An Overview of Cubist

Cubist is a tool for generating rule-based predictive models from data. Whereas its sister system See5/C5.0 produces classification models that predict categories, Cubist models predict numeric values. This short tutorial introduces Cubist's capabilities and explains how to use the system effectively.

In this tutorial, file names and Cubist input appear in blue fixed-width font while file extensions and other general forms are shown highlighted in green.


Preparing Data for Cubist

We will illustrate Cubist using a simple application -- modeling automobiles' annual fuel cost using data published in 2008 by the US Department of Energy and the US Environmental Protection Agency. Each data point concerns one automobile and the attributes or properties capture (possibly) relevant information as follows:

	Attribute                      Case 1     Case 2     Case 3   .....

	class                          SUV        VAN        CAR
	model                          HUMMER H3  GMC G1500  DODGE CHARGER
	displacement (l)               3.7        5.3        3.5
	cylinders                      4          8          6
	drive                          Auto       Auto       Auto
	gears                          4          4          5
	transmission                   4WD        RWD        4WD
	fuel                           regular    regular    regular
	charger (super- or turbo-)     none       none       none
	valves/cylinder                2          2          4
	displacement/cylinder          0.93       0.66       0.58

	annual fuel cost               2801       3900       2335

Each case has a target attribute or dependent variable -- here the estimated annual fuel cost to run the automobile -- and the other attributes provide information that may help to predict this value, although some automobiles may have unknown values for some attributes. There are only twelve attributes in this example including the target attribute, but Cubist can deal with thousands of attributes if necessary.

Cubist's job is to find how to estimate a case's target value in terms of its attribute values -- here, to relate annual fuel cost to the other information provided for the automobile. Cubist does this by building a model containing one or more rules, where each rule is a conjunction of conditions associated with a linear expression. The meaning of a rule is that, if a case satisfies all the conditions, then the linear expression is appropriate for predicting the target value. A Cubist model thus resembles a piecewise linear model, except that the rules can overlap. As we will see, Cubist can also construct multiple models and can combine rule-based models with instance-based (nearest neighbor) models.

Application files

Every Cubist application has a short name called a filestem; we will use the filestem fc2008 for this illustration. All files read or written by Cubist for an application look like filestem.extension, where filestem identifies the application and extension describes the contents of the file.

Here is a summary table of the extensions used by Cubist (to be described in later sections):

All files for an application must be kept together in one directory but several applications can share the same directory.

Names file

The first essential file is the names file (e.g. fc2008.names) that defines the attributes used to describe each case. There are two important subgroups of attributes:

The file fc2008.names looks like this:

| Data extracted from the site http://www.fueleconomy.gov provided by
| the US Department of Energy and the US Environmental Protection Agency.

fuel cost.                                | target

class:          CAR, STATION WAGON, SUV,
                PICKUP TRUCK, VAN.
model:          label.
displ:          continuous.               | engine displacement (liters/litres)
cylinders:      continuous.
drive:          Auto, Manual.
gears:          continuous.               | 'N/A' if no conventional gears
transmission:   F, R, 4.                  | front, rear, 4WD
fuel:           R, P, D, E, C.            | regular, premium, diesel, ethanol, lpg
charger:        T, S, -.                  | turbo, super, none
fuel cost:      continuous.               | estimated annual fuel cost ($)
valves/cyl:     continuous.               | valves per cylinder

displ/cyl := displ / cylinders.           | displacement per cylinder
Some of the attribute names have been abbreviated and the target attribute fuel cost appears among the others rather than at the end.

What's in a name?

Names, labels, and discrete values are represented by arbitrary strings of characters, with some fine print: Whitespace (blank lines, spaces, and tab characters) is ignored except inside a name or value and can be used to improve legibility. Unless it is escaped as above, the vertical bar `|' causes the remainder of the line to be ignored and is handy for including comments. When used in this way, `|' should not occur inside a value.

The first important entry of the names file identifies the attribute that contains the target value -- the value to be modeled in terms of the other attributes -- here, fuel cost. This attribute must be of type continuous or an implicitly-defined attribute that has numeric values (see below).

Following this entry, all attributes are defined in the order that their values will be given for each case.

Explicitly-defined attributes

The name of each explicitly-defined attribute is followed by a colon `:' and a description of the values taken by the attribute. The attribute name is arbitrary, except that each attribute must have a distinct name, and the name case weight is reserved for setting weights for individual cases. There are eight possibilities:
continuous
The attribute takes numeric values.
date
The attribute's values are dates in the form YYYY/MM/DD or YYYY-MM-DD, e.g. 1999/09/30 or 1999-09-30. Valid dates range from the year 1601 to the year 4000.
time
The attribute's values are times in the form HH:MM:SS with values between 00:00:00 and 23:59:59.
timestamp
The attribute's values are times in the form YYYY/MM/DD HH:MM:SS or YYYY-MM-DD HH:MM:SS, e.g. 1999-09-30 15:04:00. (Note that there is a space separating the date and time.)
a comma-separated list of names
The attribute takes discrete values, and these are the allowable values. The values may be prefaced by [ordered] to indicate that they are listed in a meaningful order, otherwise they will be taken as unordered. For instance, the values low, medium, high are ordered, while meat, poultry, fish, vegetables are not. If the attribute values have a natural order, it is better to declare them as ordered so that this information can be exploited by Cubist.
discrete N for some integer N
The attribute also takes discrete values, but the values are assembled from the data itself; N is the maximum number of such values. (This is not recommended, since the data cannot be checked, but it can be handy for discrete attributes with many values.)
ignore
The values of the attribute should be ignored.
label
This attribute contains an identifying label for each case, such as an account number or an order code. The value of the attribute is ignored when models are constructed, but is used when referring to individual cases. A label attribute can make it easier to locate errors in the data and also helps with cross-referencing of results to individual cases. If there are two or more label attributes, only the last is used.

Attributes defined by formulas

The name of each implicitly-defined attribute is followed by `:=' and then a formula defining the attribute value. The formula is written in the usual way, using parentheses where needed, and may refer to any attribute that has been defined before this one. Constants in the formula can be `?' (meaning unknown), `N/A' (meaning not applicable), numbers (written in decimal notation), dates, times, and discrete attribute values enclosed in string quotes `"'. The operators and functions available for use in the formula are The value of an implicitly-defined attribute is either numeric or true/false depending on the formula. This example includes one implicitly-defined attribute, displacement per cylinder (displ/cyl). This is a numeric attribute since its value is a ratio of two explicitly-defined numeric attributes. The value of a hypothetical attribute
	small := cylinders = 4 and class = "CAR".
would be either t or f since the value given by the formula is either true or false.

If the value of the formula cannot be determined for a particular case, the value of the implicitly-defined attribute is unknown. For example, consider a car with a value `?' for the cylinders attribute. The displacement per cylinder cannot then be calculated, so the implicitly-defined attribute displ/cyl would also have an unknown value.

Dates, times, and timestamps

Dates are stored by Cubist as the number of days since a particular starting point so some operations on dates make sense. Thus, if we have attributes
	d1: date.
	d2: date.
we could define
	interval := d2 - d1.
	gap := d1 <= d2 - 7.
	d1-day-of-week := (d1 + 1) % 7 + 1.
interval then represents the number of days from d1 to d2 (non-inclusive) and gap would have a true/false value signaling whether d1 is at least a week before d2. The last definition is a slightly non-obvious way of determining the day of the week on which d1 falls, with values ranging from 1 (Monday) to 7 (Sunday).

Similarly, times are stored as the number of seconds since midnight. If the names file includes

	start: time.
	finish: time.
	elapsed := finish - start.
the value of elapsed is the number of seconds from start to finish.

Timestamps are a little more complex. A timestamp is rounded to the nearest minute, but limitations on the precision of floating-point numbers mean that the values stored for timestamps from more than thirty years ago are approximate. If the names file includes

	departure: timestamp.
	arrival: timestamp.
	flight time := arrival - departure.
the value of flight time is the number of minutes from departure to arrival.

Selecting the attributes that can appear in models

An optional final entry in the names file affects the way that Cubist constructs models. This entry takes one of the forms
	attributes included:
	attributes excluded:
followed by a comma-separated list of attribute names. The first form restricts the attributes used in models to those specifically named; the second form specifies that models must not use any of the named attributes.

Excluding an attribute from models is not the same as ignoring the attribute (see `ignore' above). As an example, suppose that a numeric attribute A is defined in the data, but background knowledge suggests that only the logarithm of A should appear in models. The names file might then contain the following entries:

	   . . .
	A: continuous.
	LogA := log(A).
	   . . .
	attributes excluded: A.
In this example the attribute A could not be defined as ignore because the definition of LogA would then be invalid.

The same pattern could be used if the goal was to model the log of A rather than the value of A itself. In this case the target attribute would be given as LogA and the exclusion of A would be necessary to prevent the value of A being used in the model for LogA.

Data file

The second essential file, the application's data file (here fc2008.data), provides information on the training cases that Cubist will use to construct a model. The entry for each case consists of one or more lines that give the values for all explicitly-defined attributes. Values are separated by commas and the entry for each case is optionally terminated by a period. Once again, anything on a line after a vertical bar is ignored. (If the information for a case occupies more than one line, make sure that the line breaks occur after commas.)

The first three cases from file fc2008.data are:

SUV,HUMMER H3 4WD,3.7,5,Auto,4,4,R,-,2801,2
VAN,GMC G1500/2500 SAVANA 2WD PASS,5.3,8,Auto,4,R,E,-,3900,2
CAR,DODGE CHARGER AWD,3.5,6,Auto,5,4,R,-,2335,4
Notice that the value of the implicitly-defined attribute displ/cyl is not given for each case since it is computed from other attribute values.

Don't forget the commas between values! If you leave them out, Cubist will not be able to process your data.

A value that is missing or unknown is entered as `?'. Similarly, `N/A' denotes a value that is not applicable for a particular case.

Test and cases files (optional)

Of course, the value of predictive models lies in their ability to make accurate predictions! It is difficult to judge the accuracy of a model by measuring how well it does on the cases used in its construction; the performance of the model on new cases is much more informative.

The third kind of file used by Cubist is a test file of new cases (here fc2008.test) on which the model can be evaluated. This file is optional and has exactly the same format as the data file. In this application the 1,141 cases have been split randomly 70%:30% into data and test files containing 800 and 341 cases respectively.

Another optional file, the cases file (e.g. fc2008.cases), has the same format as the data and test files. The cases file is used primarily with the public source code described later on.

Constructing Models

Once the names, data, and optional files have been set up, everything is ready to use Cubist.

The general form of the Unix command is

        cubist -f filestem [options]
This invokes Cubist with the -f option that identifies the application name (here fc2008). If no filestem is specified using this option, Cubist uses a default filestem that is probably incorrect. (Moral: always use the -f option!)

There are several options that affect the type of model that Cubist produces and the way that it is constructed. In this section we will examine each of them, starting with the simpler situations.

Rule-based models

When Cubist is invoked with only the -f option, as
        cubist -f fc2008
it constructs a rule-based model and produces output like this:
Cubist [Release 2.08]  Mon Jun 25 13:15:27 2012
---------------------

    Options:
        Application `fc2008'

    Target attribute `fuel cost'

    Replacing unknown attribute values:
        `valves/cyl' by 3.4

Read 800 cases (12 attributes) from fc2008.data

Model:

  Rule 1: [142 cases, mean 1896.5, range 884 to 2801, est err 127.4]

    if
        class in {CAR, VAN}
        displ <= 4.6
        fuel in {R, D, C}
    then
        fuel cost = -49.2 + 163 cylinders + 1306 displ/cyl + 77 displ
                    + 46 valves/cyl

  Rule 2: [138 cases, mean 2971.4, range 2335 to 4090, est err 227.3]

    if
        displ > 4.6
        displ <= 6.2
        gears <= 6
        fuel in {R, P}
    then
        fuel cost = 1010.3 + 284 displ + 91 cylinders - 791 displ/cyl
                    + 56 valves/cyl

  Rule 3: [185 cases, mean 2202.9, range 1310 to 2999, est err 126.7]

    if
        class in {STATION WAGON, SUV, PICKUP TRUCK}
        displ <= 4.6
        fuel in {R, D, C}
    then
        fuel cost = 266.2 + 195 cylinders + 1560 displ/cyl + 33 displ
                    - 39 valves/cyl

  Rule 4: [65 cases, mean 1996.0, range 1404 to 2502, est err 122.0]

    if
        cylinders <= 4
        fuel in {P, E}
    then
        fuel cost = 28532.5 + 10271 displ - 6839 cylinders - 39685 displ/cyl

  Rule 5: [13 cases, mean 3900.0, range 3900 to 3900, est err 0.0]

    if
        class in {SUV, PICKUP TRUCK}
        displ <= 5
        fuel = E
    then
        fuel cost = 3900

  Rule 6: [16 cases, mean 3414.6, range 2999 to 3900, est err 60.0]

    if
        class in {SUV, PICKUP TRUCK}
        displ > 5
        fuel = E
    then
        fuel cost = -6038.9 + 29563 displ/cyl - 1929 displ

  Rule 7: [176 cases, mean 2357.1, range 1876 to 3460, est err 148.4]

    if
        class in {CAR, STATION WAGON}
        displ <= 4.6
        cylinders > 4
        fuel in {P, E}
    then
        fuel cost = -309.5 + 376 cylinders - 281 displ + 2293 displ/cyl

  Rule 8: [11 cases, mean 3803.2, range 3545 to 3900, est err 116.2]

    if
        class = VAN
        displ > 4.6
        fuel = E
    then
        fuel cost = 3652.1 + 25 cylinders + 20 displ - 88 displ/cyl

  Rule 9: [10 cases, mean 3628.9, range 2502 to 4500, est err 467.3]

    if
        displ > 6.2
    then
        fuel cost = 7497.8 - 502 displ - 400 displ/cyl + 18 valves/cyl

  Rule 10: [15 cases, mean 3024.2, range 2646 to 3460, est err 203.8]

    if
        displ > 4.6
        gears > 6
    then
        fuel cost = 2144.6 + 1293 displ - 7403 displ/cyl - 334 valves/cyl

  Rule 11: [26 cases, mean 2661.6, range 1800 to 3213, est err 233.6]

    if
        class in {SUV, VAN}
        displ <= 4.6
        cylinders > 4
        fuel in {P, E}
    then
        fuel cost = 1025 + 362 displ + 59 cylinders

  Rule 12: [3 cases, mean 2032.7, range 1999 to 2100, est err 67.3]

    if
        displ > 4.6
        gears = N/A
    then
        fuel cost = 1969 + 5 displ


Evaluation on training data (800 cases):

    Average  |error|              142.6
    Relative |error|               0.31
    Correlation coefficient        0.94


        Attribute usage:
          Conds  Model

           96%           fuel
           92%    98%    displ
           71%           class
           33%    98%    cylinders
           20%           gears
                  95%    displ/cyl
                  61%    valves/cyl


Evaluation on test data (341 cases):

    Average  |error|              152.6
    Relative |error|               0.35
    Correlation coefficient        0.92


Time: 0.0 secs
The first part identifies the version of Cubist, the run date, the options with which the system was invoked, and the attribute that contains the target value.

Now we come to the training data. Some attribute values might be missing; if so, Cubist replaces them by the most probable values. Missing values of continuous attributes are replaced by the mean of the known values for that attribute, while the replacement for missing discrete values is the most frequent attribute value. Any such replacements are noted on the output. Here valves/cyl is the only explicitly-defined attribute whose value is missing for some cases in fc2008.data; those cases are given the average value (a rather unrealistic 3.4). The same values are also used to replace missing values in any test cases, although the messages are not repeated.

Cubist constructs a model from the 800 training cases in the file fc2008.data, and this appears next. A model consists of a list of rules, each of the form

    if conditions then linear formula
A rule indicates that, whenever a case satisfies all the conditions, the linear formula is appropriate for predicting the value of the target attribute. (If two or more rules apply to a case, then the values are averaged to arrive at a final prediction.)

Although the order of the rules does not affect the value predicted by a model, Cubist presents them in decreasing order of importance. The first rule makes the greatest contribution to the model's accuracy on the training data; the last rule has the least impact.

Each rule also carries some descriptive information: the number of training cases that satisfy the rule's conditions, their target values' mean and range, and a rough estimate of the expected error magnitude of predictions made by the rule. Within the linear formula, the attributes are ordered in decreasing relevance to the result.

Let's illustrate all this on Rule 1 above. There are three conditions:

        class in {CAR, VAN}
        displ <= 4.6
        fuel in {R, D, C}
Among the 800 training cases there are 142 that satisfy all three conditions; their fuel costs range from $884 to $2801 with an average value of $1896.50. Cubist finds that the target value of these or other cases satisfying the conditions can be modeled by the formula
        fuel cost = -49.2 + 163 cylinders + 1306 displ/cyl + 77 displ + 46 valves/cyl
with an estimated error of $127.40. For cases covered by this rule, cylinders has the most effect on fuel cost, displ/cyl and displ a lesser effect, and valves/cyl the least effect.

There is a point worth noting about Rule 5:

  Rule 5: [13 cases, mean 3900.0, range 3900 to 3900, est err 0.0]

    if
        class in {SUV, PICKUP TRUCK}
        displ <= 5
        fuel = E
    then
        fuel cost = 3900
The formula predicts a constant value for these cases. In rules like this, the constant value may differ from the mean, which may appear odd! This is not an error -- under the default option settings, Cubist attempts to minimize average error magnitude, and so uses the median target value of the cases covered by the rule rather than the mean. This can be altered by invoking the option for unbiased rules, described later.

The next section covers the evaluation of this model shown in the second part of the output. Before we leave this output, though, the final line states the elapsed time for the run. For small applications such as this, with only a few training cases and a handful of attributes, a model is produced quite quickly. Model construction can take much longer for larger applications with many thousands of cases and tens or hundreds of attributes. The progress of Cubist on long runs can be monitored by examining the last few lines of the temporary file filestem.tmp (e.g. fc2008.tmp). This file displays the stage that Cubist has reached and, for most stages, gives an indication of the fraction of the stage that has been completed.

Evaluation

Models constructed by Cubist are evaluated on the training data from which they were generated, and also on a separate file of unseen test cases if this is present. (Evaluation by cross-validation is discussed elsewhere.) Results on the cases in fc2008.data are:

Evaluation on training data (800 cases):

    Average  |error|              142.6
    Relative |error|               0.31
    Correlation coefficient        0.94
The average error magnitude is straightforward enough. The relative error magnitude is the ratio of the average error magnitude to the error magnitude that would result from always predicting the mean value; for useful models, this should be less than 1! The correlation coefficient measures the agreement between the cases' actual values of the target attribute and those values predicted by the model.

Usually, as in this example, the results cover all training cases. When there are more than 20,000 of them and composite models (see later) are used, the evaluation covers only a random sample of 10,000 training cases and this fact is noted in the output.

For some applications, particularly those with many attributes, it may be useful to know how individual attributes contribute to the model. This is shown in the next section:

        Attribute usage:
          Conds  Model

           96%           fuel
           92%    98%    displ
           71%           class
           33%    98%    cylinders
           20%           gears
                  95%    displ/cyl
                  61%    valves/cyl
The first column shows the approximate percentage of cases for which the named attribute appears in a condition of an applicable rule, while the second column gives the percentage of cases for which the attribute appears in the linear formula of an applicable rule. The second entry, for example, says that displ is used in the condition part of rules that cover 92% of cases and in the formulas of rules that cover 98% of cases. Attributes for which both these values are less than 1% are not shown.

If a test file is present, Cubist produces a summary similar to that for the training cases:

Evaluation on test data (341 cases):

    Average  |error|              152.6
    Relative |error|               0.35
    Correlation coefficient        0.92
Cubist also generates a file filestem.pred (here fc2008.pred) that shows the actual and predicted value for each test case. The first few lines of this file generated from the run above are:
(Default value 2419.5)

   Actual  Predicted    Case
    Value      Value
 --------  ---------    ----
     1751     1736.2    JEEP PATRIOT 2WD
     1680     1593.8    SUZUKI SX4 SEDAN
     1357     1513.1    TOYOTA COROLLA
     2367     2323.3    PORSCHE CARRERA 4 S TARGA
Notice that each case is identified by its value of the label attribute; if there is no such attribute, the case number in the .test file is used instead.

Unbiased rules

In its default mode, Cubist tries to minimize the average absolute error of the values predicted for new cases. As a consequence, the rules that Cubist generates may be biased -- the mean predicted value for the training cases covered by a rule may differ from their mean value.

Suppose, for instance, that we have to summarize the values 1, 2, and 12 by a single number. If we choose the mean value 5, the average absolute error over these values would be 14/3. Choosing the median value 2, however, the average absolute error becomes 11/3. Even though it gives lower absolute error, the choice of 2 is biased since the prediction (2) is lower than the mean of the values (5).

The option -u instructs Cubist to make each rule approximately unbiased, with the downside that average absolute error is usually slightly higher. This option is recommended for applications where the training cases have a preponderance of a single target value (such as zero) because unbiased rules tend to give a finer gradation of predicted values.

When the fuel cost application is run with the option for unbiased rules, the only effect is to change the constant in the formula for each rule and the rule's estimated error. The rule that shows the greatest difference is

  Rule 8: [11 cases, mean 3803.2, range 3545 to 3900, est err 169.0]

    if
        class = VAN
        displ > 4.6
        fuel = E
    then
        fuel cost = 3555.3 + 25 cylinders + 20 displ - 88 displ/cyl
The constant term in the formula changes from 3652.1 to 3555.3 and the estimated error is greater. This shows that the original rule was strongly biased towards higher values. For this application, unbiased rules have a slightly greater error of 153.0 on the unseen test cases.

Finally, because cases can be covered by different numbers of rules, the use of unbiased rules does not guarantee that the entire model is unbiased.

Composite models

For some applications, the predictive accuracy of a rule-based model can be improved by combining it with an instance-based or nearest-neighbor model. The latter predicts the target value of a new case by finding the n most similar cases in the training data, and averaging their target values.

Cubist employs an unusual method for combining rule-based and instance-based models. Cubist finds the n training cases that are "nearest" (most similar) to the case in question. Then, rather than averaging their target values directly, Cubist first adjusts these values using the rule-based model. Here's how it works:

Suppose that x is the case whose unknown target value is to be predicted, and y is one of x's nearest neighbors in the training data. The target value of y is known: let us call it T(y). The rule-based model can be used to predict target values for any case, so let its predictions for x and y be M(x) and M(y) respectively. The model then predicts that the difference between the target values of x and y is M(x)-M(y). The value of x predicted by neighbor y is adjusted to reflect this difference, so that Cubist uses T(y)+M(x)-M(y) instead of y's raw target value. (This is described in more detail in the paper "Combining instance-based and model-based learning", Proceedings of the Tenth International Conference on Machine Learning, pages 236-243, Morgan Kaufmann Publishers, San Francisco, 1993.)

The option -i instructs Cubist to use composite models of this type. Alternatively, the option -a allows the decision regarding which kind of model to use -- rule-based or composite -- to be left to Cubist itself. In the latter case, Cubist derives from the training data a heuristic estimate of the accuracy of each type of model, and chooses the form that appears more accurate. The derivation of these estimates requires quite a lot of computation, so leaving the decision to Cubist can result in a noticeable increase in the time required to build a model.

Now for the value of n, the number of nearest neighbors to be used. The option -n neighbors sets the number directly; the allowable range is from 1 to 9. If the value is not specified in this way, Cubist will choose an appropriate value in the range.

To continue the illustration: when Cubist is allowed to choose a model type on the basis of the 800 training cases and the number of nearest neighbors is not specified, it opts for a composite model using a single nearest neighbor. The rule-based model itself is unchanged, but the composite model gives different results on the training and test cases, the latter being

Evaluation on test data (341 cases):

    Average  |error|              101.1
    Relative |error|               0.23
    Correlation coefficient        0.95
The performance of the composite model on the test cases in fc2008.test thus improves upon that of the default rule-based model, average error magnitude falling from 152.6 to 101.1.

Nearest neighbor models are adversely affected by the presence of irrelevant attributes. All attributes are taken into account when evaluating the similarity of two cases and irrelevant attributes introduce a random factor into this measurement. As a result, composite models are most effective when the number of attributes is relatively small and all attributes are relevant to the prediction task.

Committee models

In addition to the composite rule-based/nearest neighbor models discussed above, Cubist can also generate committee models made up of several rule-based models. Each member of the committee predicts the target value for a case and the members' predictions are averaged to give a final prediction.

The first member of a committee model is always exactly the same as the model generated without the committee option. The second member is a rule-based model designed to correct the predictions of the first member; if the first member's prediction is too low for a case, the second member will attempt to compensate by predicting a higher value. The third member tries to correct the predictions of the second member, and so on. The recommended number of members is five, a value that balances the benefits of the committee approach against the cost of generating extra models.

The option -C members causes Cubist to construct a model committee and specifies the number of committee members. When this option is invoked with five members, the results show a smaller improvement than that obtained with composite models:

Evaluation on test data (341 cases):

    Average  |error|              141.2
    Relative |error|               0.32
    Correlation coefficient        0.94

Committee models are of most benefit when the initial model is reasonably accurate, so they are more useful for fine-tuning good models than for overcoming the deficiencies of poor models. Finally, committee models can be used in conjunction with composite models.

Simplicity-accuracy trade-off

Cubist employs heuristics that try to simplify models without substantially reducing their predictive accuracy. In some applications, however, it might be desirable to generate more concise models -- for instance, when the models must be very easy to understand. Of course, over-simplified models usually have lower predictive accuracy so there is a trade-off between simplicity and utility.

The complexity of a model can be controlled by restricting the number of rules that it may contain (the default value being 500 rules). The option -r rules sets the maximum number of rules that may be used in a model. For the fc2008 application, setting the maximum number of rules to 5 gives a simpler model:

  Rule 1: [142 cases, mean 1896.5, range 884 to 2801, est err 127.4]

    if
        class in {CAR, VAN}
        displ <= 4.6
        fuel in {R, D, C}
    then
        fuel cost = -49.2 + 163 cylinders + 1306 displ/cyl + 77 displ
                    + 46 valves/cyl

  Rule 2: [166 cases, mean 2998.8, range 1999 to 4500, est err 264.3]

    if
        displ > 4.6
        fuel in {R, P}
    then
        fuel cost = 1822.4 + 320 displ - 1721 displ/cyl + 89 valves/cyl
                    + 39 cylinders

  Rule 3: [40 cases, mean 3679.2, range 2999 to 3900, est err 222.0]

    if
        displ > 4.6
        fuel = E
    then
        fuel cost = 4218.9 - 7008 displ + 55286 displ/cyl

  Rule 4: [185 cases, mean 2202.9, range 1310 to 2999, est err 126.7]

    if
        class in {STATION WAGON, SUV, PICKUP TRUCK}
        displ <= 4.6
        fuel in {R, D, C}
    then
        fuel cost = 266.2 + 195 cylinders + 1560 displ/cyl + 33 displ
                    - 39 valves/cyl

  Rule 5: [267 cases, mean 2298.8, range 1404 to 3460, est err 171.2]

    if
        displ <= 4.6
        fuel in {P, E}
    then
        fuel cost = -1661.1 + 581 cylinders - 733 displ + 5358 displ/cyl
The downside in this example is that the average error magnitude on the test cases increases from 152.6 to 167.0.

Extrapolation

The extrapolation parameter controls the extent to which predictions made by Cubist's models can fall outside the range of values seen in the training data. Extrapolation is inherently more risky than interpolation, where predictions must lie between the lowest and highest observed value.

The option -e extrapolation sets this extrapolation factor in the form of a percentage. Each rule records the highest and lowest target value of the training cases satisfying that rule's conditions. When the target value of a new case is predicted using the rule, the value computed from the linear formula may fall outside this range. The extrapolation parameter limits the degree to which new values can lie above or below the values seen in the training data, expressed as a percentage of the range (the default being 5%).

For example, the lowest target value among the 142 training cases covered by Rule 1 above is 884 and the highest is 2801. The range is therefore 1917 and, under the default extrapolation limit of 5%, the value predicted by this rule for a new case cannot be lower than 788.15 (884 - 95.85) or higher than 2896.85 (2801 + 95.85). Any computed value that lies outside these bounds is changed to the nearer bound. If the linear formula associated with Rule 1 were to predict a value of 600, say, then this would be adjusted to 788.15.

Extrapolation may be constrained even further in two situations. When all the training cases covered by a rule have target values greater than or equal to zero, the rule will never predict a value less than zero. This restriction prevents Cubist from making silly predictions such as negative fuel costs. Similarly, when a rule covers cases whose target values are all less than or equal to zero, the predicted value from the rule will never be positive.

Sampling from large datasets

Even though Cubist is relatively fast, building models from a large number of cases can take an inconveniently long time. Cubist incorporates a facility to extract a random sample from a dataset, construct a model from the sample, and then test the model on a disjoint collection of cases. By using a smaller set of training cases in this way, the process of generating a model is expedited, but at the cost of a possible reduction in the model's predictive performance.

The option -S x has two consequences. Firstly, a random sample containing x% of the cases in the application's data file is used to construct the model. Secondly, the model is evaluated on a non-overlapping set of test cases consisting of another (disjoint) sample of the same size as the training set (if x is less than 50%), or all cases that were not used in the training set (if x is greater than or equal to 50%).

As an example, suppose that the application's data file contains 100,000 cases. If a sample of 10% is used, the model will be constructed from a sample of 10,000 cases and tested on a disjoint sample of 10,000 cases. Alternatively, selecting sampling with 60% will cause the model to be constructed from 60,000 cases and tested on the remaining 40,000 cases.

By default, the random sample changes every time that a model is constructed, so that successive runs of Cubist with sampling will usually produce different results. This re-sampling can be avoided by the option -I seed that uses the integer seed to initialize the sampling. Runs with the same value of the seed and the same sampling percentage will always use the same training cases.

Cross-validation trials

As we saw earlier, the performance of a model on the training cases from which it was constructed gives a poor estimate of its accuracy on new cases. The true predictive accuracy of the model can be estimated by sampling, as above, or by using a separate test file; either way, the classifier is evaluated on cases that were not used to build it. However, this estimate can be unreliable unless the numbers of cases used to build and evaluate the model are both large. If the cases in fc2008.data and fc2008.test were to be shuffled and divided into new training and test sets, Cubist would probably construct a different model whose accuracy on the test cases might vary considerably.

One way to get a more reliable estimate of predictive accuracy is by f-fold cross-validation. The cases (including those in the test file, if it exists) are divided into f blocks of roughly the same size and target value distribution. For each block in turn, a model is constructed from the cases in the remaining blocks and tested on the cases in the hold-out block. In this way, each case is used just once as a test case. The accuracy of a model produced from all the cases is estimated by averaging results on the hold-out cases.

The option -X f runs such a f-fold cross-validation. For example, the command

	cubist -f fc2008 -X 10
selects 10-fold cross-validation. After reporting on the model produced at each fold, the output shows a summary like this:
Summary:

    Average  |error|              145.4
    Relative |error|               0.32
    Correlation coefficient        0.93
The file filestem.pred once again contains a case-by-case record of the actual and predicted values on the unseen cases.

As with sampling above, each cross-validation run will normally use a different random division of the data into blocks, unless this is prevented by using the -I option.

The cross-validation procedure can be repeated for different random partitions of the cases into blocks. The average error from these distinct cross-validations is then an even more reliable estimate of the error of the model produced from all the cases. A shell script and associated programs for carrying out multiple cross-validations are included with Cubist. The shell script xval is invoked with any combination of Cubist options and some further options that describe the cross-validations themselves:

If detailed results are retained via the +d option, they appear in files named filestem.oi[+suffix] where i is the cross-validation number (0 to repeats-1). A summary of the cross-validations is written to file filestem.res[+suffix].

As an example, the command

	xval -f fc2008 -a R=10 +new

runs ten complete 10-fold cross-validations (and so constructs 100 models in all), allowing Cubist to choose between rule-based and composite models, and gives the following results in file fc2008.res+new:

Summary:
--------

    Average  |error|              101.3
    Relative |error|               0.23
    Correlation coefficient        0.95

Since a single cross-validation fold uses only part of the application's data, running a cross-validation does not cause a model to be saved. To save a model for later use, simply run Cubist without employing cross-validation.

Weighting individual cases

By default, all training cases are treated equally when a model is constructed. In some applications, however, it may be desirable to assign different importance to the cases. Cubist achieves this by recognizing an optional attribute that gives the weight of each case. The attribute name must be case weight and it must have numeric values. The relative weight assigned to each case is its value of this attribute divided by the average value; if the value is undefined ("?"), not applicable ("N/A"), or is less than or equal to zero, the case's relative weight is set to 1.

The case weight attribute itself is not used in the model!

To illustrate the idea, let us suppose that we wish our model to be relatively more accurate on cars rather than other vehicle types. We might add a case weight attribute of type continuous to fc2008.names and add an extra value to each case in the .data file, 5 for cars and 1 for other vehicles. This means that the importance of a training case for a car is five times that of cases for other vehicle types. Cubist will now attempt to minimize weighted error, so cars should have more influence on the new model. (Note: we must also add an extra value to each case in the .test file, since we have increased the number of attributes. This value is not used.)

The initial model gives an average absolute error of 161.0 for the cars in the unseen test cases. With the case-weighted model, this error drops to 153.1.

A cautionary note: The use of case weighting does not guarantee that the model will be more accurate for unseen cases with higher weights. Predictive accuracy on more important cases is likely to be improved only when cases with similar values of the predictor attributes also have similar values of the case weight attribute, i.e. when relatively important cases "clump together." Without this property, case weighting can introduce an unhelpful element of randomness into the model generation process.

Linux GUI

Linux users who have installed a recent version of Wine can invoke a slightly simplified version of the user interface of the Windows version. The executable program gui starts the graphical user interface whose main window has five buttons:

Locate Data
invokes a browser to find the files for your application, or to change the current application;
Build Model
selects the type of model to be constructed and sets other options;
Stop
interrupts the model-generating process;
Review Output
re-displays the output from the most recent model (if any), saved automatically in a file filestem.out; and
Cross-Reference
shows how cases in training or test data relate to (parts of) a model and vice versa.
For more details on these, please see the Windows tutorial.

The graphical interface calls Cubist directly, so use of the GUI has minimal impact on the time taken to construct a Cubist model.

Please note: Cubist should be run for the first time from the command-line interface, not the GUI. The first run installs the licence ID; after that has been done, Cubist can be used from either interface.

Linking to Other Programs

The most recent model generated by Cubist is saved in file filestem.model. Free C source code is available to read these model files and to make predictions with them, enabling you to use Cubist models in other programs.

As an example, the source includes a program sample.c that reads a saved model file and then prints the value predicted by the model for each case in a cases file. This sample program is intended to illustrate methods for interacting with the model.

The program expects to find the following files:

There are several options that control the format of the output:

The optional error bounds are estimated heuristically so that the absolute error should be less than E for about 95% of cases. A summary at the end of the output shows the actual percentage of cases whose true value is known and lies within the given bounds.

As an example, we use the original model for fc2008 and copy the .test file into fc2008.cases. When the -e option is selected, the (abbreviated) output looks like this:

Case                 Given  Predicted
 ID                  Value      Value

 JEEP PATRIOT 2       1751     1736.2 +- 316.8
 SUZUKI SX4 SED       1680     1593.8 +- 318.5
 TOYOTA COROLLA       1357     1513.1 +- 318.5
     . . . . .
*BUGATTI VEYRON       4500     3353.8 +- 1168.2
     . . . . .
 HONDA ELEMENT        1911     1905.4 +- 316.8
 FORD F150 STX        2801     2902.0 +- 568.2

319 / 341 (94%) cases within error bounds
The asterisk in the first column of the Bugatti case indicates that its value for one or more of the attributes used in the model lies outside the range observed in the training data, so the predicted value is suspect. (It has 16 cylinders, whereas none of the training cases has more than 12.)

Click here to download a gzipped tar file containing the C source code. Please see the comments at the beginning of sample.c for information on compiling the program.


Appendix: Summary of Options

© RULEQUEST RESEARCH 2013 Last updated May 2013


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