EvalMetrics.jl

Utility package for scoring binary classification models. Performance measures for general classification tasks can be found in MLJ.jl.

Installation

Execute the following command in Julia Pkg REPL (EvalMetrics.jl requires julia 1.0 or higher)

(v1.6) pkg> add EvalMetrics

Usage

Quickstart

The fastest way of getting started is to use a simple binary_eval_report function in the following way:

julia> using EvalMetrics, Random

julia> Random.seed!(123);

julia> targets = rand(0:1, 100);

julia> scores = rand(100);

julia> binary_eval_report(targets, scores)
Dict{String, Real} with 8 entries:
  "precision@fpr0.05"          => 0.6
  "recall@fpr0.05"             => 0.0576923
  "accuracy@fpr0.05"           => 0.49
  "au_prcurve"                 => 0.507454
  "samples"                    => 100
  "true negative rate@fpr0.05" => 0.958333
  "au_roccurve"                => 0.469952
  "prevalence"                 => 0.52

julia> binary_eval_report(targets, scores, 0.001)
┌ Warning: The closest lower feasible false positive rate to some of the required values (0.001) is 0.0!
└ @ EvalMetrics .../EvalMetrics.jl/src/thresholds.jl:152
Dict{String, Real} with 8 entries:
  "recall@fpr0.001"             => 0.0
  "au_prcurve"                  => 0.507454
  "samples"                     => 100
  "precision@fpr0.001"          => 1.0
  "au_roccurve"                 => 0.469952
  "accuracy@fpr0.001"           => 0.48
  "prevalence"                  => 0.52
  "true negative rate@fpr0.001" => 1.0

Confusion Matrix

The core the package is the ConfusionMatrix structure, which represents the confusion matrix in the following form

	Actual positives	Actual negatives
Predicted positives	tp (# true positives)	fp (# false positives)
Predicted negatives	fn (# false negatives)	tn (# true negatives)
	p (# positives)	n (# negatives)

The confusion matrix can be calculated from targets and predicted values or from targets, scores, and one or more decision thresholds

julia> thres = 0.6;

julia> predicts  = scores .>= thres;

julia> cm1 = ConfusionMatrix(targets, predicts)
┌────────────┬───────────┬───────────┐
│ Tot = 100  │ Actual    │ Actual    │
│            │ positives │ negatives │
├────────────┼───────────┼───────────┤
│ Prediced   │ 19        │ 20        │
│ positives  │           │           │
├────────────┼───────────┼───────────┤
│ Prediced   │ 33        │ 28        │
│ negatives  │           │           │
└────────────┴───────────┴───────────┘

julia> cm2 = ConfusionMatrix(targets, scores, thres)
┌────────────┬───────────┬───────────┐
│ Tot = 100  │ Actual    │ Actual    │
│            │ positives │ negatives │
├────────────┼───────────┼───────────┤
│ Prediced   │ 19        │ 20        │
│ positives  │           │           │
├────────────┼───────────┼───────────┤
│ Prediced   │ 33        │ 28        │
│ negatives  │           │           │
└────────────┴───────────┴───────────┘

julia> cm3 = ConfusionMatrix(targets, scores, thres)
┌────────────┬───────────┬───────────┐
│ Tot = 100  │ Actual    │ Actual    │
│            │ positives │ negatives │
├────────────┼───────────┼───────────┤
│ Prediced   │ 19        │ 20        │
│ positives  │           │           │
├────────────┼───────────┼───────────┤
│ Prediced   │ 33        │ 28        │
│ negatives  │           │           │
└────────────┴───────────┴───────────┘

julia> cm4 = ConfusionMatrix(targets, scores, [thres, thres])
2-element Vector{ConfusionMatrix{Int64}}:
 ConfusionMatrix{Int64}(52, 48, 19, 28, 20, 33)
 ConfusionMatrix{Int64}(52, 48, 19, 28, 20, 33)

The package provides many basic classification metrics based on the confusion matrix. The following table provides a list of all available metrics and its aliases

Classification metric	Aliases
true_positive
true_negative
false_positive
false_negative
true_positive_rate	sensitivity, recall, hit_rate
true_negative_rate	specificity, selectivity
false_positive_rate	fall_out, type_I_error
false_negative_rate	miss_rate, type_II_error
precision	positive_predictive_value
negative_predictive_value
false_discovery_rate
false_omission_rate
threat_score	critical_success_index
accuracy
balanced_accuracy
error_rate
balanced_error_rate
f1_score
fβ_score
matthews_correlation_coefficient	mcc
quant
positive_likelihood_ratio
negative_likelihood_ratio
diagnostic_odds_ratio
prevalence

Each metric can be computed from the ConfusionMatrix structure

julia> recall(cm1)
0.36538461538461536

julia> recall(cm2)
0.36538461538461536

julia> recall(cm3)
0.36538461538461536

julia> recall(cm4)
2-element Vector{Float64}:
 0.36538461538461536
 0.36538461538461536

The other option is to compute the metric directly from targets and predicted values or from targets, scores, and one or more decision thresholds

julia> recall(targets, predicts)
0.36538461538461536

julia> recall(targets, scores, thres)
0.36538461538461536

julia> recall(targets, scores, thres)
0.36538461538461536

julia> recall(targets, scores, [thres, thres])
2-element Vector{Float64}:
 0.36538461538461536
 0.36538461538461536

User defined classification metrics

It may occur that some useful metric is not defined in the package. To simplify the process of defining a new metric, the package provides the @metric macro and apply function.

import EvalMetrics: @metric, apply

@metric MyRecall

apply(::Type{MyRecall}, x::ConfusionMatrix) = x.tp/x.p

In the previous example, macro @metric defines a new abstract type MyRecall (used for dispatch) and a function myrecall (for easy use of the new metric). With defined abstract type MyRecall, the next step is to define a new method for the apply function. This method must have exactly two input arguments: Type{MyRecall} and ConfusionMatrix. If another argument is needed, it can be added as a keyword argument.

apply(::Type{Fβ_score}, x::ConfusionMatrix; β::Real = 1) =
    (1 + β^2)*precision(x)*recall(x)/(β^2*precision(x) + recall(x))

It is easy to check that the myrecall metric returns the same outputs as the recall metric defined in the package

julia> myrecall(cm1)
0.36538461538461536

julia> myrecall(cm2)
0.36538461538461536

julia> myrecall(cm3)
0.36538461538461536

julia> myrecall(cm4)
2-element Vector{Float64}:
 0.36538461538461536
 0.36538461538461536

julia> myrecall(targets, predicts)
0.36538461538461536

julia> myrecall(targets, scores, thres)
0.36538461538461536

julia> myrecall(targets, scores, thres)
0.36538461538461536

julia> myrecall(targets, scores, [thres, thres])
2-element Vector{Float64}:
 0.36538461538461536
 0.36538461538461536

Label encodings

Different label encodings are considered common in different machine learning applications. For example, support vector machines use 1 as a positive label and -1 as a negative label. On the other hand, it is common for neural networks to use 0 as a negative label. The package provides some basic label encodings listed in the following table

Encoding	positive label(s)	negative label(s)
OneZero(::Type{T})	one(T)	zero(T)
OneMinusOne(::Type{T})	one(T)	-one(T)
OneTwo(::Type{T})	one(T)	2*one(T)
OneVsOne(::Type{T}, pos::T, neg::T)	pos	neg
OneVsRest(::Type{T}, pos::T, neg::AbstractVector{T})	pos	neg
RestVsOne(::Type{T}, pos::AbstractVector{T}, neg::T)	pos	neg

The current_encoding function can be used to verify which encoding is currently in use (by default it is OneZero encoding)

julia> enc = current_encoding()
OneZero{Float64}:
   positive class: 1.0
   negative class: 0.0

One way to use a different encoding is to pass the new encoding as the first argument

julia> enc_new = OneVsOne(:positive, :negative)
OneVsOne{Symbol}: 
   positive class: positive 
   negative class: negative

julia> targets_recoded = recode.(enc, enc_new, targets);

julia> predicts_recoded = recode.(enc, enc_new, predicts);

julia> recall(enc, targets, predicts)
0.36538461538461536

julia> recall(enc_new, targets_recoded, predicts_recoded)
0.36538461538461536

The second way is to change the current encoding to the one you want

julia> set_encoding(OneVsOne(:positive, :negative))
OneVsOne{Symbol}: 
   positive class: positive 
   negative class: negative

julia> recall(targets_recoded, predicts_recoded)
0.36538461538461536

Decision thresholds for classification

The package provides a thresholds(scores::RealVector, n::Int) , which returns n decision thresholds which correspond to n evenly spaced quantiles of the given scores vector. The default value of n is length(scores) + 1. The thresholds function has two keyword arguments reduced::Bool and zerorecall::Bool

• If reduced is true (default), then the function returns min(length(scores) + 1, n) thresholds.
• If zerorecall is true (default), then the largest threshold is maximum(scores)*(1 + eps()) otherwise maximum(scores).

The package also provides some other useful utilities

• threshold_at_tpr(targets::AbstractVector, scores::RealVector, tpr::Real) returns the largest threshold t that satisfies true_positive_rate(targets, scores, t) >= tpr
• threshold_at_tnr(targets::AbstractVector, scores::RealVector, tnr::Real) returns the smallest threshold t that satisfies true_negative_rate(targets, scores, t) >= tnr
• threshold_at_fpr(targets::AbstractVector, scores::RealVector, fpr::Real) returns the smallest threshold t that satisfies false_positive_rate(targets, scores, t) <= fpr
• threshold_at_fnr(targets::AbstractVector, scores::RealVector, fnr::Real) returns the largest threshold t that satisfies false_negative_rate(targets, scores, t) <= fnr

All four functions can be called with an encoding of type AbstractEncoding as the first parameter to use a different encoding than default.

Evaluation curves

Functionality for measuring performance with curves is implemented in the package as well. For example, a precision-recall (PR) curve can be computed as follows:

julia> scores = [0.74, 0.48, 0.23, 0.91, 0.33, 0.92, 0.83, 0.61, 0.68, 0.09];

julia> targets = collect(1:10 .>= 3);

julia> prcurve(targets, scores)
([1.0, 0.875, 0.75, 0.625, 0.625, 0.5, 0.375, 0.375, 0.25, 0.125, 0.0],
 [0.8, 0.7777777777777778, 0.75, 0.7142857142857143, 0.8333333333333334, 0.8, 0.75, 1.0, 1.0, 1.0, 1.0])

All possible calls:

• prcurve(targets::AbstractVector, scores::RealVector) returns all length(target) + 1 points
• prcurve(enc::AbstractEncoding, target::AbstractVector, scores::RealVector) makes different encodings possible
• prcurve(targets::AbstractVector, scores::RealVector, thres::RealVector) uses provided threshols to compute individual points
• prcurve(enc::AbstractEncoding, target::AbstractVector, scores::RealVector, thres::RealVector)
• prcurve(cms::AbstractVector{<:ConfusionMatrix})

We can also compute area under the curve using the auc_trapezoidal function which uses the trapezoidal rule as follows:

julia> auc_trapezoidal(prcurve(targets, scores)...)
0.8595734126984128

However, a convenience function au_prcurve is provided with exactly the same signature as prcurve function. Moreover, any curve(PRCurve, args...) or auc(PRCurve, args...) call is equivalent to prcurve(args...) and au_prcurve(args...), respectively.

Besides PR curve, Receiver operating characteristic (ROC) curve is also available out of the box with analogical definitions of roccurve and au_roccurve.

All points of the curve, as well as area under curve scores are computed using the highest possible resolution by default. This can be changed by a keyword argument npoints

julia> length.(prcurve(targets, scores))
(11, 11)

julia> length.(prcurve(targets, scores; npoints=9))
(9, 9)

julia> au_prcurve(targets, scores)
0.8595734126984128

julia> au_prcurve(targets, scores; npoints=9)
0.8826388888888889

Plotting

For plotting purposes, EvalMetrics.jl provides recipes for the Plots library:

julia> using Plots; pyplot()
julia> using Random, MLBase; Random.seed!(42);
julia> scores = sort(rand(10000));
julia> targets = scores .>= 0.99;
julia> targets[MLBase.sample(findall(0.98 .<= scores .< 0.99), 30; replace = false)] .= true;
julia> targets[MLBase.sample(findall(0.99 .<= scores .< 0.995), 30; replace = false)] .= false;

Then, any of the following can be used:

• prplot(targets::AbstractVector, scores::RealVector) to use the full resolution:

julia> prplot(targets, scores)

• prplot(targets::AbstractVector, scores::RealVector, thresholds::RealVector) to specify thresholds that will be used
• prplot!(enc::AbstractEncoding, targets::AbstractVector, scores::RealVector) to use a different encoding than default
• prplot!(enc::AbstractEncoding, targets::AbstractVector, scores::RealVector, thresholds::RealVector)

Furthermore, one can use vectors of vectors like [targets1, targets2] and [scores1, scores2]) to plot multiple curves at once. The calls stay the same:

julia> prplot([targets, targets], [scores, scores .+ rand(10000) ./ 5])

For ROC curve use rocplot analogically:

julia> rocplot(targets, scores)

julia> rocplot([targets, targets], [scores, scores .+ rand(10000) ./ 5])

'Modifying' versions with exclamation marks prplot! and rocplot! work as well.

The appearance of the plot can be changed in exactly the same way as with Plots library. Therefore, keyword arguments such as xguide, xlims, grid, fill can all be used:

julia> prplot(targets, scores; xguide="RECALL", fill=:green, grid=false, xlims=(0.8, 1.0))

julia> rocplot(targets, scores, title="Title", label="experiment", xscale=:log10)

Here, limits on x axis are appropriately changed, unless overridden by using xlims keyword argument.

julia> rocplot([targets, targets], [scores, scores .+ rand(10000) ./ 5], label=["a" "b";])

By default, plotted curves have 300 points, which are sampled to retain as much information as possible. This amounts to sampling false positive rate in case of ROC curves and true positive rate in case of PR curves instead of raw thresholds. The number of points can be again changed by keyword argument npoints:

julia> prplot(targets, scores; npoints=Inf, label="Original") 
julia> prplot!(targets, scores; npoints=10, label="Sampled (10 points)") 
julia> prplot!(targets, scores; npoints=100, label="Sampled (100 points)") 
julia> prplot!(targets, scores; npoints=1000, label="Sampled (1000 points)") 
julia> prplot!(targets, scores; npoints=5000, label="Sampled (5000 points)") 

Note that even though we visuallize smaller number of points, the displayed auc score is computed from all points. In case when logarithmic scale is used, the sampling is also done in logarithmic scale.

Other than that, diagonal keyword indicates the diagonal in the plot, and aucshow toggles, whether auc score is appended to a label:

julia> rocplot(targets, scores; aucshow=false, label="a", diagonal=true)

User-defined curves

PR and ROC curves are available out of the box. Additional curve definitions can be provided in the similar way as new metrics are defined using macro @curve and defining apply function, which computes a point on the curve. For instance, ROC curve can be defined this way:

julia> import EvalMetrics: @curve, apply 

julia> @curve MyROCCurve

julia> apply(::Type{MyROCCurve}, cms::AbstractVector{ConfusionMatrix{T}}) where T <: Real =
    (false_positive_rate(cms), true_positive_rate(cms))

julia> myroccurve(targets, scores) == roccurve(targets, scores)
true

In order to be able to sample from x axis while plotting, sampling_function and lowest_metric_value must be provided as well.