The package implements a collection of Generalized Elastic Net (GELnet) solvers, as outlined in the following publication: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004790 This vignette covers the basic usage of the gelnet package. The interface to the solvers was designed to be very flexible, by allowing the user to specify a large number of parameters. At the same time, nearly all of the parameters are given reasonable default values, making the interface easy to use if the additional flexibility is not required.
Let \(X\) be a samples-by-features data matrix and \(y\) be a column vector of labels. Typically, \(X\) and \(y\) are determined by the prediction task at hand, but for the purposes of this tutorial, we are going to generate them randomly:
Let’s further assume that we are given a feature-feature relationship matrix \(A\). If working with genomic features, \(A\) might be the adjacency of a gene-gene interaction network. Again, we are going to generate a random matrix for the purposes of this tutorial:
We are now going to utilize the GELnet toolkit to learn a linear regression model, such that the model weights are more similar for the features that share an interaction on \(A\). As discussed in the manuscript, this can be achieved by formulating a feature-feature penalty matrix using either the graph Laplacian or \((I-D)\), where \(D\) is the graph diffusion matrix and \(I\) is the identity matrix. The gelnet package provides a function to compute the graph Laplacian from the adjacency. Here, we utilize the normalized Laplacian to keep the penalty term on the same scale as the traditional ridge regression:
The model can now be learned via
## Training a linear regression model
## Running linear regression optimization with L1 = 0.100000, L2 = 1.000000
## f = 0.395022 after iteration 7where we set the L1-norm and L2-norm penalties to 0.1 and 1, respectively. The response for new samples is computed via the dot product with the weights:
## [,1]
## [1,] -0.218991921
## [2,] 0.112029376
## [3,] 0.031200391
## [4,] 0.240312629
## [5,] 0.176214063
## [6,] -0.015642708
## [7,] -0.546951047
## [8,] 0.007920018
## [9,] 0.138883811
## [10,] -0.328687091Linear regression is one of the three types of prediction problems supported by the package. The other two are binary logistic regression and one-class logistic regression. The latter is outlined in the following paper: http://psb.stanford.edu/psb-online/proceedings/psb16/sokolov.pdf
The package recognizes the problem type based on the class of the \(y\) argument. To train a binary predictor, we have to provide \(y\) as a two-level factor, where the first level is treated as the positive class.
## Training a logistic regression model
## Treating TRUE as the positive class
## Running logistic regression optimization with L1 = 0.100000, L2 = 1.000000
## Iteration 1: f = 0.693147
## Iteration 2: f = 0.608485
## Iteration 3: f = 0.608078If we were to score the training data using this model, we can observe that the positive samples are receiving higher scores than the negative ones
## scores labels
## 1 -0.89901216 FALSE
## 2 -0.83224434 FALSE
## 3 -1.18637134 FALSE
## 4 -0.67708084 FALSE
## 5 -0.77444729 FALSE
## 6 -0.98702967 FALSE
## 7 -0.35902005 TRUE
## 8 0.04331193 TRUE
## 9 -0.98796815 FALSE
## 10 -0.34270976 TRUE
## 11 -0.79347006 FALSE
## 12 -1.02688260 FALSE
## 13 -0.14885098 TRUE
## 14 -0.42396270 FALSE
## 15 -0.73966627 FALSE
## 16 -0.11734756 TRUE
## 17 -0.92818971 FALSE
## 18 -0.30958543 TRUE
## 19 -0.80571106 FALSE
## 20 -0.41120076 TRUEHowever, if there is class imbalance, the scores will tend to be skewed towards the class with more samples. This can be addressed by using an additional flag when training the model
## Training a logistic regression model
## Treating TRUE as the positive class
## Running logistic regression optimization with L1 = 0.100000, L2 = 1.000000
## Iteration 1: f = 0.693147
## Iteration 2: f = 0.641172
## Iteration 3: f = 0.641149## scores labels
## 1 -0.4102139 FALSE
## 2 -0.3455457 FALSE
## 3 -0.7191678 FALSE
## 4 -0.1364620 FALSE
## 5 -0.2854379 FALSE
## 6 -0.5344566 FALSE
## 7 0.2123350 TRUE
## 8 0.6414655 TRUE
## 9 -0.5065721 FALSE
## 10 0.2381333 TRUE
## 11 -0.2852109 FALSE
## 12 -0.5505074 FALSE
## 13 0.4646070 TRUE
## 14 0.1366465 FALSE
## 15 -0.2343444 FALSE
## 16 0.4680499 TRUE
## 17 -0.4338957 FALSE
## 18 0.2748091 TRUE
## 19 -0.3068224 FALSE
## 20 0.1778635 TRUETraditionally, the loss function for logistic regression is averaged over \(n\), the number of samples. This causes every sample to make the same contribution to the loss, which is what causes the skew towards the larger class. By using the balanced flag, the problem is reformulated slightly such that the loss is averaged over the positive and negative samples separately, and then the mean of both averages is used as the overall loss.
Finally, we can build a one-class logistic regression model using just the positive samples. To train a one-class model we simply provide NULL for the \(y\) argument:
## Training a one-class model
## Iteration 1 : f = 0.6931472
## Iteration 2 : f = 0.5214702
## Iteration 3 : f = 0.520937The model can now be used as a detector that recognizes the positive samples
## scores labels
## 1 -0.4102139 FALSE
## 2 -0.3455457 FALSE
## 3 -0.7191678 FALSE
## 4 -0.1364620 FALSE
## 5 -0.2854379 FALSE
## 6 -0.5344566 FALSE
## 7 0.2123350 TRUE
## 8 0.6414655 TRUE
## 9 -0.5065721 FALSE
## 10 0.2381333 TRUE
## 11 -0.2852109 FALSE
## 12 -0.5505074 FALSE
## 13 0.4646070 TRUE
## 14 0.1366465 FALSE
## 15 -0.2343444 FALSE
## 16 0.4680499 TRUE
## 17 -0.4338957 FALSE
## 18 0.2748091 TRUE
## 19 -0.3068224 FALSE
## 20 0.1778635 TRUE