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Classification and Regression Trees

# Overview

As part of tools for statistical modelling, EST includes methods for automatically building decision trees and decision lists from features data to predict both fixed classed (classification) or gaussians (regression). Wagon is the basic program that provide this facility.

The construction of CARTs (classification and regression trees) is best described in [1] and has become a common basic method for building statistical models from simple feature data. CART is powerful because it can deal with incomplete data, multiple types of features (floats, enumerated sets) both in input features and predicted features, and the trees it produces often contain rules which are humanly readable.

Decision trees contain a binary question (yes/no answer) about some feature at each node in the tree. The leaves of the tree contain the best prediction based on the training data. Decision lists are a reduced form of this where one answer to each question leads directly to a leaf node. A tree's leaf node may be a single member of some class, a probability density function (over some discrete class), a predicted mean value for a continuous feature or a gaussian (mean and standard deviation for a continuous value).

Theorectically the predicted value may be anything for which a function can defined that can give a measure of impurity for a set of samples, and a distance measure between impurities.

The basic algorithm is given a set of samples (a feature vector) find the question about some feature which splits the data minimising the mean "impurity" of the two partitions. Recursively apply this splitting on each partition until some stop criteria is reached (e.g. a minimum number of samples in the partition.

The basic CART building algorithm is a greedy algorithm in that it chooses the locally best choice which is suboptimal but a full search for a fully optimized set of question would be computationallly very expensive (and often not much better). Although there are pathological cases in most data sets this greediness is not a problem.

The basic building algorithm starts with a set of feature vectors representing samples, at each stage all possible questions for all possibles features are asked about the data finding out how the question splits the data. A measurement of impurity of each partitioning is made and the question that generates the least impure partitions is selected. This process is applied recursively on each sub-partions recursively until some stop criteria is met (e.g. a minimal number of samples in a partition).

## Impurities

The impurity of a set of samples is designed capture how similar the samples are to each other. The smaller the number the less impure the sample set is.

For sample sets with continuous predictees Wagon uses the variance times number of sample points. The variance alone could be used by this overly favour very small sample sets. As the test that uses the impurity is trying to minimise it over a partitioning of the data, multiple each part with the number of samples will encourage larger partitions, which we have found lead to better decision trees in general.

For sample sets with discrete predictees Wagon uses the entropy times number of sample points. Again the number of sample points is used in so that small sample set are not unfairly favoured. The entropy for a sample set is calculated as:

Other impurity measure could be used if required. For example an experimental cluster technique used for unit selection actually used impurity calculated as the mean euclidean distance between all vectors of parameters in the sample set. However the above two are more standard measures.

## Question forming

Wagon has to automatically form questions about each feature in the data set.

For discrete features questions are build for each member of the set, e.g. if feature n has value x. Our implementation does not currently support more complex questions which could achieve better results (though at the expense of training time). Questions about features being some subset of the class members may give smaller trees. If the data requires distinction of values a, b and c, from d e and f, our method would require three separate questions, while if subset questions could be formed this could be done in one step which would not only give a smaller tree but also not unecessarily split the samples for a, b and c. In general subset forming is exponential on the number items in the class though there are techniques that can reduce this with heuristics. However these are currently not supported. Note however the the tree formalism produced but Wagon does support such questions (with the operator "in") but Wagon will never produce these question, though other tree building techniques (e.g. by hand) may use this form of question.

For continuous features Wagon tries to find a partition of the range of the values that best optimizes the average impurity of the partitions. This is currently done by linearly splitting the range into a predefined subparts (10 by default) and testing each split. This again isn't optimal but does offer reasonably accuracy without requiring vast amounts of computation.

## Tree Building criteria

There are many ways to constrain the tree building algorithm to help build the "best" tree. Wagon supports many of these (though there are almost certainly others that is does not.

In the most basic forms of the tree building algorithm a fully exhaustive classifcation of all samples would be achieved. This, of course is unlikely to be good when given samples that are not contained within the training data. Thus the object is to build a classification/regression tree that will be most suitable for new unseen samples. The most basic method to achieve this is not to build a full tree but require that there are at least n samples in a partition before a question split is considered. We refer to that as the stop value. A number like 50 as a stop value will often be good, but depending of the amount of data you have, the distribution of it, etc various stop value may produce more general trees.

A second method for building "good" trees is to hold out some of the training data and build a (probably over-trained) tree with a small stop value. Then prune the tree back to where it best matches the held out data. This can often produce better results than a fixed stop value as this effectively allows the stop value to vary through different parts of the tree depending on how general the prediction is when compared against held out data.

It is often better to try to build more balanced trees. A small stop value may cause the tree building algorithm to find small coherent sets of samples with very specific questions. The result tree becomes heavily lop-sided and (perhaps) not optimal. Rather than having the same literal stop value more balanced trees can built if the stop value is defined to be some percentage of the number of samples under consideration. This percentage we call a balance factor. Thus the stop value is then the largest of the defined fixed stop value or the balance factor times the number of samples.

To some extent the multiplication of the entropy (or variance) by the number of samples in the impurity measure is also a way to combat imbalance in tree building.

A good technique we have found is to build trees in a stepwise fashion. In this case instead of considering all features in building the best tree. We increment build trees looking for which individual feature best increases the accuracy of the build tree on the provided test data. Unlike within the tree building process where we are looking for the best question over all features this technique limits which features are available for consideration. It first builds a tree using each and only the features provided looking for which individual feature provides the best tree. The selecting that feature is builds n-1 trees with the best feature from the first round with each of the remaining features. This process continues until no more features add to the accuracy or some stopping criteria (percentage improved) is not reached.

This technique is also a greedy technique but we've found that when many features are presented, especially when some are highly correlated with each other, stepwise building produces a significantly more robust tree on external test data. It also typically builds smaller trees. But of course there is a cost in computation time.

While using the stepwise option each new feature added is printed out. Care should be taking in interpreting what this means. It does not necessarily give the order and relative importance of the features, but may be useful if showing which features are particualrly important to this build.

Stepwise tests each success tree against the specified test set, (balance, held out and stop options are respected for each build). As this is using the test set which optimizing the tree, it is not valid to view the specified test set as a genuine test set. Another externally held test set should be used to test the accuracy of generated tree.

Another technique that has been shown to work in tree building is called random forests". In these technique multiple trees are build with some random subset of the features presented, then the multiple trees are combined by some technique (e.g. simple average). This often works much better. Although doesn't offer building randon forests as a direct option, it can be (and is) used to do this. You can vary the features in in the description file by marking them with *ignore</em> or you can use the <em>dropout</em> options <em>-dof</em> and <em>dos</em> which specify a probability of ignoring a feature or sample at easy stage of the build process. Its not clear if the <em>-dof</em> or <em>-dos</em> options are useful, we have only fully investigated random forests in the voice building process through externally selecting features, but we added these internal options (with trendier names) as part of our investigation. @subsection cart-formats Data format The input data for wagon (and some other model building tools in the Edinburgh Speech Tools library), should consist of feature vectors, and a description of the fields in these vectors. @subsubsection estwagon-featvectors Feature vectors A feature vector is a file with one sample per line, with feature value as white space separated tokens. If your features values conatin whitespace then you must quote them using double quotes. The (Festival) program <tt>dumpfeats</tt> is specifically designed to generate such files from databases of utterances but these files may be generated from any data source. Each vector must have the same number of features (and in the same order. Features may be specified as "ignored" in the description (or in actual use) so it is common that data files contain more features than are always used in model building. By default the first feature in a data file is the predictee, though at least in wagon) the predictee field can be named at tree building time to be other than the first field. Features can be discrete of continuous but at present must be single valued, "multi-valued" or "list-valued" features are not currently supported. Note this means that a feature in different samples may have different values but in a particular sample a particular feature can only have one value.

A type example is:

    0.399 pau sh  0   0     0 1 1 0 0 0 0 0 0
    0.082 sh  iy  pau onset 0 1 0 0 1 1 0 0 1
    0.074 iy  hh  sh  coda  1 0 1 0 1 1 0 0 1
    0.048 hh  ae  iy  onset 0 1 0 1 1 1 0 1 1
    0.062 ae  d   hh  coda  1 0 0 1 1 1 0 1 1
    0.020 d   y   ae  coda  2 0 1 1 1 1 0 1 1
    0.082 y   ax  d   onset 0 1 0 1 1 1 1 1 1
    0.082 ax  r   y   coda  1 0 0 1 1 1 1 1 1
    0.036 r   d   ax  coda  2 0 1 1 1 1 1 1 1

Note is it common to have thousands, even hundreds of thousands of samples in a data file, and the number of features can often be in the hundreds, though can also be less than ten depending on the what it describes.

### Data descriptions

A data file also requires a description file which names and classifies the features in a datafiles. Features must haves names so they can be refered to in the decision tree (or other model output) and also be classified into their type. The basic types available for features are:

- continuous for features that range over reals (e.g. duration of phones)
- categorial for features with a pre-defined list of possible values (e.g. phone names)
- string for features with an open class of discrete values (e.g. words)
- vectors like floats but as vectors of floats, (e.g. MFCC data)

The data description consists of a parenthesized list of feature descriptions. Each feature description consists of the feature name and its type (and/or possible values). Feature names, by convention, should be features names in the sense for features (and pathnames) used throughout the utterance structures in the Edinburgh Speech Tools. The expected method to use models generated from features sets in the Edinburgh Speech Tools is to apply them to items. In that sense having a feature name be a feature of an item (or relatve) pathname will avoid having the extra step of extracting features into a separated table before applying the model. However it should also be stated that to wagon these names are arbitrary tokens and their semantic irrelevant at training time.

A typical description file would look like this, this is one suitable for the data file given above

    ((segment_duration float)
     ( name  aa ae ah ao aw ax ay b ch d dh dx eh el em en er ey f g
        hh ih iy jh k l m n nx ng ow oy p r s sh t th uh uw v w y z zh pau )
     ( n.name 0 aa ae ah ao aw ax ay b ch d dh dx eh el em en er ey f g
        hh ih iy jh k l m n nx ng ow oy p r s sh t th uh uw v w y z zh pau )
     ( p.name 0 aa ae ah ao aw ax ay b ch d dh dx eh el em en er ey f g
        hh ih iy jh k l m n nx ng ow oy p r s sh t th uh uw v w y z zh pau )
     (position_type 0 onset coda)
     (pos_in_syl float)
     (syl_initial 0 1)
     (syl_final   0 1)
     (R:Sylstructure.parent.R:Syllable.p.syl_break float)
     (R:Sylstructure.parent.syl_break float)
     (R:Sylstructure.parent.R:Syllable.n.syl_break float)
     (R:Sylstructure.parent.R:Syllable.p.stress 0 1)
     (R:Sylstructure.parent.stress 0 1)
     (R:Sylstructure.parent.R:Syllable.n.stress 0 1)
    )

There are also a number of special symbols that may be used in a description file. If the type (first toke after the name) is ignore the feature will be ignored in the model building process. You may also specified features to ignore at tree building time but it is often convenient to explicitly ignore feature(s) in the description file.

For open categorial features the token _other_ should appear as the first in the list of possible values. This actually allows features to have a partially closed set and and open set.

A description file can't be generated automatically from a data set though an approximation is possible. Particularly its is not possible to automatically decied if a feature value is continous of that its example values happen to look like numbers. The script make_wagon_desc takes a datafile and file containing only the names of the features, and the name of the description file it will create. This is often a useful first pass though it almost certainly must be hand editted afterwards.

### Tree format

The generated tree files are written as Lisp s-expressions as this is by far the easiest external method to represent trees. Even if the trees are read by something other than Lisp it is easy to write a reader for such a format. The syntax of a tree is

    TREE ::= LEAF | QUESTION-NODE

    QUESTION-NODE ::= "(" QUESTION YES-NODE NO-NODE ")"

    YES-NODE ::= TREE

    NO-NODE ::= TREE

    QUESTION ::= "(" FEATURENAME "is" VALUE ")" |
                 "(" FEATURENAME "=" FLOAT ")" |
                 "(" FEATURENAME "<" FLOAT ")" |
                 "(" FEATURENAME ">" FLOAT ")" |
                 "(" FEATURENAME "matches" REGEX ")" |
                 "(" FEATURENAME "in" "(" VALUE0 VALUE1 ... ")" ")"

    LEAF ::= "(" STDDEV MEAN ")" |
             "(" "(" VALUE0 PROB0 ")" "(" VALUE1 PROB1 ")" ... MOSTPROBVAL ")" |
             any other lisp s-expression

Note that not all of the question types are generated by Wagon but they are supported by the interpreters.

The leaf nodes differ depending on the type of the predictee. For continuous predictees (regression trees) the leaves consist of a pair of floats, the stddev and mean. For discrete predictees (classification trees) the leaves are a probability density function for the members of the class. Also the last member of the list is the most probable value. Note that in both case the last value of the leaf list is the answer desired in many cases.

Here's a small example tree

       NOT WRITTEN

# Functions

# Programs

The following exectutable programs are provided for the building and testing of decision trees

- wagon_manual: is the core building program
- wagon_test_manual: applies a trained treee to data and tests its accuracy.