ArrowModel Blog

Agile Scoring

Agile Scoring Blog

Predictive Modeling from the Trenches

Reject Inference
Sep 05, 2011 | /jeff | Link

ArrowModel doesn't do reject inference, at least not automatically. This post tries to explain why.

Let's look at an example. Suppose we are building acquisition score for a bank. The bank does not give loans to everybody who asks for one. Instead it has a credit policy, which for this example will be very simple: approve everyone with FICO > 640. FICO here is a generic risk score. The bank is buying it from a credit bureau.

The problem is, both training and validation samples are censored - they have no people with FICO below 640. A model trained and validated on such samples might fail spectacularly when used in credit policy instead of FICO - after all, it never saw the really risky cases (rejects).

Reject inference is supposed to solve this problem. There are several methods. For example, parceling, where rejects are assigned good/bad outcome randomly with probability that corresponds to their FICO. Or fuzzy reject inference, where each reject is added twice, once as good, and once as bad, with weights equal to probabilities for this FICO band.

Unfortunately there is no way to know how well these methods work before rolling out the new model (it was trained and validated on inferences, not real data), and by then it's already too late.

So we usually recommend to do 1/N testing instead: approve some accounts below existing score cuts anyway and use their performance, oversampling if needed.

To those unable or unwilling to do so we recommend to keep exising score cut in policy and add new score cut as overlay.

Finally, reject inference can still be done in ArrowModel by feeding it data with rejects added manually.

Site translations
Oct 18, 2010 | /jeff | Link

The main ArrowModel site now has French and Russian translations. Really. More i18n is underway.

New hosting
Oct 18, 2010 | /jeff | Link ArrowModel moved to new hosting. We apologize for the downtime and inconvenience.

Spanish translation
Oct 18, 2010 | /jeff | Link

ArrowModel speaks Spanish, too.

New Hosting
Oct 17, 2010 | /jeff | Link changed hosting. Some things might still be broken, but we'll fix them as soon as we can.

ArrowModel can now import Excel files on Windows
Oct 10, 2009 | /jeff | Link

It was always possible to get the data from Microsoft Excel to ArrowModel by exporting it to CSV first, but now it's even easier because ArrowModel can import Excel files directly. The data is taken from the first worksheet. First row should contain variable names.

Unfortunately this doesn't work on Mac OS X because in contrast to Windows there is no pre-installed Excel ODBC driver.

Britney Spears Algorithm
Oct 01, 2009 | /jeff | Link

One of the neat things ArrowModel does for you is automatic creation of dummy variables from categorical predictors. For example, if the dataset contains a field sex with two possible values, M and F, ArrowModel will create a dummy variable that is 1 if sex=F and 0 otherwise.

Or maybe the other way, depending on the distribution.

Getting that distribution for each categorical variable by brute force can take a while.

Luckily, other people have encountered this problem before. American Scientist article calls it the Britney Spears problem because it appears, among other things, in finding the most popular search terms.

A stream algorithm similar the the one described in that article is implemented in ArrowModel starting with build 1167.

There are also a couple of bugfixes, most related to handling of large number of predictors. If you're working with 10,000 variables or more, get the new beta.

Important: Make sure to uninstall previous version of ArrowModel before installing the new one. One of the fixes is in a third-party component (QtSql4.dll). Its internal version remains the same, though, so ArrowModel installer does not realize that this DLL needs to be updated.

Folded root
Oct 01, 2009 | /jeff | Link ArrowModel build 1207 is available now. Changes:

  • Reference ROC. ArrowModel now makes it easy to compare ROC, KS, and AUC of the new model to a reference (another score) on Validate step. This is handy when developing a replacement for existing score.
  • Folded fourth root transformation. In addition to winsorisation, ArrowModel now offers another variable transformation: sign(x)*sqrt(sqrt(abs(x))). By default it is applied to variables with high skewness or kurtosis before feeding them into logistic regression. The choice of transformation can be overridden on the Explore step.
  • Autosave mode. Normally ArrowModel creates a temporary copy of the file you are working on, and all changes are applied to this copy. Original file is modified only when you choose to save the changes. When working with large datasets the overhead of copying (in terms of time and disk space) becomes noticeable. To speed things up, ArrowModel now provides an option to open file in place. To enable it, go to View→Console and check "Autosave". Caution: with autosave enabled, changes take effect immediately and there is no way to exit without saving.
  • Minor bugfixes and major performance improvements.

  • The Unreasonable Effectiveness of Data by Peter Norvig et al.
    May 06, 2009 | /jeff | Link

    "...invariably, simple models and a lot of data trump more elaborate models based on less data."

    Full article (PDF)

    Wir sprechen Deutsch
    May 06, 2009 | /jeff | Link

    ArrowModel site in German.

    Memory management
    May 06, 2009 | /jeff | Link

    When working with large datasets ArrowModel has to decide what to keep in memory and what to leave on the hard drive. Most of the time this decision can be left to the operating system, but there are always exceptions.

    After running some benchmarks we tweaked this algorithm a little to improve performance. Now this is not an exact science. There are just too many moving parts. So if you noticed that build 1153 is faster or slower than the one you used before, please let us know. The changes only affect handling of large datasets (with design matrix over 256 MB).

    When three hundred zeroes is not enough
    May 06, 2009 | /jeff | Link

    One of our beta testers encountered an interesting and rare corner case. On a large dataset with very strong predictors ArrowModel decision tree algorithm did not pick the best top split. Situation improved a couple of levels down, but the tree still was suboptimal.

    A quick look through the log revealed that p values of several predictors were very close to zero. In fact, so close to zero that they were indistinguishable from each other. ArrowModel picked the first one among them, which wasn't the best.

    This was interesting on many levels. First, floating point underflow is a very rare problem. After all, smallest absolute non-zero double precision float is approximately 10-308. Which leads me to the next point: our beta testers are great. There's no way we could have caught it on existing test cases. You need real-world data to tickle bugs like that.

    Anyway, the problem is now resolved. When p values are very low, exponents in χ2 distribution approximation are compared instead. We tried different approximations and jumped through some hoops to handle degrees of freedom and Bonferroni adjustments correctly, but now everything should work fine.

    Bottom line: if you build decision trees on large datasets with multiple strong predictors, get the new beta.

    More Data Beats Better Algorithms
    May 06, 2009 | /jeff | Link

    Anand Rajaraman argues that more data > better algorithm. Hear, hear.

    Good software is still required to crunch all this additional data, though.

    ArrowModel 0.2
    May 06, 2009 | /jeff | Link

    Second beta of ArrowModel is out. Registered users can download it now.

    If you are not a registered user, but would like to give ArrowModel a try, please sign up.

    Highlights of the new version include:

    ARM file format remains unchanged. You will be getting warning messages when opening models created with previous versions, but they should work.

    Thank you for the feedback and support.

    New site design
    May 06, 2009 | /jeff | Link

    We're rolling out the redesigned ArrowModel web site today.

    Second beta (build 888)
    May 06, 2009 | /jeff | Link

    Second beta of ArrowModel is out. The biggest new feature is ODBC connectivity.

    Decision trees
    May 06, 2009 | /jeff | Link

    Decision trees are not as popular in credit scoring (most typical ArrowModel use case) as logistic regression, which is why we didn't initially plan to have them in version 1.

    But it turns out that many beta testers wanted trees, and we listened.

    The bad news is that adding a rather large new feature so late in development cycle is guaranteed to push back the release date. And it did.

    The good news is that ArrowModel can now do decision trees, and it's still 2008!

    If you are a beta tester, download build 1101 now. If you are not a beta tester yet, consider signing up. It's free.

    Algorithm used by ArrowModel to create decision trees is based on venerable CHAID, and it's fast.


    Not Quite Normal
    Jul 26, 2008 | /jeff | Link

    A lot of statistical magic relies on the premise that stuff is normally distributed.

    Normal distribution
    Normal distribution

    The normal distribution has nice properties that make things easy analytically, but chances are that, most of the time, you'll see distributions that look like this:

    Not quite normal distribution
    Not quite normal distribution

    Of course I'm generalizing and there are exceptions, but it's clear that the good old normal distribution belongs on the endangered species list.

    There are several reasons why:

    So what is the poor modeler to do?

    There are more elaborate ways of dealing with not quite normally distributed data such as Johnson's SU functions and multivariate adaptive regression splines (MARS) which this margin is too narrow to contain.

    I'm new at predictive modeling. Help!
    Jul 26, 2008 | /jeff | Link

    It's true that there does not seem to be a lot of information on scoring and predictive modeling available online, and that many articles are written in rather heavy language, peppered with statistical jargon. But don't panic. To help you navigate the unchartered waters, here are some good places to start.

    There are also a few exceptionally good books. My favorites are:

    Finally, these two classes by the SAS Institute are worth taking:

    Kolmogorov-Smirnov Test
    Jul 26, 2008 | /jeff | Link

    One of the most widely (mis)used measures of scorecard performance is the Kolmogorov-Smirnov test (KS), colloquially known as the vodka test. In this post, I'll explain what KS is, and show a way to calculate it in SQL.

    Given two samples of a continuous random variable, the two sample K-S test is used answer the following question: did these two samples come from the same distribution or didn't they? The idea is simply to compute the largest absolute difference between the two empirical cumulative distributions and to conclude that there is a significant difference if the difference is large enough.

    Consider a risk score that predicts the probability of a customer defaulting (we'll call that 'going bad'). KS is the greatest difference between the cumulative distribution functions of the scores of the good and the bad populations:

    KS = maxs|B(s) - G(s)|,


    KS is often multiplied by 100 for convenience. In many contexts 40 is considered to be a good KS.

    Let's try an example. Start with the table t that contains initial data:

    Column Description
    id Unique identifier
    s Score
    outcome 1 is bad, 0 is good

    The following query calculates the KS:

    SELECT MAX(cdf.b - cdf.g) * 100                            "KS"
    FROM ( SELECT a.s                                          "s"
                , SUM(distr.bad_cnt) /
                  ( SELECT COUNT(*) FROM t WHERE outcome = 1 ) "b"
                , SUM(distr.good_cnt) /
                  ( SELECT COUNT(*) FROM t WHERE outcome = 0 ) "g"
           FROM ( SELECT DISTINCT s FROM t ) a
           JOIN (
                  SELECT s                "s"
                       , SUM(outcome)     "bad_cnt"
                       , SUM(1 - outcome) "good_cnt"
                  FROM t
                  GROUP BY s 
                ) distr
             ON distr.s <= a.s
             GROUP BY a.s 
         ) cdf

    The easiest way to understand how the query works is by decomposing it into smaller pieces. In this case there are five uncorrelated subqueries.

    This subquery returns distribution of goods and bads by score:

    SELECT s                "s"
         , SUM(outcome)     "bad_cnt"
         , SUM(1 - outcome) "good_cnt"
    FROM t
    GROUP BY s

    Note how it relies on the fact that outcome can be either 0 or 1.

    This subquery returns the list of all possible score values:


    This subquery returns the total number of bads:

    SELECT COUNT(*) FROM t WHERE outcome = 1

    This subquery returns the total number of goods:

    SELECT COUNT(*) FROM t WHERE outcome = 0

    Finally, this subquery (abbreviated for clarity) makes the distributions cumulative:

    SELECT a.s                              "s"
         , SUM(distr.bad_cnt) / total_bad   "b"
         , SUM(distr.good_cnt) / total_good "g"
    FROM a
    JOIN distr
      ON distr.s <= a.s
    GROUP BY a.s

    Note that it is rather inefficient because the join results in a partial Cartesian product. There's a better way to do the cumulation if your flavor of SQL supports online analytical processing (OLAP) functions:

    SELECT s                                                  "s"
         , SUM(FLOAT(bad_cnt)) OVER (ORDER BY s) / total_bad  "b"
         , SUM(FLOAT(good_cnt)) OVER (ORDER BY s) / total_bad "g"
    FROM distr

    Now the only thing left to do is to pick the maximum difference. This is the KS.

    ArrowModel beta FAQ
    Jul 26, 2008 | /jeff | Link

    ArrowModel goes through its first beta testing. Here are some of the frequently asked questions so far:

    [UPDATE 5/4] Pictures added to illustrate the differences in ROC curves.

    Receiver Operating Characteristic
    Jul 26, 2008 | /jeff | Link

    ROC curves were first used during World War II to graphically show the separation of radar signals from background noise. They are commonly used to graphically show the added value of any predictive model. To plot the receiver operating characteristic, or ROC curve, one plots B(s) vs. G(s) for all values of s. This curve goes from (0, 0) to (1, 1). The curve of an ideal model (complete separation) goes through (0, 1), while the curve of a totally useless model (no separation) is a straight diagonal line. The curve looks like a banana, hence the nickname banana chart.

    Very strong separation Weak separation
    Excellent model Mediocre model

    The KS query from this post can be easily modified to return coordinates of the points on the ROC curve:

    SELECT s
         , cdf.b "Sensitivity"
         , cdf.g "1-Specificity"
    FROM ( SELECT a.s                                          "s"
                , SUM(distr.bad_cnt) /
                  ( SELECT COUNT(*) FROM t WHERE outcome = 1 ) "b"
                , SUM(distr.good_cnt) /
                  ( SELECT COUNT(*) FROM t WHERE outcome = 0 ) "g"
           FROM ( SELECT DISTINCT s FROM t ) a
           JOIN (
                  SELECT s                "s"
                       , SUM(outcome)     "bad_cnt"
                       , SUM(1 - outcome) "good_cnt"
                  FROM t
                  GROUP BY s 
                ) distr
             ON distr.s <= a.s
             GROUP BY a.s 
         ) cdf

    In the context of an ROC plot, B(s) is often called sensitivity or true positive fraction, and G(s) is called 1-specificity or false positive fraction.

    Information Value
    Jul 26, 2008 | /jeff | Link

    Deciding which predictors to use is one of the key steps in model building. A good place to start is to examine predictors individually to see how good they are in a univariate sense.

    Information value is a metric that is often used to tell how good a predictor is. Let's follow the calculations step by step.

    1. Start by ranking the data by the predictor in question. The number of ranks is not very critical and, in most cases, deciles will do just fine.
    2. Calculate the total number of goods (total_good_ct)
      and the total number of bads (total_bad_ct);
    3. For each rank
      • Calculate the number of goods (good_ct)
        and the number of bads (bad_ct);
      • good_pct = good_ct / total_good_ct,
        bad_pct = bad_ct / total_bad_ct;
      • diff_pct = good_pct - bad_pct;
      • info_odds = good_pct / bad_pct;
      • Weight of evidence: woe = log(info_odds);
      • Information value: inf_val = diff_pct * woe;
    4. Finally, sum up inf_val for all the ranks. This is the predictor's information value.

    As you can see, the information value for each rank reflects log odds, but the order of ranks does not have any effect. This nicely takes care of nonlinearity and outliers.

    Ordering predictors by information value and taking the top N is a tempting strategy, but not a very prudent approach. The predictors selected this way can turn out to be redundant, regression is rather sensitive to outliers, and we haven't done anything about nonlinearity yet. But it's a good way to screen out the least likely candidates.