # [1] "matrix"
class(pr2) # [1] "dgeMatrix"
# we can also use predict() with glmnet object
pr22 <- predict(fit, x[-ind.train, ], type='link', s = cv.fit$lambda.min) all.equal(pr[, 1], pr22[, 1]) # [1] TRUE pr3 <- predict(cv.fit, x[-ind.train, ], type = "response", s = "lambda.min") range(pr3) # relative risk [1] 0.05310623 19.80143519  ## glmnet 4.0: family ## Timing nvec <- c(100, 400) pvec <- c(1000, 5000) for(n in nvec) for(p in pvec) { nzc = trunc(p/10) # 1/10 variables are non-zeros x = matrix(rnorm(n * p), n, p) beta = rnorm(nzc) fx = x[, seq(nzc)] %*% beta/3 hx = exp(fx) ty = rexp(n, hx) tcens = rbinom(n = n, prob = 0.3, size = 1) # censoring indicator y = cbind(time = ty, status = 1 - tcens) # y=Surv(ty,1-tcens) with library(survival) foldid = sample(rep(seq(10), length = n)) o <- system.time(fit1_cv <- cv.glmnet(x, y, family = "cox", foldid = foldid) ) cat(sprintf("n=%1d, p=%5d, time=%8.3f\n", n, p, o['elapsed'])) }  Running on core i7 2.6GHz, macbook pro 2018. Unit is seconds. p ∖ n 100 400 1000 5 58 5000 9 96 Xeon E5-2680 v4 @ 2.40GHz p ∖ n 100 400 1000 7 98 5000 13 125 ## All interactions ## Algorithm ### warm start ### Gradient descent ### Cyclic coordinate descent ### Quadratic programming ## Other ## Release ## Discussion # Prediction ## ROC # LASSO vs Least angle regression # R packages related to glmnet ## caret • coefficients from glmnet and caret are different for the same lambda. • the exact lambda you specified was not used by caret. the coefficients are interpolated from the coefficients actually calculated. • when you provide lambda to the glmnet call the model returns exact coefficients for that lambda • library(caret) set.seed(0) train_control = trainControl(method = 'cv', number = 10) grid = 10 ^ seq(5, -2, length = 100) tune.grid = expand.grid(lambda = grid, alpha = 0) ridge.caret = train(x[train, ], y[train], method = 'glmnet', trControl = train_control, tuneGrid = tune.grid) ridge.caret$bestTune

ridge_full <- train(x, y,
method = 'glmnet',
trControl = trainControl(method = 'none'),
tuneGrid = expand.grid(
lambda = ridge.caret$bestTune$lambda, alpha = 0)
)
coef(ridge_full$finalModel, s = ridge.caret$bestTune$lambda)  • R caret train glmnet final model lambda values not as specified • trainControl() • method - "boot", "boot632", "optimism_boot", "boot_all", "cv", "repeatedcv", "LOOCV", "LGOCV", ... • number - Either the number of folds or number of resampling iterations • repeats - For repeated k-fold cross-validation only: the number of complete sets of folds to compute • search - Either "grid" or "random", describing how the tuning parameter grid is determined. • seeds - an optional set of integers that will be used to set the seed at each resampling iteration. This is useful when the models are run in parallel. • train(). A long printout comes from the foreach loop when the nominalTrainWorkflow() function is called. • method - A string specifying which classification or regression model to use. Some examples are knn, lm, nnet, rpart, glmboost, ... Possible values are found using names(getModelInfo()). • metric - ifelse(is.factor(y_dat), "Accuracy", "RMSE"). By default, possible values are "RMSE" and "Rsquared" for regression and "Accuracy" and "Kappa" for classification. If custom performance metrics are used (via the summaryFunction argument in trainControl, the value of metric should match one of the arguments. • maximize - ifelse(metric %in% c("RMSE", "logLoss", "MAE"), FALSE, TRUE) • Return object from train() • results: matrix. Each row = one parameter (alpha, lambda) • finalModel$lambda: vector of very long length instead of what we specify. What is this?
• finalModel$lambdaOpt • finalModel$tuneValue$alpha, finalModel$tuneValue$lambda • finalModel$a0
• finalModel$beta • Regularization: Ridge, Lasso and Elastic Net. Notice the repeatedcv method. If repeatedcv is selected, the performance RMSE is computed by the average of (# CV * # reps) RMSEs for each (alpha, lambda). Then the best tuning parameter (alpha, lambda) is selected with the minimum RMSE in performance. See the releated code at Line 679, Line 744,Line 759 where trControl$selectionFunction()=best().
library(glmnet)  # for ridge regression
library(dplyr)   # for data cleaning

data("mtcars")
# Center y, X will be standardized in the modelling function
y <- mtcars %>% select(mpg) %>% scale(center = TRUE, scale = FALSE) %>% as.matrix()
X <- mtcars %>% select(-mpg) %>% as.matrix()
dim(X)  # [1] 32 10

library(caret)
# Set training control
train_control <- trainControl(method = "repeatedcv",
number = 3,    # 3-fold CV, 2 times
repeats = 2,
search = "grid",  # "random"
verboseIter = TRUE)

# Train the model
set.seed(123)    # seed for reproducibility
elastic_net_model <- train(mpg ~ .,
data = cbind(y, X),
method = "glmnet",
preProcess = c("center", "scale"),
tuneLength = 4,  # 4 alphas x 4 lambdas
trControl = train_control)

# Check multiple R-squared
y_hat_enet <- predict(elastic_net_model, X)
cor(y, y_hat_enet)^2

names(elastic_net_model)
#  [1] "method"       "modelInfo"    "modelType"    "results"      "pred"
#  [6] "bestTune"     "call"         "dots"         "metric"       "control"
# [11] "finalModel"   "preProcess"   "trainingData" "resample"     "resampledCM"
# [16] "perfNames"    "maximize"     "yLimits"      "times"        "levels"
# [21] "terms"        "coefnames"    "xlevels"
elastic_net_model$bestTune # alpha and lambda names(elastic_net_model$finalModel)
#  [1] "a0"          "beta"        "df"          "dim"         "lambda"
#  [6] "dev.ratio"   "nulldev"     "npasses"     "jerr"        "offset"
# [11] "call"        "nobs"        "lambdaOpt"   "xNames"      "problemType"
# [16] "tuneValue"   "obsLevels"   "param"
length(elastic_net_model$finalModel$lambda)
# [1] 95
length(elastic_net_model$finalModel$lambdaOpt)
# [1] 1
dim(elastic_net_model$finalModel$beta)
# [1] 10 95
which(elastic_net_model$finalModel$lambda == elastic_net_model$finalModel$lambdaOpt)
# integer(0)      <=== Weird, why

• For the Repeated k-fold Cross Validation. The final model accuracy is taken as the mean from the number of repeats.
• Penalized Regression Essentials: Ridge, Lasso & Elastic Net -> Using caret package. tuneLength was used. Note that the results element gives the details for each candidate of parameter combination.
# Load the data
data("Boston", package = "MASS")
# Split the data into training and test set
set.seed(123)
training.samples <- Boston$medv %>% createDataPartition(p = 0.8, list = FALSE) train.data <- Boston[training.samples, ] set.seed(123) # affect CV samples, not affect tuning parameter set cv_5 <- trainControl(method = "cv", number = 5) # number of folds model <- train( medv ~., data = train.data, method = "glmnet", trControl = cv_5, tuneLength = 10 # 10 alphas x 10 lambdas ) model # one row per parameter combination model$results
# Best tuning parameter
model\$bestTune


## biglasso

biglasso: Extend Lasso Model Fitting to Big Data in R