Thursday, March 10, 2016

Re-Coding Categorical Variables

Categorical variables as candidate predictors pose a distinct challenge to the analyst, especially when they exhibit high cardinality (a large number of distinct values). Numerical models (for instance linear regression and most neural networks) cannot accept these variables directly as inputs, since operations between categories and numbers are not defined.

It is sometimes advantageous (even necessary) to re-code such variables as one or more numeric dummy variables, with each new variable containing a 0 or 1 value indicating the presence (1) or absence (0) of one distinct value. This often works well with very small numbers of distinct values. However, as cardinality increases, dummy variables quickly become inconvenient: they add potentially many parameters to the model, while reducing the average number of observations available for fitting each of those parameters.

One solution is to convert each individual categorical variable to a new, numeric variable representing a local summary (typically the mean) of the target variable. An example would be replacing the categorical variable, State, with a numeric variable, StateNumeric, representing the mean of the target variable within each state.

Unfortunately, there are often insufficient observations to reliably calculate summaries for all possible categories. To overcome this limitation, one might select either the n most frequent categories, or the n categories with the largest and smallest means. Though both of those techniques sometimes work well, they are, in reality, crude attempts to discover the categories with the most significant differences from the global mean.

A more robust method which I've found useful is to select only those categories whose mean target values are at least a minimum multiple of standard errors away from the global mean. The detailed procedure is as follows:

1. Calculate the global mean (the mean of the target variable, across all observations)
2. For each category…
  • a. Calculate the local mean and local standard error of the mean for the target variable (“local” meaning just for the observations in this category).
  • b. Calculate the absolute value of the difference between the local mean and global mean, and divide it by the local standard error.
  • c. If the calculation from B is greater than some chosen threshold (I usually use 3.0, but this may be adjusted to suit the context of the problem), then this category is replaced in the new variable by its local mean.
3. All variables not chosen in 2c to get their own value are collected in an “other” category, and all such observations are assigned the mean target for the “other” group.

The threshold can be judgmentally adjusted to suit the need of the problem, but I normally stay within the arbitrary range of 2.0 to 4.0. Missing values can be treated as their own category for the purposes of this algorithm.

Note that this procedure seeks the accumulation of two kinds of evidence that categories are significantly different from the global mean: category frequency and difference between category local mean and the global mean. Some categories may exhibit one and not the other. A frequent category may not get its own numeric value in the new variable in this procedure, if its local mean is too close to the global mean. At the same time, less frequent categories might be assigned their own numeric value, if their local mean is far enough away from the global mean. Notice that this procedure does not establish a single, minimum distance from the global mean for values to be broken away from the herd.

Though using a multiple of the standard error is only an approximation to a rigorous significance test, it is close enough for this purpose. Despite some limitations, this technique is very quick to calculate and I’ve found that it works well in practice: The number of variables remains the same (one new variable for each raw one) and missing values are handled cleanly.

Quick Refresher on Standard Errors

For numeric variables, the standard error of the mean of a sample of values is the standard deviation of those values divided by the square root of the number of values. In MATLAB, this would be:

StandardError = std(MyData) / sqrt(n)

For binary classification (0/1) variables, the standard error of a sample of values is the square root of this whole mess: the mean times 1 minus the mean, over the number of values. In MATLAB, this is:

p = mean(MyData); StandardError = sqrt( (p * (1 - p) ) / n)

Final Thoughts

Data preparation is an important part of this work, often being identified as the part of the process which takes the most time. This phase of the work offers tremendous opportunity for the analyst to be creative, and for the entire process to wring the most information out of the raw data.

This is why it is so strange that so little published material from experts is dedicated to it. Except in specialized fields, such as signal and image processing, one generally only finds bits of information on data preparation here and there in the literature. One exception is Dorian Pyle's book, "Data Preparation for Data Mining" (ISBN-13: 978-1558605299), which is entirely focused on this issue.


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