Tuesday, September 30, 2008

A question about decision trees

In your experience with decision trees, do you prefer to use a small set of core variables in order to make the model more elegant and/or understandable? At what point do you feel a tree has grown too large and complicated? What are the indicators that typically tell you that you need to do some pruning?
Thank you!

Elegance and ease of understanding may or may not be important depending on your model's intended purpose. There are certainly times when it is important to come up with a small set of simple rules. In our book Mastering Data Mining we give an example of a decision tree model used to produce rules that were printed on a poster next to a printing press so the press operators could avoid a particular printing defect. When a decision tree is used for customer segmentation, it is unlikely that your marketing department is equipped to handle more than a handful of segments and the segments should be described in terms of a few famiar variables. In both of these cases, the decision tree is meant to be descriptive.

On the other hand, many (I would guess most) decision trees are not intended as descriptions; they are intended to produce scores of some kind. If the point of the model is to give each prospect a probability of response, then I see no reason to be concerned about having hundreds or even thousands of leaves so long as each one receives sufficient training records that the proportion of responders at the leaf is a statistically confident estimate of the response probability. A very nice feature of decision tree models is that one need not grok the entire tree in order to interpret any particular rule it generates. Even in a very complex tree, the path from the root to a particular leaf of interest gives a fairly simple description of records contained in that leaf.

For trees used to estimate some continuous quantity, an abundance of leaves is very desirable. As estimators, regression trees have the desirable quality of never making truly unreasonable estimates (as a linear regression, for example, might do) because every estimate is an average of a large number of actual observed values. The downside is that it cannot produce any more distinct values than it has leaves. So, the more leaves the better.

The need for pruning usually arises when leaves are allowed to become too small. This leads to splits that are not statistically significant. Apply each split rule to your training set and a validation set drawn from the same population. You should see the same distribution of target classes in both training and validation data. If you do not, your model has overfit the training data. Many software tools have absurdly low default minimum leaf sizes--probably because they were developed on toy datasets such as the ubiquitous irises. I routinely set the minimum leaf size to something like 500 so overfitting is not an issue and pruning is unnecesary.

I have focused on the number of leaves rather than the number of variables since I think that is a better measure of tree complexity. You actually asked about the number of variables. I recommend a two-stage approach. In the first stage, do not worry about how many variables there are or which variable from each family of related variables gets picked by the model. One of the great uses for a decision tree is to pick a small subset of useful variables out of hundreds or thousands of candidates. At a later stage, look at the variables that were picked and think about what concept each of them is getting at. Then pick a set of variables that express those concepts neatly and perhaps even elegantly. You might find, for example, that the customer ID is a good predictor and appears in many rules because customer IDs were assigned serially and long-time customers behave differently than recent customers. Even though this makes perfect sense, it would be hard to explain so you would replace it with a more transparent indication of customer tenure such as "months since first purchase."

Monday, September 29, 2008

Three Questions

Hi Gordon & Michael,

I have a few questions, hope you can help me!

1. While modeling, if we don’t have a very specific client requirement, at what accuracy should we usually stop? Should we stop at 75%, or 80%? Are there standard accuracy requirements based on the industry? For example, in drug research & development, model accuracy is required to be very high.

2. What is the best approach for selecting records/training dataset when the client doesn’t have info on the cut-off/valid ranges for certain numeric columns? If it’s something like Age, there is no problem. But when it’s client/business specific columns, it’s not that easy to figure out the valid ranges. What I usually do for such problems is – 1. do some research on the web to have an understanding on all the values that the specific column can take 2. see the data distribution of that column and select values based on the percentiles. E.g if values from 10 to 60 (for that column) represent 80% of all the records, I exclude all records having values outside this range. Is this a good approach? Are there other alternatives?

3. Generally, I see model accuracy (predictive/risk/churn models) getting better when I recode/transform continuous variables into categorical variables through binning/grouping. But this also results in loss of information. How do we strike a balance here? I believe the business/domain should only decide whether I should use continuous or categorical values, and not the statistics. Is that correct?

Will check your blog regularly for the answersJ


Romakanta Irungbam

These three questions have something in common: There is no single right answer since so much depends on the business context (in the first two cases) or the modeling context (in the third case).

First Question

My statement about no right answers is especially true of the question regarding accuracy. There are contexts where a 95% error rate is perfectly acceptable. I am thinking of response modeling for direct mail. If a model is used to choose people likely to respond to an offer and only 5% of those chosen actually respond, then the error rate is 95%. How could that be acceptable? Well, if a 4% response rate is required for profitability and the response rate for a randomly selected control group is 3% then the model--despite its apparently terrible error rate--has heroically turned a money-losing campaign into a profitable one. Success is measured in dollars (or rupees or yen, but you know what I mean) not by error rates.

In other contexts, much better accuracy is required. A model for credit-card fraud cannot afford a high false-positive rate because this will result in legitimate transactions not being approved. The result is unhappy card holders canceling their accounts. Even if your client cannot provide an explicit requirement for accuracy, you may be able to derive one from the business context.

Absent any other constraints, I tend to stop trying to improve a model when I reach the point of diminishing returns. When a large effort on my part yields only a minor improvement, my time will probably be better spent on some other problem.

Second Question

This question is really about when to throw out data. I see know reason to discard data just because it happens to be in the tails of the distribution. To use your example where 80% of the records have values between 10 and 60, it may be that all the best customers have a value of 75 or more. It may make sense to throw out records which contain clearly impossible values, but even in that case, I would want to understand how the impossible values were generated. If all the records with impossibly high ages were generated in the same geographic region or from the same distribution channel, throwing them out will bias your sample.

Often, unusual values have some fairly simple explanation. When looking at loyalty card data for a supermarket, we found that there were a few cards that had seemingly impossibly large numbers of orders. The explanation was that when people checked out without their card and were therefore in danger of missing out on a discount, the nice checkout lady took pity on them and used her own card to get them the discount. Understanding that mechanism meant we could safely ignore data for those cards since they did not represent the actual shopping habits of any real customer.

Third Question

Whether or not binning continuous variables is helpful or harmful will depend very much on the particular modeling algorithm you are using and on how the binning is performed. I do not agree that, as a general rule, models are improved by binning continuous variables. As you note, this process destroys information. As an extreme example, suppose you have a relationship that is completely determined by a continuous (or discrete, but with small increments) relationship--a tax of a constant amount per liter, say. The more accurately you can measure the number of liters sold, the more accurately you can estimate the tax revenue. In such a case, binning could only be harmful.

When binning tends to be helpful is when the relationship between the explanatory variable and the thing you are trying to explain is more complex than the particular modeling technique you have chosen can handle. For example, you have chosen a linear model and the relationship is non-linear. I once modeled household penetration for my local newspaper, the Boston Globe. One of my explanatory variables was distance from Boston. Clearly, this should have some effect, but there is only a low level of linear correlation. This is because penetration goes up as a function of distance as you travel out to the first ring of suburbs where penetration is highest, but then goes down again as you continue to travel farther from Boston. So a linear model could not make good use of the untransformed variable, but it could make use of three variables in the form within_three, three_to_ten, and beyond_ten (assuming that 3 and 10 are the right bin boundaries). Of course, binning is not the only transformation that could help and linear models are not the only choice of model.

Friday, September 5, 2008

Sorting Cells in Excel Using Formulas, Part 2

In a previous post, I described how to create a new table in Excel from an existing table where the cells in the new table are sorted by some column in the existing table. In addition, the new table is automatically updated when the values in the original table are modified.

The previously described approach, alas, has some shortcomings:
  • Only one column can be used for the sort key.
  • The column must be numeric.
  • The column cannot have any duplicate values.
This post generalizes on the earlier method by fixing these problems.

If you are interested in this post, you may be interested in my book Data Analysis Using SQL and Excel.

Overview of Simpler Method

The simpler method described in the earlier post recognizes that creating a live sorted table connect to another table consists of the following steps:
  1. Ranking the rows in the table by the column to be sorted.
  2. Using the rank with the OFFSET() function to create the resulting table.
For Step (1), the method uses the built-in RANK() function provided by Excel. This introduces the limitations described above, because RANK() only works on numeric values and produces the same value for duplicates.

The key to fixing these problems is to replace the RANK() function with more general purpose functions.

Instead of RANK()

RANK() determines whether a value is the largest, second largest, third largest, or so on with respect to a list (or smallest, if we are going in the opposite order, which is determined by an optional third argument). One way to think of what it does is that it sorts the values in the list and determines the position of the original value.

An alternative but equivalent way of thinking about the calculation is that it tells us how many values are larger than (or smaller than) the given value. This alternative definition suggests other ways of arriving at the same rankings, such as:

....=COUNTIF(data!B$2:B$55, ">="&data!B2)

This formula can be placed alongside the original table (or anywhere else) and then copied down. It works by counting the number of values that are less than or equal to each value. The resulting ranking is from smallest value to largest value. To reverse the order, simply use "<=" instead. This solves one of the original problems, because the COUNTIF() function works with string data as well as numeric data.

An almost equivalent formulation is to use array functions.

....{=SUM(if(data!B$2:B$55>=data!B2, 1, 0)}

(If you are not familiar with array functions, check out Excel documentation or Data Analysis Using SQL and Excel.)

This is very similar to the COUNTIF() method, although the array functions have one advantage. The conditional logic can be more complicated, so we can do the ranking by multiple columns at the same time.

Using our own version of the rank function fixes two of the three problems. At this point, duplicates still get the same rank value.

Handling Duplicates

The problem with duplicate values is that all these methods assign the same ranking when two different rows have the same value. This makes it impossible to distinguish between the two rows, so one will be included in the sorted table multiple times.

The solution is to fix this problem by adding an offset. If the highest value is repeated multiple times, then all of those rows will have a ranking equal to the number of duplicates. In the following little table, the second column contains the rankings as calculated by either of the above two methods (RANK() does not work because the first column is not numeric):

a 3

a 3

a 3

b 5

b 5

What we want, though, is to have distinct values in the second column:

a 1

a 2

a 3

b 4

b 5

The solution is to subtract a value from the calculated ranking. This is the number of values that we have already seen that are equal to the value in question. Once again, this can be accomplished with either COUNTIF() or array functions:

....=COUNTIF(data!B$2:B$55, ">="&data!B2) + COUNTIF(data!B$2:B2, "="&data!B2)-1


....{=SUM(IF(data!B$2:B$55>=data!B2, 1, 0)) + SUM(IF(data!B$2:B2=data!B2, 1, 0))-1}

These formulations consist of two parts. The first part calculates a ranking, where duplicates get the same value. The second part subtracts the number of duplicates already seen in the list. For the simple example above, the results are actually:

a 3

a 2

a 1

b 5

b 4

This works just as well, although it does not preserve the original ordering.

Note that these formulas are all structured so they can be copied down cells and continue working.

What It All Looks Like Together

This method is perhaps best explained by seeing an example. The file sort-in-place.xls contains random information about the fifty states (latitude, longitude, population, and capital for example) on the "data" worksheet. The "data-sorted" worksheet shows the states abbreviations by rank order for each of the columns. For instance, for the size column Alaska is first, followedy by Texas, California, and Montana. For the population columns, the ordering is California, Texas, New York, and Florida. This worksheet using the rankings on the "ranking-countif()" worksheet.

The three worksheets called "ranking-" illustrate the three different methods of doing the rankings -- using RANK(), using COUNTIF(), and using array functions. Note that the RANK() method cannot handle text columns, so it does not work in this case.

If you like, you can change the data on the "data" tab and see the rankings change on the sorted tab. Voila! A sorted table connected by formulas to the original table!