Comparison of neuro-fuzzy systems for classification of transcranial Doppler signals with their chaotic invariant measures

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Abstract

Transcranial Doppler (TCD) is a non-invasive diagnosis method which is used in diagnosis of various brain diseases by measuring the blood flow velocities in brain arteries. In this study, chaos analysis of the TCD signals recorded from the middle arteries of the temporal region of the brain of 82 patients and 24 healthy people was investigated. Among 82 patients, 20 of them had cerebral aneurism, 10 patients had brain hemorrhage, 22 patients had cerebral oedema and the remaining 30 patients had brain tumour. It was found that all of the TCD signals represented nonlinear dynamics and had an underlying low-level determinism. All of the TCD signals were passed through the nonlinearity tests which involved the application of surrogate data method. The maximum Lyapunov exponent (λ1) which is the strongest quantitative indicator of chaos was found to be positive for all TCD signals. The correlation dimension (D2) was found as greater than 2 and as fractional value for all TCD signals. This result indicates that the nonlinear dynamics of the TCD signals corresponds to a strange attractor in phase space which implies a non-ergodic dissipative system having low-level chaotic behaviour. Besides, the values of λ1 and D2 were approximately the same for the TCD signals of the patients having the same brain disease. Relying on this observation, these two chaotic invariant measures were divided into training and test subsets including 52 and 54 subjects, respectively. For comparison purposes, the training set was used to build two different neuro-fuzzy models, namely ANFIS and NEFCLASS. The rule base of the NEFCLASS model was created by applying the samples in the training subset for 1000 epochs. On the other hand, the ANFIS model was trained for 250 epochs until the convergence error has decreased to 0.42 × 10−5. The ANFIS model achieved better classification accuracy than the NEFCLASS model for the samples in the test set. The classification accuracy of the ANFIS model after training was 94.40% whilst this value was found as 88.88% for the NEFCLASS model.

Introduction

TCD study of the adult intracerebral circulation is used to evaluate intracranial stenoses, cerebral arteriovenous malformations, cerebral vasospasm and cerebral hemodynamics in general (Evans, McDicken, Skidmore, & Woodcock, 1989). To make intra-brain blood flow measurements requires sophisticated imaging methods. The Doppler principle utilized in medicine is different from the classical Doppler method. Since the targets in the human body do not emit radiation, it is necessary to transmit a signal into the body and then observe the frequency changes between that transmitted signal and the received signals reflecting from the targets. Under these conditions, there is a shift in the ultrasound frequency given byfd=ft-fr=2ftvcosθ/c,where ft and fr are the transmitted and received ultrasound frequencies, ν is the velocity of the target, c is the velocity of the sound in the medium, and θ is the angle between ultrasound beam and the direction of motion of the target.

The TCD signals may be regarded as a series of discrete values varying in time. This kind of series is called a time series. If a time series produced by a nonlinear system is analysed by linear methods such as power analysis, linear transformations or parametric linear modelling, the critical features of it remains undetected and most parts of this time series can be incorrectly evaluated as random noise. Chaos theory (nonlinear time series analysis) provides some necessary tools to quantitatively analyse an experimental time series such as a TCD signal. The values such as maximum Lyapunov exponent and correlation dimension are some of the quantitative measures of chaos. For example, if the maximum Lyapunov exponent is found to be positive, then we say that there is an underlying chaotic behaviour in the system that generates the nonlinear time series. This means that the system has a picture of chaotic attractor embedded in the phase space which includes nearby orbits that diverges from each other in time. According to the Takens theorem (Takens, 1981), if the real dimension of the attractor is DA, then choosing the embedding dimension DE as DE > 2DA prevents the orbits from crossing each other resulting in a non-periodic flow. The chaotic attractors generally have a correlation dimension which is a fractional value greater than 2. On the other hand, in order to apply nonlinear time series analysis (chaos theory) methods, it must be proven that the TCD signals are both nonlinear and stationary.

The adaptive neuro-fuzzy inference system (ANFIS) (Jang, 1993) is a specific kind of neuro-fuzzy classifier approach of which successful implementations were reported in biomedical engineering (Belal et al., 2002). In this study, an ANFIS which has two inputs and one output has been trained with the chaotic invariant measures belonging to different classes. The output of ANFIS corresponds to the kind of disease. Since there are four different types of patients and also healthy subjects, the output will belong to one of five different classes according to the input.

NEFCLASS is able to learn fuzzy rules and fuzzy sets by simple heuristics. The main idea of NEFCLASS is to quickly create interpretable fuzzy classifiers. The learning algorithm of NEFCLASS has two stages: rule (structure) learning and parameter learning. Rule learning is achieved via a variation of the approach by Wang and Mendel (1992). Each input space corresponding to a feature is partitioned by a given number of initial fuzzy sets. Then, by processing the training data only once, the antecedents for the rules are created by checking which areas of the input space contain data. An evaluation procedure then creates a rule base by assigning appropriate consequents (class labels) to the discovered antecedents and selects only a certain number of rules with good performance (Nauck and Kruse, 1997, Nauck et al., 1996, Nauck et al., 1999). The NEFCLASS model used in this study had two input nodes corresponding to the chaotic invariant measures and five output nodes which represent the four different brain diseases and the healthiness.

In this study, it has been shown that all the TCD signals are not only nonlinear, but also stationary to an acceptable level. Two most strong chaos indicators, namely the maximal Lyapunov exponent and the correlation dimension were calculated for each signal. The results confirm that all of the signals are chaotic.

Both of the ANFIS and NEFCLASS models were evaluated against the test set and the performance measurements were reported to compare these two different neuro-fuzzy classification approaches.

Section snippets

Hardware

The hardware of the system used for this study involves a 2 MHz ultrasound transducer, analog Doppler unit (Multi Doppler Transducer XX, DWL Gmb, Uberlingen, Germany), analog/digital interface board (Sound Blaster Pro-16), and PIII 600 MHz microprocessor PC with printer. The Doppler unit is also equipped with imaging software that makes it possible to focus the sample volume at a desired location in the temporal region.

The analog Doppler unit can work in both continuous and pulse wave modes. The

Experimental results and discussion

In this study, the TCD signals belonging to 82 patients and 24 healthy people were recorded. Among 82 patients, 20 of them had cerebral aneurism, 10 patients had brain hemorrhage, 22 patients had cerebral oedema and the remaining 30 patients had brain tumour. It was found that both nonlinearity and stationarity existed for all the TCD time series. This proved that the methods which would be used to estimate the chaotic quantities like correlation dimension and Lyapunov exponents could be

Conclusion

In this study, the experimental results indicate that the TCD signals represent nonlinear dynamics having an underlying low-level determinism. All of the TCD signals were passed through the nonlinearity tests successfully and the maximum Lyapunov exponent (λ1) was found to be positive for all the TCD signals. The correlation dimension (D2) was found as greater than 2 and as fractional for all TCD signals. These results indicate that the nonlinear dynamics of the TCD signals corresponds to a

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