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I read from literature that the following two methods can be used for feature selection prior to model development: 1. Correlation factor between target and feature variables (select those features that have correlation > threshold) 2. Lasso

Which of the above two methods is preferred?

In one of the exercises I did, Lasso retained some features which have a lower correlation than the features it dropped. In other words, the above two methods didn't result in the same set of features selected. How do we explain this?

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  • $\begingroup$ Lasso acts on the conditional (i.e., partial) correlation between features and the target, whereas the correlation method acts on the marginal correlation between the features and the target. The partial correlation is more relevant for prediction since you will be using all the variables you end up including in future predictions, so I would expect the lasso method to be a better choice. $\endgroup$ – Noah Nov 25 '19 at 7:19
  • $\begingroup$ Selecting features based on correlations is dubious, (the whole correlation does not equal causation) because 1) the correlation may not be linear or monotonous (Pearson / Spearman), 2) there may be intercorrelation between the variables, which you will not identify with a correlation coefficient. $\endgroup$ – user2974951 Nov 26 '19 at 12:23
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A couple things:

...following two methods can be used for feature selection prior to model development

Those are actually part of the model development and should be cross validated. What most people do is look at the correlations, select only the most correlated with the output, and then move on to do cross validation etc. That's wrong for two reasons.

  1. Correlation measures strength of a linear relationship. If the effect of the variable is non linear, your correlation may not pick up on this. Here is a concrete example. When I worked for a marketing company, most of the customers we dealt with were in their late 30s to early 40s. They spent the most money, and people younger and older spent less because young people typically didn't have as much money or interest in our products. So the effect of age kind of looked like a concave function. If you simulate something like x = rnorm(1000); y = -0.2*x^2 + rnorm(1000, 0, 0.5) (here x has a concave relationship to y) the correlation is low even though x can explain 75% of the variation observed in y. If you removed features based on their correlation, surely you would not select this very important feature.
  1. Had you had different training data, you might have picked different features. So when you fit models in the cross validation step, you need to repeat the selection of features based on the correlations. Same thing with the lasso. In every cross validation refit, you need to fit the lasso, select the features, then refit a model with the selected features.

Which of the above two methods is preferred?

I don't think either are very good to be honest. Correlation is myopic for the reasons I've described. Lasso is better, but there is no reason to think the features it selects are the "best" features, nor is there a reason to think that the features it selects would be selected had you had different data. Here is a code example to demonstrate that

library(tidyverse)
library(glmnet)

S = 0
N = 1000
p = 100
mu = rep(0, p)
betas = rnorm(p, 2, 2)*rbinom(p, 1, 0.10)
while(max(abs(S))<0.9){
  S = rethinking::rlkjcorr(1, p)
}


do_glmnet<-function(){
  X = MASS::mvrnorm(N, mu, S)
  y  = X %*% betas + rnorm(N, 0, 2.5)
  cvmodel = cv.glmnet(X,y, alpha = 1)
  model = glmnet(X, y, alpha = 1)
  coef(model, cvmodel$lambda.1se) %>% 
    as.matrix() %>% 
    t() %>% 
    as_tibble()
  
  
}

results = map_df(1:100, ~do_glmnet())

results %>% 
  summarise_all(~mean(abs(.)>0) )

In that example, I generate data from a sparse linear model. Some variables are selected every time (those are the variables with real effects) but you can see that some variables with 0 effect are sometimes selected and sometimes not selected.

The absolute best way to select features is to use your knowledge about the data generating process to determine what is important and what is not. If you can't do that, use lasso to trade off variance for a but of bias, but don't select out features. Just keep the entire fit in the model. I saw Trevor Hastie speak at a zoom talk the other day and he showed us an example of which LASSO with all features performed better than selecting features with LASSO and then refitting the full model. I can't say that is the case for every problem, but it was pretty compelling evidence. Let me see if I can find a link to the talk.

That being said, I'm open to seeing numerical experiments that show that selection via glmnet does better than just putting everything into glmnet and not selecting. That just hasn't been the story I've seen.

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I think by saying correlation you are referring to SIS, developed by Jianqing Fan and Jinchi Lv. Actually, the logic behind the two methods is different. LASSO does the selection by using a penalized loss function and sparsity of the variables is required. Normally, for ultra-high dimensional data, we perform SIS first and reduce the dimension to a relatively small amount, and then perform LASSO to further reduce the number of variables that enter the final model.

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  • $\begingroup$ Thanks... by Correlation I meant Pearson correlation factor between the target (output) and input variables. $\endgroup$ – Lakshmi Srinivasan Nov 26 '19 at 15:38

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