Abstract
This vignette lists the available metrics in cvms, along
with their formulas.
Contact the author at r-pkgs@ludvigolsen.dk
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).
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 | |
| \({\scriptstyle \operatorname{IQR}(x)}\) | Interquartile Range | \({\scriptstyle \operatorname{quantile}(x, 3/4) - \operatorname{quantile}(x, 1/4)}\) |
| \(\lvert x \rvert\) | 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 | \({\scriptstyle \sqrt{RSE} }\) |
| 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 | \({\scriptstyle \sum_{i=1}^{n}\lvert\hat{y}_{i}-y_{i}\rvert}\) |
| Total Squared Error | TSE | \({\scriptstyle \sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^2}\) |
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 TPWith 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 | \({\scriptstyle 1 - Sensitivity}\) | |
| False Positive Rate | \({\scriptstyle 1 - Specificity}\) | |
| False Discovery Rate | \({\scriptstyle 1 - Pos Pred Value}\) | |
| False Omission Rate | \({\scriptstyle 1 - Neg Pred Value}\) | |
| Kappa | For Kappa, the counts
(TP, TN, FP, FN) are
normalized to percentages (summing to 1). Then: \({\scriptstyle p_{observed} = TP + TN}\)
\({\scriptstyle p_{expected} = (TN +
FP)(TN + FN) + (FN + TP)(FP + TP)}\) \(Kappa = \frac{p_{observed} - p_{expected}}{1 -
p_{expected}}\) |
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 (
\({\scriptstyle y_{o,c} = 1 \text{ if } y_{o}
= c \text{ else } 0}\) and \({\scriptstyle \hat{y}_{o,c} = 1 \text{ if }
\hat{y} _{o} = c \text{ else } 0}\) ) 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 | \({\scriptstyle \frac{1}{\lvert C \rvert}\sum_{c}^{C} metric_{c}}\) | |
| Support | \({\scriptstyle support_c = \lvert \{ o \in O : o=c \} \rvert \quad \forall c \in C}\)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 | \({\scriptstyle \frac{N
\operatorname{Tr}(C)-\sum_{k l} \tilde{\mathcal{C}}_{k}
\hat{\mathcal{C}}_{l}}{\sqrt{N^{2}-\sum_{k l}
\tilde{\mathcal{C}}_{k}\left(\hat{\mathcal{C}}^{\mathrm{T}}\right)_{l}}
\sqrt{N^{2}-\sum_{k l}\left(\tilde{C}^{\mathrm{T}}\right)_{k}
\hat{C}_{l}}} }\) \(N\): 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. |
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 |