# Metrics used in deep learning

Last updated on：3 months ago

As a researcher in deep learning, we have to make use of metrics to measure our models’ performance. I’ll introduce the most seen metrics in this blog.

# Error matrices

**Accuracy**= (true positives + true negatives) / (total examples)**Precision**= (true positives) / (true positives + false positives)**Recall**(or**Sensitivity**) = (true positives) / (true positives + false negatives)**F1**score (F score) = $2\frac{PR}{P+R}$ or $\frac{2}{1/P+1/R}$

**Positive and negative are your judgement result. True or False means your judgement is right or wrong.**

## Metrics Rate

**Rate** means the proportion of an indicator. The fraction of metrics rate is depended on the rate name.

Rate name | Denominator | Numerator | Formula |
---|---|---|---|

True positive rate/TPR |
TP + FN | TP | $\frac{TP}{TP + FN}$ |

False positive rate/FPR |
FP + TN | FP | $\frac{FP}{FP + TN}$ |

True negative rate/TNR |
TN + FP | TN | $\frac{TN}{TN + FP}$ |

False negative rate/FNR |
FN + TP | FN | $\frac{FN}{FN + TP}$ |

**Denominator** is the **total of the rate name situation in judgement (positive or negative)**. For instance, True positive means ground truth and judgement are right. We get all positive judgement item, that is TP $+$ FN. Meanwhile, **the numerator is the rate name itself**. True positive is TP.

**The ground truth and judgement are both different** in two rate name means we can sum up them to 1.

$$\text{FPR} + \text{TNR} = 1$$

$$\text{TPR} + \text{FNR} = 1$$

## Name

**TPR: Sensitivity, Recall**

FPR: Fall out

**TNR: Specificity**, **selectivity**

FNR: Miss rate

# ROC

A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the **diagnostic ability of a binary classifier system** as its discrimination threshold is varied.

ROC curve is **TPR-FPR** curve, which means that we need to compute TPR and FPR first.

**AUC is area under curve**. It can be obtained by calculate the area among ROC curve, $x$ axis, and $y = 1$ axis.

I guess you may have a question about plotting a curve. We get the TPR and FPR result in evaluating process, which can just be plotted in one point. How do we plot this curve? Actually, we need to set a **threshold** to define **how large possibility** is taken as a positive judgement.

For multi classes, we plot each class ROC one by one. First binarize the other class and the interested class, then take it as a two class curve plotting problem.

```
from sklearn.metrics import roc_curve, auc
import matplotlib.pyplot as plt
import numpy as np
from sklearn import metrics
y = np.array([1, 1, 2, 2])
scores = np.array([0.1, 0.4, 0.35, 0.8])
fpr, tpr, thresholds = metrics.roc_curve(y, scores, pos_label=2)
>>> fpr
array([ 0. , 0.5, 0.5, 1. ])
>>> tpr
array([ 0.5, 0.5, 1. , 1. ])
>>> thresholds
array([ 0.8 , 0.4 , 0.35, 0.1 ])
auc = metrics.auc(fpr, tpr)
>>> auc
0.75
plt.figure()
lw = 2
plt.plot(fpr, tpr, color='darkorange',
lw=lw, label='ROC curve (area = %0.2f)' % auc)
plt.plot([0, 1], [0, 1], color='navy', lw=lw, linestyle='--')
plt.xlim([0.0, 1.0])
plt.ylim([0.0, 1.05])
plt.xlabel('False Positive Rate')
plt.ylabel('True Positive Rate')
plt.title('Receiver operating characteristic example')
plt.legend(loc="lower right")
plt.show()
```

# Reference

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