R help - Apr 2015 - glmnet: converting coefficients back to original scale

Dear R-help,

I'm having trouble understanding how glmnet converts its coefficient
estimates back to the original scale. Here's an example with ridge
regression:

########################################################################
library(glmnet)
set.seed(1)

n <- 20  # sample size

d <- data.frame(x1 = rnorm(n, 1, 1), x2 = rnorm(n, 10, 2), y = rnorm(n, 1,
2))

# Sample means
mx1 <- mean(d$x1)
mx2 <- mean(d$x2)
my <- mean(d$y)

# Scaling factors ("1/n standard deviations")
sx1 <- sd(d$x1)*sqrt((n - 1)/n)
sx2 <- sd(d$x2)*sqrt((n - 1)/n)
sy <- sd(d$y)*sqrt((n - 1)/n)

# Scaled variables
d$x1s <- (d$x1 - mx1)/sx1
d$x2s <- (d$x2 - mx2)/sx2
d$ys <- (d$y - my)/sy

lam <- 0.5  # lambda value

# Using scaled variables (same result for standardize=TRUE and
standardize=FALSE)
glmnet1 <- glmnet(as.matrix(d[, c("x1s", "x2s")]), d$ys,
alpha=0, lambda = lam)

# Using unscaled variables
glmnet2 <- glmnet(as.matrix(d[, c("x1", "x2")]), d$y,
alpha=0, lambda=lam)

coef(glmnet2)
##                     s0
## (Intercept)  2.5658491
## x1           0.3471199
## x2          -0.1703715

# I want to calculate the glmnet2 coef estimates from the glmnet1 coef
estimates.
# The following attempts are based on rearrangement of
# (y - my)/sy = beta1*(x1 - mx1)/sx1 + beta2*(x2 - mx2)/sx2

my - coef(glmnet1)["x1s", "s0"]*mx1*sy/sx1 -
coef(glmnet1)["x2s",
"s0"]*mx2*sy/sx2
# 2.430971
# Not the same as coef(glmnet2)["(Intercept)", "s0"]

coef(glmnet1)["x1s", "s0"]*sy/sx1
# 0.3096897
# Not the same as coef(glmnet2)["x1", "s0"]

coef(glmnet1)["x2s", "s0"]*sy/sx2
# -0.1524043
# Not the same as coef(glmnet2)["x2", "s0"]

######################################################################

I can apply a similar method (with centring of y instead of
standardisation) to successfully get the coefficient estimates on the
original scale given by lm.ridge in the MASS package. I would
appreciate any help anyone can give on where I'm going wrong with
glmnet.

Thanks,
Mark

This is interesting, can you post your lm.ridge solution as well?  I
suspect in glmnet, you need to use model.matrix with intercept, that
could be the reason.

-m

Thanks for your reply Mehmet. I've found that the problem was that I
didn't scale the lambda value. My original example did not follow the
instruction not to give a single lambda value, but that in itself
wasn't the problem. Example shown below.

library(glmnet)
library(MASS)

set.seed(1)
n <- 20

d <- data.frame(x1 = rnorm(n, 1, 1),
                x2 = rnorm(n, 10, 2),
                y = rnorm(n, 1, 2))

# Sample means
mx1 <- mean(d$x1)
mx2 <- mean(d$x2)
my <- mean(d$y)

# Scaling factors
sx1 <- sd(d$x1)*sqrt((n-1)/n)
sx2 <- sd(d$x2)*sqrt((n-1)/n)
sy <- sd(d$y)*sqrt((n-1)/n)

# Scaled variables
d$x1s <- (d$x1 - mx1)/sx1
d$x2s <- (d$x2 - mx2)/sx2
d$ys <- (d$y - my)/sy

# Centred y
d$yc <- d$y - my

lam <- 1  # lambda value for lm.ridge

lmr1 <- lm.ridge(y ~ x1 + x2, data=d, lambda=lam)
lmr2 <- lm.ridge(yc ~ x1s + x2s, data=d, lambda=lam)

coef(lmr1)

my - coef(lmr2)["x1s"]*mx1/sx1 - coef(lmr2)["x2s"]*mx2/sx2
# same as coef(lmr1)[1]

coef(lmr2)["x1s"]/sx1  # same as coef(lmr1)["x1"]
coef(lmr2)["x2s"]/sx2  # same as coef(lmr1)["x2"]

glmnet1 <- glmnet(as.matrix(d[, c("x1", "x2")]), d[,
"y"], alpha=0)
glmnet2 <- glmnet(as.matrix(d[, c("x1s", "x2s")]), d[,
"ys"], alpha=0)

# Note: glmnet1$lambda is glmnet2$lambda*sy

ind <- 80  # index of lambda values to look at

coef(glmnet1)[, ind]

my - coef(glmnet2)["x1s", ind]*mx1*sy/sx1 -
  coef(glmnet2)["x2s", ind]*mx2*sy/sx2
# same as coef(glmnet1)["(Intercept)", ind]

coef(glmnet2)["x1s", ind]*sy/sx1
# same as coef(glmnet1)["x1", ind]

coef(glmnet2)["x2s", ind]*sy/sx2
# same as coef(glmnet1)["x2", ind]


On Sat, Apr 4, 2015 at 6:03 AM, Suzen, Mehmet <mehmet.suzen at
physics.org> wrote:
> This is interesting, can you post your lm.ridge solution as well?  I
> suspect in glmnet, you need to use model.matrix with intercept, that
> could be the reason.
>
> -m