Li S.Z., Jain A.K. (eds.) Encyclopedia of Biometrics

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Speech Parametrization

▶ Speech Analysis

Speech Processing

Speech processing is a technology that operates on the

stream of speech.

▶ Speaker Recognition, Standardization

Speech Production

LAURA DOCIO-FERNAND EZ,CARMEN GARCIA-MATEO

University of Vigo, Vigo, Spain

Synonyms

Speech system; Sound generation

Definition

Speech production is the process of uttering articulated

sounds or words, i.e., how humans generate meaningful

speech. It is a complex feedback process in which also

hearing, perception, and information processing in the

nervous system and the brain is involved.

Speaking is in essence the by-product of a necessar y

bodily process, the expulsion from the lungs of air

charged with carbon dioxide after it has fulﬁlled its

function in respiration. Most of the time one breathes

out silently; but it is possible, by contracting and relax-

ing the vocal tract to change the characteristics of the

air expelled from the lungs.

Introduction

Speech is one of the most natural forms of communi-

cation for human beings. Researchers in speech

technolog y are working on developing systems with

the ability to understand speech and speak with a

human being .

Human–computer interaction is a discipline con-

cerned with the design, evaluation, and implementation

of the most natural interactive computing systems for

human use [1]. Computers with this kind of ability are

gradually becoming a reality today, through the success

of speech synthesis, speech recognition, and other

related speech technologies. However, in order to give

them functions that are much closer to those of human

beings, one must learn more about the mechanisms by

which speech is produced and perceived, and develop

speech information processing technologies that make

use of these functions.

However, progress in advanced computer speech

interfaces is limited in part due to incomplete knowl-

edge of the physics of speech production. For compu-

ter generated speech output, this means limitations in

the naturalness and intelligibility of synthetic speech.

The generation of human speech involves a re-

markably complex process. In modeling the process

of human speech production one may recognize two

principal stages:

1. Formation in the mind of thoughts to be expressed

as well as the choice of words to be used. The

message is organized on the linguistic level and

structured grammatically and phonologically.

2. The string of phonemes is converted into a set of

continuous signals controlling the musculature of

the various articulators. This results in a highly

complex integrated movement sequence in which

generally participate all the articulators, the lips,

the tongue, the mandible, etc. Finally, the physical

interaction of the vibrating vocal cords and the

moving articulatory structure produces a continu-

ous acoustic signal perceived as speech.

Speech production is an activity embodied in a com-

plex physical system. It is produced by a cooperation of

lungs, glottis (with vocal cords), and articulation tract

(mouth and nose cavity). The speaker produces a

speech signal in the form of pressure waves that travel

from the speaker’s head to the listener’s ears. This

signal consists of variations in pressure as a function

of time and is usually measured directly in front of the

mouth, the primary sound source. The amplitude var-

iations correspond to deviations from atmospheric

pressure caused by traveling waves.

1290

Speech Parametrization

An audible speech signal is produced by moving

the vocal articulators to modify and ‘‘sculpt’’ the

source of sound energy in the vocal tract (e.g ., air

turbulence or vocal fold vibration). The signal is non-

stationary changing characteristics as the muscles of

the vocal tract contract and relax. Since spe aker wishes

to produce a

▶ sound sequence corresponding to the

message to be conveyed, most major vocal tract move-

ments have a voluntary basis. For each sound, there is a

positioning for each of the vocal tract articulators:

vocal cords, tongue, lips, teeth, velum, and jaw. Sounds

are typically divided into two broad classes: vowels,

which allow unrestricted airﬂow in the vocal tract;

and consonants, which restrict airﬂow at some point

and have a weaker intensity than vowels.

The most common sound generation sources are

quasi-periodic vibration of the vocal cords and tur bu-

lent noise generated by the passage of air through a

narrow constriction, usually in the oral cavity. More

rarely, sounds are generated by plosive release of air

(following the buildup of pressure behind an obstruc-

tion in the vocal tract), implosion (following the crea-

tion of a vacuum behind an obstruction in the vocal

tract), and clicks created by, the action of the tongue

pulling away from the roof of the mouth.

Depending on the type of excitation, i.e., sound

generation sources, two types of sounds are produced:

voiced and unvoiced sounds.

▶ Voiced sounds are pro-

duced by forcing air through the glottis or an op ening

between the vocal folds. Then the vocal folds vibrate,

they interrupt the air stream and produce a quasi-

periodic pressure wave that excite the vocal tract. An

example of voiced sound is the vowel ‘‘a’’ in cut, or ‘‘ee’’

in ‘‘beet.’’

▶ Unvoiced sounds are generated by form-

ing a constriction at some point along the vocal tract

and forcing air through the constriction to produce

turbulence. Vocal folds do not vibrate in this case. An

example of unvoiced sound is ‘‘s’’ as in ‘‘six.’’

The frequency at which vocal folds open and close

is called the fundamental frequency.

A major focus of speech production research is in

modeling articulatory–acoustic relationships of speech

sounds. Physically and physiologically based models

for speech acoustics are particularly important for

developing high-quality speech synthesis and low bit

rate (articulatory) coding. Signiﬁcant progress has

been recently made towards developing improved

articulatory-acoustic models.

The Human Speech Production

Mechanism

Figure 1 shows an illustration of the human speech

production system. The gross anatomical components

of the systems are the lungs, trachea, larynx (orga n

of speech production), pharyngeal cavity (throat),

buccal cavity (mouth), and nasal cavity (nose).

Speech Production. Figure 1 Schematic view of human speech production mechanism.

Speech Production

1291

The pharyngeal and buccal cavities are usually grouped

and referred to as the vocal tract, and the nasal cavity is

often called nasal tract. Accordingly, the vocal tract

begins at the output of the larynx, and terminates at

the input of the lips. The nasal tract begins at the velum

and ends at the nostrils of the nose. There are many

other anatomical components that contribute to the

production of speech, such as the vocal folds (or

cords), tongue, lips, teeth, and jaw. These are referred

to as articulators and move to different positions in

order to produce various speech sounds. Due to the

physical constraints of the vocal tract, the positions of

the articulators can only change slowly with time and

individual realizations of a phone are strongly inﬂu-

enced by previous and future phones in an utterance.

This phenomenon is known as coarticulation and is

important for both accurate speech analysis and natu-

ral speech production.

The speaker-speciﬁc characteristics of speech are

due to differences in physiological and behavio ral

aspects of the speech production system in humans.

The main physiological aspect of the human speech

production system is the vocal tract shape.

The process of human speech production can be

summarized as follows. While speaking, the air pushed

out from the lungs (the main energy source) travels

into the trachea, then up into the glottis, where it is

periodically interrupted by the movement of the vocal

cords. The tension of the vocal cords is adjusted by the

larynx so that the chords vibrate in an oscillatory

fashion, resulting in the production of voiced speech.

During unvoiced speech, constrictions within the vocal

tract (oral cavities – mouth, throat, etc.) force air

through the constriction to produce turbulence.

Speech production can be viewed as a ﬁltering

operation (source-ﬁlter model of speech production

[2]) in which the three main cavities of the speech

production system (vocal and nasal tracts) comprise

the main acoustic ﬁlter. The ﬁlter is excited by the

organs below it (

▶ glottal excitation), and is loaded at

its main output by a radiation impedance due to the

lips (

▶ lip-radiatio n effect). The articulators, most of

which are associated with the ﬁlter itself, are used to

change the properties of the system, its form of excita-

tion, and its output loading over time. Figure 2 shows

this model. The source which excites the ﬁlter may be

either periodic, resulting in voiced speech, or noisy and

aperiodic, causing unvoicing speech.

The basic assumption of the model is that the source

signal produced at the glottal level is linearly ﬁltered

Speech Production. Figure 2 Block diagram of human speech production system.

1292

Speech Production

through the vocal tract. The resulting sound is emit-

ted to the surrounding air through radiation loading

(lips). The model assumes that source and ﬁlter are

independent of each other. Although recent ﬁndings

show some interaction between the vocal tract and a

glottal source [3], Fant’s theor y of speech production

is still used as a framework for the descript ion of

the human voice.

An articulatory representation of the speech produc-

tion system has certain attractive properties which

might be exploited to help in modeling speech. Articu-

lation is the term used for all actions of the organs of the

vocal tract that effect modiﬁcations of the signal gener-

ated by the voice source. This modiﬁcation results in

speech events which can be identiﬁed as vowels, con-

sonants or other phonological units of a language.

Speech articulators move relatively slowly and smoothly,

and their movements are continuous. The mouth can-

not jump instantaneously from one conﬁguration to

a completely different one. Using speech production

knowledge could help to improve speech processing

methods by providing useful constraints. Suggested

applications include, for example, automatic speech

recognition, low bit-rate speech coding, speech analysis

and synthesis, and animated talking heads.

Modeling the Human Speech Production

System

Speech is transmitted between humans in the acoustic

domain, and fortunately, it can be easily measured and

recorded as acoustic representation. Underlyingly,

however, the acoustic speech signal is the product of

events in a speaker’s articulatory system, and there

has long been interest in ways to exploit the under-

lying articulator y information for speech techno-

logy. Increasingly detailed and sophisticated models

about how speech is generated in human speech

production system has been developed in the past

30 years [4].

The most relevant models of the speech production

mechanism belong to acoustic modeling, that is based

on the acoustic theory of speech production [5].

According to this theory, speech waveform is consid-

ered to be the output of a resonant network (namely

the vocal tract ﬁlter) that is excited by sound sources

placed at the glottis. The main sections of the speech

production mechanism, namely the voice source, vocal

tract, and radiation effects, are likely to be linearly

modeled in a noncoupled manner following a source-

ﬁlter arrangement. The assumption that the source and

the ﬁlter can be separately modeled probably holds for

most of the cases. However, this assumptio n is ques-

tionable for low frequencies, because the nonlinear

coupling may produce damping of the ﬁrst formant.

It is also disputable for unvoiced speech (excitation is

due to turbulence originating at constrictions on the

vocal tract itself).

As described in the previous section, the human

speech production can be illustrated by a simple source-

ﬁlter model (Fig. 3). Here the lungs are replaced by a

DC source, the vocal cords by an impulse generator,

and the articulation tract by a linear ﬁlter system. A

noise generator produces the unvoiced excitation. In

practice, all sounds have a mixed excitation, which

Speech Production. Figure 3 A general model for speech production.

Speech Production

1293

means that the excitation consists of voiced and un-

voiced portions. Of course, the relation between these

portions varies strongly with the sound being generated.

In general, the source-ﬁlter model is related to

linear prediction. Based on this model, a further sim-

pliﬁcation can be made ( Fig. 4). A ‘‘hard’’ switch which

only selects between voiced and unvoiced excitation is

used. The ﬁlter, representing the articulation tract, is a

simple recursive digital ﬁlter; its resonance behavior

(frequency response) is deﬁned by a set of ﬁlter coefﬁ-

cients. Since the computation of the coefﬁcients is based

on the mathematical optimization procedure of Linear

Prediction Coding, they are called Linear Prediction

Coding Coefﬁcients or LPC coefﬁcients and the com-

plete model is the so-called LPC Vocoder (Vocoder is a

concatenation of the terms ‘voice’ and ‘coding’) [6].

Speech can be modeled as the response of linear

time varying system with appropriate excitation. For

voiced speech, the excitation can be approximated by a

pulse train in which the pulses appear according to the

instantaneous pitch rate. If a single pitch period is

analyzed at a time, an analysis known as ‘‘pitch syn-

chronous analysis,’’ only one pulse occurs somewhere

in the period.

This model has a great advantage. Since the main

parameters of the speech production, namely the pitch

and the artic ulation characteristics, expressed by the

LPC coefﬁcients, are directly accessible, the audible

voice characteristics can be widely inﬂuenced. Also

the number of ﬁlter coefﬁcients can be varied to inﬂu-

ence the sound char acteristics, above all, the formant

characteristics.

Articulatory models of speech production mechan-

isms aim at modeling the physical, anatomical, and

physiological functioning of the organs involved in

human voice production. In this approach, the system

is modeled instead of the signal or its acoustic charac-

teristics. Modeling the process at the articulatory level

can be expected to be simpler because the articulators

respond to muscular forces with predictable changes in

their position and rates of movement.

It is worth noting that, globally, the speech signal is

a nonstationary signal. Then, all the systems in such

speech production models should be time-varying,

with their parameters changing in accordance to the

sound to be produced. The classic source-ﬁlter linear

model is satisfactory only as a ﬁrst approximation of

the overall nonlinear process of speech production,

and only for short time frames, on which the signal is

quasi-stationary.

The variability of the speech signal originates in

the speciﬁc dynamics of the articulatory apparatus.

It is well-known that the phon atory system is a time-

varying system, and consequently speech signal is

nonstationary. A large class of nonstationary and

nonlinear processes is involved in speech production.

Applications

To provide a compact computational model for speech

production that can be beneﬁcial to a wide range of

areas in speech signal processing.

The speech production models which have been

derived can be applied in almost all ﬁelds of speech

processing like speech synthesis, speech analysis, speech

and speaker recognition, and also in speech coding.

Speech Production. Figure 4 A simplified discrete time model for speech production.

1294

Speech Production

Summary

Decades of research have gone into the technical and

quantitative understanding of human speech produc-

tion mechanisms.

This communication focuses on the human speech

production mechanism. Speech is produced through

the careful movement and positioning of the vocal-

tract articulators in response to an excitation signal

that may be periodic at the glottis, or noiselike due to

a major constriction along the vocal tract, or a

combination.

Related Entries

▶ Biometrics, Overview

▶ Speaker Features

▶ Speech Analysis

References

1. Hewett, T., Baecker, R., Card, S., Carey, T., Gasen, J., Mantei, M.,

Perlman, G., Strong, G., Verplank, W.: Chapter 2: Human–

computer interaction. In: B. Heﬂey (ed.) ACM SIGCHI

Curricula for Human–Computer Interaction. ACM, (2007)

2. Fant, G.: Acoustic Theory of Speech Production, 1st edn. Mou-

ton, The Hague (1960)

3. Fant, G.: Glottal ﬂow: models and interaction. J. Phon. 14,

393–399 (1986)

4. Kent, R.D., Adams, S.G., Turner, G.S.: Models of speech produc-

tion. In: Principles of Experimental Phonetics, pp. 2–45. N.J.

Lass, Mosby (1996)

5. Burrows, T.L.: Speech Processing with Linear and Neural Net-

work Models (1996)

6. Deller, J.R., Proakis, j.G., Hansen, J.H.L.: Discrete-Time Proces-

sing of Speech Signals, 1st edn. Macmillan, New York (1993)

Speech Recognition

Speech recognition is a non-biometric technology that

enables a machine to recognize the words a speaker

speaks.

▶ Remote Authentication

▶ Speaker Recognition, Standardization

Speech Spectral Envelope

The speech spectral envelope is referred as an approxi-

mation to the frequency response of the vocal tract

obtained from the speech signal. The speech spectral

envelope contains the most discriminative information

available in speech to distinguish phonemes and

speakers.

▶ Speech Analysis

Speech Synthesis

▶ Voice Sample Synthesis

Speech System

▶ Speech Production

Speed

▶ Signature Recognition

SPME

SPME is a simple, solvent-free headspace extrac-

tion technique in which volatile and semi-volatile

organic compounds adhere to the ﬁber either by

adsorption or absorption depending on the ﬁber

type. The ﬁber itself is constructed out of a station-

ary phase-like material which is selected for use

based on the functionality of the substances present

for extraction.

▶ Odor Biometrics

SPME

1295

Spoofing

Spooﬁng entails the presentation of an artifact

designed to imitate a legitimate biometric so as to

defeat or circumvent a biometric system or process.

▶ Anti-spoofing

▶ Fraud Reduction, Overview

Spoofing Countermeasures

▶ Liveness Detection: Iris

Spoof-resistance

▶ Liveness Detection: Iris

Stand Off

The distance between a biometric capture device and

the subject from whom the biometric is captured.

▶ Iris Device

▶ Iris on the Move

Statistical Models

▶ Deformable Models

Statistical Signal Processing

▶ Fusion, Decision-Level

Steganography

Hiding a secret message within a larger, typically unre-

lated cover message suc h that unintended recipients do

not suspect the presence of the hidden message. In

steganography, the cover message ser ves only as a

mechanism to transport the secret me ssage; it may be

discarded or destroyed after the secret message has

been extracted and communicated to the intended

recipient. An example might include a spy posing as

a tourist sending government secrets via images of

tourist venues.

▶ Iris Digital Watermarking

Strain Gauge

Strain gauge is a device used to measure deformation

(strain) of an object. It is a sensor whose resistance

varies with applied force; it converts force, pressure,

tension, weig ht, etc., into a change in electrical resis-

tance which can then be measured.

▶ Digitizing Table t

Stream of Speech

Stream of speech is the ﬂow of sounds, words, and

utterances produced by a human speaker.

▶ Speaker Recognition Standardization

Strength of Voice Evidence

The strength of the forensic evidence of voice is the

result of the interpretation of the evidence, expressed

in terms of the likelihood ratio of two alternative

hypotheses: H

– the suspecte d speaker is the source

of the questioned recording, H

– the speaker at the

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Spoofing

origin of the questioned recording is not the suspected

speaker.

▶ Voice, Forensic Evidence of

Stroke

Single movement of a pen during the signing process.

The number of strokes and the stroke order are very

important features for on-line signature veriﬁcation.

Strokes are in general delimited by a change in the pen

up/down status.

▶ Signature Recognition

Structural and Functional Anatomy

▶ Anatomy of Face

Structural Model

Describes the fundamental structures that interlink

and constitute a body and speciﬁes their constraints:

e.g., torso and upper leg are linked by the hip, upper

leg and lower leg are linked by the knee, and the knee

constrains the degree of freedom of the lower leg.

Therefore, it also deﬁnes the dependency of one body

segment to another. Structural model is normally used

together with a motion model to guide feature (of

motion) extraction process. It may also consist of

information such as length, thickness, area, or volume.

▶ Gait Recognition, Model-Based

Structural Risk

The error associated with the nature of the classiﬁer.

Structural risk minimization tunes the complexity

of the classiﬁcation function in order to optimize

the generalization. The true risk combines structural

risk with empirical risk, which is dependent on the

training set.

▶ Support Vector Machine

Structure Tensor Field

The matrix ﬁeld obtained by outer products of the

gradients.

▶ Fingerprint Features

Structure-from-Motion

The process of recovering the three-dimensional struc-

ture of an object by the analysis of subsequent two-

dimensional images of the object in motion. Although

this term most commonly refers to the recovery of the

structure of a rigid object (i.e., the distances between

all points on the object remain constant), it still applies

to piecewise rigid objects such as the human body

where only some subsets of points remain rigid.

▶ Psychology of Gait and Action Recognition

Subject Interaction Time

▶ Operational Times

Super-Resolution

The techniques that form an enhanced-resolution

image by fusing together multiple low-resolution

Super-Resolution

1297

and/or learning from high-resolution training images

are known as super-resolution. Super-resolution can

be performed in either frequency or spatial domain.

▶ Face Sample Quality

Super-Resolution for Iris

▶ Iris Super-Resolution

Supervised

Supervised are the class labels of all data w hich are

known. Algorithms are designed for separation of

data of different classes by additionally using labeled

information.

▶ Linear Dimension Reduction

Supervised Learning

JONG KYOU NG KIM,KYE-HYEON KIM,SEUNGJIN CHOI

Department of Computer Science, Pohang University

of Science and Technology, Korea

Synonyms

Classiﬁcation

Introduction

Supervised learning is to ﬁnd a model for an unknown

target function, using a given set of training examples –

pairs of a data point and a target function value. The

fundamental assumption of learning by examples is

that similar data points tend to have similar function

values. In this chapter, learning tasks are focused

on classiﬁcation, in which the aim is to assign each

data point to one of a ﬁnite number of function

values, called class labels. Then, the assumption can

be restated that a group of similar data points form a

meaningful pattern , which corresponds to the same

class label.

Biometric recognition systems require two different

classiﬁcation problems of veriﬁcation and identiﬁca-

tion, depending on the application context [1]. In

the veriﬁcation problems, we need to verify a claimed

identity by comparing an input feature vector extrac-

ted from the biometric data with the corresponding

template feature set. The veriﬁcation problem can be

restated as binary classiﬁcation, in which the learned

classiﬁer takes the feature vector as an input and deter-

mines whether the claimed identity is true or false.

The goal of identiﬁcation problems, however, is to rec-

ognize a person’s identity by searching templates

corresponding to all the users enrolled in the biometric

database. The identiﬁcation problem may be stated

as classiﬁcation with K þ 1 classes fC

; ...; C

; C

Kþ1

where K is the number of the users enrolled in the

system and the last class C

Kþ1

indicates the unidenti-

ﬁed user.

A variety of approaches for classiﬁcation can be

grouped into parametric and nonparametric, based

on the assumptions about class-conditional densities

[2]. If the class-conditional densities are speciﬁed with

a functional form of distributions, which are charac-

terized completely with a ﬁnite number of parameters,

we have a parametric method for classiﬁcation. In

contrast, if we model the class-conditional densities

without any assumptions regarding the functional

form of the underlying distribution, non-parametric

algorithms are derived. The methods that directly con-

struct the decision boundaries without referring to the

class-conditional densities are also considered as non-

parametric approaches.

Parametric Approach

Suppose we have a D-dimensional real-valued feature

vector x ¼ (x

,...x

)

. The goal of classiﬁcation is to

predict the class label for a new value of x given a training

set of D. According to Bayesian decision theory, an

optimal decision boundary can be obtained from

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Super-Resolution for Iris

posterior probabilities for classes. Using Bayes’ theorem,

the posterior probabilities pðC

jxÞ can be written as

pðC

jxÞ¼

pðxjC

ÞpðC

pðxjC

ÞpðC

expða

; ð 1Þ

where pðxjC

Þ is the class-conditional density repre-

senting the probability of a feature vector x given class

, pðC

Þ is the prior probability of class C

, and a

deﬁned by

¼ ln pðxjC

ÞpðC

Þ: ð2Þ

In parametric approaches to classiﬁcation, we directly

model the class-conditional density with a parametric

form of probability distribution (e.g., multivariate

Gaussian). Many parametric methods for classiﬁcation

have been proposed based on different assumptions for

pðxjC

Þ [3, 4, 5] (see Table 1):

1. Linear discriminant analysis (LDA) uses Gaussian

distributions Nðm

; S

Þ for the class-conditional

densities with a common covariance matrix S

¼S.

2. Quadratic discriminant analysis (QDA) also uses

Gaussian distributions and does not assume equal

covariance matrices.

3. In a naive Bayes classiﬁer, we assume that the dis-

tributions of the feature values x

,...x

are condi-

tionally independent given the class label, pðxjC

Þ¼

i¼1

pðx

Þ. In parametric approach, the one-

dimensional densities pðx

Þ are usually Gaussian

for continuous features or multinomial for discrete

features.

The parameters of each class-conditional densities can

be estimated from a training set D with a maximum

likelihood approach.

Linear Discriminant Analysis

We assume that all class-conditional densities pðxjC

are Gaussian with the same covariance matrix S. Then

the density for class C

is given by

pðxjC

Þ¼

ð2pÞ

D=2

jSj

1=2

exp 

ðx m

1

ðx m



ð3Þ

From (1), we have

ðxÞ¼w

x þw

; ð4Þ

where

¼ S

1

: ð5Þ

¼

1

þ ln pðC

Þ: ð6Þ

We see that the equal covariance matrices make a

(x)

to be linear in x, and the resulting decision boundaries

will also be linear. As a special case of LDA, the nearest-

neighbor classiﬁer can be obtained, when S¼s

I. If the

prior probabilities pðC

Þ are equal, we assign a feature

vector x to the class C

with the minimum Euclidean

distance jjx  m

, which is equivalent to the opti-

mum decision rule based on the maximum posterior

probability. Anoth er extension of LDA could be

obtained by allowing for mixtures of Gaussians for

the class-conditional densities instead of the single

Gaussian. Mixture discriminant analysis (MDA) [6]

incorporates the Gaussian mixture distribution for

the class-conditional densities to provide a richer

class of density models than the single Gaussian.

The class-conditional density for class C

has the

form of the Gaussian mixture model, pðxjC

Þ¼

r¼1

Nðxjm

; SÞ, where the mixing coefﬁcients

must satisfy p

 0 together with

r¼1

¼ 1.

In this model, the same covariance matrix S is used

within and between classes. The Gaussian mixture

model allows for more complex decision boundaries

although it does not guarantee the global optimum of

maximum likelihood estimates.

Quadratic Discriminant Analysis

If the covariance matrices S

are not assumed to be

equal, then we get quadratic functions of x for a

(x)

ðxÞ¼

ðx m

1

ðx m

Þ

ln jS

jþln pðC

Þ:

ð7Þ

Supervised Learning. Table 1 Comparison among

parametric methods for classification

Method pðxjC

Number of

parameters

Decision

boundary

LDA Nðm

; SÞ (K1)

(Dþ1)

Linear

QDA Nðm

; S

Þ (K1)

(D(Dþ3)∕

2þ1)

Quadratic

Naive Bayes

classifier with

multinomial

distributions

i¼1

pðx

Þ (K1)

(Dþ1)

Linear

Supervised Learning

1299