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Open AcceResearchExploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder DetectionMax A Little*1,2,3, Patrick E McSharry1, Stephen J Roberts2, Declan AE Costello4 and Irene M Moroz3
Address: 1Systems Analysis, Modelling and Prediction Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK, 2Pattern Analysis Research Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK, 3Applied Dynamical Systems Research Group, Oxford Centre for Industrial and Applied Mathematics, Mathematics Institute, University of Oxford, Oxford OX1 3JP, UK and 4Milton Keynes General Hospital, Standing Way, Eaglestone, Milton Keynes, Bucks MK6 5LD, UK

Email: Max A Little* - [email protected]; Patrick E McSharry - [email protected]; Stephen J Roberts - [email protected]; Declan AE Costello - [email protected]; Irene M Moroz - [email protected]

* Corresponding author

AbstractBackground: Voice disorders affect patients profoundly, and acoustic tools can potentiallymeasure voice function objectively. Disordered sustained vowels exhibit wide-ranging phenomena,from nearly periodic to highly complex, aperiodic vibrations, and increased "breathiness". Modellingand surrogate data studies have shown significant nonlinear and non-Gaussian random propertiesin these sounds. Nonetheless, existing tools are limited to analysing voices displaying nearperiodicity, and do not account for this inherent biophysical nonlinearity and non-Gaussianrandomness, often using linear signal processing methods insensitive to these properties. They donot directly measure the two main biophysical symptoms of disorder: complex nonlinearaperiodicity, and turbulent, aeroacoustic, non-Gaussian randomness. Often these tools cannot beapplied to more severe disordered voices, limiting their clinical usefulness.

Methods: This paper introduces two new tools to speech analysis: recurrence and fractal scaling,which overcome the range limitations of existing tools by addressing directly these two symptomsof disorder, together reproducing a "hoarseness" diagram. A simple bootstrapped classifier thenuses these two features to distinguish normal from disordered voices.

Results: On a large database of subjects with a wide variety of voice disorders, these newtechniques can distinguish normal from disordered cases, using quadratic discriminant analysis, tooverall correct classification performance of 91.8 ± 2.0%. The true positive classificationperformance is 95.4 ± 3.2%, and the true negative performance is 91.5 ± 2.3% (95% confidence).This is shown to outperform all combinations of the most popular classical tools.

Conclusion: Given the very large number of arbitrary parameters and computational complexityof existing techniques, these new techniques are far simpler and yet achieve clinically usefulclassification performance using only a basic classification technique. They do so by exploiting theinherent nonlinearity and turbulent randomness in disordered voice signals. They are widelyapplicable to the whole range of disordered voice phenomena by design. These new measurescould therefore be used for a variety of practical clinical purposes.

Published: 26 June 2007

BioMedical Engineering OnLine 2007, 6:23 doi:10.1186/1475-925X-6-23

Received: 6 March 2007Accepted: 26 June 2007

This article is available from: http://www.biomedical-engineering-online.com/content/6/1/23

© 2007 Little et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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BackgroundVoice disorders arise due to physiological disease or psy-chological disorder, accident, misuse of the voice, or sur-gery affecting the vocal folds and have a profound impacton the lives of patients. This effect is even more extremewhen the patients are professional voice users, such assingers, actors, radio and television presenters, for exam-ple. Commonly used by speech clinicians, such as sur-geons and speech therapists, are acoustic tools, recordingchanges in acoustic pressure at the lips or inside the vocaltract. These tools [1], amongst others, can provide poten-tially objective measures of voice function. Althoughacoustic examination is only one tool in the completeassessment of voice function, such objective measurementhas many practical uses in clinical settings, augmentingthe subjective judgement of voice function by clinicians.These measures find uses, for example, in the evaluationof surgical procedures, therapy, differential diagnosis andscreening [1,2], and often augment subjective voice qual-ity measurements, for example the GRB (Grade, Rough-ness and Breathiness) scale. [3] These objective measurescan be used to portray a 'hoarseness" diagram for clinicalapplications [4], and there also exists a variety of tech-niques for automatically screening for voice disordersusing these measures [5-7].

Phenomenologically, normal and disordered sustainedvowel speech sounds exhibit a large range of behaviour.This includes nearly periodic or regular vibration, aperiodicor irregular vibration and sounds with no apparent vibra-tion at all. All can be accompanied by varying degrees ofnoise which can be described as "breathiness". Voice dis-orders therefore commonly exhibit two characteristic phe-nomena: increased vibrational aperiodicity and increasedbreathiness compared to normal voices [4].

In order to better characterise the vibrational aperiodicityaspects of voice disorders, Titze [8] introduced a typology(extended with subtypes [1]). Type I sounds are those thatare nearly periodic: coming close to perfect periodicity.Type II are those that show qualitative dynamical changesand/or modulating frequencies or subharmonics. Thethird class, Type III are those sounds that appear to haveno periodic pattern at all. They have no single, obvious ordominant period and can described as aperiodic. Normalvoices can usually be classed as Type I and sometimesType II, whereas voice disorders commonly lead to allthree types of sounds. This is because voice disorders oftencause the breakdown of stable periodicity in voice produc-tion. The breathiness aspect of disordered voices is oftendescribed as dominating high-frequency noise. Althoughthis original typology covered sounds of only apparentlydeterministic origin, a very large proportion of voice sig-nals seen in the clinic are so noisy as to be better consid-ered stochastic rather than deterministic [2]. Methods that

are based upon theories of purely deterministic nonlineardynamical systems, although they can be appropriate forsounds of deterministic origin covered by the originaltypology, cannot in principle be applied to such noise-dominated sounds, that is, to sounds that would be bettermodelled as stochastic processes rather than determinis-tic. This makes it impossible to characterise the full rangeof signals encountered in the clinic. For these reasons, inthis paper, when we refer to Type III sounds we includerandom, noise-like sounds (which, in keeping with origi-nal typology, have no apparent periodic structure, by vir-tue of their randomness).

There exists a very large number of approaches to theacoustic measurement of voice function. The most popu-lar of these are the perturbation measures jitter and shimmerand variants, and harmonics-to-noise ratios (HNR) [1,4].The effect of a variety of voice disorders on these measureshas been tested under both experimental and clinical con-ditions [4,9], showing that different measures respond todifferent disorders in different ways [10]. For example, insome disorders, jitter will increase with severity of the dis-order, and for other disorders jitter is unaffected. Thus,although these measures can have value in measuring cer-tain limited aspects of voice disorders such as speech pitchvariability, there is no simple relationship between theextent or severity of voice disorder and these measures[4,10]. Therefore, they cannot be used to directly quantifythe two main biophysical symptoms of voice disorders:increasingly severe aperiodicity and breath noise, a quan-tification required to differentiate normal from disor-dered voices.

Another limitation of existing measures is that they areonly properly applicable when near periodicity holds: inTitze's typology only Type I sounds satisfy this property[1]. The behaviour of the algorithms for other sound typesis not known theoretically and limited only to experimen-tal results and informal arguments [4]. The source of thislimitation is that they make extensive use of extraction ofthe pitch period, or fundamental frequency (defined as theinverse of the pitch period) from the acoustic signal [1].

Popular pitch period extraction techniques include zero-crossing detection, peak-picking and waveform matching[1]. The concept of pitch period is only valid for Type Isounds, therefore, application of these methods basedupon periodicity analysis, to any other type of sound isproblematic [6]. Type II and III have therefore receivedmuch less attention in the literature [8], such that thereexist few methods for characterising these types, despitethe fact that they exist in great abundance in clinical set-tings. This precludes the proper use of these tools on alarge number of disordered voice cases, limiting the relia-bility of the analysis, since in fact some algorithms will



not produce any results at all for Type II and III sounds,even though they contain valuable clinical information[2]. Another reason for the limitations of these methods isthat they are based upon classical linear signal processingmethods that are insensitive to the inherent biophysicalnonlinearity and non-Gaussianity in speech [1]. These lin-ear methods include autocorrelation, the discrete Fouriertransform, linear prediction analysis and cepstral process-ing [11].

Since standardised, reliable and reproducible results fromacoustic measures of voice function are required for clini-cal applications, these limitations of perturbation meth-ods are problematic in clinical practice [2]. It is clear thatthere is a clinical need for reliable tools that can character-ise all types of disordered voice sounds for a variety ofclinical applications, regardless of whether they satisfy therequirements of near periodicity, or contain significantnonlinearity, randomness or non-Gaussianity [8].

Furthermore, analysis techniques are complicated by theuse of any arbitrary algorithmic parameters whose choiceaffects the analysis method – changing these parameterscan change the analysis results. The values of such param-eters must be chosen in order to apply the algorithms, andto be principled, it is better, when making that choice, tohave a theoretical framework to apply which offers spe-cific guidance on that choice. Often however, no such gen-eral theory exists, and therefore these values must bechosen by experimental and empirical evaluation alone[12]. There exists the danger then that these parametersare highly "tuned" to the particular data set used in anyone study, limiting the reproducibility of the analysis ondifferent data sets or clinical settings. It is necessary there-fore to reduce the number arbitrary parameters toimprove the reproducibility of these measures.

To address these limitations, empirical investigations andtheoretical modelling studies have been conducted whichhave lead to the suggestion that nonlinear dynamical sys-tems theory is a candidate for a unified mathematicalframework modelling the dynamics seen in all types ofdisordered vowel speech sounds [1,8,13]. The motivationfor the introduction of a more general model than theclassical linear model is the principle of parsimony: themore general model explains more phenomena (moretypes of speech) with a smaller number of assumptionsthan the classical linear model [12,14].

These suggestions have lead to growing interest in apply-ing tools from nonlinear time series analysis to speech sig-nals in order to attempt to characterise and exploit thesenonlinear phenomena [1,13]. For normal Type I speechsounds, fractal dimensions, Lyapunov exponents and bis-pectral methods have all been applied, also finding evi-

dence to support the existence of nonlinearity [15,16].Extracting dynamical structure using local linear predic-tors, neural networks and regularised radial basis func-tions have all been used, with varying reported degrees ofsuccess. Algorithms for calculating the correlation dimen-sion have been applied, which was successful in separat-ing normal from disordered subjects [17]. Correlationdimension and second-order dynamical entropy meas-ures showed statistically significant changes before andafter surgical intervention for vocal fold polyps [18], andLyapunov exponents for disordered voices were found toconsistently higher than those for healthy voices [19]. Itwas also found that jitter and shimmer measurementswere less reliable than correlation dimension analysis onType I and unable to characterise Type II and (non-ran-dom) Type III sounds [20]. Mixed results were found forfractal dimension analysis of sounds from patients withneurological disorders, for both acoustic and electroglot-tographic signals [21]. Instantaneous nonlinear AM andFM formant modulations were shown effective at detect-ing muscle tension dysphonias [22]. For the automatedacoustic screening of voice disorders, higher-order statis-tics lead to improved normal/disordered classificationperformance when combined with several standard per-turbation methods [7].

In order to categorise individual voice signals into classesfrom the original typology (excluding severely turbulent,noise-like sounds), recent studies have found that byapplying correlation dimension measurements to signalsof these types, it was possible, over a sample of 122 stablevowel phonations, to detect a statistically significant dif-ference between the three different classes [23,24]. Thisprovides further evidence in favour of the appropriatenessof nonlinear signal analysis techniques for the analysis ofdisordered voices.

These studies show that nonlinear time series methodscan be valuable tools for the analysis of voice disorders, inthat they can analyse a much broader range of speechsounds than perturbation measures, and in some cases arefound to be more reliable under conditions of high noise.However, very noisy, breathy signals have so far receivedlittle attention from nonlinear time series analysisapproaches, despite these promising successes. Commonapproaches such as correlation dimension, Lyapunovexponent calculation and predictor construction requirethat the scaling properties of the embedded attractor arenot destroyed by noise, and for thus very noisy, breathysignals, there is the possibility that such nonlinearapproaches may fail. There are also numerical, theoreticaland algorithmic problems associated with the calculationof nonlinear measures such as Lyapunov exponents orcorrelation dimensions for real speech signals, castingdoubt over the reliability of such tools [21,25-27]. For



example, correlation dimension analysis shows high sen-sitivity to the variance of signals in general, and it is there-fore necessary to check that changes in correlationdimension are not due simply to changes in variance [28].Therefore, as with classical perturbation methods, currentnonlinear approaches cannot yet directly measure the twomost important biophysical aspects of voice disorder.

A limitation of deterministic nonlinear time series analy-sis techniques for random, very noisy signals is that theimplicit, deterministic, nonlinear dynamical model,which is ordinarily assumed to represent the nonlinearoscillations of the vocal folds [29] is no longer appropri-ate. This is because randomness is an inherent part of thebiophysics of speech production [30,31]. Thus there is aneed to expand the nonlinear dynamical systems frame-work to include a random component, such that randomvoice signals and breath noise can also be fully character-ised within the same, unified framework.

This paper therefore introduces a new, framework modelof speech production that splits the dynamics into bothdeterministic nonlinear and stochastic components. Theoutput of this model can then be analysed using methodsthat are able to characterise both nonlinearity and ran-domness. The deterministic component of the model canexhibit both periodic and aperiodic dynamics. It is pro-posed to characterise this component using recurrenceanalysis. The stochastic components can exhibit statisticaltime dependence or autocorrelation, which can be ana-lysed effectively using fractal scaling analysis. This paperreports the replication of the "hoarseness" diagram [4]illustrating the extent of voice disorder, and demonstrates,using a simple quadratic classifier, how these new meas-ures may be used to screen normal from disordered voicesfrom a large, widely-used database of patients. It alsodemonstrates that these new measures achieve superiorclassification performance overall when compared toexisting, classical perturbation measures, and the derivedirregularity and noise measures of Michaelis [4].

MethodsIn this section we first discuss in detail the evidence thatsupports the development of a new stochastic/determinis-tic model of speech production.

The classical linear theory of speech production bringstogether the well-developed subjects of classical linear sig-nal processing and linear acoustics (where any nonlinear-ities in the constitutive relationships between dynamicvariables of the fluid are considered small enough to benegligible) to process and analyse speech time series [11].The biophysical, acoustic assumption is that the vocaltract can be modelled as a linear resonator driven by thevibrating vocal folds that are the source of excitation

energy [11]. However, extensive modelling studies, exper-imental investigations and analysis of voice signals haveshown that dynamical nonlinearity and randomness arefactors that should not be ignored in modelling speechproduction.

For the vocal folds, there are two basic, relevant modelcomponents to consider. The first is the vocal fold tissue,and the second is the air flowing through that structure.The vocal folds, during phonation, act as a vibrating valve,disrupting the constant airflow coming from the lungsand forming it into regular puffs of air. In general the gov-erning equations are those of fluid dynamics coupled withthe elastodynamics of a deformable solid. In oneapproach to solving the problem, the airflow is modelledas a modified quasi-one-dirnensional Euler system whichis coupled to the vocal fold flow rate, and the vocal foldsare modelled by a lumped two mass system [32]. A widelyused and simplified model is the two-mass model inIshizaka [33], further simplified in asymmetric [29,34]and symmetric versions [35]. These models demonstrate aplausible physical mechanism for phonation as nonlinearoscillation: dynamical forcing from the lungs supplies theenergy needed to overcome dissipation in the vocal foldtissue and vocal tract air. The vocal folds themselves aremodelled as elastic tissue with nonlinear stress-strain rela-tionship. Classical nonlinear vocal fold models coupledwith linear acoustic vocal tract resonators appear toaccount for the major part of the mechanisms of audiblespeech, but from these mechanisms an important compo-nent is missing: that of aspiration, or '"breath" noise [36].Such noise is produced when the air is forced through anarrow constriction at sufficiently high speeds that "tur-bulent" airflow is generated, which in turn producesnoise-like pressure fluctuations [37]. Aspiration noise isan unavoidable, involuntary consequence of airflow fromthe lungs being forced through the vocal organs, and canbe clearly heard in vowel phonation [38]. Certain voicepathologies are accompanied by a significant increase insuch aspiration noise, which is perceived as increased"breathiness" in speech [4]. This noise is therefore animportant part of sound generation in speech.

One significant deficiency in the classical linear acousticmodels is due to the assumptions about fluid flow uponwhich their construction is based [39]. These classical lin-ear models make very many simplifying assumptionsabout the airflow in the vocal organs, for example, that theacoustic limit [40] holds in which the fluid is nearly in astate of uniform motion. Similarly, the Bernoulli's equationassuming energy conservation along streamlines, uponwhich many classical vocal fold models rely, applies onlyif the fluid is assumed inviscid and irrotational [41,42].The important point for this study is that these classicalassumptions forbid the development of complicated,



"turbulent" fluid flow motion, in which the flow followsconvoluted paths of rapidly varying velocity, with eddiesand other irregularities at all spatial scales [43]. Thisbreakdown of laminar flow occurs at high Reynoldsnumber, and for the relevant length scales of a few centi-metres in the vocal tract and for subsonic air flow speedstypical of speech [38], this number is very large (of order105), indicating that airflow in the vocal tract can beexpected to be turbulent. Under certain assumptions, tur-bulent structures, and vortices in particular (fluid particlesthat have rotational motion), can be shown to be a sourceof aeroacoustic sound [37]. Turbulence is a highly com-plex dynamical phenomenon and any point measure-ment such as acoustic pressure taken in the fluid will losemost of the information about the dynamics of the fluid.Consequently, even if turbulence has a purely determinis-tic origin, it is reasonable to model any one single dynam-ical variable measured at a point in space as a randomprocess [43].

There are two broad classes of physically plausible math-ematical models of the effects of aeroacoustic noise gener-ation in speech. The first involves solving numerically thefull partial differential equations of gas dynamics (e.g. theNavier-Stokes equations), and the second uses the theoryof vortex sound [37]. For example, the study of Zhao [44]focused on the production of aspiration noise generatedby vortex shedding at the top of the vocal folds, simulatedover a full vocal fold cycle. This study demonstrates thatthe computed sound radiation due to vorticity containssignificant high frequency fluctuations when the folds arefully open and beginning to close. On the basis of theseresults, it can be expected that if the folds do not closecompletely during a cycle (which is observed in cases ofmore "breathy" speech), the amplitude of high frequencynoise will increase.

The second class of models, which makes use of Lighthill'sacoustic analogy [37], arc based around the theory of vortexsound generated in a cylindrical duct, [37] where, essen-tially, each vortex shed at an upstream constriction acts asa source term for the acoustic wave equation in the duct,as the vortex is convected along with the steady part of theairflow. The resulting source term depends upon not onlythe attributes of the vortex itself, but also upon themotion of the vortex through the streamlines of the flow[31,37]. One model that uses this approach involves thenumerical simulation of two overall components: themean steady flow field and the acoustic wave propagationin the vocal tract [38]. Vortices are assumed to be shed atrandom intervals at constrictions at particular locations inthe vocal tract, for example, at the vocal folds or betweenthe roof of the mouth and the tongue. Each vortex con-tributes to the acoustic source term at each spatial gridpoint. Here, an important observation is that simulated

pressure signals from this model are stochastic processes[45], i.e. a sequence of random variables. It is also notice-able from the spectra of simulated pressure signals thatalthough the signals are stochastic, they exhibit significantnon-zero autocorrelation since the spectral magnitudesare not entirely constant, leading to statistical self-similar-ity in these signals [15,43].

Other potential sources of biophysical fluctuation includepulsatile blood flow, muscle twitch and other variabilityand randomness in the neurological and biomechanicalsystems of the larynx [30]. The typical nonlinear dynami-cal behaviour of models of the vocal folds, such as period-doubling (subharmonics), bifurcations [29], and transi-tions to irregular vibration [35] have been observed inexperiments with excised larynxes, a finding that helps tosupport the modelling hypothesis that speech is an inher-ently nonlinear phenomena [29,46]. Similarly, models ofturbulent, random aeroacoustic aspiration noise havebeen validated against real turbulent airflow inducedsound generated in acoustic duct experiments [38]. Suchstudies show that the models of vortex shedding at ran-dom intervals are plausible accounts for the dynamicalorigins of breath noise in phonation.

Complementing modelling and experimental studies, thefinal source of evidence for nonlinearity and randomnessin speech signals comes from studies of voice pressure sig-nals. Using surrogate data analysis, it has been shown thatnonlinearity and/or non-Gaussianity is an important fea-ture of Type I sounds [47-49]. Nonlinear bifurcationshave been observed in excised larynx experiments [46],and period-doubling bifurcations and chaos-like featureshave been observed in signals from patients with organicand functional dysphonias [13]. Aspiration noise hasbeen observed and measured as a source of randomness invoiced phonation, in both normal [50] and disorderedspeech sounds [1,4]. The fractal, self-similarity propertiesof aspiration noise as a turbulent sound source have alsobeen observed and quantified in normal [15] and disor-dered speech [27].

Taken as a whole, these modelling, simulation, validationand signal analysis studies suggest that during voiced pho-nation there will be a combination of both deterministicand stochastic elements, the deterministic componentattributable to the nonlinear movements of the vocal foldtissue and bulk of the air in the vocal tract, and the sto-chastic component the high frequency aeroacoustic pres-sure fluctuations caused by vortex shedding at the top ofthe vocal folds, whose frequency and intensity is modu-lated by the bulk air movement (and other sources of bio-physical fluctuation). During voiced phonation, thedeterministic oscillations will dominate in amplitudeover the noise component which will show high fre-



quency fluctuations around this oscillation. Duringunvoiced or breathy pathological phonation, the turbu-lent noise component will dominate over any determinis-tic motion.

In order to capture all these effects in one unified model,we introduce a continuous time, two component dynam-ical model that is taken to generate the measured speechsignal. The state of the system at time t ∈ � is representedby the vector u(t) of size d. Then the equation of motionthat generates the speech signal is the following vector sto-chastic differential equation, commonly known as a Lan-gevin equation [25].

(t) = f (u (t)) + ε (t), (1)

where ε(t) is a vector of stochastic forcing terms. It is notnecessary to assume that these fluctuations are independ-ent and identically distributed (i.i.d.). The function f : �d

→ �d is unknown and represents the deterministic part ofthe dynamics. Given an initial condition vector u(0) thena solution that satisfies equation (1) is called a trajectory.Ensembles of trajectories can be shown to satisfy the prop-erties of a stochastic process with finite memory (a higher-order Markov chain) [25].

Of importance to both deterministic and stochastic sys-tems is the notion of recurrence in state space. Recurrenttrajectories are those that return to a given region of statespace [51]. Recurrence time statistics provide useful infor-mation about the properties of both purely deterministicdynamical systems and stochastic systems [51]. Recur-rence analysis forms the basis of the method of recurrenceplots in nonlinear time series analysis [25].

In the context of the model (1), state-space recurrence isdefined as:

u(t) ⊂ B(u(t + δt), r), (2)

where B(u, r) is a closed ball of small radius r > 0 aroundthe point u in state-space, and u(t) ⊄ B(u(t + s), r) for 0 <s<δt. Each different t ∈ � may be associated with a differentδt, called the recurrence time. We will define aperiodic asrecurrent in the sense of (2) but not periodic. Periodicity isrecovered from the definition of recurrence in the specialcase when r = 0 and δt is the same for all t, so that the sys-tem vector u(t) assumes the same value after a time inter-val of δt:

u(t) = u(t + δt), (3)

for all t ∈ �. Then δt is the period of the system. Therefore,although these concepts of periodicity and aperiodicityare mutually exclusive, both are special cases of the more

general concept of recurrence. The requirement of perio-dicity is central to many linear signal processing methods(the basis of the Fourier transform, for example), but ape-riodicity is a common feature of many voice disorders. Itcan be seen therefore that in order to characterise theinherent irregularity of disordered speech sounds, werequire more general processing techniques that candirectly account for such departures from strict periodic-ity. Using this analysis, nearly periodic speech sounds ofType I can be described as recurrent for some small r > 0,with δt nearly the same for each t. Type II sounds are moreirregular than Type I, and for the same r, the δt will assumea wider range of values than for Type I. Similarly. Type IIIsounds which are highly irregular and aperiodic, will havea large range of values of δt again for the same r.

Similarly, of importance in the analysis of stochastic proc-esses is scaling analysis [25]. Stochastic time series inwhich the individual measurements in time are notentirely independent of earlier time instants are often self-similar, that is, when a sub-section of the time series isscaled up by a certain factor, it has geometrical, approxi-mate or statistical similarity to the whole time series [25].Such self-similarity is a property of fractals. [43] The toolsof dimension measurement and scaling analysis may beused to characterise the self-similarity in signals such asspeech. As discussed above, theoretical models of aeroa-coustic turbulent sound generation in speech predict ran-domly-timed impulses convolved with an individualimpulse response for each vortex that induces autocorre-lated random noise sequences [31], so that turbulentspeech signals at one time instant are not independent ofearlier time instants. It has also been found experimen-tally that changes in the statistical time dependence prop-erties of turbulent noise in speech, as measured by aparticular fractal dimension of the graph of the speech sig-nal, are capable of distinguishing classes of phonemesfrom each other [15]. As introduced above, disorderedvoices are often accompanied by increased "breathiness",due in part to the inability of the vocal folds to close prop-erly, so that air escapes through the partial constriction ofthe vocal folds creating increased turbulence in the airflow[31]. Thus scaling analysis (and more general graphdimension measures) could be useful for characterisingvocal fold disorders.

Recent experiments have shown that the use of recurrenceanalysis coupled with scaling analysis can distinguishhealthy from disordered voices on a large database ofrecordings with high accuracy [27]. These techniques arecomputationally simple and furthermore substantiallyreduce the number of arbitrary algorithmic parametersrequired, compared to existing classical measures, thusleading to increased reproducibility and reliability.

u



Time-Delay State-Space Recurrence AnalysisIn order to devise a practical method for applying the con-cept of recurrence defined earlier and measuring theextent of aperiodicity in a speech signal, we can make useof time-delay embedding [25]. Measurements of the outputof the system (1) are assumed to constitute the acousticsignal, sn:

sn = h(u(nΔt)), (4)

where the measurement function h : �d→ � projects thestate vector u(t) onto the discrete-time signal at timeinstances nΔt where Δt is the sampling time, and n ∈ � isthe time index.

From these sampled measurements, we then construct ofan m-dimensional time delay embedding vector:

sn = [sn, sn-τ,...sn-(m-1)τ]T. (5)

Here τ is the embedding time delay and m is the embeddingdimension.

We will make use of the approach introduced in Ragwitz[52] to optimise the embedding parameters m and τ suchthat the recurrence analysis produces results that are closeas possible to known, analytically derived results uponcalibration with known signals. (We use this as an alterna-tive to common techniques for finding embeddingparameters, such as false-nearest neighbours, whichexplicitly require purely deterministic dynamics [25,52]).Note that, under this approach, for very noisy signals, wewill not always resolve all signals without self-intersec-tions. However, in the context of this study, achieving acompletely non-intersecting embedding is not necessary.For very high-dimensional deterministic or stochastic sys-tems, any reconstruction with self-intersections due toinsufficiently high embedding dimension can be consid-ered as a different stochastic system in the reconstructedstate-space. We can then analyze the stochastic recurrenceof this reconstructed system. This recurrence, albeit differ-ent from the recurrence properties in the original system,is often sufficient to characterize the noisy end of the scaleof periodicity and aperiodicity.

Figure 1 shows the signals sn for one normal and one dis-ordered voice example (Kay Elemetrics Disordered VoiceDatabase). Figure 2 shows the result of applying the aboveembedding procedure for the same speech signals.

In order to investigate practically the recurrence time sta-tistics of the speech signal, we can make use of the methodof close returns [53], originally designed for studying recur-rence in deterministic, chaotic dynamics. Here we adopt

this method to study stochastic dynamics as well as deter-ministic dynamics. In this method, a small, closed ball

B( , r) of radius r > 0 is placed around the embedded

data point . Then the trajectory is followed forward in

time ... until it has left this ball. i.e. until

> r for some j > 0, where ||·|| is the Eucli-

dean distance. Subsequently, the time n1 at which the tra-

jectory first returns to this same ball is recorded (i.e. the

first time when ≤ r), and the difference of these

two times is the (discrete) recurrence time T = n1 - n0. This

procedure is repeated for all the embedded data points sn,

forming a histogram of recurrence times R(T). This histo-gram is normalised to give the recurrence time probabilitydensity:

where Tmax is the maximum recurrence time found in theembedded state space. The choice of r is critical to capturethe properties of interest to this study. For example, if thetrajectory is a nearly periodic (Type I), we require that r is

sn0

sn0

s sn n0 01 2+ +,

s sn n j0 0− +

s sn n0 1−

P TR T

R iiT

( )( )

( ),

max=

=∑ 1

(6)

Selected normal and disordered speech signal examplesFigure 1Selected normal and disordered speech signal exam-ples. Discrete-time signals from (a) one normal (JMC1NAL) and (b) one disordered (JXS01AN) speech signal from the Kay Elernetrics database. For clarity only a small section is shown (1500 samples).

200 400 600 800 1000 1200 1400-1

-0.5

0

0.5

1

n (samples)

s n

(a)

200 400 600 800 1000 1200 1400-1

-0.5

0

0.5

1

n (samples)

s n

(b)



large enough to capture all the recurrences, but not toolarge to find recurrences that are due to spurious intersec-tions of B(u, r) with other parts of the trajectory, violatingthe conditions for proper recurrence. The appropriatechoice of embedding delay τ has a role to play: selecting τtoo small means that any trajectory lies close to a diagonalin the reconstructed state-space, potentially causing spuri-ous recurrences. Thus τ must be chosen large enough toavoid spurious recurrences. Similarly, if τ is too large thenthe points in the embedded state-space fill a large cloudwhere recurrences will be difficult to find without usingan inappropriately large value of r. There will be an opti-mum value of τ which traditionally is set with reference toautocorrelation or time-delayed mutual information esti-mates, for more details see [25].

In order to understand the behaviour of this algorithm(partly for optimising the embedding parameters), we

consider two extreme forms that the density (6) mayassume. The first is the ideal limiting case in which therecurrence distance r tends to zero for a periodic trajec-tory. The recurrence time probability density is:

where k is the period of the trajectory. In the secondextreme case we consider a purely random, uniform i.i.d.signal which is normalised to the range [-1, 1]. The recur-rence probability density is approximately uniform:

Proofs for the results in equations (7) and (8) are given inthe Appendix.

We can then optimise m, τ and r such that for a syntheticsignal of perfect periodicity. P(T) is determined using theclose returns method such that it is as close as possible tothe theoretical expression (7). This optimisation is carriedout by a straightforward systematic search of values ofthese parameters m = 2, 3...10, τ = 2,3... 50, and for r =0.02, 0.04,...0.5, on a perfectly periodic test signal. Thissearch can be considered as a scan for the optimumparameter values through all points on a three-dimen-sional parameter cube with m, τ and r as the co-ordinatesof that cube.

All disordered voice speech signals will lie somewhere inbetween the extremes of perfect periodicity and completerandomness. Thus it will be useful to create, from therecurrence time probability density, a straightforward slid-ing scale so that normal and disordered voice signals canbe ranked alongside each other. This depends upon a sim-ple characterisation of the recurrence probability densityP(T). One simple measure of any probability density is theentropy which measures the average uncertainty in thevalue of the discrete-valued density p(i), i = 1, 2... M [25]:

with units of nats (by convention. 0 ln 0 is taken to bezero). This measure can then rank disordered voice signalsby representing the uncertainty in the period of the disor-dered voice signal in the following way. For perfectly peri-odic signals the recurrence probability density entropy(RPDE) is:

P TT k

( ) ,==⎧

⎨⎩

1

0

if

otherwise(7)

P TT

( ) .max

≈ 1(8)

H p i p ii

M= −

=∑ ( )ln ( ),

1

(9)

H P i P ii

T

per = − ==∑ ( )ln ( ) .max

1

0 (10)

Selected time-delay embedded speech signalsFigure 2Selected time-delay embedded speech signals. Time-delay embedded discrete-time signals from (a) one normal (JMC1NAL) and (b) one disordered (JXS01AN) speech signal from the Kay Elernetrics database. For clarity only a small section is shown (1500 samples). The embedding dimension is m = 3 and the time delay is τ = 7 samples.

-0.50

0.5-0.5 0 0.5

-0.5

0

0.5

sn

sn +

(a)

s n +

2

-0.50

0.5-0.5 0 0.5

-0.5

0

0.5

sn

sn +

(b)

s n +

2



since P(k) = 1 and the rest are zero. Conversely, for thepurely stochastic, uniform i.i.d. case (derived in theappendix), the uniform density can be taken as a goodapproximation, so that the RPDE is:

in units of nats. The entropy scale H therefore ranges fromHper, representing perfectly periodic examples of Type Isounds, to Hiid as the most extreme cases of noise-likeType III sounds. In practice, all sounds will lie somewherein between these extremes.

However, the entropy of a probability density is maxi-mum for the uniform density, so that Hiid is the maximumvalue that H can assume. Thus, in addition to ranking sig-nals on a scale of aperiodicity, we can know precisely thetwo extremes of that scale. For different sampling times Δtthe value Tmax will change. Therefore, we can normalisedthe RPDE scale for subsequent usage:

a unit less quantity that assumes real values in the range[0,1].

The method of close returns, upon which this technique isbased, was originally designed to characterise determinis-tic, chaotic systems [53]. In this case, if the chaotic systemis ergodic and has evolved past any transients, then therecurrence properties of the system are independent of theinitial conditions and initial time, i.e. they are invariantsof the system. Similarly, if the system is purely stochasticand ergodic, then it has a stationary distribution. Again,after any transient phase, the recurrence properties will beinvariant in the above sense. Thus the derived measure Hwill also be an invariant of the system. We note that tradi-tional jitter and shimmer measurements do not share thisinvariance property, in this sense they do not give a relia-ble description for chaotic or Type III time series. Often,when stable phonation is initiated in speech, the vocalfolds will take some time to settle into a pattern of stableperiodic or chaotic oscillation. The behaviour of speechsignals during this "settling time" is similar to the tran-sient behaviour typically observed in nonlinear dynami-cal systems. In this study, we make use of voice signalswhich are stable phonations, and we discard any of thesetransient phases. Thus, to a reasonable approximation Hcan be considered as an invariant of the dynamics of thespeech organs.

H P i P i Ti

T

iid = − ==∑ ( )ln ( ) ln ,max

max1

(11)

HP i P i

HiT

normiid

=− =∑ ( )ln ( )

,max1 (12)

State-space recurrence analysis for a periodic signalFigure 3State-space recurrence analysis for a periodic signal. Demonstration of results of time-delayed state-space recur-rence analysis applied to a perfectly periodic signal (a) cre-ated by taking a single cycle (period k = 134 samples) from a speech signal and repeating it end-to-end many times. The signal was normalised to the range [-1, 1]. (b) All values of P(T) are zero except for P(133) = 0.1354 and P(134) = 0.8646 so that P(T) is properly normalised. This analysis is also applied to (c) a synthesised, uniform i.i.d. random signal on the range [-1, 1], for which (d) the density P(T) is fairly uni-form. For clarity only a small section of the time series (1000 samples) and the recurrence time (1000 samples) is shown. Here, Tmax = 1000. The length of both signals was 18088 sam-ples. The optimal values of the recurrence analysis parame-ters were found at r = 0.12, m = 4 and τ = 35.

200 400 600 800 1000-1

0

1

s n

(a)

n (samples)

200 400 600 800 1000-1

0

1

s n

(c)

n (samples)

200 400 600 800 10000

0.5

1

P(T

)

Hnorm

= 0.06

(b)

T (samples)

200 400 600 800 10000

0.05

0.1

P(T

)

Hnorm

= 0.91

(d)

T (samples)



Figure 3 shows the result of this recurrence analysis,applied to a synthetic, perfectly periodic signal created bytaking a single cycle from a speech signal and repeating itend-to-end many times. It also shows the analysis appliedto a synthesised, uniform, i.i.d. random signal (on the sig-nal range [-1, 1]) after optimising m, τ and r by griddedsearch. Even though exact results are impossible to obtaindue to the approximation inherent to the algorithm andonly finite-length signals, the figure shows that a closematch is obtainable between the theoretical, predictedresults and the simulated results.

Detrended Fluctuation AnalysisIn order to investigate the second aspect of disorderedvoices, that of increased breath noise, we require a practi-cal approach to applying the scaling ideas introducedabove. Detrended fluctuation analysis is one straightfor-ward technique for characterising the self-similarity of thegraph of a signal from a stochastic process [54].

It is designed to calculate the scaling exponent α in nonsta-tionary time series (where the statistics such as mean, var-iance and autocorrelation properties might change withtime), and has been shown to produce robust resultswhen there are slowly moving trends in the data. Thesewill naturally include low frequency vibrations [54],which are similar in nature to the nonlinear vibrations ofthe vocal folds described by the function f in the model(1). Thus this technique can be used to characterise theproperties of only the stochastic part ε(t) of the model (1).

In this technique, the scaling exponent α is a measure ofthe ratio of the logarithm of the fluctuations or verticalheight of the graph to the logarithm of the horizontalwidth of a chosen time window over which that verticalheight is measured. The scaling exponent is calculated asthe slope of a log-log graph of a range of different horizon-tal time window sizes against, the vertical height of thesignal in those time windows. Mathematically, for self-similar signals with positive scaling exponent α the self-similarity property of the graph of the signal sn shouldhold on all time scales, but we are limited by the finiteamplitude range of physical measurements to a certainmaximum scale. Thus the signal is first integrated in orderto create a new stochastic process that exhibits self-simi-larity over a large range of time scales (then, for example,a purely independent, stochastic process will result in aself-similar random walk). First, the time series sn is inte-grated:

for n = 1,2...N, where N is the number of samples in thesignal sn. Then, yn is divided into windows of length L sam-

ples. A least-squares straight line local trend is calculated byanalytically minimising the squared error E2 over theslope and intercept parameters a and b:

Next, the root-mean-square deviation from the trend, thefluctuation, is calculated over every window at every timescale:

This process is repeated over the whole signal at a range ofdifferent window sizes L, and a log-log graph of L againstF(L) is constructed. A straight line on this graph indicatesself-similarity expressed as F(L) ∝ Lα . The scaling expo-nent α is calculated as the slope of a straight line fit to thelog- log graph of L against F(L) using least-squares asabove. For a more in-depth presentation and discussion ofself-similarity in signals in general, and further informa-tion about DFA, please see Kantz, Hu [25,54].

We are assuming that the signal, as the measured outputof the new model, represents a combination of determin-istic and stochastic dynamics. The deterministic part ofthe dynamics, dictated by the function f in equation (1)will result in slower changes in the signal sn taking placeover a relatively long time scale. Similarly, the stochasticfluctuations in the signal indicated changes taking placeover a much shorter time scale. Since the goal of DFA is toanalyse the faster changing, stochastic properties of thesignal, only a limited range of window sizes is investi-gated, over which the stochastic component of the signalexhibits self-similarity as indicated by a straight-line onthe log-log graph of window size against fluctuation. Asan example, Type III would include some speech signalsthat are actually chaotic, where the chaos is due to slow,nonlinear dynamics in the vocal fold tissue and airflow,the characteristic time of this nonlinear oscillation will bemuch longer than the window sizes over which the scalingexponent is estimated. Thus, the nature of the chaoticoscillation will not affect the scaling exponent, which willrespond only to any random fluctuations occurring on amuch shorter time scale.

The resulting scaling exponent can assume any numberon the real line. However, it would be more convenient torepresent this scaling exponent on a finite sliding scalefrom zero to one, as we have done for the RPDE measure.Thus we need a mapping function g : � → [0, 1]. One suchfunction finding common use in statistical and pattern

y sn jj

n=

=∑

1

, (13)

argmin ( ) .,a b

n nn

LE y a b2 2

1

= − −=∑ (14)

F LL

y a bn nn

L( ) ( ) .= − −

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

∑1 2

1

12

(15)



recognition applications is the logistic function g(x) = (1 +exp(-x))-1 [55], so that the normalised scaling exponent is:

Therefore, each sound will lie somewhere between theextremes of zero and one on this scale, according to theself-similarity properties of the stochastic part of thedynamics. As will be shown later, speech sounds forwhich αnorm in is closer to one are characteristic of generalvoice disorder.

Application to Examples of Normal and Disordered VoicesIn this section we will apply these two new measures tosome examples of normal and disordered voices, as a lim-ited demonstration of characterising the extent of aperio-dicity and breathiness in these signals. Figure 4 shows theRPDE value Hnorm calculated on the same two speech sig-nals from the Kay database as shown in figure 1. Note thatthe second, disordered example is of Type III and shows

significantly irregular vibration, which is detected by alarge increase in Hnorm.

Similarly, figure 5 shows two more speech examples, onenormal and one disordered from the same database andthe corresponding values of the scaling exponent α andαnorm. In these cases, the disordered example is extremely"breathy", and this turbulent noise is detected by anincrease in the scaling exponent.

Quadratic Discriminant AnalysisIn order to test the effectiveness of these two measures inpractice, one approach is to set up a classification task toseparate normal control subjects from disordered exam-ples using these measures alone. Here we choose one ofthe simplest approaches, quadratic discriminant analysis(QDA), which allows separation by modelling the dataconditional upon each class, here the normal (class C1)and disordered (class C2) cases, using joint Gaussianprobability density functions [55]. For a J × K data matrixv = vjk of observations consisting of the measures j = 1, 2for RPDE and DFA respectively, and all subjects k, theselikelihood densities are parameterised by the mean andcovariance matrices of the data set:

μ = E[v], C = E [(v - μ) (v - μ)T], (17)

where E is the expectation operator, and μ is the mean vec-tor formed from the means of each row of v. The class like-lihoods are:

for classes i = 1, 2 and an arbitrary observation vector w. Itcan be shown that, given these Gaussian class models, themaximum likelihood regions of the observation space �J

are separated by a decision boundary which is a (hyper-)conic section calculated from the difference of log-likeli-hoods for each class, which is the unique set of pointswhere each class is equally likely [55]. The maximum like-lihood classification problem is then solved using thedecision rule that l(w) ≥ 0 assigns w to class C1, and l(w)< 0 assigns it to class C2, where:

ααnorm =

+ −1

1 exp( ). (16)

f CC iJ

i iT

i i( | ) ( ) | | exp ( ) ( ) ,/ /w C w C w= − − −⎡⎣⎢

⎤⎦⎥

− − −212

2 1 2 1π μμ μμ

(18)

l AT( ) ,w w A w A w= + +12 2 1 0 (19)

A C C A C C2 11

21

1 1 11

2 21= − = −− − − −, ,μμ μμT T (20)

RPDE analysis resultsFigure 4RPDE analysis results. Results of RPDE analysis carried out on the two example speech signals from the Kay data-base as shown in figure 1. (a) Normal voice (JMC1NAL), (b) disordered voice (JXS01AN). The values of the recurrence analysis parameters were the same as those in the analysis of figure 3. The normalised RPDE value Hnormis larger for the disordered voice.

200 400 600 800 10000

0.5

1

P(T

)

Hnorm

= 0.14

(a)

200 400 600 800 10000

0.05

0.1

P(T

)

Hnorm

= 0.89

(b)

T (samples)

T (samples)



In order to avoid the problem of overfitting, where theparticular chosen separation model shows good perform-ance on the training data but fails to generalise well to new,unseen data, the classifier results require validation.

In this paper, we make use of bootstrap resampling for vali-dation [55]. In the bootstrap approach, the classifier istrained on K cases selected at random with replacementfrom the original data set of K cases. This trial resamplingprocesses is repeated many times and the mean classifica-tion parameters E[A2], E[A1], E[A0] are selected as theparameters that would achieve the best performance onentirely novel data sets.

Bootstrap training of the classifier involves calculating

and (the observations) for each speech

sample k in the database, (where the superscript k denotesthe measure for the k-th subject). Then, K random selec-

tions of these values with replacement and

form the entries of the vector v1k = and v2k =

Then the mean vectors for each class μ1 and μ2 and covar-

iance matrices C1, C2 for the whole selection are calcu-

lated. Next, for each subject, the decision function isevaluated:

Subsequently, applying the decision rule assigns the sub-ject k into either normal or disordered classes. Then theperformance of the classifier can be evaluated in terms ofpercentage of true positives (when a disordered subject iscorrectly assigned to the disordered class C1) and true neg-atives (when a normal subject is correctly assigned to thenormal class C2). The overall performance is the percent-age of correctly classified subjects, in both classes. Thisbootstrap trial process of creating random selections ofthe measures, calculating the class mean vectors and cov-ariance matrices, and then evaluating the decision func-tion on all the measures to obtain the classificationperformance is repeated many times. Assuming that theperformance percentages are normally distributed, thenthe 95% confidence interval of the classification perform-ance percentages can be calculated. The best classificationboundary can then be taken as the mean boundary overallall the trials.

A T T0 1 2 1 1

11 2 2

12

12

12

12

12

= − + − +− −ln | | ln | | .C C C Cμμ μμ μμ μμ

(21)

Hknorm αnorm

k

′H knorm ′αnorm

k

′H knorm ′αnorm

k

l l Hkk k T( ) ([ , ] ).w = norm normα (22)

DFA analysis resultsFigure 5DFA analysis results. Results of scaling analysis carried out on two more example speech signals from the Kay database. (a) Normal voice (GPG1NAL) signal, (c) disordered voice (RWR14AN). Discrete-time signals sn shown over a limited range of n for clarity. (b) Logarithm of scaling window sizes L against the logarithm of fluctuation size F(L) for normal voice in (a). (d) Logarithm of scaling window sizes L against the log-arithm of fluctuation size F(L) for disordered voice in (b). The values of L ranged from L = 50 to L = 100 in steps of five. In (b) and (d), the dotted line is the straight-line fit to the loga-rithms of the values of L and F(L) (black dots). The values of α and the normalised version αnorm show an increase for the disordered voice.

200 400 600 800 1000-1

0

1

s n

(a)

n (samples)

200 400 600 800 1000-1

0

1

s n

(c)

n (samples)

1.699 1.7959 1.8751 1.942 20

= 0.15

norm = 0.54

log10L

(b)

0.24

0.40

0.510.60

log 1

0F(L

)

(d)

1.699 1.7959 1.8751 1.942 20

= 1.75

norm = 0.85

0.24

0.40

0.510.60

log 1

0F(L

)

log10L



Efficient implementations of the algorithms described inthis paper written in C with Matlab MEX interface accom-pany this paper: close returns [see Additional files 1 and2] and detrended fluctuation analysis [see Additional files3, 4 and 5].

Algorithms for Classical TechniquesFor the purposes of comparison, we calculate the classicalmeasures of jitter, shimmer and HNR (Noise-to-Harmon-ics Ratio) [1]. There are many available algorithms for cal-culating this quantity, in this study we make use of thealgorithms supplied in the software package Praat [56].These measures are based on an autocorrelation methodfor determining the pitch period (see Boersma [57] for adetailed description of the method).

We also use the methods described in Michaelis [4]. Thisfirst requires calculating the measures EPQ (Energy Pertur-bation Quotient), PPQ (Pitch Perturbation Quotient),GNE (Glottal to Noise Excitation Ratio) and the mean cor-relation coefficient between successive cycles, measureswhich require the estimation of the pitch period using thewaveform matching algorithm (see Titze [58] for adetailed description of this algorithm). The EPQ, PPQ,GNE and correlation coefficients are calculated over suc-cessive overlapping "frames" of the speech signal. Eachframe starts at a multiple of 0.26 seconds, and is 0.5 sec-onds long. For each frame, the EPQ, PPQ, GNE and corre-lation coefficients are combined into a pair of componentmeasures, called Irregularity and Noise. We use the aver-age of the Irregularity and Noise components over allthese frames [4].

Classification Test DataThis study makes use of the Kay Elemetrics disorderedvoice-database (KayPENTAX Model 4337, New Jersey,USA), which contains 707 examples of disordered andnormal voices from a wide variety of organic, neurologicaland traumatic voice disorders. This represents examples ofall three types of disordered voice speech signals (Types I,II and III). There are 53 control samples from normal sub-jects. Each speech sample in the database was recorded

under controlled, quiet acoustic conditions, and is onaverage around two seconds long, 16 bit uncompressedPCM audio. Some of the speech samples were recorded at50 kHz and then downsampled with anti-aliasing to 25kHz. Used in this study are sustained vowel phonations,since this controls for any significant nonstationarity duechanges in the position of the articulators such as thetongue and lips in running speech, which would have anadverse effect upon the analysis methods. For calculatingthe Irregularity and Noise components, the signals areresampled with anti-aliasing to 44.1 kHz.

ResultsFigure 6 shows hoarseness diagrams after Michaelis [4]constructed using the speech data and RPDE and DFAmeasures, the derived irregularity and noise componentsof Michaelis, along with the same diagrams using twoother combinations of the three classical perturbationmeasures for direct comparison. The three classical meas-ures are jitter, shimmer and NHR (Noise-to-HarmoniesRatio) [1]. The normalised RPDE, DFA scaling exponentsand derived irregularity and noise components are calcu-lated for each of the K = 707 speech signals. Where the tra-ditional perturbation algorithms did not fail to produce aresult, the traditional perturbation values were calculatedfor a smaller subset of the subjects, see [1] for details ofthese traditional algorithms. Also shown in figure 6 is theresult of the classification task applied to the dataset; thebest classification boundary is calculated using bootstrapresampling over 1000 trials. Table 1 summarises all theclassification performance results for the classificationtasks on the hoarseness diagrams of figure 6. The RPDEparameters were the same as for figure 3, and the DFAparameters were the same as for figure 5.

DiscussionAs shown in table 1, of all the combinations of the newand traditional measures, and derived irregularity andnoise components, the highest overall correct classifica-tion performance of 91.8 ± 2.0% is achieved by the RPDE/DFA pair. The combination of jitter and shimmer leads tothe next highest performance. These results confirm that,

Table 1: Summary of disordered voice classification results. Summary of disordered voice classification task performance results, for several different combinations of the new measures, the derived irregularity (Irreg) and noise (Noise) components of Michaelis [4], and traditional perturbation measures, jitter (Jitt), shimmer (Shim) and noise-to-harmonics ratio (NHR). The RPDE parameters were the same as for figure 4. and the DFA parameters were the same as for figure 5.

Combination Subjects True Positive True Negative Overall

RPDE/DFA 707 95.4 ± 3.2% 91.5 ± 2.3% 91.8 ± 2.0%Jitt/Shim 685 86.9 ± 6.9% 81.0 ± 4.7% 81.4 ± 3.9%Shim/NHR 684 91.4 ± 5.9% 79.8 ± 4.7% 80.7 ± 4.0%Irreg/Noise 707 78.4 ± 6.2% 90.5 ± 4.9% 79.3 ± 5.5%Jitt/NHR 684 93.2 ± 7.4% 75.0 ± 5.5% 76.4 ± 4.8%



compared under the same, simple classifier approach, thenew nonlinear measures are more accurate on averagethan traditional measures or the derived irregularity andnoise components. We will now discuss particular aspectsof these results in comparison with traditional perturba-tion measures.

Feature DimensionalityThe curse of dimensionality afflicts all challenging data anal-ysis problems [55]. In pattern analysis tasks such as auto-mated normal/disordered separation, increasing the sizeof the feature vector (in this case, the number of measures

J in the classifier vector v) does not necessarily increase theperformance of the classifier in general. This is because thevolume of the feature space (the space spanned by the pos-sible values of the measures) grows exponentially with thenumber of features. Therefore, the limited number ofexamples available to train the classifier occupy anincreasingly small volume in the feature space, providinga poor representation of the mapping from features toclasses that the classifier must learn [55]. Therefore thenew measures help to mitigate this problem of dimen-sionality, since only these two new measures are requiredto obtain good separation performance. By comparison,

"Hoarseness" diagramsFigure 6"Hoarseness" diagrams. "Hoarseness" diagrams illustrating graphically the distinction between normal (blue '+' symbols) and disordered (black '+' symbols) on all speech examples from the Kay Elemetrics dataset, for (a) the new measures return period density entropy (RPDE) (horizontal axis) and detrended fluctuation analysis (DFA) (vertical axis), (b) for the irregularity (horizontal) and noise (vertical) components of Michaelis [4], (c) for classical perturbation measures jitter (horizontal) and noise-to-harmonics ratio (NHR) (vertical) and (d) shimmer (horizontal) against NHR (vertical). The red dotted line shows the best normal/disordered classification task boundary over 1000 bootstrap trials using quadratic discriminant analysis (QDA). The values of the RPDE and DFA analysis parameters were the same those in the analysis of figures 3 and 5 respectively. The logarithm of the classical perturbation measures was used to improve the classification performance with QDA.



we need to calculate four different measures in orderobtain the irregularity and noise components [4].

Feature Redundancy – Information ContentIt is also important to use as few features as possiblebecause in practice, increasing the number of featurescauses excessive data to be generated that may well con-tain redundant (overlapping) information. The actual,useful information contained in these vectors has a muchsmaller dimensionality. For clinical purposes, it is impor-tant that only this useful data is presented. This effect ofredundant information for the traditional measures canbe clearly seen in figure 6, where combinations of pairs of(the logarithms of) these measures are seen to clusteraround a line or curve in the feature space, indicating highpositive correlation between these measures. Traditionalmeasures occupy an effectively one-dimensional object inthis two-dimensional space. The irregularity and noisecomponents occupy more of the area of the feature spacethan traditional measures, and the new measures arespread evenly over the same space.

Arbitrary Parameters – ReproducibilityMinimising the number of arbitrary parameters used tocalculate these measures is necessary to avoid selecting anexcessively specialised set of parameters that leads, forexample, to good normal/disordered separation on a par-ticular data set but does not generalise well to new data.

Many parameters are required for the algorithms used incalculating traditional perturbation measures [4,5,7]. Forexample, the waveform matching algorithm [1] requiresthe definition of rough markers, upper and lower pitchperiod limits, low-pass filter cutoff frequencies, band-width and order selection parameters, and the number ofpitch periods for averaging should these pitch period lim-its be exceeded [58]. Similarly, in just one of the noisemeasures (glottal-to-noise excitation ratio) used inMichaelis [4], we can determine explicitly at least fourparameters relating to linear prediction order, bandpassfilter number, order, cutoff selection, and time lag rangeparameters. There are two additional parameters for thelength and starting sample of the part of the signalselected for analysis.

Our new measures require only five arbitrary parametersthat must be chosen in advance: the length of the speechsignal N, the maximum recurrence time Tmax, and thelower value, upper value and increment of the DFA inter-val lengths L. We have also shown, using analytical results,that we can calibrate out the dependence upon the statespace close recurrence radius r, the time-delay reconstruc-tion dimension d and the reconstruction delay τ.

Interpretation of ResultsWe have found, in agreement with Titze [8] and Carding[2], that perturbation measures cannot be obtained for allthe speech sounds produced by subjects (see table 1). Thislimits the clinical usefulness of these traditional measures.By contrast, the new measures presented in this chapter donot suffer from this limitation and are capable of measur-ing, by design, all types of speech signals.

Taking into account the classification performance achiev-able using a simple classifier, the number of these meas-ures that need to be combined to achieve effectivenormal/disordered separation, the number of arbitraryparameters used to calculate the measures, and the inde-pendence of these measures, traditional approaches andderived irregularity and noise components are seen to beconsiderably more complex than the new measures devel-oped in this paper. The results of the classification com-parison with traditional measures and the irregularity andnoise components suggest that, in order to reach the clas-sification performance of the new measures, we will eitherneed much more complex classifiers, or need to combinemany more classical features together [5-7]. It is thereforenot clear that traditional approaches capture the essentialbiomechanical differences between normal and disorderedvoices in the most parsimonious way, and an excessivelycomplicated relationship exists therefore between the val-ues of these measures and extent of the voice disorder. Asa final comment, we note that the classical perturbationmeasures were, for the majority of signals, able to producea result regardless of the type of the signal. This is consist-ent with the findings of other studies [4], where for TypeII/III and random noise signals, the correct interpretationof these measures breaks down. Therefore, although it isno longer possible in these cases to assign a coherentmeaning to the results produced by these measures, thisdoes not per se mean that there is not some as yetunknown connection between disorder and the thesemeasures. For this reason, we do not discard the results ofthese measures for Type II/III and random cases.

Limitations of the New MeasuresThere are certain limitations to the new measures in clini-cal practice. These measures rely upon sustained vowelphonation, and sometimes subjects experience difficultyin producing such sounds, which limits the applicability.Also, at the beginning of a sustained vowel phonation, thevoice of many subjects may require some time to settleinto a more stable vibration. As such, discarding thebeginning of the phonation is usually a prerequisite (butthis does not adversely affect the applicability of thesemethods). Nonetheless, the extent of breathiness inspeech is not usually affected in this way. In practice werequire that the subject maintains a constant distancefrom the microphone when producing speech sounds;



this can be achieved, for example, with the use of head-mounted microphones. We note that these limitationsalso apply to existing measures.

Possible Improvements and ExtensionsThere are several improvements that could be made tothese new measures. Firstly, every arbitrary parameterintroduces extra variability that affects the reliability of theresults. Much as it has been possible to calibrate out thedependence upon the RPDE parameters using analyticalresults, a theoretical study of the DFA interval lengthsbased upon typical sustained phonation recurrence peri-ods could reveal values that would be found for all possi-ble speech signals. These would be related to the samplingtime Δt. The particular choice of normalisation function gfor the scaling exponent might affect the classification per-formance, and better knowledge of the possible range ofα values using theoretical studies of the DFA algorithmwould be useful for this. It should also be possible toincrease the recurrence time precision of the RPDE analy-sis by interpolating the state space orbits around the timesof close recurrence n0, n1. It should then be possible toachieve the same high-resolution as waveform matchingtechniques which would make RPDE competitive for thedetailed analysis of Type I periodic sounds.

ConclusionIn this study, in order to directly characterise the two mainbiophysical factors of disordered voices: increased nonlin-ear, complex aperiodicity and non-Gaussian, aeroacousticbreath noise, we have introduced recurrence and scalinganalysis methods. We introduced a new, combined non-linear/stochastic signal model of speech production thatis capable, in principle, of producing the wide variation inbehaviour of normal and disordered voice examples. Toexploit the output of this model in practice, and hence alltypes of normal and disordered voices, we explored theuse of two nonlinear measures: the recurrence period den-sity entropy and detrended fluctuation analysis.

Our results show that, when the assumptions of themodel hold under experimental conditions (in that thespeech examples are sustained vowels recorded underquiet acoustic conditions), we can directly characterise thetwo main factors of aperiodicity and breath noise in dis-ordered voices and thus construct a "hoarseness" diagramshowing the extent of normality/disorder from a speechsignal. The results also show that on average, over allbootstrap resampling trials, these two measures alone arecapable of distinguishing normal subjects from subjectswith all types of voice disorder, with better classificationperformance than existing measures.

Furthermore, taking into account the number of arbitraryparameters in algorithms for calculating existing perturba-

tion measures, and the number of these existing measuresthat need to be combined to perform normal/disorderedseparation, we have shown that existing approaches areconsiderably more complex. We conclude that the nonlin-earity and non-Gaussianity of the biophysics of speechproduction can be exploited in the design of signal analy-sis methods and screening systems that are better ablecharacterise the wide variety of biophysical changes aris-ing from voice disease and disorder. This is because ulti-mately the biophysics of speech production generate thewidely varying phenomenology.

Appendix – Mathematical ProofsPeriodic Recurrence Probability DensityWe consider the purely deterministic case, i.e. when themodel of equation (1) has no forcing term ε(t). Thus themeasured time series is purely deterministic and points inthe time series follow each other in an exactly prescribedsequence. When the measured, trajectory sn is a purelyperiodic orbit of finite period k steps, there is an infinitesequence of points {rn}, n ∈ Z in the reconstructed statespace with rn = rn+k, and rn ≠ rn+j for 0 <j <k.

Picking any point s in the reconstructed state-space, thereare two cases to consider. In the first case, if s = rn for somen, then s is not the same as any other points in the peri-odic orbit except for rn+k, so that the trajectory returns withcertainty for the first time to this point after k time steps.This certainty, with the requirement the that probabilityof first recurrence is normalised for T = 1, 2... implies that:

In the second case when s ≠ rn for any n, the trajectorynever intersects the point so that there are also never anyfirst returns to this point. All the points in the recon-structed space form a disjoint partition of the whole space.Thus the probability of recurrence to the whole space isthe sum of the probability of recurrence to each point inthe space separately, appropriately weighted to satisfy therequirement that the probability of first recurrence to thewhole space is normalised However, only the k distinctpoints of the periodic orbit contribute to the total proba-bility of first recurrence to the whole space. Therefore, theprobability of first recurrence is:

Uniform i.i.d. Stochastic Recurrence Probability DensityConsider the purely stochastic case when the nonlinearterm f is in equation (1) is zero and the stochastic forcingterm is an i.i.d. random vector. Then the measured trajec-tory sn is also a stochastic, i.i.d. random vector. Since all

P T rr k

s( ) .= ==⎧

⎨⎩

1

0

if

otherwise(23)

P Tk

P T rr k

ii

k( ) ( ) .= = =

=⎧⎨⎩=

−∑1 1

00

1

rif

otherwise(24)



the tune series are normalised to the range [-1,1] theneach member of the measurement takes on a value fromthis range. Then the trajectories sn occupy the recon-structed state-space which is the region [-1, 1]m, and eachco-ordinate sn is i.i.d. uniform. We form a equal-sized par-tition of this space into Nm (hyper)-cubes, denoting eachcubical region R. The length of the side of each cube R isΔs = 2/N. Then the probability of finding the trajectory inthis cube is PR = Δsm/2m. Since the co-ordinates sn are uni-form i.i.d., then the probability of first recurrence of timeT to this region R is geometric [51]:

This is properly normalised for T = 1, 2.... However, werequire the probability of first recurrence to all possiblecubes. The cubes are a disjoint partition of the total recon-struction space [-1, 1]m. Thus the probability of recurrenceto the whole space is the sum of the probability of recur-rence to each cube separately, appropriately weighted tosatisfy the requirement that the probability of recurrenceto the whole space is normalised. Since the probability offirst recurrence to each cube R, PR(T) is the same, the prob-ability of recurrence to all cubes is:

For small cube side lengths Δs and close returns algorithmradius r, the first recurrence probability determined by theclose returns algorithm is then:

Similarly, for small close returns radius r and/or for largeembedding dimensions m, 1 - rm/2m ≈ 1 so that:

Note that for fixed m and r this expression is constant.Since the close returns algorithm can only measure recur-

rence periods over a limited range 1 ≤ T ≤ Tmax, and wenormalise the recurrence histogram R(T) over this rangeof T, then the probability of first recurrence is the uniformdensity:

which is proportional to the expression rm/2m above.Thus, up to a scale factor, for a uniform i.i.d. stochastic sig-nal, the recurrence probability density is uniform.

Authors' contributionsMAL lead the conceptual design of the study, developedthe mathematical methods, wrote the software and dataanalysis tools, and prepared and analysed the data. PEMparticipated in the conceptual design of the study and themathematical methods. SJR participated in the discrimi-nant analysis of the data. DAEC participated in the datapreparation. IMM contributed to the development of themathematical methods. All authors read and approvedthe manuscript.

Additional material

P T P Ps s

R R RT

m

m

m

m

T

( ) [ ] .= − = −⎡

⎣⎢⎢

⎤

⎦⎥⎥

−−

12

12

11

Δ Δ

(25)

P Ts

P T Ns

P Tm

m Ri

Nm

m

m R

m

( ) ( ) ( )= ==∑ Δ Δ

2 21

(26)

= − = −⎡

⎣⎢⎢

⎤

⎦⎥⎥

−−

2

21

21

21

1m

m

m

m R RT

m

m

m

m

T

s

sP P

s s

ΔΔ Δ Δ

[ ] .

(27)

P Ts s r rm

m

m

m

T m

m

m

m

T

( ) .= −⎡

⎣⎢⎢

⎤

⎦⎥⎥

≈ −⎡

⎣⎢⎢

⎤

⎦⎥⎥

− −Δ Δ2

12 2

12

1 1

(28)

P Trm

m( ) .≈

2(29)

Additional file 1Close returns algorithm implemented in C with Matlab MEX interface. Standard ASCII text file format.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S1.c]

Additional file 2Close returns algorithm compiled as a DLL for Windows. Standard Win-dows DLL format.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S2.dll]

Additional file 3Efficient implementation of the detrended fluctuation analysis (DFA) algorithm written in C with Matlab MEX interface. Core C code. Stand-ard ASCII text file format.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S3.m]

Additional file 4DFA algorithm core compiled as a DLL for Windows. Standard Windows DLL format.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S4.c]

P TT

( ) ,max

≈ 1(30)


http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S1.c

http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S2.dll

http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S3.m

http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S4.c


AcknowledgementsMax Little acknowledges the financial support of the Engineering and Phys-ical Sciences Research Council, UK, and wishes to thank Prof. Gesine Rein-ert (Department of Statistics, Oxford University, UK), Mr Martin Burton (Radcliffe Infirmary, Oxford, UK) and Prof. Adrian Fourcin (University Col-lege London, London, UK) for valuable discussions and comments on early drafts of this paper.

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Additional file 5Matlab function wrapper for above DFA algorithm implementation. Standard ASCII Matlab script file format. Type 'help fastdfa.m' at the Matlab command prompt for usage instructions.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-6-23-S5.dll]


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