Compute metric of the following one-dimensional binary classification setup: any x value not more than threshold value is classified as "negative"; if strictly greater - "positive". Classification metrics are computed based on two pdqr-functions: f, which represents the distribution of values which should be classified as "negative" ("true negative"), and g - the same for "positive" ("true positive").

summ_classmetric(f, g, threshold, method = "F1")

summ_classmetric_df(f, g, threshold, method = "F1")

## Arguments

f A pdqr-function of any type and class. Represents distribution of "true negative" values. A pdqr-function of any type and class. Represents distribution of "true positive" values. A numeric vector of classification threshold(s). Method of classification metric (might be a vector for summ_classmetric_df()). Should be one of "TPR", "TNR", "FPR", "FNR", "PPV", "NPV", "FDR", "FOR", "LR+", "LR-", "Acc", "ER", "GM", "F1", "OP", "MCC", "YI", "MK", "Jaccard", "DOR" (with possible aliases, see Details).

## Value

summ_classmetric() returns a numeric vector, of the same length as threshold, representing classification metrics for different threshold values.

summ_classmetric_df() returns a data frame with rows corresponding to threshold values. First column is "threshold" (with threshold values), and all other represent classification metric for every input method (see Examples).

## Details

Binary classification setup used here to compute metrics is a simplified version of the most common one, when there is a finite set of already classified objects. Usually, there are N objects which are truly "negative" and P truly "positive" ones. Values N and P can vary, which often results in class imbalance. However, in current setup both N and P are equal to 1 (total probability of f and g).

In common setup, classification of all N + P objects results into the following values: "TP" (number of truly "positive" values classified as "positive"), "TN" (number of negatives classified as "negative"), "FP" (number of negatives falsely classified as "positive"), and "FN" (number of positives falsely classified as "negative"). In current setup all those values are equal to respective "rates" (because N and P are both equal to 1).

Both summ_classmetric() and summ_classmetric_df() allow aliases to some classification metrics (for readability purposes).

Following classification metrics are available:

• Simple metrics:

• True positive rate, method "TPR" (aliases: "TP", "sensitivity", "recall"): proportion of actual positives correctly classified as such. Computed as 1 - as_p(g)(threshold).

• True negative rate, method "TNR" (aliases: "TN", "specificity"): proportion of actual negatives correctly classified as such. Computed as as_p(f)(threshold).

• False positive rate, method "FPR" (aliases: "FP", "fall-out"): proportion of actual negatives falsely classified as "positive". Computed as 1 - as_p(f)(threshold).

• False negative rate, method "FNR" (aliases: "FN", "miss_rate"): proportion of actual positives falsely classified as "negative". Computed as as_p(g)(threshold).

• Positive predictive value, method "PPV" (alias: "precision"): proportion of output positives that are actually "positive". Computed as TP / (TP + FP).

• Negative predictive value, method "NPV": proportion of output negatives that are actually "negative". Computed as TN / (TN + FN).

• False discovery rate, method "FDR": proportion of output positives that are actually "negative". Computed as FP / (TP + FP).

• False omission rate, method "FOR": proportion of output negatives that are actually "positive". Computed as FN / (TN + FN).

• Positive likelihood, method "LR+": measures how much the odds of being "positive" increase when value is classified as "positive". Computed as TPR / (1 - TNR).

• Negative likelihood, method "LR-": measures how much the odds of being "positive" decrease when value is classified as "negative". Computed as (1 - TPR) / TNR.

• Combined metrics (for all, except "error rate", bigger value represents better classification performance):

• Accuracy, method "Acc" (alias: "accuracy"): proportion of total number of input values that were correctly classified. Computed as (TP + TN) / 2 (here 2 is used because of special classification setup, TP + TN + FP + FN = 2).

• Error rate, method "ER" (alias: "error_rate"): proportion of total number of input values that were incorrectly classified. Computed as (FP + FN) / 2.

• Geometric mean, method "GM": geometric mean of TPR and TNR. Computed as sqrt(TPR * TNR).

• F1 score, method "F1": harmonic mean of PPV and TPR. Computed as 2*TP / (2*TP + FP + FN).

• Optimized precision, method "OP": accuracy, penalized for imbalanced class performance. Computed as Acc - abs(TPR - TNR) / (TPR + TNR).

• Matthews correlation coefficient, method "MCC" (alias: "corr"): correlation between the observed and predicted classifications. Computed as (TP*TN - FP*FN) / sqrt((TP+FP) * (TN+FN)) (here equalities TP+FN = 1 and TN+FP = 1 are used to simplify formula).

• Youden’s index, method "YI" (aliases: "youden", "informedness"): evaluates the discriminative power of the classification setup. Computed as TPR + TNR - 1.

• Markedness, method "MK" (alias: "markedness"): evaluates the predictive power of the classification setup. Computed as PPV + NPV - 1.

• Jaccard, method "Jaccard": accuracy ignoring correct classification of negatives. Computed as TP / (TP + FP + FN).

• Diagnostic odds ratio, method "DOR" (alias: "odds_ratio"): ratio between positive and negative likelihoods. Computed as "LR+" / "LR-".

summ_separation for computing optimal separation threshold (which is symmetrical with respect to f and g).

Other summary functions: summ_center(), summ_distance(), summ_entropy(), summ_hdr(), summ_interval(), summ_moment(), summ_order(), summ_prob_true(), summ_pval(), summ_quantile(), summ_roc(), summ_separation(), summ_spread()

## Examples

d_unif <- as_d(dunif)
d_norm <- as_d(dnorm)
t_vec <- c(0, 0.5, 0.75, 1.5)

summ_classmetric(d_unif, d_norm, threshold = t_vec, method = "F1")
#> [1] 0.4000000 0.3412008 0.3069521 0.1252455summ_classmetric(d_unif, d_norm, threshold = t_vec, method = "Acc")
#> [1] 0.2500000 0.4042686 0.4883134 0.5334032
summ_classmetric_df(
d_unif, d_norm, threshold = t_vec, method = c("F1", "Acc")
)
#>   threshold        F1       Acc
#> 1      0.00 0.4000000 0.2500000
#> 2      0.50 0.3412008 0.4042686
#> 3      0.75 0.3069521 0.4883134
#> 4      1.50 0.1252455 0.5334032
# Using method aliases
summ_classmetric_df(
d_unif, d_norm, threshold = t_vec, method = c("TPR", "sensitivity")
)
#>   threshold        TPR sensitivity
#> 1      0.00 0.50000000  0.50000000
#> 2      0.50 0.30853717  0.30853717
#> 3      0.75 0.22662682  0.22662682
#> 4      1.50 0.06680635  0.06680635