# The available metrics in cvms

#### 2021-10-05

Abstract

This vignette lists the available metrics in cvms, along with their formulas.

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## Introduction

cvms has a large set of metrics for model evaluation. In this document, we list the metrics and their formulas.

Some of the metrics in the package are computed with external packages. These are listed at the bottom.

Some of the metrics are disabled by default to avoid cluttering the output tibble. These can be enabled in the metrics argument. This argument takes a list of named booleans, like list("Accuracy" = FALSE, "Weighted F1" = TRUE). This can be generated with the helper functions gaussian_metrics(), binomial_metrics(), and multinomial_metrics(). If, for instance, we only wish to calculate the RMSE metric for our regression model, we can use either list("all" = FALSE, "RMSE" = TRUE) or gaussian_metrics(all = FALSE, rmse = TRUE).

## Gaussian Metrics

The metrics used to evaluate regression tasks (like linear regression):

Symbol Denotes Formula
$$y$$ Targets
$$\hat{y}$$ Predictions
$$\bar{y}$$ Average target
$$n$$ Number of observations
$${\operatorname{IQR}(x)}$$ Interquartile Range $${\operatorname{quantile}(x, 3/4) - \operatorname{quantile}(x, 1/4)}$$
$$\lvert x \rvert$$ Absolute value of $$x$$
Metric name Abbreviation Formula
Root Mean Square Error RMSE $$\sqrt{\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^2}{n}}$$
Mean Absolute Error MAE $$\frac{\sum_{i=1}^{n}\lvert\hat{y}_{i}-y_{i}\rvert}{n}$$
Root Mean Square Log Error RMSLE $$\sqrt{\frac{\sum_{i=1}^{n}(\ln{(\hat{y}_{i}+1)}-\ln{(y_{i}+1))^2}}{n}}$$
Mean Absolute Log Error MALE $$\frac{\sum_{i=1}^{n}\lvert\ln{(\hat{y}_{i}+1)}-\ln{(y_{i}+1)\rvert}}{n}$$
Relative Absolute Error RAE $$\frac{\sum_{i=1}^{n}\lvert\hat{y}_{i}-y_{i}\rvert}{\sum_{i=1}^{n}\lvert y_{i}-\bar{y}\rvert}$$
Relative Squared Error RSE $$\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^2}{\sum_{i=1}^{n}(y_{i} - \bar{y})^2}$$
Root Relative Squared Error RRSE $${\sqrt{RSE} }$$
Mean Absolute Percentage Error MAPE $$\frac{1}{n}\sum_{i=1}^{n} \lvert \frac{\hat{y}_{i}-y_{i}}{y_{i}} \rvert$$
Normalized RMSE
(by target range)
NRMSE(RNG) $$\frac{RMSE}{\max{y}-\min{y}}$$
Normalized RMSE
(by target IQR)
NRMSE(IQR) $$\frac{RMSE}{\operatorname{IQR}(y)}$$
Normalized RMSE
(by target STD)
NRMSE(STD) $$\frac{RMSE}{\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(y_i-\bar{y})^2}}$$
Normalized RMSE
(by target mean)
NRMSE(AVG) $$\frac{RMSE}{\bar{y}}$$
Mean Square Error MSE $$\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^2}{n}$$
Total Absolute Error TAE $${\sum_{i=1}^{n}\lvert\hat{y}_{i}-y_{i}\rvert}$$
Total Squared Error TSE $${\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^2}$$

## Binomial Metrics

The metrics used to evaluate binary classification tasks:

Based on a confusion matrix, we first count the True Positives (TP), True Negatives (TN), False Positives (FP), and False Negatives (FN). Below, 1 is the positive class.

#>           Target
#> Prediction 0  1
#>          0 TN FN
#>          1 FP TP

With these counts, we can calculate the following metrics. Note, that the Kappa metric normalizes the counts to percentages.

Metric name(s) Abbreviation Formula
Accuracy $$\frac{TP + TN}{TP + TN + FP + FN}$$
Balanced Accuracy $$\frac{Sensitivity + Specificity}{2}$$
Sensitivity,
Recall,
True Positive Rate
$$\frac{TP}{TP + FN}$$
Specificity,
True Negative Rate
$$\frac{TN}{TN + FP}$$
Positive Predictive Value,
Precision
Pos Pred Value $$\frac{TP}{TP + FP}$$
Negative Predictive Value Neg Pred Value $$\frac{TN}{TN + FN}$$
F1 score $$2 \cdot \frac{Pos Pred Value \cdot Sensitivity}{Pos Pred Value + Sensitivity}$$
Matthews Correlation Coefficient MCC $$\frac{TP \cdot TN - FP \cdot FN}{\sqrt{(TP + FP)(TP + FN)(TN + FP) (TN + FN)}}$$
Note: When the denominator is 0, we set it to 1 to avoid NaN.
Detection Rate $$\frac{TP}{TP + FN + TN + FP}$$
Detection Prevalence $$\frac{TP + FP}{TP + FN + TN + FP}$$
Prevalence $$\frac{TP + FN}{TP + FN + TN + FP}$$
Threat Score $$\frac{TP}{TP + FN + FP}$$
False Negative Rate $${1 - Sensitivity}$$
False Positive Rate $${1 - Specificity}$$
False Discovery Rate $${1 - Pos Pred Value}$$
False Omission Rate $${1 - Neg Pred Value}$$
Kappa For Kappa, the counts (TP, TN, FP, FN) are normalized to percentages (summing to 1). Then:
$${p_{observed} = TP + TN}$$
$${p_{expected} = (TN + FP)(TN + FN) + (FN + TP)(FP + TP)}$$
$$Kappa = \frac{p_{observed} - p_{expected}}{1 - p_{expected}}$$

## Multinomial Metrics

We have four types of metrics for the multiclass classification evaluation:

Overall metrics simply look at whether a prediction is correct or not. Currently, cvms only has the Overall Accuracy.

The Macro/Average metrics are based on one-vs-all evaluations of each class. In a one-vs-all evaluation, we set all predictions and targets for the current class to 1 and all others to 0 ( $${y_{o,c} = 1 \text{ if } y_{o} = c \text{ else } 0}$$ and $${\hat{y}_{o,c} = 1 \text{ if } \hat{y} _{o} = c \text{ else } 0}$$ ) and perform a binomial evaluation. Once done for all classes, we average the results. Note that this is sometimes referred to as one-vs-rest, as it is the current class against the rest of the classes.

With a few exceptions (AUC and MCC), the metrics in the multinomial outputs that share their name with the binomial metrics are macro metrics. AUC and MCC instead have specific multiclass variants.

The Weighted metrics are averages, similar to the macro metrics, but weighted by the Support for each class.

Metric name Abbreviation Formula
Overall Accuracy $$\frac{Correct}{Correct + Incorrect}$$
Macro Metric $${\frac{1}{\lvert C \rvert}\sum_{c}^{C} metric_{c}}$$
Support $${support_c = \lvert \{ o \in O : o=c \} \rvert \quad \forall c \in C}$$
I.e., a count of the class in the target column.
$$C$$: the set of classes; $$O$$: the observations.
$$\lvert x \rvert$$ denotes length of $$x$$.
Weighted metric $$\frac{\sum_{c}^{C} metric_{c} \cdot support_{c}}{\sum_{c}^{C} support_{c}}$$
Multiclass MCC MCC $${\frac{N \operatorname{Tr}(C)-\sum_{k l} \tilde{\mathcal{C}}_{k} \hat{\mathcal{C}}_{l}}{\sqrt{N^{2}-\sum_{k l} \tilde{\mathcal{C}}_{k}\left(\hat{\mathcal{C}}^{\mathrm{T}}\right)_{l}} \sqrt{N^{2}-\sum_{k l}\left(\tilde{C}^{\mathrm{T}}\right)_{k} \hat{C}_{l}}} }$$
$$N$$: number of samples
$$C$$: a $$K \times K$$ confusion matrix
$$Tr(C)$$: Number of correct predictions
$$\tilde{\mathcal{C}}_{k}$$: $$k$$th row of $$C$$ ; $$\hat{C}_{l}$$: $$l$$th column of $$C$$
$$C^{T}$$: $$C$$ transposed
Note: When the computation is NaN, we return 0.
Code was ported from scikit-learn.
Gorodkin, J. (2004). Comparing two K-category assignments by a K-category correlation coefficient. Computational biology and chemistry, 28(5-6), 367-374.

## External metrics

These metrics are calculated by other packages:

Metric name Abbreviation Package::Function Used in
Aikake Information Criterion AIC stats::AIC Shared
Corrected Aikake Information Criterion AICc MuMIn::AICc Shared
Bayesian Information Criterion BIC stats::BIC Shared
Aikake Information Criterion AIC stats::AIC Shared
Marginal R-squared r2m MuMIn::r.squaredGLMM Gaussian
Conditional R-squared r2c MuMIn::r.squaredGLMM Gaussian
ROC curve ROC pROC::roc Binomial
Area Under the Curve AUC pROC::roc Binomial
Multiclass ROC curve ROC pROC::multiclass.roc Multinomial
Multiclass Area Under the Curve AUC pROC::multiclass.roc Multinomial