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Quality Metrics for Multi-class Classification Algorithms

For \(l\) classes \(C_1, \ldots, C_l\), given a vector \(X = (x_1, \ldots, x_n)\) of class labels computed at the prediction stage of the classification algorithm and a vector \(Y = (y_1, \ldots, y_n)\) of expected class labels, the problem is to evaluate the classifier by computing the confusion matrix and connected quality metrics: precision, error rate, and so on.

QualityMetricsId for multi-class classification is confusionMatrix.

Details

Further definitions use the following notations:

Notations for Quality Metrics for Multi-class Classification Algorithms

\(\text{tp}_i\)	true positive	the number of correctly recognized observations for class \(C_1\)
\(\text{tn}_i\)	true negative	the number of correctly recognized observations that do not belong to the class \(C_1\)
\(\text{fp}_i\)	false positive	the number of observations that were incorrectly assigned to the class \(C_1\)
\(\text{fn_i}\)	false negative	the number of observations that were not recognized as belonging to the class \(C_1\)

The library uses the following quality metrics for multi-class classifiers:

Definitions of Quality Metrics for Multi-class Classification Algorithms

Quality Metric	Definition
Average accuracy	\(\frac {\sum _{i = 1}^{l} \frac {\text{tp}_i + \text{tn}_i}{\text{tp}_i + \text{fn}_i + \text{fp}_i + \text{tn}_i}}{l}\)
Error rate	\(\frac {\sum _{i = 1}^{l} \frac {\text{fp}_i + \text{fn}_i}{\text{tp}_i + \text{fn}_i + \text{fp}_i + \text{tn}_i}}{l}\)
Micro precision (\(\text{Precision}_\mu\))	\(\frac {\sum _{i = 1}^{l} \text{tp}_i} {\sum _{i = 1}^{l} (\text{tp}_i + \text{fp}_i)}\)
Micro recall (\(\text{Recall}_\mu\))	\(\frac {\sum _{i = 1}^{l} \text{tp}_i} {\sum _{i = 1}^{l} (\text{tp}_i + \text{fn}_i)}\)
Micro F-score (\(\text{F-score}_\mu\))	\(\frac {(\beta^2 + 1)(\text{Precision}_\mu \times \text{Recall}_\mu)}{\beta^2 \times \text{Precision}_\mu + \text{Recall}_\mu}\)
Macro precision (\(\text{Precision}_M\))	\(\frac {\sum _{i = 1}^{l} \frac {\text{tp}_i}{\text{tp}_i + \text{fp}_i}}{l}\)
Macro recall (\(\text{Recall}_M\))	\(\frac {\sum _{i = 1}^{l} \frac {\text{tp}_i}{\text{tp}_i + \text{fn}_i}}{l}\)
Macro F-score (\(\text{F-score}_M\))	\(\frac {(\beta^2 + 1)(\text{Precision}_M \times \text{Recall}_M)}{\beta^2 \times \text{Precision}_M + \text{Recall}_M}\)

For more details of these metrics, including the evaluation focus, refer to [Sokolova09].

The following is the confusion matrix:

Confusion Matrix for Multi-class Classification Algorithms

	Classified as Class \(C_1\)	\(\ldots\)	Classified as Class \(C_i\)	\(\ldots\)	Classified as Class \(C_l\)
Actual Class \(C_1\)	\(c_{11}\)	\(\ldots\)	\(c_{1i}\)	\(\ldots\)	\(c_{1l}\)
\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)
Actual Class \(C_i\)	\(c_{i1}\)	\(\ldots\)	\(c_{ii}\)	\(\ldots\)	\(c_{il}\)
\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)
Actual Class \(C_l\)	\(c_{l1}\)	\(\ldots\)	\(c_{li}\)	\(\ldots\)	\(c_{ll}\)

The positives and negatives are defined through elements of the confusion matrix as follows:

\[\text{tp}_i = c_{ii}\]

\[\text{fp}_i = \sum _{n = 1}^{l} c_{ni} - \text{tp}_i\]

\[\text{fn}_i = \sum _{n = 1}^{l} c_{in} - \text{tp}_i\]

\[\text{tn}_i = \sum _{n = 1}^{l} \sum _{k = 1}^{l} c_{nk} - \text{tp}_i - \text{fp}_i - \text{fn}_i\]

Batch Processing

Algorithm Input

The quality metric algorithm for multi-class classifiers accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Algorithm Input for Quality Metrics for Multi-class Classification Algorithms (Batch Processing)

Input ID	Input
predictedLabels	Pointer to the \(n \times 1\) numeric table that contains labels computed at the prediction stage of the classification algorithm. This input can be an object of any class derived from NumericTable except PackedSymmetricMatrix, PackedTriangularMatrix, and CSRNumericTable.
groundTruthLabels	Pointer to the \(n \times 1\) numeric table that contains expected labels. This input can be an object of any class derived from NumericTable except PackedSymmetricMatrix, PackedTriangularMatrix, and CSRNumericTable.

Algorithm Parameters

The quality metric algorithm has the following parameters:

Algorithm Parameters for Quality Metrics for Multi-class Classification Algorithms (Batch Processing)

Parameter	Default Value	Description
algorithmFPType	float	The floating-point type that the algorithm uses for intermediate computations. Can be float or double.
method	defaultDense	Performance-oriented computation method, the only method supported by the algorithm.
nClasses	\(0\)	The number of classes (\(l\)).
useDefaultMetrics	true	A flag that defines a need to compute the default metrics provided by the library.
beta	\(1\)	The \(\beta\) parameter of the F-score quality metric provided by the library.

Algorithm Output

The quality metric algorithm calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Algorithm Output for Quality Metrics for Multi-class Classification Algorithms (Batch Processing)

Result ID	Result
confusionMatrix	Pointer to the \(\text{nClasses} \times \text{nClasses}\) numeric table with the confusion matrix. Note By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.
multiClassMetrics	Pointer to the \(1 \times 8\) numeric table that contains quality metrics, which you can access by an appropriate Multi-class Metrics ID: averageAccuracy - average accuracy errorRate - error rate microPrecision - micro precision microRecall - micro recall microFscore - micro F-score macroPrecision - macro precision macroRecall - macro recall macroFscore - macro F-score Note By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.

Examples

C++ (CPU)Java*

Batch Processing:

• svm_multi_class_metrics_dense_batch.cpp

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