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## Confusion Matrix

• A confusion matrix is a table that is often used to describe the performance of a classification model or classifiers on a set of test data for which the true values are known.

• The confusion matrix itself is relatively simple to understand, but the related terminology can be confusing.

• In short, it is a technique that summarizes the performance of classification algorithms.

• Calculating a confusion matrix can give a better idea of what the classification model is getting right and what types of errors it is making.

• As it shows the errors in the classification model in the form of a matrix, therefore, also called an Error Matrix.

Basic Features of Confusion Matrix -

• The confusion matrix is divided into two dimensions such as Predicted values and Actual values along with the Total values (n).

• Predicted values are those values, which are predicted by the classification model, and Actual values are the true values that are coming from the real observations.

There are 4 types of values represented by the confusion matrix such as follows:

A] True Positive (TP) -

• The predicted value matches the actual value.
• The actual value was positive and the model predicted a positive value.

B] True Negative (TN) -

• The predicted value matches the actual value.
• The actual value was negative and the model predicted a negative value.

C] False Positive (FP) -

• The predicted value was falsely predicted.
• The actual value was negative but the model predicted a positive value.
• This is also known as the Type 1 Error.

D] False Negative (FN) -

• The predicted value was falsely predicted.
• The actual value was positive but the model predicted a negative value.
• This is also known as the Type 2 Error.

Calculations using Confusion Matrix -

Several calculations can be performed by using the confusion matrix.

1] Accuracy

• This shows overall, how often is the classifier correct?
• It defines how often the model predicts the correct output.

$$Accuracy = \frac{TP + TN}{Total\ Values\ (n)} = \frac{TP + TN}{TP + TN + FP + FN}$$

2] Misclassification Rate (Error Rate)

• This shows overall, how often is it wrong?
• It defines how often the model gives the wrong predictions.
• It is also termed as Error rate.

$$Misclassification\ Rate = \frac{FP + FN}{Total\ Values\ (n)} = \frac{FP + FN}{TP + TN + FP + FN}$$

3] Precision

• This shows when it predicts yes, how often is it correct
• It defines the accuracy of the positive class.
• It measures how likely the prediction of the positive class is correct.

$$Precision = \frac{TP}{Total\ Predicted\ YES} = \frac{TP}{TP + FP}$$

4] Recall (Sensitivity / True Positive Rate)

• This shows when it's actually yes, how often does it predict yes?
• It defines the ratio of positive classes correctly detected.
• The recall must be as high as possible.
• The Recall is also called Sensitivity or True Positive Rate.

$$Recall = \frac{TP}{Total\ Actual\ YES} = \frac{TP}{TP + FN}$$

5] F-measure

• If two models have low precision and high recall or vice versa, it is difficult to compare these models.
• Hence for this F-score is used as a performance parameter.
• This evaluates the recall and precision at the same time.

$$F-Measure = \frac{2 \times Recall \times Precision}{Recall + Precision}$$

6] False Positive Rate

• This shows when it's actually no, how often does it predict yes?

$$False\ Positive\ Rate = \frac{FP}{Total\ Actual\ NO} = \frac{FP}{FP + TN}$$

7] True Negative Rate (Specificity)

• This shows when it's actually no, how often does it predict no.
• It is the same as True Negative Rate = 1 - False Positive Rate.
• It is also called Specificity.

$$True\ Negative\ Rate = \frac{TN}{Total\ Actual\ NO} = \frac{TN}{FP + TN}$$

8] Prevalence

• This shows how often the yes condition occurs in our sample.

$$Prevalence = \frac{Total\ Actual\ YES}{Total\ Values\ (n)} = \frac{TP + FN}{TP + TN + FP + FN}$$

9] Null Error Rate

• This term is used to define how many times your prediction would be wrong if you can predict the majority class.
• It is considered as a baseline metric to compare classifiers.

$$Null\ Error\ Rate = \frac{Total\ Actual\ NO}{Total\ Values\ (n)} = \frac{FP + TN}{TP + TN + FP + FN}$$

10] Roc Curve

• The Roc curve shows the true positive rates against the false positive rate at various cut points.
• The x-axis indicates the False Positive Rate and the y-axis indicates the True Positive Rate.
• The ROC curve shows how sensitivity and specificity vary at every possible threshold.

$$ROC\ Curve = Plot\ of\ False\ Positive\ Rate\ (X-axis)\ VS\ True\ Positive\ Rate\ (Y-axis)$$

11] Cohen's Kappa

• This is essentially a measure of how well the classifier performed as compared to how well it would have performed simply by chance.
• In other words, a model will have a high Kappa score if there is a big difference between the Accuracy and the Null Error Rate.

$$k = \frac{p_o - p_e}{1 - p_e} = 1 - \frac{1 - p_o}{1 - p_e}$$

Where,

$p_o$ = Observed Values = Overall accuracy of the model.

$p_e$ = Expected Values = Measure of the model predictions and the actual class values.

Benefits of Confusion Matrix -

• It evaluates the performance of the classification models, when they make predictions on test data, and tells how good our classification model is.

• It not only tells the error made by the classifiers but also the type of errors such as it is either type-I or type-II error.

• With the help of the confusion matrix, we can calculate the different parameters for the model, such as Accuracy, Precision, Recall, F-measure, etc.