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Evaluation: From precision, recall and F-measure to ROC, informedness, markedness & correlation

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Commonly used evaluation measures including Recall, Precision, F-Measure and Rand Accuracy are biased and should not be used without clear understanding of the biases, and corresponding identification of chance or base case levels of the statistic. Using these measures a system that performs worse in the objective sense of Informedness, can appear to perform better under any of these commonly used measures. We discuss several concepts and measures that reflect the probability that prediction is informed versus chance. Informedness and introduce Markedness as a dual measure for the probability that prediction is marked versus chance. Finally we demonstrate elegant connections between the concepts of Informedness, Markedness, Correlation and Significance as well as their intuitive relationships with Recall and Precision, and outline the extension from the dichotomous case to the general multi-class case.
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Journal of Machine Learning Technologies
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011, pp-37-63
Available online at http://www.bioinfo.in/contents.php?id=51
Copyright © 2011 Bioinfo Publications 37
EVALUATION: FROM PRECISION, RECALL AND F-MEASURE TO ROC,
INFORMEDNESS, MARKEDNESS & CORRELATION
POWERS, D.M.W.
*AILab, School of Computer Science, Engineering and Mathematics, Flinders University, South Australia, Australia
Corresponding author. Email: David.Powers@flinders.edu.au
Received: February 18, 2011; Accepted: February 27, 2011
Abstract-Commonly used evaluation measures including Recall, Precision, F-Measure and Rand Accuracy are
biased and should not be used without clear understanding of the biases, and corresponding identification of chance
or base case levels of the statistic. Using these measures a system that performs worse in the objective sense of
Informedness, can appear to perform better under any of these commonly used measures. We discuss several
concepts and measures that reflect the probability that prediction is informed versus chance. Informedness and
introduce Markedness as a dual measure for the probability that prediction is marked versus chance. Finally we
demonstrate elegant connections between the concepts of Informedness, Markedness, Correlation and Significance
as well as their intuitive relationships with Recall and Precision, and outline the extension from the dichotomous case
to the general multi-class case.
KeywordsRecall and Precision, F-Measure, Rand Accuracy, Kappa,Informedness and Markedness, DeltaP,
Correlation, Significance.
INTRODUCTION
A common but poorly motivated way of evaluating
results of Machine Learning experiments is using
Recall, Precision and F-measure. These measures
are named for their origin in Information Retrieval and
present specific biases, namely that they ignore
performance in correctly handling negative examples,
they propagate the underlying marginal prevalences
and biases, and they fail to take account the chance
level performance. In the Medical Sciences, Receiver
Operating Characteristics (ROC) analysis has been
borrowed from Signal Processing to become a
standard for evaluation and standard setting,
comparing True Positive Rate and False Positive
Rate. In the Behavioural Sciences, Specificity and
Sensitivity, are commonly used. Alternate techniques,
such as Rand Accuracy and Cohen Kappa, have
some advantages but are nonetheless still biased
measures. We will recapitulate some of the literature
relating to the problems with these measures, as well
as considering a number of other techniques that
have been introduced and argued within each of
these fields, aiming/claiming to address the problems
with these simplistic measures.
This paper recapitulates and re-examines the
relationships between these various measures,
develops new insights into the problem of measuring
the effectiveness of an empirical decision system or a
scientific experiment, analyzing and introducing new
probabilistic and information theoretic measures that
overcome the problems with Recall, Precision and
their derivatives.
THE BINARY CASE
It is common to introduce the various measures in the
context of a dichotomous binary classification
problem, where the labels are by convention + and-
and the predictions of a classifier are summarized in
a four-cell contingency table. This contingency table
may be expressed using raw counts of the number of
times each predicted label is associated with each
real class, or may be expressed in relative terms.
Cell and margin labels may be formal probability
expressions, may derive cell expressions from margin
labels or vice-versa, may use alphabetic constant
labels a, b, c, d or A, B, C, D, or may
use acronyms for the generic terms for True and
False, Real and Predicted Positives and Negatives.
Often UPPER CASE is used where the values are
counts, and lower case letters where the values are
probabilities or proportions relative to N or the
marginal probabilities we will adopt this convention
throughout this paper (always written in
typewriter font), and in addition will use
Mixed Case (in the normal text font) for popular
nomenclature that may or may not correspond
directly to one of our formal systematic names. True
and False Positives (TP/FP) refer to the number of
Predicted Positives that were correct/incorrect, and
similarly for True and False Negatives (TN/FN), and
these four cells sum to N. On the other hand tp,
fp, fn, tn and rp, rn and pp, pn refer
to the joint and marginal probabilities, and the four
contingency cells and the two pairs of marginal
probabilities each sum to 1. We will attach other
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
38 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
popular names to some of these probabilities in due
course.
We thus make the specific assumptions that we are
predicting and assessing a single condition that is
either positive or negative (dichotomous), that we
have one predicting model, and one gold standard
labeling. Unless otherwise noted we will also for
simplicity assume that the contingency is non-trivial in
the sense that both positive and negative states of
both predicted and real conditions occur, so that none
of the marginal sums or probabilities is zero.
We illustrate in Table 1 the general form of a binary
contingency table using both the traditional alphabetic
notation and the directly interpretable systematic
approach. Both definitions and derivations in this
paper are made relative to these labellings, although
English terms (e.g. from Information Retrieval) will
also be introduced for various ratios and probabilities.
The green positive diagonal represents correct
predictions, and the pink negative diagonal incorrect
predictions. The predictions of the contingency table
may be the predictions of a theory, of some
computational rule or system (e.g. an Expert System
or a Neural Network), or may simply be a direct
measurement, a calculated metric, or a latent
condition, symptom or marker. We will refer
generically to "the model" as the source of the
predicted labels, and "the population" or "the world"
as the source of the real conditions. We are
interested in understanding to what extent the model
"informs" predictions about the world/population, and
the world/population "marks" conditions in the model.
Recall & Precision, Sensitivity & Specificity
Recall or Sensitivity (as it is called in Psychology) is
the proportion of Real Positive cases that are
correctly Predicted Positive. This measures the
Coverage of the Real Positive cases by the +P
(Predicted Positive) rule. Its desirable feature is that it
reflects how many of the relevant cases the +P rule
picks up. It tends not to be very highly valued in
Information Retrieval (on the assumptions that there
are many relevant documents, that it doesn't really
matter which subset we find, that we can't know
anything about the relevance of documents that aren't
returned). Recall tends to be neglected or averaged
away in Machine Learning and Computational
Linguistics (where the focus is on how confident we
can be in the rule or classifier). However, in a
Computational Linguistics/Machine Translation
context Recall has been shown to have a major
weight in predicting the success of Word Alignment
[1]. In a Medical context Recall is moreover regarded
as primary, as the aim is to identify all Real Positive
cases, and it is also one of the legs on which ROC
analysis stands. In this context it is referred to as
True Positive Rate (tpr). Recall is defined, with its
various common appellations, by equation (1):
Recall = Sensitivity = tpr = tp/rp
= TP / RP = A /(A+C) (1)
Conversely, Precision or Confidence (as it is called in
Data Mining) denotes the proportion of Predicted
Positive cases that are correctly Real Positives. This
is what Machine Learning, Data Mining and
Information Retrieval focus on, but it is totally ignored
in ROC analysis. It can however analogously be
called True Positive Accuracy (tpa), being a
measure of accuracy of Predicted Positives in
contrast with the rate of discovery of Real Positives
(tpr). Precision is defined in (2):
Precision = Confidence =tpa=tp/pp
=TP / PP = A /(A+B) (2)
These two measures and their combinations focus
only on the positive examples and predictions,
although between them they capture some
information about the rates and kinds of errors made.
However, neither of them captures any information
about how well the model handles negative cases.
Recall relates only to the +R column and Precision
only to the +P row. Neither of these takes into
account the number of True Negatives. This also
applies to their Arithmetic, Geometric and Harmonic
Means: A, Gand F=G2/A (the F-factor or F-measure).
Note that the F1-measure effectively references the
True Positives to the Arithmetic Mean of Predicted
Positives and Real Positives, being a constructed rate
normalized to an idealized value, and expressed in
this form it is known in statistics as a Proportion of
Specific Agreement as it is a applied to a specific
class, so applied to the Positive Class, it is PS+. It
also corresponds to the set-theoretic Dice Coefficient.
The Geometric Mean of Recall and Precision (G-
measure) effectively normalizes TP to the Geometric
Mean of Predicted Positives and Real Positives, and
its Information content corresponds to the Arithmetic
Table 1. Systematic and traditional notations in a binary contingency table. Shading indicates correct
(light=green) and incorrect (dark=red) rates or counts in the contingency table.
+R
R
+R
R
tp
pp
+P
A
B
A+B
fn
pn
P
C
D
C+D
rp
rn
1
A+C
B+D
N
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Copyright © 2011 Bioinfo Publications 39
Mean of the Information represented by Recall and
Precision.
In fact, there is in principle nothing special about the
Positive case, and we can define Inverse statistics in
terms of the Inverse problem in which we interchange
positive and negative and are predicting the opposite
case. Inverse Recall or Specificity is thus the
proportion of Real Negative cases that are correctly
Predicted Negative (3), and is also known as the True
Negative Rate (tnr). Conversely, Inverse Precision
is the proportion of Predicted Negative cases that are
indeed Real Negatives (4), and can also be called
True Negative Accuracy (tna):
Inverse Recall =tnr =tn/rn
=TN/RN =D/(B+D) (3)
Inverse Precision =tna =tn/pn
=TN/PN =D/(C+D) (4)
The inverse of F1 is not known in AI/ML/CL/IR but is
just as well known as PS+ in statistics,being the
Proportion of Specific Agreement for the class of
negatives, PS. Note that where as F1 is advocated
in AI/ML/CL/IR as a single measure to capture the
effectiveness of a system, it still completely ignores
TN which can vary freely without affecting the
statistic. In statistics, PS+ is used in conjunction with
PS to ensure the contingencies are completely
captured, and similarly Specificity (Inverse Recall) is
always recorded along with Sensitivity (Recall).
Rand Accuracy explicitly takes into account the
classification of negatives, and is expressible (5) both
as a weighted average of Precision and Inverse
Precision and as a weighted average of Recall and
Inverse Recall:
Accuracy =tca=tcr=tp+tn
=rptpr+rntnr =(TP+TN)/N
=pptpa+pntna =(A+D)/N (5)
Dice = F1 =tp/(tp+(fn+fp)/2)
=A/(A+(B+C)/2) (6)
=1/(1+mean(FN,FP)/TP)
Jaccard =tp/(tp+fn+fp)=TP/(N-TN)
=A/(A+B+C) = A/(N-D) (7)
=1/(1+2mean(FN,FP)/TP)
= F1 / (2 F1)
As shown in (5) Rand Accuracy is effectively a
prevalence-weighted average of Recall and Inverse
Recall, as well as a bias-weighted average of
Precision and Inverse Precision. Whilst it does take
into account TN in the numerator, the sensitivity to
bias and prevalence is an issue since these are
independent variables, with prevalence varying as we
apply to data sampled under different conditions, and
bias being directly under the control of the system
designer (e.g. as a threshold). Similarly, we can note
that one of N,FP or FN is free to vary. Whilst it
apparently takes into account TN in the
numerator,theJaccard (or Tanimoto) similarity
coefficient uses it to heuristicallydiscount the correct
classification of negatives, but it can be written (6)
independently of FN and N in a way similar tothe
effectively equivalent Dice or PS+ or F1 (7), or in
terms of them, and so is subject to bias as FN or N is
free to vary and theyfail to capture contingencies fully
without knowing inverse statisticstoo.
Each of the above also has a complementary form
defining an error rate, of which some have specific
names and importance: Fallout or False Positive Rate
(fpr) are the proportion of Real Negatives that occur
as Predicted Positive (ring-ins); Miss Rate or False
Negative Rate (fnr) are the proportion of Real
Positives that are Predicted Negatives (false-drops).
False Positive Rate is the second of the legs on
which ROC analysis is based.
Fallout =fpr =fp/rp
=FP/RP =B/(B+D) (8)
Miss Rate =fnr =fn/rn
=FN/RN =C/(A+C) (9)
Note that FN and FP are sometimes referred to as
Type I and Type II Errors, and the rates fn and fp
as alpha and beta, respectively referring to falsely
rejecting or accepting a hypothesis. More correctly,
these terms apply specifically to the meta-level
problem discussed later of whether the precise
pattern of counts (not rates) in the contingency table
fit the null hypothesis of random distribution rather
than reflecting the effect of some alternative
hypothesis (which is not in general the one
represented by +P+R or -P -Ror both).
Note that all the measures discussed individually
leave at least two degree of freedom (plus N)
unspecified and free to control, and this leaves the
door open for bias, whilst Nis needed too for
estimating significance and power.
Prevalence, Bias, Cost & Skew
We now turn our attention to the various forms of bias
that detract from the utility of all of the above surface
measures [2]. We will first note that rp represents
the Prevalence of positive cases, RP/N, and is
assumed to be a property of the population of interest
it may be constant, or it may vary across
subpopulations, but is regarded here as not being
under the control of the experimenter, and so we
want a prevalence independent measure. By
contrast, pp represents the (label) Bias of the model
[3], the tendency of the model to output positive
labels, PP/N, and is directly under the control of the
experimenter, who can change the model by
changing the theory or algorithm, or some parameter
or threshold, to better fit the world/population being
modeled. As discussed earlier, F-factor (or Dice or
Jaccard) effectively references tp (probability or
proportion of True Positives) to the Arithmetic Mean
of Bias and Prevalence (6-7). A common rule of
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
40 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
thumb, or even a characteristic of some algorithms, is
to parameterize a model so that Prevalence = Bias,
viz. rp = pp. Corollaries of this setting are Recall
= Precision (= Dice but not Jaccard), Inverse Recall
= Inverse Precision and Fallout = Miss Rate.
Alternate characterizations of Prevalence are in terms
of Odds[4] or Skew [5], being the Class Ratio cs=
rn/rp, recalling that by definition rp+rn = 1
and RN+RP = N. If the distribution is highly
skewed, typically there are many more negative
cases than positive, this means the number of errors
due to poor Inverse Recall will be much greater than
the number of errors due to poor Recall. Given the
cost of both False Positives and False Negatives is
equal, individually, the overall component of the total
cost due to False Positives (as Negatives) will be
much greater at any significant level of chance
performance, due to the higher Prevalence of Real
Negatives.
Note that the normalized binary contingency table
with unspecified margins has three degrees of
freedom setting any three nonRedundant ratios
determines the rest (setting any count supplies the
remaining information to recover the original table of
counts with its four degrees of freedom). In particular,
Recall, Inverse Recall and Prevalence, or
equivalently tpr, fpr and cs, suffice to determine all
ratios and measures derivable from the normalized
contingency table, but N is also required to
determine significance. As another case of specific
interest, Precision, Inverse Precision and Bias, in
combination, suffice to determine all ratios or
measures, although we will show later that an
alternate characterization of Prevalence and Bias in
terms of Evenness allows for even simpler
relationships to be exposed.
We can also take into account a differential value for
positives (cp) and negatives (cn) this can be
applied to errors as a cost (loss or debit) and/or to
correct cases as a gain (profit or credit), and can be
combined into a single Cost Ratio cv= cn/cp.
Note that the value and skew determined costs have
similar effects, and may be multiplied to produce a
single skew-like cost factor c = cvcs. Formulations
of measures that are expressed using tpr, fpr and cs
may be made cost-sensitive by using c = cvcs in
place of c = c
s, or can be made skew/cost-
insensitive by using c = 1[5].
ROC and PN Analyses
Flach [5] highlighted the utility of ROC analysis to the
Machine Learning community, and characterized the
skew sensitivity of many measures in that context,
utilizing the ROC format to give geometric insights
into the nature of the measures and their sensitivity to
skew. [6] further elaborated this analysis, extending it
to the unnormalized PN variant of ROC, and targeting
their analysis specifically to rule learning. We will not
examine the advantages of ROC analysis here, but
will briefly explain the principles and recapitulate
some of the results.
ROC analysis plots the rate tpr against the rate
fpr, whilst PN plots the unnormalized TP against
FP. This difference in normalization only changes the
scales and gradients, and we will deal only with the
normalized form of ROC analysis. A perfect classifier
will score in the top left hand corner
(fpr=0,tpr=100%). A worst case classifier will
score in the bottom right hand corner
(fpr=100%,tpr=0). A random classifier would
be expected to score somewhere along the positive
diagonal (tpr=fpr) since the model will throw up
positive and negative examples at the same rate
(relative to their populations these are Recall-like
scales: tpr = Recall, 1-fpr = Inverse Recall).
For the negative diagonal (tpr+cfpr=1)
corresponds to matching Bias to Prevalence for a
skew of c.
The ROC plot allows us to compare classifiers
(models and/or parameterizations) and choose the
one that is closest to (0,1) and furtherest from
tpr=fpr in some sense. These conditions for
choosing the optimal parameterization or model are
not identical, and in fact the most common condition
is to minimize the area under the curve (AUC), which
for a single parameterization of a model is defined by
Figure 1. Illustration of ROC Analysis. The main
diagonal represents chance with parallel isocost lines
representing equal cost-performance. Points above
the diagonal represent performance better than
chance, those below worse than chance. For a single
good (dotted=green) system, AUC is area under curve
(trapezoid between green line and x=[0,1] ).
The perverse (dashed=red) system shown is the same
(good) system with class labels reversed.
best
tpr
fpr
worst
good
perverse
sse0
B
sse1
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Copyright © 2011 Bioinfo Publications 41
a single point and the segments connecting it to (0,0)
and (1,1). For a parameterized model it will be a
monotonic function consisting of a sequence of
segments from (0,0) to (1,1). A particular cost model
and/or accuracy measure defines an isocost gradient,
which for a skew and cost insensitive model will be
c=1, and hence another common approach is to
choose a tangent point on the highest isocost line that
touches the curve. The simple condition of choosing
the point on the curve nearest the optimum point (0,1)
is not commonly used, but this distance to (0,1) is
given by [(-fpr)2+ (1-tpr)2], and
minimizing this amounts to minimizing the sum of
squared normalized error, fpr2+fnr2.
A ROC curve with concavities can also be locally
interpolated to produce a smoothed model following
the convex hull of the original ROC curve. It is even
possible to locally invert across the convex hull to
repair concavities, but this may overfit and thus not
generalize to unseen data. Such repairs can lead to
selecting an improved model, and the ROC curve can
also be used to return a model to changing
Prevalence and costs. The area under such a
multipoint curve is thus of some value, but the
optimum in practice is the area under the simple
trapezoid defined by the model:
AUC = (tpr-fpr+1)/2
= (tpr+tnr)/2
= 1 – (fpr+fnr)/2 (10)
For the cost and skew insensitive case, with c=1,
maximizing AUC is thus equivalent to maximizing
tpr-fpr or minimizing a sum of (absolute)
normalized error fpr+fnr. The chance line
corresponds to tpr-fpr=0, and parallel isocost
lines for c=1 have the form tpr-fpr=k. The
highest isocost line also maximizes tpr-fpr and
AUC so that these two approaches are equivalent.
Minimizing a sum of squared normalized error,
fpr2+fnr2, corresponds to a Euclidean distance
minimization heuristic that is equivalent only under
appropriate constraints, e.g. fpr=fnr, or
equivalently, Bias=Prevalence, noting that all cells are
non-negative by construction.
We now summarize relationships between the various
candidate accuracy measures as rewritten [5,6] in
terms of tpr, fpr and the skew, c, as well in
terms of Recall, Bias and Prevalence:
Accuracy = [tpr+c·(1-fpr)]/[1+c]
= 2·Recall·Prev+1-BiasPrev (11)
Precision = tpr/[tpr+c·fpr]
= Recall·Prev/Bias (12)
F-Measure F1 = 2·tpr/[tpr+c·fpr+1]
= 2·Recall·Prev/[Bias+Prev] (13)
WRacc = 4c·[tpr-fpr]/[1+c]2
= 4·[Recall-Bias]·Prev (14)
The last measure, Weighted Relative Accuracy, was
defined [7] to subtract off the component of the True
Positive score that is attributable to chance and
rescale to the range ±1. Note that maximizingWRacc
is equivalent to maximizing AUC or tpr-fpr
=2·AUC1, as c is constant. Thus WRAcc is an
unbiased accuracy measure, and the skew-
insensitive form of WRAcc, with c=1, is precisely
tpr-fpr. Each of the other measures (1012)
shows a bias in that it can not be maximized
independent of skew, although skew-insensitive
versions can be defined by setting c=1. The
recasting of Accuracy, Precision and F-Measure in
terms of Recall makes clear how all of these vary only
in terms of the way they are affected by Prevalence
and Bias.
Prevalence is regarded as a constant of the target
condition or data set (and c=[1Prev]/Prev),
whilst parameterizing or selecting a model can be
viewed in terms of trading off tpr and fpr as in
ROC analysis, or equivalently as controlling the
relative number of positive and negative predictions,
namely the Bias, in order to maximize a particular
accuracy measure (Recall, Precision, F-Measure,
Rand Accuracy and AUC). Note that for a given
Recall level, the other measures (1013) all decrease
with increasing Bias towards positive predictions.
DeltaP, Informedness and Markedness
Powers [4] also derived an unbiased accuracy
measure to avoid the bias of Recall, Precision and
Accuracy due to population Prevalence and label
bias. The Bookmaker algorithm costs wins and losses
in the same way a fair bookmaker would set prices
based on the odds. Powers then defines the concept
of Informedness which represents the 'edge' a punter
has in making his bet, as evidenced and quantified by
his winnings. Fair pricing based on correct odds
should be zero sum that is, guessing will leave you
with nothing in the long run, whilst a punter with
certain knowledge will win every time. Informedness
is the probability that a punter is making an informed
bet and is explained in terms of the proportion of the
time the edge works out versus ends up being pure
guesswork. Powers defined Bookmaker
Informedness for the general, K-label, case, but we
will defer discussion of the general case for now and
present a simplified formulation of Informedness, as
well as the complementary concept of Markedness.
Definition 1
Informedness quantifies how informed a
predictor is for the specified condition, and
specifies the probability that a prediction is
informed in relation to the condition (versus
chance).
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
42 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
Definition 2
Markedness quantifies how marked a
condition is for the specified predictor, and
specifies the probability that a condition is
marked by the predictor (versus chance).
These definitions are aligned with the psychological
and linguistic uses of the terms condition and marker.
The condition represents the experimental outcome
we are trying to determine by indirect means. A
marker or predictor (cf. biomarker or neuromarker)
represents the indicator we are using to determine
the outcome. There is no implication of causality
that is something we will address later. However there
are two possible directions of implication we will
address now. Detection of the predictor may reliably
predict the outcome, with or without the occurrence of
a specific outcome condition reliably triggering the
predictor.
For the binary case we have
Informedness = Recall + Inverse Recall 1
= tpr-fpr = 1-fnr-fpr (15)
Markedness = Precision + Inverse Precision 1
= tpa-fna = 1-fpa-fna
We noted above that maximizing AUC or the
unbiased WRAcc measure effectively maximized tpr-
fpr and indeed WRAcc reduced to this in the skew
independent case. This is not surprising given both
Powers [4] and Flach [5-7] set out to produce an
unbiased measure, and the linear definition of
Informedness will define a unique linear form. Note
that while Informedness is a deep measure of how
consistently the Predictor predicts the Outcome by
combining surface measures about what proportion of
Outcomes are correctly predicted, Markedness is a
deep measure of how consistently the Outcome has
the Predictor as a Marker by combining surface
measures about what proportion of Predictions are
correct.
In the Psychology literature, Markedness is known as
DeltaP and is empirically a good predictor of human
associative judgements that is it seems we develop
associative relationships between a predictor and an
outcome when DeltaP is high, and this is true even
when multiple predictors are in competition [8]. In the
context of experiments on information use in syllable
processing, [9] notes that Schanks [8] sees DeltaP as
"the normative measure of contingency", but propose
a complementary, backward, additional measure of
strength of association, DeltaP' aka dichotomous
Informedness. Perruchet and Peeremant [9] also
note the analog of DeltaP to regression coefficient,
and that the Geometric Mean of the two measures is
a dichotomous form of the Pearson correlation
coefficient, the Matthews' Correlation Coefficient,
which is appropriate unless a continuous scale is
being measured dichotomously in which case a
Tetrachoric Correlation estimate would be appropriate
[10,11].
Causality, Correlation and Regression
In a linear regression of two variables, we seek to
predict one variable, y, as a linear combination of the
other, x, finding a line of best fit in the sense of
minimizing the sum of squared error (in y). The
equation of fit has the form
y= y0 + rx·x where
rx= [nx·y-x·y]/[nx2-x·x] (16)
Substituting in counts from the contingency table, for
the regression of predicting +R (1) versus-R (0)
given +P (1) versus-P (0), we obtain this gradient of
best fit (minimizing the error in the real values R):
rP = [ADBC] / [(A+B)(C+D)]
= A/(A+B) C/(C+D)
= DeltaP = Markedness (17)
Conversely, we can find the regression coefficient for
predicting P from R (minimizing the error in the
predictions P):
rR = [ADBC] / [(A+C)(B+D)]
= A/(A+C) B/(B+D)
= DeltaP' = Informedness (18)
Finally we see that the Matthews correlation, a
contingency matrix method of calculating the Pearson
product-moment correlation coefficient, ρ, is defined
by
rG =[ADBC]/[(A+C)(B+D)(A+B)(C+D)]
=Correlation
[Informedness·Markedness] (19)
Given the regressions find the same line of best fit,
these gradients should be reciprocal, defining a
perfect Correlation of 1. However, both Informedness
and Markedness are probabilities with an upper
bound of 1, so perfect correlation requires perfect
regression. The squared correlation is a coefficient of
proportionality indicating the proportion of the
variance in R that is explained by P, and is
traditionally also interpreted as a probability. We can
now interpret it either as the joint probability that P
informs R and R marks P, given that the two
directions of predictability are independent, or as the
probability that the variance is (causally) explained
reciprocally. The sign of the Correlation will be the
same as the sign of Informedness and Markedness
and indicates whether a correct or perverse usage of
the information has been made take note in
interpreting the final part of (19).
Psychologists traditionally explain DeltaP in terms of
causal prediction, but it is important to note that the
direction of stronger prediction is not necessarily the
direction of causality, and the fallacy of abductive
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reasoning is that the truth of A B does not in
general have any bearing on the truth of B A.
If Pi is one of several independent possible causes
of R, PiR is strong, but R Pi is in general
weak for any specific Pi. If Pi is one of several
necessary contributing factors to R, PiR is weak
for any single Pi, but R Pi is strong. The
directions of the implication are thus not in general
dependent.
In terms of the regression to fit R from P, since there
are only two correct points and two error points, and
errors are calculated in the vertical (R) direction only,
all errors contribute equally to tilting the regression
down from the ideal line of fit. This Markedness
regression thus provides information about the
consistency of the Outcome in terms of having the
Predictor as a Marker the errors measured from the
Outcome R relate to the failure of the Marker P to be
present.
We can gain further insight into the nature of these
regression and correlation coefficients by reducing
the top and bottom of each expression to probabilities
(dividing by N2, noting that the original contingency
counts sum to N, and the joint probabilities after
reduction sum to 1). The numerator is the
determinant of the contingency matrix, and common
across all three coefficients, reducing to dtp, whilst
the reduced denominator of the regression
coefficients depends only on the Prevalence or Bias
of the base variates. The regression coefficients,
Bookmaker Informedness (B) and Markedness (M),
may thus be re-expressed in terms of Precision (Prec)
or Recall, along with Bias and Prevalence (Prev) or
their inverses (I-):
M = dtp/ [Bias · (1-Bias)]
= dtp/ [pp·pn] = dtp / pg2
= dtp / BiasG2 = dtp / EvennessP
= [PrecisionPrevalence] / IBias (20)
B = dtp/ [Prevalence · (1Prevalence)]
= dtp/ [rp·rn] = dtp / rg2
= dtp / PrevG2= dtp / EvennessR
= [Recall Bias] / IPrev
= Recall Fallout
= Recall + IRecall 1
= Sensitivity + Specificity 1
= (LR1)· (1Specificity)
= (1NLR)· Specificity
= (LR 1)· (1NLR) / (LRNLR) (21)
In the medical and behavioural sciences, the
Likelihood Ratio is LR=Sensitivity/[1Specificity], and
the Negative Likelihood Ratio is NLR=Specificity/[1
Sensitivity]. For non-negative B, LR>1>NLR, with 1
as the chance case. We also express Informedness
in these terms in (21).
The Matthews/Pearson correlation is expressed in
reduced form as the Geometric Mean of Bookmaker
Informedness and Markedness, abbreviating their
product as BookMark (BM) and recalling that it is
BookMark that acts as a probability-like coefficient of
determination, not its root, the Geometric Mean
(BookMarkG or BMG):
BMG = dtp/ [Prev · (1Prev)· Bias· (1-Bias)]
= dtp / [PrevG · BiasG]
= dtp / EvennessG
=[(RecallBias)(PrecPrev)]/(IPrev·IBias) (22)
These equations clearly indicate how the Bookmaker
coefficients of regression and correlation depend only
on the proportion of True Positives and the
Prevalence and Bias applicable to the respective
variables. Furthermore, Prev · Bias represents the
Expected proportion of True Positives (etp) relative
to N, showing that the coefficients each represent the
proportion of Delta True Positives (deviation from
expectation, dtp=tp-etp) renormalized in
different ways to give different probabilities.
Equations (20-22) illustrate this, showing that these
coefficients depend only on dtp and either
Prevalence, Bias or their combination. Note that for a
particular dtp these coefficients are minimized when
the Prevalence and/or Bias are at the evenly biased
0.5 level, however in a learning or parameterization
context changing the Prevalence or Bias will in
general change both tp and etp, and hence can
change dtp.
It is also worth considering further the relationship of
the denominators to the Geometric Means, PrevG of
Prevalence and Inverse Prevalence (IPrev = 1Prev
is Prevalence of Real Negatives) and BiasG of Bias
and Inverse Bias (IBias = 1Bias is bias to Predicted
Negatives). These Geometric Means represent the
Evenness of Real classes (EvennessR = PrevG2) and
Predicted labels (EvennessP = BiasG2). We also
introduce the concept of Global Evenness as the
Geometric Mean of these two natural kinds of
Evenness, EvennessG. From this formulation we can
see that for a given relative delta of true positive
prediction above expectation (dtp), the correlation is
at minimum when predictions and outcomes are both
evenly distributed (EvennessG = Evenness
R =
EvennessP = Prev = Bias = 0.5), and Markedness and
Bookmaker are individually minimal when Bias resp.
Prevalence are evenly distributed (viz. Bias resp.
Prev = 0.5). This suggests that setting Learner Bias
(and regularized, cost-weighted or subsampled
Prevalence) to 0.5, as sometimes performed in
Artificial Neural Network training is in fact
inappropriate on theoretical grounds, as has
Previously been shown both empirically and based on
Bayesian principles rather it is best to use
Learner/Label Bias = Natural Prevalence which is in
general much less than 0.5 [12].
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
44 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
Note that in the above equations (20-22) the
denominator is always strictly positive since we have
occurrences and predictions of both Positives and
Negatives by earlier assumption, but we note that if in
violation of this constraint we have a degenerate case
in which there is nothing to predict or we make no
effective prediction, then tp=etp and dtp=0, and
all the above regression and correlation coefficients
are defined in the limit approaching zero. Thus the
coefficients are zero if and only if dtp is zero, and
they have the same sign as dtp otherwise.
Assuming that we are using the model the right way
round, then dtp, B and M are non-negative, and
BMG is similarly non-negative as expected. If the
model is the wrong way round, then dtp, B, M and
BMG can indicate this by expressing below chance
performance, negative regressions and negative
correlation, and we can reverse the sense of P to
correct this.
The absolute value of the determinant of the
contingency matrix, dp= dtp, in these probability
formulae (20-22), also represents the sum of absolute
deviations from the expectation represented by any
individual cell and hence 2dp=2DP/N is the total
absolute relative error versus the null hypothesis.
Additionally it has a geometric interpretation as the
area of a trapezoid in PN-space, the unnormalized
variant of ROC [6].
We already observed that in (normalized) ROC
analysis, Informedness is twice the triangular area
between a positively informed system and the chance
line, and it thus corresponds to the area of the
trapezoid defined by a system (assumed to perform
no worse than chance), and any of its perversions
(interchanging prediction labels but not the real
classes, or vice-versa, so as to derive a system that
performs no better than chance), and the endpoints of
the chance line (the trivial cases in which the system
labels all cases true or conversely all are labelled
false). Such a kite-shaped area is delimited by the
dotted (system) and dashed (perversion) lines in Fig.
1 (interchanging class labels), but the alternate
parallelogram (interchanging prediction labels) is not
shown. The Informedness of a perverted system is
the negation of the Informedness of the correctly
polarized system.
We now also express the Informedness and
Markedness forms of DeltaP in terms of deviations
from expected values along with the Harmonic mean
of the marginal cardinalities of the Real classes or
Predicted labels respectively, defining DP,
DELTAP, RH, PH and related forms in terms of
their NRelative probabilistic forms defined as
follows:
etp = rp · pp; etn = rn· pn (23)
dp = tp etp = dtp
= -dtn = -(tn etn)
deltap = dtp dtn = 2dp (24)
rh = 2rp·rn / [rp+rn] = rp2/ra2
ph = 2pp·pn / [pp+pn] = pp2/pa2 (25)
DeltaP' or Bookmaker Informedness may now be
expressed in terms of deltap and rh, and DeltaP
or Markedness analogously in terms of deltap and
ph:
B = DeltaP' = [etp+dtp]/rp[efp-dtp]/rn
= etp/rp efp/rn + 2dtp/rh
= 2dp/rh = deltap/rh (26)
M = DeltaP = 2dp/ph = deltap/ph (27)
These harmonic relationships connect directly with
the previous geometric evenness terms by observing
HarmonicMean = GeometricMean2/ArithmeticMean
as seen in (25) and used in the alternative
expressions for normalization for Evenness in (26-
27). The use of HarmonicMean makes the
relationship with F-measure clearer, but use of
GeometricMean is generally preferred as a consistent
estimate of central tendency that more accurately
estimates the mode for skewed (e.g. Poisson) data
bounded below by 0 and unbounded above, and as
the central limit of the family of Lp based averages.
Viz. the Geometric (L0) Mean is the Geometric Mean
of the Harmonic (L1) and Arithmetic (L+1) Means, with
positive values of p being biased higher (toward
L+=Max) and negative values of p being biased
lower (toward L−∞=Min).
Effect of Bias and Prev on Recall and Precision
The final form of the equations (26-27) cancels out
the common Bias and Prevalence (Prev) terms, that
denormalizedtp to tpr (Recall) or tpa (Precision).
We now recast the Bookmaker Informedness and
Markedness equations to show Recall and Precision
as subject (28-29), in order to explore the affect of
Bias and Prevalence on Recall and Precision, as well
as clarify the relationship of Bookmaker and
Markedness to these other ubiquitous but iniquitous
measures.
Recall = Bookmaker (1Prevalence) + Bias
Bookmaker = (Recall-Bias)/(1Prevalence) (28)
Precision = Markedness (1-Bias) + Prevalence
Markedness = (PrecisionPrevalence)/(1-Bias) (29)
Bookmaker and Markedness are unbiased estimators
of above chance performance (relative to respectively
the predicting conditions or the predicted markers).
Equations (28-29) clearly show the nature of the bias
introduced by both Label Bias and Class Prevalence.
If operating at chance level, both Bookmaker and
Markedness will be zero, and Recall, Precision, and
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derivatives such as the F-measure, will be skewed by
the biases. Note that increasing Bias or decreasing
Prevalence increases Recall and decreases
Precision, for a constant level of unbiased
performance. We can more specifically see that the
regression coefficient for the prediction of Recall from
Prevalence is Informedness, and from Bias is +1,
and similarly the regression coefficient for the
prediction of Precision from Bias is Markedness,
and from Prevalence is +1. Using the heuristic of
setting Bias = Prevalence then sets Recall =
Precision = F1 and Bookmaker Informedness =
Markedness = Correlation. Setting Bias = 1
(Prevalence<1) may be seen to make Precision track
Prevalence with Recall = 1, whilst Prevalence = 1
(Bias<1) means Recall = Bias with Informedness = 1,
and under either condition no information is utilized
(Bookmaker Informedness = Markedness = 0).
In summary, Recall reflects the Bias plus a
discounted estimation of Informedness and Precision
reflects the Prevalence plus a discounted estimation
of Markedness. Given usually Prevalence << ½ and
Bias << ½, their complements Inverse Prevalence >>
½ and Inverse Bias >> ½ represent substantial
weighting up of the true unbiased performance in
both these measures, and hence also in F1. High
Bias drives Recall up strongly and Precision down
according to the strength of Informedness; high
Prevalence drives Precision up and Recall down
according to the strength of Markedness.
Alternately, Informedness can be viewed (21) as a
renormalization of Recall after subtracting off the
chance level of Recall, Bias, and Markedness (20)
can be seen as a renormalization of Precision after
subtracting off the chance level of Precision,
Prevalence (and Flach’s WRAcc, the unbiased form
being equivalent to Bookmaker Informedness, was
defined in this way as discussed in §2.3).
Informedness can also be seen (21) as a
renormalization of LR or NLR after subtracting off
their chance level performance. The Kappa measure
[13-16] commonly used in assessor agreement
evaluation was similarly defined as a renormalization
of Accuracy after subtracting off an estimate of the
expected Accuracy, for Cohen Kappa being the dot
product of the Biases and Prevalences, and
expressible as a normalization of the discriminant of
contingency, dtp, by the mean error rate (cf. F1;
viz. Kappa is dtp/[dtp+mean(fp,fn)]). All three
measures are invariant in the sense that they are
properties of the contingency tables that remain
unchanged when we flip to the Inverse problem
(interchange positive and negative for both conditions
and predictions). That is we observe:
Inverse Informedness = Informedness,
Inverse Markedness = Markedness,
Inverse Kappa = Kappa.
The Dual problem (interchange antecedent and
consequent) reverses which condition is the predictor
and the predicted condition, and hence interchanges
Precision and Recall, Prevalence and Bias, as well as
Markedness and Informedness. For cross-evaluator
agreement, both Informedness and Markedness are
meaningful although the polarity and orientation of the
contingency is arbitrary. Similarly when examining
causal relationships (conventionally DeltaP vs
DeltaP'), it is useful to evaluate both deductive and
abductive directions in determining the strength of
association. For example, the connection between
cloud and rain involves cloud as one causal
antecedent of rain (but sunshowers occur
occasionally), and rain as one causal consequent of
cloud (but cloudy days aren't always wet) only once
we have identified the full causal chain can we reduce
to equivalence, and lack of equivalence may be a
result of unidentified causes, alternate outcomes or
both.
The Perverse systems (interchanging the labels on
either the predictions or the classes, but not both)
have similar performance but occur below the chance
line (since we have assumed strictly better than
chance performance in assigning labels to the given
contingency matrix).
Note that the effect of Prevalence on Accuracy,
Recall and Precision has also been characterized
above (§2.3) in terms of Flach's demonstration of how
skew enters into their characterization in ROC
analysis, and effectively assigns different costs to
(False) Positives and (False) Negatives. This can be
controlled for by setting the parameter c
appropriately to reflect the desired skew and cost
tradeoff, with c=1 defining skew and cost insensitive
versions. However, only Informedness (or
equivalents such as DeltaP' and skew-insensitive
WRAcc) precisely characterizes the probability with
which a model informs the condition, and conversely
only Markedness (or DeltaP) precisely characterizes
the probability that a condition marks (informs) the
predictor. Similarly, only the Correlation (aka
Coefficient of Proportionality aka Coefficient of
Determination aka Squared Matthews Correlation
Coefficient) precisely characterizes the probability
that condition and predictor inform/mark each other,
under our dichotomous assumptions. Note the
Tetrachoric Correlation is another estimate of the
Pearson Correlation made under the alternate
assumption of an underlying continuous variable
(assumed normally distributed), and is appropriate if
we instead assume that we are dichotomizing a
normal continuous variable [11]. But in this article we
are making the explicit assumption that we are
dealing with a right/wrong dichotomy that is
intrinsically discontinuous.
Although Kappa does attempt to renormalize a
debiased estimate of Accuracy, and is thus much
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
46 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
more meaningful than Recall, Precision, Accuracy,
and their biased derivatives, it is intrinsically non-
linear, doesn't account for error well, and retains an
influence of bias, so that there does not seem that
there is any situation when Kappa would be
preferable to Correlation as a standard independent
measure of agreement [16,13]. As we have seen,
Bookmaker Informedness, Markedness and
Correlation reflect the discriminant of relative
contingency normalized according to different
Evenness functions of the marginal Biases and
Prevalences, and reflect probabilities relative to the
corresponding marginal cases. However, we have
seen that Kappa scales the discriminant in a way that
reflects the actual error without taking into account
expected error due to chance, and in effect it is really
just using the discriminant to scale the actual mean
error: Kappa is dtp/[dtp+mean(fp,fn)] =
1/[1+mean(fp,fn)/dtp] which approximates for
small error to 1-mean(fp,fn)/dtp.
The relatively good fit of Kappa to Correlation and
Informedness is illustrated in Fig. 2, along with the
poor fit of the Rank Weighted Average and the
Geometric and Harmonic (F-factor) means. The fit of
the Evenness weighted determinant is perfect and not
easily distinguishable but the separate components
(Determinant and geometric means of Real
Prevalences and Prediction Biases) are also shown
(+1 for clarity).
Significance and Information Gain
The ability to calculate various probabilities from a
contingency table says nothing about the significance
of those numbers is the effect real, or is it within the
expected range of variation around the values
expected by chance? Usually this is explored by
considering deviation from the expected values (ETP
and its relatives) implied by the marginal counts (RP,
PP and relatives) or from expected rates implied by
the biases (Class Prevalence and Label Bias). In the
case of Machine Learning, Data Mining, or other
artificially derived models and rules, there is the
further question of whether the training and
parameterization of the model has set the 'correct' or
'best' Prevalence and Bias (or Cost) levels.
Furthermore, should this determination be undertaken
by reference to the model evaluation measures
(Recall, Precision, Informedness, Markedness and
their derivatives), or should the model be set to
maximize the significance of the results?
This raises the question of how our measures of
association and accuracy, Informedness, Markedness
and Correlation, relate to standard measures of
significance.
This article has been written in the context of a
Prevailing methodology in Computational Linguistics
and Information Retrieval that concentrates on target
positive cases and ignores the negative case for the
purpose of both measures of association and
significance. A classic example is saying “water” can
only be a noun because the system is inadequate to
the task of Part of Speech identification and this
boosts Recall and hence F-factor, or at least setting
the Bias to nouns close to 1, and the Inverse Bias to
verbs close to 0. Of course, Bookmaker will then be
0 and Markedness unstable (undefined, and very
sensitive to any words that do actually get labelled
verbs). We would hope that significance would also
Figure 2. Accuracy of traditional measures.
110 Monte Carlo simulations with 11 stepped
expected Informedness levels (red) with Bookmaker-
estimated Informedness (red dot), Markedness (green
dot) and Correlation (blue dot), and showing (dashed)
Kappa versus the biased traditional measures Rank
Weighted Average (Wav), Geometric Mean (Gav) and
Harmonic Mean F1 (Fav). The Determinant (D) and
Evenness k-th roots (gR=PrevG and gP=BiasP) are
shown +1. K=4, N=128.
(Online version has figures in colour.)
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Copyright © 2011 Bioinfo Publications 47
be 0 (or near zero given only a relatively small
number of verb labels). We would also like to be able
to calculate significance based on the positive case
alone, as either the full negative information is
unavailable, or it is not labelled.
Generally when dealing with contingency tables it is
assumed that unused labels or unrepresented
classes are dropped from the table, with
corresponding reduction of degrees of freedom. For
simplicity we have assumed that the margins are all
non-zero, but the freedoms are there whether they
are used or not, so we will not reduce them or reduce
the table.
There are several schools of thought about
significance testing, but all agree on the utility of
calculating a p-value [19], by specifying some statistic
or exact test T(X) and setting p = Prob(T(X)
T(Data)). In our case, the Observed Data is
summarized in a contingency table and there are a
number of tests which can be used to evaluate the
significance of the contingency table.
For example, Fisher's exact test calculates the
proportion of contingency tables that are at least as
favourable to the Prediction/Marking hypothesis,
rather than the null hypothesis, and provides an
accurate estimate of the significance of the entire
contingency table without any constraints on the
values or distribution. The log-likelihood-based G2
test and Pearson's approximating χ2 tests are
compared against a Chi-Squared Distribution of
appropriate degree of freedom (r=1 for the binary
contingency table given the marginal counts are
known), and depend on assumptions about the
distribution, and may focus only on the Predicted
Positives.
χ2 captures the Total Squared Deviation relative to
expectation, is here calculated only in relation to
positive predictions as often only the overt prediction
is considered, and the implicit prediction of negative
case is ignored [17-19], noting that it sufficient to
count r=1 cells to determine the table and make a
significance estimate. However, χ2 is valid only for
reasonably sized contingencies (one rule of thumb is
that the expectation for the smallest cell is at least 5,
and the Yates and Williams corrections will be
discussed in due course [18,19]):
χ2+P = (TP-ETP)2/ETP+(FP-EFP)2/EFP
= DTP2/ETP + DFP2/EFP
= 2DP2/EHP, EHP
= 2ETP·EFP/[ETP+EFP]
= 2N·dp2/ehp,ehp
= 2etp·efp/[etp+efp]
= 2N·dp2/[rh·pp]= N·dp2/PrevG2/Bias
= N·B2·EvennessR/Bias = N·r2P·PrevG2/Bias
(N+PN)·r2P·PrevG2 (Bias 1)
= (N+PN)·B2·EvennessR (30)
G2 captures Total Information Gain, being N times the
Average Information Gain in nats, otherwise known
as Mutual Information, which however is normally
expressed in bits. We will discuss this separately
under the General Case. We deal with G2 for positive
predictions in the case of small effect, that is dp
close to zero, showing that G2is twice as sensitive as
χ2 in this range.
G2+P/2=TP·ln(TP/ETP) + FP·ln(FP/EFP)
=TP·ln(1+DTP/ETP)+FP·ln(1+DFP/EFP)
TP·(DTP/ETP) + FP·(DFP/EFP)
= 2N·dp2/ehp
= 2N·dp2/[rh·pp]
= N·dp2/PrevG2/Bias
= N·B2·EvennessR/Bias
= N·r2P·PrevG2/Bias
(N+PN)·r2P·PrevG2 (Bias 1)
= (N+PN)·B2·EvennessR (31)
In fact χ2 is notoriously unreliable for small N and
small cell values, and G2 is to be preferred. The Yates
correction (applied only for cell values under 5) is to
subtract 0.5 from the absolute dp value for that cell
before squaring completing the calculation [17-19].
Our result (30-1) shows that χ2 and G2 significance of
the Informedness effect increases with N as
expected, but also with the square of Bookmaker, the
Evenness of Prevalence (EvennessR = PrevG2 =
Prev·(1Prev)) and the number of Predicted
Negatives (viz. with Inverse Bias)! This is as
expected. The more Informed the contingency
regarding positives, the less data will be needed to
reach significance. The more Biased the contingency
towards positives, the less significant each positive is
and the more data is needed to ensure significance.
The Bias-weighted average over all Predictions (here
for K=2 case: Positive and Negative) is simply
KN·B2·PrevG2 which gives us an estimate of the
significance without focussing on either case in
particular.
χ2
KB = 2N·dtp2/PrevG2
= 2N·rP2 ·PrevG2
= 2N·rP2 ·EvennessR
= 2N·B2·EvennessR (32)
Analogous formulae can be derived for the
significance of the Markedness effect for positive real
classes, noting that EvennessP = BiasG2 .
χ2
KM = 2N·dtp2/BiasG2
= 2N ·rR2 · BiasG2
= 2N·M2·EvennessP (33)
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
48 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
The Geometric Mean of these two overall estimates
for the full contingency table is
χ2
KBM = 2N·dtp2/PrevG·BiasG
= 2N·rP·rR ·PrevG·BiasG
= 2N·r2
G·EvennessG= 2Nρ2·EvennessG
= 2N·B·M ·EvennessG (34)
This is simply the total Sum of Squares Deviance
(SSD) accounted for by the correlation coefficient
BMG (22) over the N data points discounted by the
Global Evenness factor, being the squared Geometric
Mean of all four Positive and Negative Bias and
Prevalence terms (EvennessG= PrevG·BiasG). The
less even the Bias and Prevalence, the more data will
be required to achieve significance, the maximum
evenness value of 0.25 being achieved with both
even bias and even Prevalence. Note that for even
bias or Prevalence, the corresponding positive and
negative significance estimates match the global
estimate.
When χ2+P or G2+P is calculated for a specific label in
a dichotomous contingency table, it has one degree
of freedom for the purposes of assessment of
significance. The full table also has one degree of
freedom, and summing for goodness of fit over only
the positive prediction label will clearly lead to a lower
χ2 estimate than summing across the full table, and
while summing for only the negative label will often
give a similar result it will in general be different. Thus
the weighted arithmetic mean calculated by χ2
KB is
an expected value independent of the arbitrary choice
of which predictive variate is investigated. This is
used to see whether a hypothesized main effect (the
alternate hypothesis, HA) is borne out by a significant
difference from the usual distribution (the null
hypothesis, H0). Summing over the entire table
(rather than averaging of labels), is used for χ2 or G2
independence testing independent of any specific
alternate hypothesis [21], and can be expected to
achieve a χ2 estimate approximately twice that
achieved by the above estimates, effectively
cancelling out the Evenness term, and is thus far less
conservative (viz. it is more likely to satisfy p<α):
χ2
BM = N·r2
G= N·ρ2= N·φ2= N·B·M (35)
Note that this equates Pearson’s Rho, ρ, with the Phi
Correlation Coefficient, φ, which is defined in terms of
the Inertia φ2=χ2/N. We now have confirmed that not
only does a factor of N connects the full contingency
G2 to Mutual Information (MI), but it also normalizes
the full approximate χ2 contingency to
Matthews/Pearson (=BMG=Phi) Correlation, at least
for the dichotomous case. This tells us moreover, that
MI and Correlation are measuring essentially the
same thing, but MI and Phi do not tell us anything
about the direction of the correlation, but the sign of
Matthews or Pearson or BMG Correlation does (it is
the Biases and Prevalences that are multiplied and
squarerooted).
The individual or averaged goodness-of-fit estimates
are in general much more conservative than full
contingency table estimation of p by the Fisher Exact
Test, but the full independence estimate can over
inflate the statistic due to summation of more than
there are degrees of freedom. The conservativeness
has to do both with distributional assumptions of the
χ2 or G2 estimates that are only asymptotically valid
as well as the approximative nature of χ2 in particular.
Also note that α bounds the probability of the null
hypothesis, but 1-α is not a good estimate of the
probabilty of any specific alternate hypothesis. Based
on a Bayesian equal probability prior for the null
hypothesis (H0, e.g. B=M=0 as population effect) and
an unspecific one-tailed alternate hypothesis (HA, e.g.
the measured B and C as true population effect), we
can estimate new posterior probability estimates for
Type I (H0 rejection, Alpha(p)) and Type II (HA
rejection, Beta(p)) errors from the posthoc
likelihood estimation [22]:
L(p) =Alpha(p)/Beta(p)
e p log(p) (36)
Alpha(p) = 1/[1+1/L(p)] (37)
Beta(p) = 1/[1+L(p)] (38)
Confidence Intervals and Deviations
An alternative to significance estimation is confidence
estimation in the statistical rather than the data
mining sense. We noted earlier that selecting the
highest isocost line or maximizing AUC or Bookmaker
Informedness, B, is equivalent to minimizing
fpr+fnr=(1-B) or maximizing tpr+tnr=(1+B),
which maximizes the sum of normalized squared
deviations of B from chance, sseB=B2 (as is seen
geometrically from Fig. 1). Note that this contrasts
with minimizing the sum of squares distance from the
optimum which minimizes the relative sum of squared
normalized error of the aggregated contingency,
sseB=fpr2+fnr2. However, an alternate
definition calculating the sum of squared deviation
from optimum is as a normalization the square of the
minimum distance to the isocost of contingency,
sseB=(1-B)2.
This approach contrasts with the approach of
considering the error versus a specific null hypothesis
representing the expectation from margins.
Normalization is to the range [0,1] like |B| and
normalizes (due to similar triangles) all orientations of
the distance between isocosts (Fig. 1). With these
estimates the relative error is constant and the
relative size of confidence intervals around the null
and full hypotheses only depend on N as |B| and |1-
B| are already standardized measures of deviation
from null or full correlation respectively (σ/µ=1). Note
however that if the empirical value is 0 or 1, these
measures admit no error versus no information or full
information resp. If the theoretical value is B=0, then
Powers DMW
Copyright © 2011 Bioinfo Publications 49
a full ±1 error is possible, particularly in the discrete
low N case where it can be equilikely and will be more
likely than expected values that are fractional and
thus likely to become zeros. If the theoretical value is
B=1, then no variation is expected unless due to
measurement error. Thus |1-B| reflects the maximum
(low N) deviation in the absence of measurement
error.
The standard Confidence Interval is defined in terms
of the Standard Error, SE =[SSE/(N(N-1))]
=[sse/(N-1)]. It is usual to use a multiplier X of
around X=2 as, given the central limit theorem applies
and the distribution can be regarded as normal, a
multiplier of 1.96 corresponds to a confidence of 95%
that the true mean lies in the specified interval around
the estimated mean, viz. the probability that the
derived confidence interval will bound the true mean
is 0.95 and the test thus corresponds approximately
to a significance test with alpha=0.05 as the
probability of rejecting a correct null hypothesis, or a
power test with beta=0.05 as the probability of
rejecting a true full or partial correlation hypothesis. A
number of other distributions also approximate 95%
confidence at 2SE.
We specifically reject the more traditional approach
which assumes that both Prevalence and Bias are
fixed, defining margins which in turn define a specific
chance case rather than an isocost line representing
all chance cases we cannot assume that any
solution on an isocost line has greater error than any
other since all are by definition equivalent. The above
approach is thus argued to be appropriate for
Bookmaker and ROC statistics which are based on
the isocost concept, and reflects the fact that most
practical systems do not in fact preset the Bias or
match it to Prevalence, and indeed Prevalences in
early trials may be quite different from those in the
field.
he specific estimate of sse that we present for
alpha, the probability of the current estimate for B
occurring if the true Informedness is B=0,
issseB0=|1-B|=1, which is appropriate for testing
the null hypothesis, and thus for defining
unconventional error bars on B=0. Conversely,
sseB2=|B|=0, is appropriate for testing deviation
from the full hypothesis in the absence of
measurement error, whilst sseB2=|B|=1
conservatively allows for full range measurement
error, and thus defines unconventional error bars on
B=M=C=1.
In view of the fact that there is confusion between the
use of beta in relation to a specific full dependency
hypothesis, B=1 as we have just considered, and the
conventional definition of an arbitrary and unspecific
alternate contingent hypothesis, B0, we designate
the probability of incorrectly excluding the full
hypothesis by gamma, and propose three possible
related kinds of correction for the sse for beta:
some kind of mean of |B| and |1-B| (the unweighted
arithmetic mean is 1/2, the geometric mean is less
conservative and the harmonic mean least
conservative), the maximum or minimum (actually a
special case of the last, the maximum being
conservative and the minimum too low an
underestimate in general), or an asymmetric interval
that has one value on the null side and another on the
full side (a parameterized special case of the last that
corresponds to percentile-based usages like box
plots, being more appropriate to distributions that
cannot be assumed to be symmetric).
The sse means may be weighted or unweighted
and in particular a self-weighted arithmetic mean
gives our recommended definition, sseB1=1-
2|B|+2B2, whilst an unweighted geometric mean
gives sseB1=[|B|-B2] and an unweighted
harmonic mean gives sseB1=|B|-B2. All of these
are symmetric, with the weighted arithmetic mean
giving a minimum of 0.5 at B=±0.5 and a maximum of
1 at both B=0 and B=±1, contrasting maximally with
sseB0and sseB2resp in these neighbourhoods,
whilst the unweighted harmonic and geometric means
having their minimum of 0 at both B=0 and B=±1,
acting like sseB0and sseB2resp in these
neighbourhoods (which there evidence zero variance
around their assumed true values). The minimum at
B=±0.5 for the geometric mean is 0.5 and for the
harmonic mean, 0.25.
For this probabilistic |B| range, the weighted
arithmetic mean is never less than the arithmetic
mean and the geometric mean is never more than
the arithmetic mean. These relations demonstrate the
complementary nature of the weighted/arithmetic and
unweighted geometric means. The maxima at the
extremes is arguably more appropriate in relation to
power as intermediate results should calculate
squared deviations from a strictly intermediate
expectation based on the theoretical distribution, and
will thus be smaller on average if the theoretical
hypothesis holds, whilst providing emphasized
differentiation when near the null or full hypothesis.
The minima of 0 at the extremes are not very
appropriate in relation to significance versus the null
hypothesis due the expectation of a normal
distribution, but its power dual versus the full
hypothesis is appropriately a minimum as perfect
correlation admits no error distribution. Based on
Monte Carlo simulations, we have observed that
setting sseB1=sseB2=1-|B| as per the usual
convention is appropriately conservative on the
upside but a little broad on the downside, whilst the
weighted arithmetic mean, sseB1=1-2|B|+2B2, is
sufficiently conservative on the downside, but
unnecessarily conservative for high B.
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
50 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
Note that these two-tailed ranges are valid for
Bookmaker Informedness and Markedness that can
go positive or negative, but a one tailed test would be
appropriate for unsigned statistics or where a
particular direction of prediction is assumed as we
have for our contingency tables. In these cases a
smaller multiplier of 1.65 would suffice, however the
convention is to use the overlapping of the confidence
bars around the various hypotheses (although usually
the null is not explicitly represented).
Thus for any two hypotheses (including the null
hypothesis, or one from a different contingency table
or other experiment deriving from a different theory or
system) the traditional approach of checking that
1.95SE (or 2SE) error bars don’t overlap is rather
conservative (it is enough for the value to be outside
the range for a two-sided test), whilst checking
overlap of 1SE error bars is usually insufficiently
conservative given that the upper represents
beta<alpha. Where it is expected that one will be
better than the other, a 1.65SE error bar including the
mean for the other hypothesis is enough to indicate
significance (or power=1-beta) corresponding to
alpha (or beta) as desired.
The traditional calculation of error bars based on Sum
of Squared Error is closely related to the calculation
of Chi-Squared significance based on Total Squared
Deviation, and like it are not reliable when the
assumptions of normality are not approximated, and
in particular when the conditions for the central limit
theorem are not satisfied (e.g. N<12 or cell-count<5).
They are not appropriate for application to
probabilistic measures of association or error. This is
captured by the meeting of the X=2 error bars for the
full (sseB2) and null (sseB0) hypotheses at N=16
(expected count of only 4 per cell).
Here we have considered only the dichotomous case
but discuss confidence intervals further below, in
relation to the general case.
SIMPLE EXAMPLES
Bookmaker Informedness has been defined as the
Probability of an informed decision, and we have
shown identity with DeltaP' and WRAcc, and the
close relationship (10, 15) with ROC AUC. A system
that makes an informed (correct) decision for a target
condition with probability B, and guesses the
remainder of the time, will exhibit a Bookmaker
Informedness (DeltaP') of B and a Recall of
B·(1Prev) + Bias. Conversely a proposed marker
which is marked (correctly) for a target condition with
probability M, and according to chance the remainder
of the time, will exhibit a Markedness (DeltaP) of M
and a Precision of M·(1-Bias) + Prev. Precision and
Recall are thus biased by Prevalence and Bias, and
variation of system parameters can make them rise or
fall independently of Informedness and Markedness.
Accuracy is similarly dependent on Prevalence and
Bias:
2·(B·(1Prev)·Prev+Bias·Prev)+1-(Bias+Prev),
and Kappa has an additional problem of non-linearity
due to its complex denominator:
B·(1Prev)·Prev / (1-Bias·Prev-(Bias+Prev)/2).
It is thus useful to illustrate how each of these other
measures can run counter to an improvement in
overall system performance as captured by
Informedness. For the examples in Table 2 (for
N=100) all the other measure rise, some quite
considerably, but Bookmaker actually falls. Table 2
also illustrates the usage of the Bookmaker and
Markedness variants of the χ2 statistic versus the
standard formulation for the positive case, showing
also the full K class contingency version (for K=2 in
this case).
Note that under the distributional and approximative
assumptions for χ2 neither of these contingencies
differ sufficiently from chance at N=100 to be
Table 2. Binary contingency tables.Colour coding highlights example counts of correct (light green) and incorrect
(dark red) decisions with the resulting Bookmaker Informedness (B=WRacc=DeltaP'), Markedness (C=DeltaP),
Matthews Correlation (C), Recall, Precision, Rand Accuracy, Harmonic Mean of Recall and Precision (F=F1),
Geometric Mean of Recall and Precision (G), Cohen Kappa (κ),andχ2 calculated using Bookmaker (χ 2+P), Markedness
(χ 2+R) and standard (χ 2) methods across the positive prediction or condition only, as well as calculated across the
entire K=2 class contingency, all of which are designed to be referenced to alpha (α) according to the χ2 distribution,
with the latter more reliable due to taking into account all contingencies. Single-tailed threshold is shown for α =0.05.
68.0%
32.0%
χ2@α=0.05
3.85
76.0%
56
20
76
B
19.85%
Recall
82.35%
F
77.78%
χ2+P
1.13
χ2KB
1.72
24.0%
12
12
24
M
23.68%
Precision
73.68%
G
77.90%
χ2+R
1.61
χ2KM
2.05
68
32
100
C
21.68%
Rand Acc
68.00%
κ
21.26%
χ2
1.13
χ2KBM
1.87
60.0%
40.0%
χ2@α=0.05
3.85
42.0%
30
12
42
B
20.00%
Recall
50.00%
F
58.82%
χ2+P
2.29
χ2
KB
1.92
58.0%
30
28
58
M
19.70%
Precision
71.43%
G
59.76%
χ2
+R
2.22
χ2KM
1.89
60
40
100
C
19.85%
Rand Acc
58.00%
κ
18.60%
χ 2
2.29
χ2KBM
1.91
Powers DMW
Copyright © 2011 Bioinfo Publications 51
significant to the 0.05 level due to the low
Informedness Markedness and Correlation, however
doubling the performance of the system would suffice
to achieve significance at N=100 given the Evenness
specified by the Prevalences and/or Biases).
Moreover, even at the current performance levels the
Inverse (Negative) and Dual (Marking) Problems
show higher χ2 significance, approaching the 0.05
level in some instances (and far exceeding it for the
Inverse Dual). The KB variant gives a single
conservative significance level for the entire table,
sensitive only to the direction of proposed implication,
and is thus to be preferred over the standard versions
that depend on choice of condition.
Incidentally, the Fisher Exact Test shows significance
to the 0.05 level for both the examples in Table 2.
This corresponds to an assumption of a
hypergeometric distribution rather than normality
viz. all assignments of events to cells are assumed to
be equally likely given the marginal constraints (Bias
and Prevalence). However it is in appropriate given
the Bias and Prevalence are not specified by the
experimenter in advance of the experiment as is
assumed by the conditions of this test. This has also
been demonstrated empirically through Monte Carlo
simulation as discussed later. See [22] for a
comprehensive discussion on issues with significance
testing, as well as Monte Carlo simulations.
PRACTICAL CONSIDERATIONS
If we have a fixed size dataset, then it is arguably
sufficient to maximize the determinant of the
unnormalized contingency matrix, DT. However this
is not comparable across datasets of different sizes,
and we thus need to normalize for N, and hence
consider the determinant of the normalized
contingency matrix, dt. However, this value is still
influenced by both Bias and Prevalence.
In the case where two evaluators or systems are
being compared with no a priori preference, the
Correlation gives the correct normalization by their
respective Biases, and is to be preferred to Kappa.
In the case where an unimpeachable Gold Standard
is employed for evaluation of a system, the
appropriate normalization is for Prevalence or
Evenness of the real gold standard values, giving
Informedness. Since this is constant, optimizing
Informedness and optimizing dtare equivalent.
More generally, we can look not only at what
proposed solution best solves a problem, by
comparing Informedness, but which problem is most
usefully solved by a proposed system. In a medical
context, for example, it is usual to come up with
potentially useful medications or tests, and then
explore their effectiveness across a wide range of
complaints. In this case Markedness may be
appropriate for the comparison of performance across
different conditions.
Recall and Informedness, as biased and unbiased
variants of the same measure, are appropriate for
testing effectiveness relative to a set of conditions,
and the importance of Recall is being increasingly
recognized as having an important role in matching
human performance, for example in Word Alignment
for Machine Translation [1]. Precision and
Markedness, as biased and unbiased variants of the
same measure, are appropriate for testing
effectiveness relative to a set of predictions. This is
particularly appropriate where we do not have an
appropriate gold standard giving correct labels for
every case, and is the primary measure used in
Information Retrieval for this reason, as we cannot
know the full set of relevant documents for a query
and thus cannot calculate Recall.
However, in this latter case of an incompletely
characterized test set, we do not have a fully
specified contingency matrix and cannot apply any of
the other measures we have introduced. Rather,
whether for Information Retrieval or Medical Trials, it
is assumed that a test set is developed in which all
real labels are reliably (but not necessarily perfectly)
assigned. Note that in some domains, labels are
assigned reflecting different levels of assurance, but
this has lead to further confusion in relation to
possible measures and the effectiveness of the
techniques evaluated [1]. In Information Retrieval,
the labelling of a subset of relevant documents
selected by an initial collection of systems can lead to
relevant documents being labelled as irrelevant
because they were missed by the first generation
systems so for example systems are actually
penalized for improvements that lead to discovery of
relevant documents that do not contain all specified
query words. Thus here too, it is important to develop
test sets that of appropriate size, fully labelled, and
appropriate for the correct application of both
Informedness and Markedness, as unbiased versions
of Recall and Precision.
This Information Retrieval paradigm indeed provides
a good example for the understanding of the
Informedness and Markedness measures. Not only
can documents retrieved be assessed in terms of
prediction of relevance labels for a query using
Informedness, but queries can be assessed in terms
of their appropriateness for the desired documents
using Markedness, and the different kinds of search
tasks can be evaluated with the combination of the
two measures. The standard Information Retrieval
mantra that we do not need to find all relevant
documents (so that Recall or Informedness is not so
relevant) applies only where there are huge numbers
of documents containing the required information and
a small number can be expected to provide that
information with confidence. However another kind of
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
52 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
Document Retrieval task involves a specific and
rather small set of documents for which we need to
be confident that all or most of them have been found
(and so Recall or Informedness are especially
relevant). This is quite typical of literature review in a
specialized area, and may be complicated by new
developments being presented in quite different forms
by researchers who are coming at it from different
directions, if not different disciplinary backgrounds.
THE GENERAL CASE
So far we have examined only the binary case with
dichotomous Positive versus Negative classes and
labels.
It is beyond the scope of this article to consider the
continuous or multi-valued cases, although the
Matthews Correlation is a discretization of the
Pearson Correlation with its continuous-valued
assumption, and the Spearman Rank Correlation is
an alternate form applicable to arbitrary discrete value
(Likert) scales, and Tetrachoric Correlation is
available to estimate the correlation of an underlying
continuous scale [11]. If continuous measures
corresponding to Informedness and Markedness are
required due to the canonical nature of one of the
scales, the corresponding Regression Coefficients
are available.
It is however, useful in concluding this article to
consider briefly the generalization to the multi-class
case, and we will assume that both real classes and
predicted classes are categorized with K labels, and
again we will assume that each class is non-empty
unless explicitly allowed (this is because Precision is
ill-defined where there are no predictions of a label,
and Recall is ill-defined where there are no members
of a class).
Generalization of Association
Powers [4] derives Bookmaker Informedness (41)
analogously to Mutual Information & Conditional
Entropy (39-40) as a pointwise average across the
contingency cells, expressed in terms of label
probabilities PP(l), where PP(l) is the probability of
Prediction l, and label-conditioned class probabilities
PR(c|l) , where P
R(c|l) is the probability that the
Prediction labeled l is actually of Real class c, and in
particular PR(l|l) = Precision(l), and where we use the
delta functions as mathematical shorthands for
Boolean expressions interpreted algorithmically as in
C, with true expressions taking the value 1 and false
expressions 0, so that δ|c-l| (c = l) represents a Dirac
measure (limit as δ→0); |c-l| (c l) represents its
logical complement (1 if c l and 0 if c = l)).
MI(R||P) =l PP(l) c PR(c|l) [log(PR(c|l))/PR(c)] (39)
H(R|P) =l PP(l) c PR(c|l) [log(PR(c|l))] (40)
B(R|P) =l PP(l) c PR(c|l) [PP(l)/(PR(l) |c-l|)] (41)
We now define a binary dichotomy for each label l
with l and the corresponding c as the Positive cases
(and all other labels/classes grouped as the Negative
case). We next denote its Prevalence Prev(l) and its
dichotomous Bookmaker Informedness B(l), and so
can simplify (41) to
B(R|P) = l Prev(l) B(l) (42)
Analogously we define dichotomous Bias(c) and
Markedness(c) and derive
M(P|R) = c Bias(c) M(c) (43)
These formulations remain consistent with the
definition of Informedness as the probability of an
informed decision versus chance, and Markedness as
its dual. The Geometric Mean of multi-class
Informedness and Markedness would appear to give
us a new definition of Correlation, whose square
provides a well defined Coefficient of Determination.
Recall that the dichotomous forms of Markedness
(20) and Informedness (21) have the determinant of
the contingency matrix as common numerators, and
have denominators that relate only to the margins, to
Prevalence and Bias respectively. Correlation,
Markedness and Informedness are thus equal when
Prevalence = Bias. The dichotomous Correlation
Coefficient would thus appear to have three factors, a
common factor across Markedness and
Informedness, representing their conditional
dependence, and factors representing Evenness of
Bias (cancelled in Markedness) and Evenness of
Prevalence (cancelled in Informedness), each
representing a marginal independence.
In fact, Bookmaker Informedness can be driven
arbitrarily close to 0 whilst Markedness is driven
arbitrarily close to 1, demonstrating their
independence in this case Recall and Precision will
be driven to or close to 1. The arbitrarily close hedge
relates to our assumption that all predicted and real
classes are non-empty, although appropriate limits
could be defined to deal with the divide by zero
problems associated with these extreme cases.
Technically, Informedness and Markedness are
conditionally independent once the determinant
numerator is fixed, their values depend only on their
respective marginal denominators which can vary
independently. To the extent that they are
independent, the Coefficient of Determination acts as
the joint probability of mutual determination, but to the
extent that they are dependent, the Correlation
Coefficient itself acts as the joint probability of mutual
determination.
These conditions carry over to the definition of
Correlation in the multi-class case as the Geometric
Mean of Markedness and Informedness once all
numerators are fixed, the denominators demonstrate
marginal independence.
Powers DMW
Copyright © 2011 Bioinfo Publications 53
We now reformulate the Informedness and
Markedness measures in terms of the Determinant of
the Contingency and Evenness, generalizing (20-22).
In particular, we note that the definition of Evenness
in terms of the Geometric Mean or product of biases
or Prevalences is consistent with the formulation in
terms of the determinants DET and det
(generalizing dichotomous DP=DTP and dp=dtp)
and their geometric interpretation as the area of a
parallelogram in PN-space and its normalization to
ROC-space by the product of Prevalences, giving
Informedness, or conversely normalization to
Markedness by the product of biases. The
generalization of DET to a volume in high
dimensional PN-space and det to its normalization
by product of Prevalences or biases, is sufficient to
guarantee generalization of (20-22) to K classes by
reducing from KD to SSD so that BMG has the form
of a coefficient of proportionality of variance:
M [det / BiasGK]2/K
= det2/K / EvennessP+ (44)
B [det / PrevGK ]2/K
= det2/K / EvennessR+ (45)
BMG det2/K / [PrevG · BiasG]
= det2/K / EvennessG+ (46)
We have marked the Evenness terms in these
equations with a trailing plus to distinguish them from
other usages, and their definitions are clear from
comparison of the denominators. Note that the
Evenness terms for the generalized regressions (44-
45) are not Arithmetic Means but have the form of
Geometric Means. Furthermore, the dichotomous
case emerges for K=2 as expected. Empirically (Fig.
3), this generalization matches well near B=0 or B=1,
but fares less well in between the extremes,
suggesting a mismatched exponent in the heuristic
conversion of K dimensions to 2. Here we set up the
Monte Carlo simulation as follows: we define the
diagonal of a random perfect performance
contingency table with expected N entries using a
random uniform distribution, we define a random
chance level contingency table setting margins
independently using a random binormal distribution,
then distributing randomly across cells around their
expected values, we combine the two (perfect and
chance) random contingency tables with respective
weights I and (1-I), and finally increment or
decrement cells randomly to achieve cardinality N
which is the expected number but is not constrained
by the process for generating the random (perfect
and chance) matrices. This procedure was used to
ensure Informedness and Markedness estimates
retain a level of independence; otherwise they tend to
correlate very highly with overly uniform margins for
higher K and lower N (conditional independence is
lost once the margins are specified) and in particular
Informedness, Markedness, Correlation and Kappa
would always agree perfectly for either I=1 or
perfectly uniform margins. Note this use of
Informedness to define a target probability of an
informed decision followed by random inclusion or
deletion of cases when there is a mismatch versus
the expected number of instances N the preset
Informedness level is thus not a fixed preset
Informedness but a target level that permits jitter
around that level, and in particular will be an
Figure 3. Determinant-based estimates of correlation.
110 Monte Carlo simulations with 11 stepped expected
Informedness levels (red line) with Bookmaker-
estimated Informedness (red dots), Markedness (green
dot) and Correlation (blue dot), with significance (p+1)
calculated using G2, X2, and Fisher estimates, and
Correlation estimates calculated from the Determinant of
Contingency using two different exponents, 2/K (DB &
DM) and 1/[3K-2] (DBa and DMa). The difference
between the estimates is also shown.
Here K=4, N=128, X=1.96, α=β=0.05.
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
54 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
overestimate for the step I=1 (no negative counts
possible) which can be detected by excess deviation
beyond the set Confidence Intervals for high
Informedness steps.
In Fig. 3 we therefore show and compare an alternate
exponent of 1/(3K-2) rather than the exponent of 2/K
shown in (44 to 45). This also reduces to 1 and
hence the expected exact correspondence for K=2.
This suggests that what is important is not just the
number of dimensions, but the also the number of
marginal degrees of freedom: K+2(K-1), but
although it matches well for high degrees of
association it shows similar error at low informedness.
The precise relationship between Determinant and
Correlation, Informedness and Markedness for the
general case remains a matter for further
investigation. We however continue with the use of
the approximation based on 2/K.
The EvennessR (Prev.IPrev) concept corresponds to
the concept of Odds (IPrev/Prev), where
Prev+IPrev=1, and Powers [4] shows that (multi-
class) Bookmaker Informedness corresponds to the
expected return per bet made with a fair Bookmaker
(hence the name). From the perspective of a given
bet (prediction), the return increases as the
probability of winning decreases, which means that
an increase in the number of other winners can
increase the return for a bet on a given horse
(predicting a particular class) through changing the
Prevalences and thus EvennessR and the Odds. The
overall return can thus increase irrespective of the
success of bets in relation to those new wins. In
practice, we normally assume that we are making our
predictions on the basis of fixed (but not necessarily
known) Prevalences which may be estimated a priori
(from past data) or post hoc (from the experimental
data itself), and for our purposes are assumed to be
estimated from the contingency table.
Generalization of Significance
In relation to Significance, the single class χ+P2 and
G+P2 definitions both can be formulated in terms of
cell counts and a function of ratios, and would
normally be summed over at least (K1)2 cells of a K-
class contingency table with (K1)2 degrees of
freedom to produce a statistic for the table as a
whole. However, these statistics are not independent
of which variables are selected for evaluation or
summation, and the p-values obtained are thus quite
misleading, and for highly skewed distributions (in
terms of Bias or Prevalence) can be outlandishly
incorrect. If we sum log-likelihood (31) over all K2
cells we get N·MI(R||P) which is invariant over
Inverses and Duals.
The analogous Prevalence-weighted multi-class
statistic generalized from the Bookmaker
Informedness form of the Significance statistic, and
the Bias-weighted statistic generalized from the
Markedness form, extend Eqns 32-34 to the K>2
case by probability-weighted summation (this is a
weighted Arithmetic Mean of the individual cases
targeted to r=K-1 degree of freedom):
χ2KB = KN·B2·EvennessR– (47)
χ2KM = KN·M2·EvennessP– (48)
χ2KBM= KN·B·M·EvennessG– (49)
For K=2 and r=1, the Evenness terms were the
product of two complementary Prevalence or Bias
terms in both the Bookmaker derivations and the
Significance Derivations, and (30) derived a single
multiplicative Evenness factor from a squared
Evenness factor in the numerator deriving from
dtp2, and a single Evenness factor in the
denominator. We will discuss both these Evenness
terms in the a later section. We have marked the
Evenness terms in (47-49) with a trailing minus to
distinguish them from forms used in (20-22,44-46).
One specific issue with the goodness-of-fit approach
applied to K-class contingency tables relates to the
up to (K1)2 degrees of freedom, which we focus on
now. The assumption of independence of the counts
in (K1)2 of the cells is appropriate for testing the null
hypothesis, H0, and the calculation versus alpha,
but is patently not the case when the cells are
generated by K condition variables and K prediction
variables that mirror them. Thus a correction is in
order for the calculation of beta for some specific
alternate hypothesis HA or to examine the significance
of the difference between two specific hypotheses HA
and HB which may have some lesser degree of
difference.
Whilst many corrections are possible, in this case
correcting the degrees of freedom directly seems
appropriate and whilst using r = (K1)2 degrees of
freedom is appropriate for alpha, using r = K1
degrees of freedom is suggested for beta under the
conditions where significance is worth testing, given
the association (mirroring) between the variables is
almost complete. In testing against beta, as a
threshold on the probability that a specific alternate
hypothesis of the tested association being valid
should be rejected. The difference in a χ2 statistic
between two systems (r = K1) can thus be tested
for significance as part of comparing two systems (the
Correlation-based statistics are recommended in this
case). The approach can also compare a system
against a model with specified Informedness (or
Markedness). Two special cases are relevant here,
H0, the null hypothesis corresponding to null
Informedness (B = 0: testing alpha with r =
(K1)2), and H1, the full hypothesis corresponding to
full Informedness (B = 1: testing beta with r =
K1).
Equations 47-49 are proposed for interpretation under
r = K1 degrees of freedom (plus noise) and are
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hypothesized to be more accurate for investigating
the probability of the alternate hypothesis in question,
HA (beta).
Equations 50-52 are derived by summing over the
(K1) complements of each class and label before
applying the Prevalence or bias weighted sum across
all predictions and conditions. These measures are
thus applicable for interpretation under r = (K1)2
degrees of freedom (plus biases) and are
theoretically more accurate for estimating the
probability of the null hypothesis H0 (alpha). In
practice, the difference should always be slight (as
the cumulative density function of the gamma
distribution χ2 is locally near linear in r see Fig. 4)
reflecting the usual assumption that alpha and
beta may be calculated from the same distribution.
Note that there is no difference in either the formulae
nor r when K=2.
χ2XB = K(K1)·N·B2·EvennessR– (50)
χ2XBM = K(K1)·N·B·M·EvennessG– (52)
Equations 53-55 are applicable to naïve unweighted
summation over the entire contingency table, but also
correspond to the independence test with r = (K1)2
degrees of freedom, as well as slightly
underestimating but asymptotically approximating the
case where Evenness is maximum in (50-52) at
1/K2. When the contingency table is uneven,
Evenness factors will be lower and a more
conservative p-value will result from (50-52), whilst
summing naively across all cells (53-55) they can
lead to inflated statistics and underestimated p-
values. However, they are the equations that
correspond to common usage of the χ2 and G2
statistics as well as giving rise implicitly to Cramer’s V
= [χ2/N(K-1)]1/2 as the corresponding estimate of
the Pearson correlation coefficient, ρ, so that
Cramer’s V is thus also likely to be inflated as an
estimate of association where Evenness is low. We
however, note these, consistent with the usual
conventions, as our definitions of the conventional
forms of the χ2 statistics applied to the multiclass
generalizations of the Bookmaker
accuracy/association measures:
χ2B = (K1)·N·B2 (53)
χ2M = (K1)·N·M2 (54)
χ2BM = (K1)·N·B·M (55)
Note that Cramer’s V calculated from standard full
Figure 4. Chi-squared against degrees of freedom cumulative density isocontours
(relative to α = 0.05: cyan/yellow boundary of p/α=1=1E0)
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56 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
contingency χ2 and G2 estimates tends vastly
overestimate the level of association as measured by
Bookmaker and Markedness or constructed
empirically. It is also important to note that the full
matrix significance estimates (and hence Cramer’s V
and similar estimates from these χ2 statistics) are
independent of the permutations of predicted labels
(or real classes) assigned to the contingency tables,
and that in order to give such an independent
estimate using the above family of Bookmaker
statistics, it is essential that the optimal assignment of
labels is made perverse solutions with suboptimal
allocations of labels will underestimate the
significance of the contingency table as they clearly
do take into account what one is trying to
demonstrate and how well we are achieving that goal.
The empirical observation concerning Cramer’s V
suggests that the strict probabilistic interpretation of
the multiclass generalized Informedness and
Markedness measures (probability of an informed or
marked decision), is not reflected by the traditional
correlation measures, the squared correlation being a
coefficient of proportionate determination of variance
and that outside of the 2D case where they match up
with BMG, we do not know how to interpret them as a
probability. However, we also note that Informedness
and Markedness tend to correlate and are at most
conditionally independent (given any one cell, e.g
given tp), so that their product cannot necessarily be
interpreted as a joint probability (they are
conditionally dependent given a margin, viz.
prevalence rp or bias pp: specifying one of B or M
now constrains the other; setting bias=prevalence, as
a common heuristic learning constraint, maximizes
correlation at BMG=B=M).
We note further that we have not considered a
tetrachoric correlation, which estimates the
regression of assumed underlying continuous
variables to allow calculation of their Pearson
Correlation.
Sketch Proof of General Chi-squared Test
The traditional χ2 statistic sums over a number of
terms specified by r degrees of freedom, stopping
once dependency emerges. The G2 statistic derives
from a log-likelihood analysis which is also
approximated, but less reliably, by the χ2 statistic. In
both cases, the variates are assumed to be
asymptotically normal and are expected to be
normalized to mean µ=0, standard deviation σ=1, and
both the Pearson and Matthews correlation and the
χ2 and G
2 significance statistics implicitly perform
such a normalization. However, this leads to
significance statistics that vary according to which
term is in focus if we sum over r rather than K2. In
the binary dichotomous case, it makes sense to sum
over only the condition of primary focus, but in the
general case it involves leaving out one case (label
and class). By the Central Limit Theorem, summing
over (K-1)2 such independent z-scores gives us a
normal distribution with σ=(K-1).
We define a single case χ2+lP from the χ2+P (30)
calculated for label l = class c as the positive
dichotomous case. We next sum over these for all
labels other than our target c to get a (K-1)2 degree
of freedom estimate χ2-lXP given by
χ2-lXP = cl χ2+lP= c χ2+cPχ2+lP (56)
We then perform a Bias(l) weighted sum over χ2-lXP
to achieve our label independent (K-1)2 degree of
freedom estimate χ2XB as follows (substituting from
equation 30 then 39):
χ2XB =lBias(l) · [N·B2·EvennessR(l)/Bias(l) – χ2+lP]
=K · χ2KBχ2KB= (K-1) · χ2KB
=K(K-1) ·N·B2·EvennessR (57)
This proves the Informedness form of the generalized
(K-1)2 degree of freedom χ2 statistic (42), and
defines EvennessR as the Arithmetic Mean of the
individual dichotomous EvennessR(l) terms
(assuming B is constant). The Markedness form of
the statistic (43) follows by analogous (Dual)
argument, and the Correlation form (44) is simply the
Geometric Mean of these two forms. Note however
that this proof assumes that B is constant across all
labels, and that assuming the determinant det is
constant leads to a derivative of (20-21) involving a
Harmonic Mean of Evenness as discussed in the next
section.
The simplified (K-1) degree of freedom χ2K statistics
were motivated as weighted averages of the
dichotomous statistics, but can also be seen to
approximate the χ2X statistics given the observation
that for a rejection threshold on the null hypothesis
H0, alpha< 0.05, the χ2 cumulative isodensity lines
are locally linear in r (Fig. 4). Testing differences
within a beta threshold as discussed above, is
appropriate using the χ2K series of statistics since
they are postulated to have (K-1) degrees of
freedom. Alternately they may be tested according to
the χ2X series of statistics given they are postulated to
differ in (K-1)2 degrees of freedom, namely the
noise, artefact and error terms that make the cells
different between the two hypotheses (viz. that
contribute to decorrelation). In practice, when used to
test two systems or models other than the null, the
models should be in a sufficiently linear part of the
isodensity contour to be insensitive to the choice of
statistic and the assumptions about degrees of
freedom. When tested against the null model, a
relatively constant error term can be expected to be
introduced by using the lower degree of freedom
model. The error introduced by the Cramer’s V (K-1
degree of freedom) approximation to significance
from G2 or χ2 can be viewed in two ways. If we start
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with a G2 or χ2 estimate as intended by Cramer we
can test the accuracy of the estimate versus the true
correlation, markedness and informedness as
illustrated in Fig. 5. Note that we can see here that
Cramer’s V underestimates association for high levels
of informedness, whilst it is reasonably accurate for
lower levels. If we use (53) to (55) to estimate
significance from the empirical association measures,
we will thus underestimate significance under
conditions of high association viz. it the test is more
conservative as the magnitude of the effect
increases.
Generalization of Evenness
The proof that the product of dichotomous Evenness
factors is the appropriate generalization in relation to
the multiclass definition of Bookmaker Informedness
and Markedness does not imply that it is an
appropriate generalization of the dichotomous usage
of Evenness in relation to Significance, and we have
seen that the Arithmetic rather the Geometric Mean
emerged in the above sketch proof. Whilst in general
one would assume that Arithmetic and Harmonic
Means approximate the Geometric Mean, we argue
that the latter is the more appropriate basis, and
indeed one may note that it not only approximates the
Geometric Mean of the other two means, but is much
more stable as the Arithmetic and Harmonic means
can diverge radically from it in very uneven situations,
and increasingly with higher dimensionality. On the
other hand, the Arithmetic Mean is insensitive to
evenness and is thus appropriate as a baseline in
determining evenness. Thus the ratios between the
means, as well as between the Geometric Mean and
the geometric mean of the Arithmetic and Harmonic
means, give rise to good measures of evenness.
On geometric grounds we introduced the Determinant
of Correlation, det, generalizing dp, and
representing the volume of possible deviations from
chance covered by the target system and its
perversions, showing its normalization to and
Informedness-like statistic is EvennessP+ the product
of the Prevalences (and is exactly Informedness for
K=2). This gives rise to an alternative dichotomous
formulation for the aggregate false positive error for
an individual case in terms of the K-1 negative
cases, using a ratio or submatrix determinant to
submatrix product of Prevalences. This can be
extended to all K cases while reflecting K-1 degrees
of freedom, by extending to the full contingency
matrix determinant, det, and the full product of
Prevalences, as our definition of another form of
Evenness, EvennessR# being the Harmonic Mean of
the dichotomous Evenness terms for constant
determinant:
χ2KB = KN·det2/K / EvennessR# (58)
χ2KM = KN·det2/K / EvennessP# (59)
χ2KBM = KN·det2/K / EvennessG# (60)
Recall that the + form of Evenness is exemplified by
EvennessR+ = [ΠlPrev(l)]2/K =PrevG (61)
Figure 5. Illustration of significance and Cramer’s V.
110Monte Carlo simulations with 11 stepped expected
Informedness (red) levels with Bookmaker-
estimated Informedness (red dots), Markedness (green
dot) and Correlation (blue dot), with significance (p+1)
calculated using G2, X2, and Fisher estimates, and
(skewed) Cramer’s V Correlation estimates calculated
from both G2 and X2. Here K=4, N=128, X=1.96,
α=β=0.05.
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
58 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
and that the relationship between the three forms of
Evenness is of the form
EvennessR– = EvennessR+/ EvennessR# (62)
where the + form is defined as the squared Geometric
Mean (44-46), again suggesting that the form is
best approximated as an Arithmetic Mean (47-49).
The above division by the Harmonic Mean is
reminiscent of the Williams’ correction which divides
the G2 values by an Evenness-like term q=1+(a2-
1)/6Nr where a is the number of categories for a
goodness-of-fit test, K [18-20] or more generally,
K/PrevH [17] which has maximum K when
Prevalence is even, and r=K-1 degrees of freedom,
but for the more relevant usage as an independence
test on a complete contingency table with r=(K-
1)2 degrees of freedom it is given by a2-
1=(K/PrevH-1)·(K/BiasH-1) where PrevH
and BiasH are the Harmonic Means across the K
classes or labels respectively [17-23].
In practice, any reasonable excursion from Evenness
will be reflected adequately by any of the means
discussed, however it is important to recognize that
the + form is actually a squared Geometric Mean and
is the product of the other two forms as shown in (62).
An uneven bias or Prevalence will reduce all the
corresponding Evenness forms, and compensate
against reduced measures of association and
significance due to lowered determinants.
Whereas broad assumptions and gross accuracy
within an order of magnitude may be acceptable for
calculating significance tests and p-values [23], it is
clearly not appropriate for estimate the strength of
associations. Thus the basic idea of Cramer’s V is
flawed given the rough assumptions and substantial
errors associated with significance tests. It is thus
better to start with a good measure of association,
and use analogous formulae to estimate significance
or confidence.
Generalization of Confidence
The discussion of confidence generalizes directly to
the general case, with the approximation using
Bookmaker Informedness1, or analogously
Markedness, applying directly (the Informedness form
is again a Prevalence weighted sum, in this case of a
sum of squared versus absolute errors), viz.
CIB2= X · [1-|B|] / [2 E· (N-1)] (63)
CIM2= X · [1-|B|] / [2 E· (N-1)] (64)
CIC2= X · [1-|B|] / [ 2 E· (N-1)] (65)
1 Informedness may be dichotomous and relates in this
form to DeltaP, WRacc and the Gini Coefficient as
discussed below. Bookmaker Informedness refers to the
polychotomous generalization based on the Bookmaker
analogy and algorithm [4].
In Equations 63-65 Confidence Intervals derived from
the sse estimates of §2.8 are subscripted to show
those appropriate to the different measures of
association (Bookmaker Informedness, B;
Markedness, M, and their geometric mean as a
symmetric measure of Correlation, C). Those shown
relate to beta (the empirical hypothesis based on
the calculated B, giving rise to a test of power), but
are also appropriate both for significance testing the
null hypothesis (B=0) and provide tight (0-width)
bounds on the full correlation (B=1) hypothesis as
appropriate to its signification of an absence of
random variation and hence 100% power (and
extending this to include measurement error,
discretization error, etc.)
The numeric subscript is 2 as notwithstanding the
different assumptions behind the calculation of the
confidence intervals (0 for the null hypothesis
corresponding to alpha=0.05, 1 for the alternate
hypothesis corresponding to beta=0.05 based on
the weighted arithmetic model, and 2 for the full
correlation hypothesis corresponding to
gamma=0.05 for practical purposes it is reasonable
to use |1-B| to define the basic confidence interval
for CIB0, CIB1 and CIB2, given variation is due solely
to unknown factors other than measurement and
discretization error. Note that all error, of whatsoever
kind, will lead to empirical estimates B<1.
If the empirical (CIB1) confidence intervals include
B=1, the broad confidence intervals (CIB2) around a
theoretical expectation of B=1 would also include the
empirical contingency it is a matter of judgement
based on an understanding of contributing error
whether the hypothesis B=1 is supported given non-
zero error. In general B=1 should be achieved
empirically for a true correlation unless there are
measurement or labelling errors that are excluded
from the informedness model, since B<1 is always
significantly different from B=1 by definition (1-B=0
unaccounted variance due to guessing).
None of the traditional confidence or significance
measures fully account for discretization error (N<8K)
or for the distribution of margins, which are ignored by
traditional approaches. To deal with discretization
error we can adopt an sse estimate that is either
constant independent of B, such as the unweighted
arithmetic mean, or a non-trivial function that is non-
zero at both B=0 and B=1, such as the weighted
arithmetic mean which leads to:
CIB1= X ·[1-2|B|+2B2] / [2 E· (N-1)] (66)
CIM1= X · [1-2|B|+2B2] / [2 E· (N-1)] (67)
CIC1= X ·[1-2|B|+2B2] / [ 2 E· (N-1)] (68)
Substituting B=0 and B=1 into this gives equivalent
CIs for the null and full hypothesis. In fact it is
sufficient to use the B=0 and 1 confidence intervals
based on this variant since for X=2 they overlap at
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N<16. We illustrate such a marginal significance case
in Fig. 6, where the large difference between the
significance estimates is clear with Fisher showing
marginal significance or better almost everywhere, G2
for B>~0.6, χ2 for B>~0.8. >~95% of Bookmaker
estimates are within the confidence bands as
required (with 100% bounded by the more
conservative lower band), however our B=0 and B=1
confidence intervals almost meet showing that we
cannot distinguish intermediate B values other than
B=0.5 which is marginal. Viz. we can say that this
data seems to be random (B<0.5) or informed
(B>0.5), but cannot be specific about the level of
informedness for this small N (except for
B=0.5±0.25).
If there is a mismatch of the marginal weights
between the respective prevalences and biases, this
is taken to contravene our assumption that
Bookmaker statistics are calculated for the optimal
assignment of class labels. Thus we assume that
any mismatch is one of evenness only, and thus we
set the Evenness factor E=PrevG*BiasG*K2. Note
that the difference between Informedness and
Markedness also relates to Evenness, but
Markedness values are likely to lie outside bounds
attached to Informedness with probability greater than
the specified beta. Our model can thus take into
account distribution of margins provided the optimal
allocation of predictions to categories (labelling) is
assigned.
The multiplier X shown is set from the appropriate
(inverse cumulative) Normal or Poisson distribution,
and under the two-tailed form of the hypothesis,
X=1.96 gives alpha, beta and gamma of 0.05.
A multiplier of X=1.65 is appropriate for a one-tailed
hypotheses at 0.05 level. Significance of difference
from another model is satisfied to the specified level if
the specified model (including null or full) does not lie
in the confidence interval of the alternate model.
Power is adequate to the specified level if the
alternate model does not lie in the confidence interval
of the specified model. Figure 7 further illustrates the
effectiveness of the 95% empirical and theoretical
confidence bounds in relation to the significance
achievable at N=128 (K=5).
EXPLORATION AND FUTURE WORK
Powers Bookmaker Informedness has been used
extensively by proponent and his students over the
last 10 years, in particular in the PhD Theses and
other publications relating to AudioVisual Speech
Recognition [25-26] and EEG/Brain Computer
Interface [27-28], plus Matlab scripts that are
available for calculating both the standard and
Bookmaker statistics2 (these were modified by the
present author to produce the results presented in
this paper). The connection with DeltaP was noted in
the course of collaborative research in
Psycholinguistics, and provides an important
2 http://www.mathworks.com/matlabcentral/fileexchange/
5648-informedness-of-a-contingency-matrix
Figure 6. Illustration of significance and confidence.
110 Monte Carlo simulations with 11 stepped
expected Informedness levels (red line) with
Bookmaker-estimated Informedness (red dots),
Markedness (green dot) and Correlation (blue dot),
with significance (p+1) calculated using G2, X2, and
Fisher estimates, and confidence bands shown for
both the theoretical Informedness and the B=0 and
B=1 levels (parallel almost meeting at B=0.5). The
lower theoretical band is calculated twice, using both
CIB1andCIB2. Here K=4, N=16, X=1.96, α=β=0.05.
Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation
60 Journal of Machine Learning Technology
ISSN: 2229-3981 & ISSN: 2229-399X, Volume 2, Issue 1, 2011
Psychological justification or confirmation of the
measure where biological plausibility is desired. We
have referred extensively to the equivalence of
Bookmaker Informedness to ROC AUC, as used
standardly in Medicine, although AUC has the
apparent form of an undemeaned probability based
on a parameterized classifier or a series of classifiers,
and B is a demeaned renormalized kappa-like form
based on a single fully specified classifier.
The Informedness measure has thus proven its worth
across a wide range of disciplines, at least in its
dichotomous form. A particular feature of the major
PhD studies that used Informedness, is that they
covered different numbers of classes (exercising the
multi-class form of Bookmaker as implemented in
Matlab), as well as a number of different noise and
artefact conditions. Both of these aspects of their
work meant that the traditional measures and
derivatives of Recall, Precision and Accuracy were
useless for comparing the different runs and the
different conditions, whilst Bookmaker gave clear
unambiguous, easily interpretable results which were
contrasted with the traditional measures in these
studies.
The new χ2KB, χ2KM and χ2KBM,χ2XB, χ2XM and
χ2XBM correlation statistics were developed
heuristically with approximative sketch
proofs/arguments, and have only been investigated to
date in toy contrived situations and the Monte Carlo
simulations in Figs 2, 3, 5, 6 and 7. In particular,
whilst they work well in the dichotomous state, where
they demonstrate a clear advantage over χ2
traditional approaches, there has as yet been no no
application to our multi-class experiments and no
major body of work comparing new and conventional
approaches to significance. Just as Bookmaker (or
DeltaP') is the normative measure of accuracy for a
system against a Gold Standard, so is χ2XB the
proposed χ2 significance statistic for this most
common situation in the absence of a more specific
model (noting that x = x2 for dichotomous data in
{0,1}). For the cross-rater or cross-system
comparison, where neither is normative, the BMG
Correlation is the appropriate measure, and
correspondingly we propose that χ2KBM is the
appropriate χ2 significance statistic. To explore these
thoroughly is a matter for future research. However,
in practice we tend to recommend the use of
Confidence Intervals as illustrated in Figs 4 and 5,
since these give a direct indication of power versus
the confidence interval on the null hypothesis, as well
as power when used with confidence intervals on an
alternate hypothesis.
Furthermore, when used on the empirical mean
(correlation, markedness or informedness), the
overlap of the interval with another system, and vice-
versa, give direct indication of both significance and
power of the difference between them. If a system
occurs in another confidence interval it is not
significantly different from that system or hypothesis,
and if it is it is significantly different. If its own
confidence interval also avoids overlapping the
alternate mean this mutual significance is actually a
Figure 7. Illustration of significance and confidence.
110 Monte Carlo simulations with 11 stepped expected
Informedness levels (red line) with Bookmaker-
estimated Informedness (red dots), Markedness (green
dot) and Correlation (blue dot), with significance (p+1)
calculated using G2, X2, and Fisher estimates, and
confidence bands shown for both the theoretical
Informedness and the B=0 and B=1 levels (parallel
almost meeting at B=0.5). The lower theoretical band
is calculated twice, using both CIB1andCIB2. Here K=5,
N=128, X=1.96, ==0.05.
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reflection of statistical power at a complementary
level. However, as with significance tests, it is
important to avoid reading to avoid too much into
non-overlap of interval and mean (not of intervals) as
the actual probabilities of the hypotheses depends
also on unknown priors.
Thus whilst our understanding of Informedness and
Markedness as performance measure is now quite
mature, particularly in view of t