Performance Measures for Classification

Why different performance measures?


To date, we have focussed on accuracy: How often is my classifier correct on new data?

Depending on how the classifier will be applied, however, other measures may be more appropriate.

Review: Error Rate / Accuracy

Compute the proportion that were correctly or incorrectly classified.

\[\mathrm{Accuracy} = n^{-1} \sum_{i=1}^n 1(\hat{y}_i = y_i)\]

\[\mathrm{Error Rate} = n^{-1} \sum_{i=1}^n 1(\hat{y}_i \ne y_i)\]

Imbalanced classes


  • Suppose in the true population, 95% are negative.
  • Classifier that always outputs negative is 95% accurate. This is the baseline accuracy.
  • Accuracy is not a useful measure.

Literature on learning from unbalanced classes:

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5299216 http://link.springer.com/article/10.1007%2Fs10115-013-0670-6 http://www.computer.org/csdl/proceedings/icdm/2012/4905/00/4905a695-abs.html http://www.computer.org/csdl/proceedings/icdm/2011/4408/00/4408a754-abs.html

Example: 50% Positive, 50% Negative

npos <- 500; nneg <- 500; set.seed(1)
df <- rbind(data.frame(x=rnorm(npos,mupos), y=1),data.frame(x=rnorm(nneg,muneg),y=-1)); df$y <- as.factor(df$y)
sep <- tune(svm,y~x,data=df,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
df$ypred <- predict(sep$best.model)
ggplot(df,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

Example: 5% Positive, 95% Negative

npos <- 50; nneg <- 950; set.seed(1)
df <- rbind(data.frame(x=rnorm(npos,mupos), y=1),data.frame(x=rnorm(nneg,muneg),y=-1)); df$y <- as.factor(df$y)
rsep <- tune(svm,y~x,data=df,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
df$ypred <- predict(rsep$best.model);
ggplot(df,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

Example: Upsampling

newneg <- df %>% filter(y == 1) %>% sample_n(900,replace=T); dfupsamp <- rbind(df,newneg)
upsep <- tune(svm,y~x,data=dfupsamp,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
dfupsamp$ypred <- predict(upsep$best.model);
ggplot(dfupsamp,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

Upsampling: Accuracy

df$upsampred <- predict(upsep$best.model,df)
mean(df$y == df$ypred)
## [1] 0.95
mean(df$y == df$upsampred)
## [1] 0.694

No upsampling: 95% Accuracy

Upsampled: 68% Accuracy

So why do you like the upsampled classifier better?

Definitions:
True/False Positives/Negatives


True class
Total population class positive class negative
Predicted
class
Predicted class
positive
True positive False positive
(Type I error)
Predicted class
negative
False negative
(Type II error)
True negative


Careful! A “false positive” is actually a negative and a “false negative” is actually a positive.

Precision and Recall, F-measure

Precision =
Σ True positiveΣ Predicted positive
Recall =
Σ True positiveΣ Class positive


  • In Information Retrieval, typically very few positives, many negatives. (E.g. billion webpages, dozen relevant to search query.) Focus is on correctly identifying positives.
  • Recall: What proportion of the positives in the population do I correctly capture?
  • Precision: What proportion of the instances I labeled positive are actually positive?

\[\mbox{F-measure} = 2 \frac{\mathrm{Precision}\cdot\mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}\] https://en.wikipedia.org/wiki/F1_score

F-measure Example

\[\mbox{F-measure} = 2 \frac{\mathrm{Precision}\cdot\mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}\] https://en.wikipedia.org/wiki/F1_score

For the “always predict -1” classifier, recall = 0, precision = 0, so
F-measure = 0.

For the classifier learned from up-sampled data,

prec <- sum(df$y == 1 & df$upsampred == 1) / sum(df$upsampred == 1)
recall <- sum(df$y == 1 & df$upsampred == 1) / sum(df$y == 1)
F1.upsamp <- 2 * prec*recall / (prec + recall)
print(F1.upsamp)
## [1] 0.2072539

NOTE that F-measure is not “symmetric”; it depends on the definition of the positive class. Typically used when positive class is rare but important e.g. information retrieval.

Sensitivity and Specificity,
Balanced Accuracy


Sensitivity =
Σ True positiveΣ Class positive
Specificity =
Σ True negativeΣ Class negative


  • Sensitivity: What proportion of the positives in the population do I correctly label?
  • Specificity: What proportion of the negatives in the population do I correctly label?

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

https://en.wikipedia.org/wiki/Accuracy_and_precision#In_binary_classification
Note: Sensitivity is same as Recall.

Balanced accuracy Example

  • Sensitivity: What proportion of the positives in the population do I correctly label?
  • Specificity: What proportion of the negatives in the population do I correctly label?

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

For “always predict -1” classifier, sensitivity = 0, specificity = 1, balanced accuracy = 0.5.

For the classifier learned from up-sampled data,

sens <- sum(df$y == 1 & df$upsampred == 1) / sum(df$y == 1)
spec <- sum(df$y == -1 & df$upsampred == -1) / sum(df$y == -1)
bal.acc.upsamp <- 0.5*(sens + spec)
print(bal.acc.upsamp)
## [1] 0.7442105

Many Measures

https://en.wikipedia.org/wiki/Evaluation_of_binary_classifiers

True class
Total population class positive class negative Prevalence = Σ class positiveΣ Total population
Predicted
class
Predicted class
positive
True positive False positive
(Type I error)
Positive predictive value (PPV), Precision = Σ True positiveΣ Predicted positive False discovery rate (FDR) = Σ False positiveΣ Predicted positive
Predicted class
negative
False negative
(Type II error)
True negative False omission rate (FOR) = Σ False negativeΣ Predicted negative Negative predictive value (NPV) = Σ True negativeΣ Predicted negative
Accuracy (ACC) = Σ True positive
+ Σ True negative
Σ Total population
True positive rate (TPR), Sensitivity, Recall = Σ True positiveΣ Class positive False positive rate (FPR), Fall-out = Σ False positiveΣ Class negative Positive likelihood ratio (LR+) = TPRFPR Diagnostic odds ratio (DOR) = LR+LR−
False negative rate (FNR), Miss rate = Σ False negativeΣ Class positive True negative rate (TNR), Specificity (SPC) = Σ True negativeΣ Class negative Negative likelihood ratio (LR−) = FNRTNR

Cost sensitivity

Sensitivity =
Σ True positiveΣ Class positive
Specificity =
Σ True negativeΣ Class negative

Recall:

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

What if e.g. false positives are more costly than false negatives?

Let \(\mathrm{P}\) and \(\mathrm{N}\) the proportions of positives and negatives in the population.

\[\mathrm{FNRate} = (1 - \mathrm{Sensitivity}), \mathrm{FPRate} = (1 - \mathrm{Specificity})\]

\[\mbox{NormExpectedCost} = c_{\mathrm{FP}}\cdot\mathrm{FPRate}\cdot\mathrm{P}+ c_{\mathrm{FN}}\cdot\mathrm{FNRate}\cdot\mathrm{N}\]

http://www.csi.uottawa.ca/~cdrummon/pubs/pakdd08.pdf

Receiver operating characteristic (ROC)


  • Suppose classifier can rank inputs according to “how positive” they appear to be.
  • E.g., can use probability from Logistic Regression, or \(w^{\mathsf T}x + b\) for SVM.
  • By adjusting the “threshold” value for deciding an instance is positive, we can obtain different false positive rates. Low threshold gives higher false positives (but higher true negatives), high threshold gives lower false positives (but higher false negatives.)
  • ROC curve: Try all possible cutoffs, plot FPR on \(x\)-axis, TPR on \(y\)-axis.

https://en.wikipedia.org/wiki/Receiver_operating_characteristic

Reading an ROC Curve

Think: “If I fix FPR at 0.4, what is my TPR?”

Obviously, higher is better. Random guessing gives an ROC curve along \(y = x\).

If the area under the curve (AUC) is 1, we have a perfect classifier. AUC of 0.5 is pretty bad.

Very common measure of classifier performance, especially when classes are imbalanced.

ROC example

library(ROCR)
## Loading required package: gplots
## 
## Attaching package: 'gplots'
## The following object is masked from 'package:stats':
## 
##     lowess
preds <- attr(predict(upsep$best.model,df,decision.values = TRUE),"decision.values")
plot(performance(prediction(preds,df$y),"tpr","fpr"))