# Prediction Error Cost-Benefit Matrix

(Redirected from Cost-Benefit Matrix)

A Prediction Error Cost-Benefit Matrix is a symmetric matrix that defines the misclassification costs of a classification task.

**Context:**- It can be an Input to a Classification Problem (and allow predictive modelers to describe the costs and the benefits associated with each possible prediction).
- It can be an Identity Matrix (optimizes for Accuracy).
- When the cost-benefit matrix has new non-default values assigned, the model optimizes the net benefit (profit) associated with each prediction.
- It can be used to Optimize Return on Investment.

**Counter-Example(s)****See:**Outlier Detection

## References

### 2016

- (Branco et al., 2016) ⇒ Paula Branco, Luís Torgo, and Rita P. Ribeiro. (2016). “A Survey of Predictive Modeling on Imbalanced Domains.” ACM Computing Surveys (CSUR) 49, no. 2
- QUOTE: ... In order to perform a modification on a selected algorithm, it is essential to understand why it fails when the distribution is skewed. Also, some of the adaptations assume that a cost/cost-benefit matrix is known for different error types, which is frequently not the case. ...

### 2015

- https://github.com/podopie/DAT18NYC/blob/master/classes/13-expected_value_cost_benefit_analysis.ipynb
- QUOTE: Objectives
- Extend our metrics into business application
- Identify and understand a cost benefit matrix
- Using a confusion matrix with a cost benefit matrix to solve for expected value

- QUOTE: Objectives

The technique we'll use is a cost benefit matrix. This is very similar to our confusion matrix:

- \begin{bmatrix}TP & FP\\FN & TN\end{bmatrix}

Which to find probabilities turns into this:

- \begin{bmatrix}p(TP) & p(FP)\\p(FN) & p(TN)\end{bmatrix}

and our cost benefit matrix will be somewhat similar:

- \begin{bmatrix}b(TP) & c(FP)\\c(FN) & b(TN)\end{bmatrix}

where:

- [math]b[/math] represents benefit (the benefits of accurately predicting positives and negatives), while...
- [math]c[/math] represents cost (the costs of misclassifying positives and negatives).

To simplify we'll associate benefits to be positive value and costs to be negative value:

- \begin{bmatrix}v(TP) & -v(FP)\\-v(FN) & v(TN)\end{bmatrix}

... ...