# designecoc

Coding matrix for reducing error-correcting output code to binary

## Description

returns
the coding matrix `M`

= designecoc(`K`

,`name`

)`M`

that reduces the error-correcting
output code (ECOC) design specified by `name`

and `K`

classes
to a binary problem. `M`

has `K`

rows
and *L* columns, with each row corresponding to a
class and each column corresponding to a binary learner. `name`

and `K`

determine
the value of *L*.

You can view or customize `M`

, and then specify
it as the coding matrix for training an ECOC multiclass classifier
using `fitcecoc`

.

returns
the coding matrix with additional options specified by one or more `M`

= designecoc(`K`

,`name`

,`Name,Value`

)`Name,Value`

pair
arguments.

For example, you can specify the number of trials when generating a dense or sparse, random coding matrix.

## Examples

### Train ECOC Classifiers Using a Custom Coding Design

Consider the `arrhythmia`

data set. There are 16 classes in the study, 13 of which are represented in the data. The first class indicates that the subject did not have arrhythmia, and the last class indicates that the subject's arrhythmia state was not recorded. Suppose that the other classes are ordinal levels indicating the severity of arrhythmia. Train an ECOC classifier using a custom coding design specified by the description of the classes.

Load the `arrhythmia`

data set.

load arrhythmia K = 13; % Number of distinct classes

Construct a coding matrix that describes the nature of the classes.

OrdMat = designecoc(11,'ordinal'); nOM = size(OrdMat); class1VSOrd = [1; -ones(11,1); 0]; class1VSClass16 = [1; zeros(11,1); -1]; OrdVSClass16 = [0; ones(11,1); -1]; Coding = [class1VSOrd class1VSClass16 OrdVSClass16,... [zeros(1,nOM(2)); OrdMat; zeros(1,nOM(2))]]

`Coding = `*13×13*
1 1 0 0 0 0 0 0 0 0 0 0 0
-1 0 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 0 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 0 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1
-1 0 1 1 1 1 -1 -1 -1 -1 -1 -1 -1
-1 0 1 1 1 1 1 -1 -1 -1 -1 -1 -1
-1 0 1 1 1 1 1 1 -1 -1 -1 -1 -1
-1 0 1 1 1 1 1 1 1 -1 -1 -1 -1
-1 0 1 1 1 1 1 1 1 1 -1 -1 -1
-1 0 1 1 1 1 1 1 1 1 1 -1 -1
⋮

Train an ECOC classifier using the custom coding design `Coding`

and specify that the binary learners are decision trees.

Mdl = fitcecoc(X,Y,'Coding',Coding,'Learner','Tree');

Estimate the in-sample classification error.

genErr = resubLoss(Mdl)

genErr = 0.1460

### Choose Among Several Random Coding Designs

If you request a random coding matrix by specifying `sparserandom`

or `denserandom`

, then, by default, `designecoc`

generates 10,000 random matrices. Then, it chooses the matrix with the largest, minimal, pair-wise row distances based on the Hamming measure. You can specify to generate more matrices to increase the chance of obtaining a better one, or you can generate several coding matrices, and then see which performs best.

Load the `arrhythmia`

data set. Reserve the observations classified into class 16 (i.e., those that do not have an arrhythmia classification) as new data.

```
load arrhythmia
oosIdx = Y == 16;
isIdx = ~oosIdx;
Y = categorical(Y(isIdx));
tabulate(Y)
```

Value Count Percent 1 245 56.98% 2 44 10.23% 3 15 3.49% 4 15 3.49% 5 13 3.02% 6 25 5.81% 7 3 0.70% 8 2 0.47% 9 9 2.09% 10 50 11.63% 14 4 0.93% 15 5 1.16%

K = numel(unique(Y));

Generate four random coding design matrices such that the first two are dense and the second two are sparse. Specify to find the best out of 20,000 variates.

rng(1); % For reproducibility Coding = cell(4,1); % Preallocate for coding matrices CodingTypes = {'denserandom','denserandom','sparserandom','sparserandom'}; for j = 1:4; Coding{j} = designecoc(K,CodingTypes{j},'NumTrials',2e4); end

`Coding`

is a 4-by-1 cell array, where each cell is a coding design matrix. The matrices have `K`

rows, but the number of columns (i.e., binary learners) might vary.

Train and cross validate ECOC classifiers using the 15-fold cross validation. Specify that each ECOC classifier be trained using a classification tree, and the random coding matrix stored in `Coding`

.

Mdl = cell(4,1); % Preallocate for the ECOC classifiers for j = 1:4; Mdl{j} = fitcecoc(X(isIdx,:),Y,'Learners','tree',... 'Coding',Coding{j},'KFold',15); end

Warning: One or more of the unique class values in GROUP is not present in one or more folds. For classification problems, either remove this class from the data or use N instead of GROUP to obtain nonstratified partitions. For regression problems with continuous response, use N.

Warning: One or more of the unique class values in GROUP is not present in one or more folds. For classification problems, either remove this class from the data or use N instead of GROUP to obtain nonstratified partitions. For regression problems with continuous response, use N.

Warning: One or more of the unique class values in GROUP is not present in one or more folds. For classification problems, either remove this class from the data or use N instead of GROUP to obtain nonstratified partitions. For regression problems with continuous response, use N.

`Mdl`

is a 4-by-1 cell array of `ClassificationPartitionedECOC`

models. Several classes have low relative frequency in the data, and so there is a chance that, during cross validation, some in-sample folds will not train using observations from those classes.

Estimate the 15-fold classification error for each classifier.

genErr = nan(4,1); for j = 1:4; genErr(j) = kfoldLoss(Mdl{j}); end genErr

`genErr = `*4×1*
0.2233
0.2116
0.2186
0.2209

Though the generalization error is still high, the best performing model, based solely on the out-of-sample classification error, is the model that used the coding design `Coding{3}`

.

You can try to improve the generalization error by tuning some parameters of the binary learners. For example, you can specify to use the twoing rule or deviance for the split criterion, rather than the default Gini's diversity index. You might also specify to use surrogate splits since there are missing values in the data.

## Input Arguments

`K`

— Number of classes

positive integer

Number of classes, specified as a positive integer.

`K`

specifies the number of rows of the coding
matrix `M`

.

**Data Types: **`single`

| `double`

`name`

— Coding design name

`'binarycomplete'`

| `'denserandom'`

| `'onevsall'`

| `'onevsone'`

| `'sparserandom'`

| ...

Coding design name, specified as a value in the following table. The table summarizes the coding schemes.

Value | Number of Binary Learners | Description |
---|---|---|

`'allpairs'` and `'onevsone'` | K(K – 1)/2 | For each binary learner, one class is positive, another is negative, and the software ignores the rest. This design exhausts all combinations of class pair assignments. |

`'binarycomplete'` | $${2}^{(K-1)}-1$$ | This design partitions the classes into all binary combinations, and does not ignore any
classes. For each binary learner, all class assignments are
`–1` and `1` with at least one positive
class and one negative class in the assignment. |

`'denserandom'` | Random, but approximately 10 log_{2}K | For each binary learner, the software randomly assigns classes into positive or negative classes, with at least one of each type. For more details, see Random Coding Design Matrices. |

`'onevsall'` | K | For each binary learner, one class is positive and the rest are negative. This design exhausts all combinations of positive class assignments. |

`'ordinal'` | K – 1 | For the first binary learner, the first class is negative and the rest are positive. For the second binary learner, the first two classes are negative and the rest are positive, and so on. |

`'sparserandom'` | Random, but approximately 15 log_{2}K | For each binary learner, the software randomly assigns classes as positive or negative with probability 0.25 for each, and ignores classes with probability 0.5. For more details, see Random Coding Design Matrices. |

`'ternarycomplete'` | $$\left({3}^{K}-{2}^{(K+1)}+1\right)/2$$ | This design partitions the classes into all ternary combinations. All class assignments are
`0` , `–1` , and `1` with
at least one positive class and one negative class in each assignment. |

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'NumTrials',1000`

specifies to generate `1000`

random
matrices.

`NumTrials`

— Number of random coding matrices to generate

`10000`

(default) | positive integer

Number of random coding matrices to generate, specified as the
comma-separated pair consisting of `'NumTrials'`

and
a positive integer.

The software:

Generates

`NumTrials`

matrices, and selects the one with the maximal, pair-wise row distance.Ignores

`NumTrials`

for all values of`name`

except`'denserandom'`

and`'sparserandom'`

.

**Example: **`'NumTrials',1000`

**Data Types: **`single`

| `double`

## Output Arguments

`M`

— Coding matrix

numeric matrix

Coding matrix that reduces an ECOC scheme to binary, returned
as a numeric matrix. `M`

has `K`

rows
and *L* columns, where *L* is the
number of binary learners. Each row corresponds to a class and each
column corresponds to a binary learner.

The elements of `M`

are `-1`

, `0`

,
or `1`

, and the value corresponds to a dichotomous
class assignment. This table describes the meaning of `M(i,j)`

,
that is, the class that learner `j`

assigns to observations
in class `i`

.

Value | Dichotomous Class Assignment |
---|---|

`–1` | Learner `j` assigns observations in class `i` to a negative
class. |

`0` | Before training, learner `j` removes observations
in class `i` from the data set. |

`1` | Learner `j` assigns observations in class `i` to a positive
class. |

The binary learners for designs `denserandom`

, `binarycomplete`

,
and `onevsall`

do not assign `0`

to
observations in any class.

## Tips

The number of binary learners grows with the number of classes. For a problem with many classes, the

`binarycomplete`

and`ternarycomplete`

coding designs are not efficient. However:If

*K*≤ 4, then use`ternarycomplete`

coding design rather than`sparserandom`

.If

*K*≤ 5, then use`binarycomplete`

coding design rather than`denserandom`

.

You can display the coding design matrix of a trained ECOC classifier by entering

`Mdl.CodingMatrix`

into the Command Window.You should form a coding matrix using intimate knowledge of the application, and taking into account computational constraints. If you have sufficient computational power and time, then try several coding matrices and choose the one with the best performance (e.g., check the confusion matrices for each model using

`confusionchart`

).Leave-one-out cross-validation (

`Leaveout`

) is inefficient for data sets with many observations. Instead, use*k*-fold cross-validation (`KFold`

).

## Algorithms

### Custom Coding Design Matrices

Custom coding matrices must have a certain form. The software validates a custom coding matrix by ensuring:

Every element is –1, 0, or 1.

Every column contains as least one –1 and one 1.

For all distinct column vectors

*u*and*v*,*u*≠*v*and*u*≠ –*v*.All row vectors are unique.

The matrix can separate any two classes. That is, you can move from any row to any other row following these rules:

Move vertically from 1 to –1 or –1 to 1.

Move horizontally from a nonzero element to another nonzero element.

Use a column of the matrix for a vertical move only once.

If it is not possible to move from row

*i*to row*j*using these rules, then classes*i*and*j*cannot be separated by the design. For example, in the coding design$$\left[\begin{array}{cc}1& 0\\ -1& 0\\ 0& 1\\ 0& -1\end{array}\right]$$

classes 1 and 2 cannot be separated from classes 3 and 4 (that is, you cannot move horizontally from –1 in row 2 to column 2 because that position contains a 0). Therefore, the software rejects this coding design.

### Random Coding Design Matrices

For a given number of classes *K*, the software generates random coding
design matrices as follows.

The software generates one of these matrices:

Dense random — The software assigns 1 or –1 with equal probability to each element of the

*K*-by-*L*coding design matrix, where $${L}_{d}\approx \lceil 10{\mathrm{log}}_{2}K\rceil $$._{d}Sparse random — The software assigns 1 to each element of the

*K*-by-*L*coding design matrix with probability 0.25, –1 with probability 0.25, and 0 with probability 0.5, where $${L}_{s}\approx \lceil 15{\mathrm{log}}_{2}K\rceil $$._{s}

If a column does not contain at least one 1 and one –1, then the software removes that column.

For distinct columns

*u*and*v*, if*u*=*v*or*u*= –*v*, then the software removes*v*from the coding design matrix.

The software randomly generates 10,000 matrices by default, and retains the matrix with the largest, minimal, pairwise row distance based on the Hamming measure ([2]) given by

$$\Delta ({k}_{1},{k}_{2})=0.5{\displaystyle \sum}_{l=1}^{L}\left|{m}_{{k}_{1}l}\right|\left|{m}_{{k}_{2}l}\right|\left|{m}_{{k}_{1}l}-{m}_{{k}_{2}l}\right|,$$

where
*m _{kjl}* is an element of
coding design matrix

*j*.

## References

[1] Fürnkranz, Johannes. “Round Robin
Classification.” *J. Mach. Learn. Res.*, Vol. 2, 2002, pp.
721–747.

[2] Escalera, S., O. Pujol, and P. Radeva. “Separability of ternary codes for sparse designs of error-correcting output codes.” *Pattern Recog. Lett.*, Vol. 30, Issue 3, 2009, pp. 285–297.

## Version History

**Introduced in R2014b**

## See Also

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