glmnet is an R package by Jerome Friedman, Trevor Hastie, Rob Tibshirani that fits entire Lasso or ElasticNet regularization paths for linear, logistic, multinomial, and Cox models using cyclic coordinate descent. This Julia package wraps the Fortran code from glmnet.
To fit a basic regression model:
julia> using GLMNet
julia> y = collect(1:100) + randn(100)*10;
julia> X = [1:100 (1:100)+randn(100)*5 (1:100)+randn(100)*10 (1:100)+randn(100)*20];
julia> path = glmnet(X, y)
Least Squares GLMNet Solution Path (86 solutions for 4 predictors in 930 passes):
──────────────────────────────
df pct_dev λ
──────────────────────────────
[1] 0 0.0 30.0573
[2] 1 0.152922 27.3871
[3] 1 0.279881 24.9541
:
[84] 4 0.90719 0.0133172
[85] 4 0.9072 0.0121342
[86] 4 0.907209 0.0110562
──────────────────────────────path represents the Lasso or ElasticNet fits for varying values of λ. The value of the intercept for each λ value are in path.a0. The coefficients for each fit are stored in compressed form in path.betas.
julia> path.betas
4×86 CompressedPredictorMatrix:
0.0 0.0925032 0.176789 0.253587 0.323562 0.387321 0.445416 0.498349 0.546581 0.590527 0.63057 0.667055 0.700299 … 1.33905 1.34855 1.35822 1.36768 1.37563 1.3829 1.39005 1.39641 1.40204 1.40702 1.41195
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.165771 -0.17235 -0.178991 -0.185479 -0.190945 -0.195942 -0.200851 -0.20521 -0.209079 -0.212501 -0.215883
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.00968611 -0.0117121 -0.0135919 -0.0154413 -0.0169859 -0.0183965 -0.0197951 -0.0210362 -0.0221345 -0.0231023 -0.0240649
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.110093 -0.110505 -0.111078 -0.11163 -0.112102 -0.112533 -0.112951 -0.113324 -0.113656 -0.113953 -0.11424This CompressedPredictorMatrix can be indexed as any other AbstractMatrix, or converted to a Matrix using convert(Matrix, path.betas).
One can visualize the path by
julia> using Plots, LinearAlgebra, LaTeXStrings
julia> betaNorm = [norm(x, 1) for x in eachslice(path.betas,dims=2)];
julia> extraOptions = (xlabel=L"\| \beta \|_1",ylabel=L"\beta_i", legend=:topleft,legendtitle="Variable", labels=[1 2 3 4]);
julia> plot(betaNorm, path.betas'; extraOptions...)
To predict the output for each model along the path for a given set of predictors, use predict:
julia> predict(path, [22 22+randn()*5 22+randn()*10 22+randn()*20])
1×86 Array{Float64,2}:
50.3295 47.6932 45.291 43.1023 41.108 39.2909 37.6352 36.1265 34.7519 33.4995 32.3583 31.3184 30.371 29.5077 28.7211 28.0044 … 21.3966 21.3129 21.2472 21.1746 21.1191 21.0655 21.0127 20.9687 20.9284 20.8885 20.8531 20.8218 20.7942 20.7667To find the best value of λ by cross-validation, use glmnetcv:
julia> cv = glmnetcv(X, y)
Least Squares GLMNet Cross Validation
86 models for 4 predictors in 10 folds
Best λ 0.136 (mean loss 101.530, std 10.940)
julia> argmin(cv.meanloss)
59
julia> coef(cv) # equivalent to cv.path.betas[:, 59]
4-element Array{Float64,1}:
1.1277676556880305
0.0
0.0
-0.08747434292954445julia> using RDatasets
julia> iris = dataset("datasets", "iris");
julia> X = convert(Matrix, iris[:, 1:4]);
julia> y = convert(Vector, iris[:Species]);
julia> iTrain = sample(1:size(X,1), 100, replace = false);
julia> iTest = setdiff(1:size(X,1), iTrain);
julia> iris_cv = glmnetcv(X[iTrain, :], y[iTrain])
Multinomial GLMNet Cross Validation
100 models for 4 predictors in 10 folds
Best λ 0.001 (mean loss 0.130, std 0.054)
julia> yht = round.(predict(iris_cv, X[iTest, :], outtype = :prob), digits=3);
julia> DataFrame(target=y[iTest], set=yht[:,1], ver=yht[:,2], vir=yht[:,3])[5:5:50,:]
10×4 DataFrame
│ Row │ target │ set │ ver │ vir │
│ │ Cat… │ Float64 │ Float64 │ Float64 │
├─────┼────────────┼─────────┼─────────┼─────────┤
│ 1 │ setosa │ 0.997 │ 0.003 │ 0.0 │
│ 2 │ setosa │ 0.995 │ 0.005 │ 0.0 │
│ 3 │ setosa │ 0.999 │ 0.001 │ 0.0 │
│ 4 │ versicolor │ 0.0 │ 0.997 │ 0.003 │
│ 5 │ versicolor │ 0.0 │ 0.36 │ 0.64 │
│ 6 │ versicolor │ 0.0 │ 0.05 │ 0.95 │
│ 7 │ virginica │ 0.0 │ 0.002 │ 0.998 │
│ 8 │ virginica │ 0.0 │ 0.001 │ 0.999 │
│ 9 │ virginica │ 0.0 │ 0.0 │ 1.0 │
│ 10 │ virginica │ 0.0 │ 0.001 │ 0.999 │
julia> irisLabels = reshape(names(iris)[1:4],(1,4));
julia> βs =iris_cv.path.betas;
julia> λs= iris_cv.lambda;
julia> sharedOpts =(legend=false, xlabel=L"\lambda", xscale=:log10)
julia> p1 = plot(λs,βs[:,1,:]',ylabel=L"\beta_i";sharedOpts...);
julia> p2 = plot(λs,βs[:,2,:]',title="Across Cross Validation runs";sharedOpts...);
julia> p3 = plot(λs,βs[:,3,:]', legend=:topright,legendtitle="Variable", labels=irisLabels,xlabel=L"\lambda",xscale=:log10);
julia> plot(p1,p2,p3,layout=(1,3))
julia> plot(iris_cv.lambda, iris_cv.meanloss, xscale=:log10, legend=false, yerror=iris_cv.stdloss,xlabel=L"\lambda",ylabel="loss")
julia> vline!([lambdamin(iris_cv)])
glmnet has two required parameters: the n x m predictor matrix X and the dependent variable y. It additionally accepts an optional third argument, family, which can be used to specify a generalized linear model. Currently, Normal() (least squares, default), MvNormal() Multi-response gaussian, Binomial() (logistic), Poisson() , Multinomial(), CoxPH() (Cox model) are supported.
- For linear and Poisson models,
yis a numerical vector of length n. - For logistic models,
yis either a string vector of length n or a n x 2 matrix, where the first column is the count of negative responses for each row inXand the second column is the count of positive responses. - For multinomial models,
yis either a string vector (with at least 3 unique values) or a n x k matrix, where k is the number of unique values (classes). - For Cox models,
yis a n x 2 matrix, where the first column is survival time and the second column is (right) censoring status. Note that for survival data,glmnethas another methodglmnet(X::Matrix, time::Vector, status::Vector)(glmnetcvhas a corresponding method as well).
glmnet also accepts many optional keyword parameters as described below:
weights: A vector of length n of weights.alpha: The tradeoff between lasso and ridge regression. This defaults to1.0, which specifies a lasso model.penalty_factor: A vector of length m of penalties for each predictor/column inX. This defaults to all ones, which weighs each predictor equally. To unpenalize a predictor, set the corresponding entry to zero.constraints: A 2 x m matrix specifying lower bounds (first row) and upper bounds (second row) of the predictors. By default, this is[-Inf; Inf]for each predictor inX.dfmax: The maximum number of predictors in the largest model.pmax: The maximum number of predictors in any model.nlambda: The number of values of λ along the path to consider.lambda_min_ratio: The smallest λ value to consider, as a ratio of the λ value that gives the null model (i.e., the model with only an intercept). If the number of observations (n) exceeds the number of variables (m), this defaults to0.0001, otherwise0.01.lambda: The λ values to consider. By default, this is determined fromnlambdaandlambda_min_ratio.tol: Convergence criterion, with the default value of1e-7.standardize: Whether to standardize predictors so that they are in the same units, with the default value oftrue. Beta values are always presented on the original scale.standardize_response: (only forMvNormal()), Whether to standardize the response variables so that they are the same units. defaults tofalse.intercept: Whether to fit an intercept term. The intercept is always unpenalized. The default value istrue.maxit: The maximum number of iterations of the cyclic coordinate descent algorithm. If convergence is not achieved, a warning is returned. The default value is1e6.