When should we discretize/bin continuous independent variables/features and when should not? When should we discretize/bin independent variables/features and when should not?
My attempts to answer the question:


*

*In general, we should not bin, because binning will lose information.

*Binning is actually increasing the degree of freedom of the model, so, it is possible to cause over-fitting after binning. If we have a "high bias" model, binning may not be bad, but if we have a "high variance" model, we should avoid binning.

*It depends on what model we are using. If it is a linear mode, and data has a lot of "outliers" binning probability is better. If we have a tree model, then, outlier and binning will make too much difference.


Am I right? and what else?

I thought this question should be asked many times but I cannot find it in CV only these posts
Should we bin continuous variables?
What is the benefit of breaking up a continuous predictor variable?
 A: Aggregation is substantively meaningful (whether or not the researcher is aware of that). 
One should bin data, including independent variables, based on the data itself when one wants:


*

*To hemorrhage statistical power.

*To bias measures of association. 
A literature starting, I believe, with Ghelke and Biehl (1934—definitely worth a read, and suggestive of some easy enough computer simulations that one can run for one's self), and continuing especially in the 'modifiable areal unit problem' literature (Openshaw, 1983; Dudley, 1991; Lee and Kemp, 2000) makes both these points clear.
Unless one has an a priori theory of the scale of aggregation (how many units to aggregate to) and the categorization function of aggregation (which individual observations will end up in which aggregate units), one should not aggregate. For example, in epidemiology, we care about the health of individuals, and about the health of populations. The latter are not simply random collections of the former, but defined by, for example, geopolitical boundaries, social circumstances like race-ethnic categorization, carceral status and history categories, etc. (See, for example Krieger, 2012)
References
Dudley, G. (1991). Scale, aggregation, and the modifiable areal unit problem. [pay-walled] The Operational Geographer, 9(3):28–33.
Gehlke, C. E. and Biehl, K. (1934). Certain Effects of Grouping Upon the Size of the Correlation Coefficient in Census Tract Material. [pay-walled] Journal of the American Statistical Association, 29(185):169–170.
Krieger, N. (2012). Who and what is a “population”? historical debates, current controversies, and implications for understanding “population health” and rectifying health inequities. The Milbank Quarterly, 90(4):634–681.
Lee, H. T. K. and Kemp, Z. (2000). Hierarchical reasoning and on-line analytical processing of spatial and temporal data. In Proceedings of the 9th International Symposium on Spatial Data Handling, Beijing, P.R. China. International Geographic Union.
Openshaw, S. (1983). The modifiable areal unit problem. Concepts and Techniques in Modern Geography. Geo Books, Norwich, UK.
A: Looks like you're also looking for an answer from a predictive standpoint, so I put together a short demonstration of two approaches in R


*

*Binning a variable into equal sized factors.

*Natural cubic splines.


Below, I've given the code for a function that will compare the two methods automatically for any given true signal function
test_cuts_vs_splines <- function(signal, N, noise,
                                 range=c(0, 1), 
                                 max_parameters=50,
                                 seed=154)

This function will create noisy training and testing datasets from a given signal, and then fit a series of linear regressions to the training data of two types


*

*The cuts model includes binned predictors, formed by segmenting the range of the data into equal sized half open intervals, and then creating binary predictors indicating to which interval each training point belongs.

*The splines model includes a natural cubic spline basis expansion, with knots equally spaced throughout the range of the predictor.


The arguments are


*

*signal: A one variable function representing the truth to be estimated.

*N: The number of samples to include in both training and testing data.

*noise: The amound of random gaussian noise to add to the training and testing signal.

*range: The range of the training and testing x data, data this is generated uniformly within this range.

*max_paramters: The maximum number of parameters to estimate in a model.  This is both the maximum number of segments in the cuts model, and the maximum number of knots in the splines model.


Note that the number of parameters estimated in the splines model is the same as the number of knots, so the two models are fairly compared.
The return object from the function has a few components


*

*signal_plot: A plot of the signal function.

*data_plot: A scatter plot of the training and testing data.

*errors_comparison_plot: A plot showing the evolution of the sum of squared error rate for both models over a range of the number of estiamted parameters.


I'll demonstrate with two signal functions.  The first is a sin wave with an increasing linear trend superimposed
true_signal_sin <- function(x) {
  x + 1.5*sin(3*2*pi*x)
}

obj <- test_cuts_vs_splines(true_signal_sin, 250, 1)

Here is how the error rates evolve

The second example is a nutty function I keep around just for this kind of thing, plot it and see
true_signal_weird <- function(x) {
  x*x*x*(x-1) + 2*(1/(1+exp(-.5*(x-.5)))) - 3.5*(x > .2)*(x < .5)*(x - .2)*(x - .5)
}

obj <- test_cuts_vs_splines(true_signal_weird, 250, .05)


And for fun, here is a boring linear function
obj <- test_cuts_vs_splines(function(x) {x}, 250, .2)


You can see that:


*

*Splines give overall better overall test performance when the model complexity is properly tuned for both.

*Splines give optimal test performance with much fewer estimated parameters.

*Overall the performance of splines is much more stable as the number of estimated parameters is varied.


So splines are always to be prefered from a predictive standpoint.
Code
Here's the code I used to produce these comparisons.  I've wrapped it all in a function so that you can try it out with your own signal functions. You will need to import the ggplot2 and splines R libraries.
test_cuts_vs_splines <- function(signal, N, noise,
                                 range=c(0, 1), 
                                 max_parameters=50,
                                 seed=154) {

  if(max_parameters < 8) {
    stop("Please pass max_parameters >= 8, otherwise the plots look kinda bad.")
  }

  out_obj <- list()

  set.seed(seed)

  x_train <- runif(N, range[1], range[2])
  x_test <- runif(N, range[1], range[2])

  y_train <- signal(x_train) + rnorm(N, 0, noise)
  y_test <- signal(x_test) + rnorm(N, 0, noise)

  # A plot of the true signals
  df <- data.frame(
    x = seq(range[1], range[2], length.out = 100)
  )
  df$y <- signal(df$x)
  out_obj$signal_plot <- ggplot(data = df) +
    geom_line(aes(x = x, y = y)) +
    labs(title = "True Signal")

  # A plot of the training and testing data
  df <- data.frame(
    x = c(x_train, x_test),
    y = c(y_train, y_test),
    id = c(rep("train", N), rep("test", N))
  )
  out_obj$data_plot <- ggplot(data = df) + 
    geom_point(aes(x=x, y=y)) + 
    facet_wrap(~ id) +
    labs(title = "Training and Testing Data")

  #----- lm with various groupings -------------   
  models_with_groupings <- list()
  train_errors_cuts <- rep(NULL, length(models_with_groupings))
  test_errors_cuts <- rep(NULL, length(models_with_groupings))

  for (n_groups in 3:max_parameters) {
    cut_points <- seq(range[1], range[2], length.out = n_groups + 1)
    x_train_factor <- cut(x_train, cut_points)
    factor_train_data <- data.frame(x = x_train_factor, y = y_train)
    models_with_groupings[[n_groups]] <- lm(y ~ x, data = factor_train_data)

    # Training error rate
    train_preds <- predict(models_with_groupings[[n_groups]], factor_train_data)
    soses <- (1/N) * sum( (y_train - train_preds)**2)
    train_errors_cuts[n_groups - 2] <- soses

    # Testing error rate
    x_test_factor <- cut(x_test, cut_points)
    factor_test_data <- data.frame(x = x_test_factor, y = y_test)
    test_preds <- predict(models_with_groupings[[n_groups]], factor_test_data)
    soses <- (1/N) * sum( (y_test - test_preds)**2)
    test_errors_cuts[n_groups - 2] <- soses
  }

  # We are overfitting
  error_df_cuts <- data.frame(
    x = rep(3:max_parameters, 2),
    e = c(train_errors_cuts, test_errors_cuts),
    id = c(rep("train", length(train_errors_cuts)),
           rep("test", length(test_errors_cuts))),
    type = "cuts"
  )
  out_obj$errors_cuts_plot <- ggplot(data = error_df_cuts) +
    geom_line(aes(x = x, y = e)) +
    facet_wrap(~ id) +
    labs(title = "Error Rates with Grouping Transformations",
         x = ("Number of Estimated Parameters"),
         y = ("Average Squared Error"))

  #----- lm with natural splines -------------  
  models_with_splines <- list()
  train_errors_splines <- rep(NULL, length(models_with_groupings))
  test_errors_splines <- rep(NULL, length(models_with_groupings))

  for (deg_freedom in 3:max_parameters) {
    knots <- seq(range[1], range[2], length.out = deg_freedom + 1)[2:deg_freedom]

    train_data <- data.frame(x = x_train, y = y_train)
    models_with_splines[[deg_freedom]] <- lm(y ~ ns(x, knots=knots), data = train_data)

    # Training error rate
    train_preds <- predict(models_with_splines[[deg_freedom]], train_data)
    soses <- (1/N) * sum( (y_train - train_preds)**2)
    train_errors_splines[deg_freedom - 2] <- soses

    # Testing error rate
    test_data <- data.frame(x = x_test, y = y_test)  
    test_preds <- predict(models_with_splines[[deg_freedom]], test_data)
    soses <- (1/N) * sum( (y_test - test_preds)**2)
    test_errors_splines[deg_freedom - 2] <- soses
  }

  error_df_splines <- data.frame(
    x = rep(3:max_parameters, 2),
    e = c(train_errors_splines, test_errors_splines),
    id = c(rep("train", length(train_errors_splines)),
           rep("test", length(test_errors_splines))),
    type = "splines"
  )
  out_obj$errors_splines_plot <- ggplot(data = error_df_splines) +
    geom_line(aes(x = x, y = e)) +
    facet_wrap(~ id) +
    labs(title = "Error Rates with Natural Cubic Spline Transformations",
         x = ("Number of Estimated Parameters"),
         y = ("Average Squared Error"))


  error_df <- rbind(error_df_cuts, error_df_splines)
  out_obj$error_df <- error_df

  # The training error for the first cut model is always an outlier, and
  # messes up the y range of the plots.
  y_lower_bound <- min(c(train_errors_cuts, train_errors_splines))
  y_upper_bound = train_errors_cuts[2]
  out_obj$errors_comparison_plot <- ggplot(data = error_df) +
    geom_line(aes(x = x, y = e)) +
    facet_wrap(~ id*type) +
    scale_y_continuous(limits = c(y_lower_bound, y_upper_bound)) +
    labs(
      title = ("Binning vs. Natural Splines"),
      x = ("Number of Estimated Parameters"),
      y = ("Average Squared Error"))

  out_obj
}

