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Improved Cole parameter extraction based on the least absolute deviation method

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 Physiol. Meas. 34 1239 (http://iopscience.iop.org/0967-3334/34/10/1239) View the table of contents for this issue, or go to the journal homepage for more

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PHYSIOLOGICAL MEASUREMENT

Physiol. Meas. 34 (2013) 1239–1252

doi:10.1088/0967-3334/34/10/1239

Improved Cole parameter extraction based on the least absolute deviation method Yuxiang Yang 1 , Wenwen Ni 1 , Qiang Sun 2 , He Wen 3 and Zhaosheng Teng 3 1 Department of Precision Instrumentation Engineering, Xi’an University of Technology, Xi’an, People’s Republic of China 2 Department of Electronic Engineering, Xi’an University of Technology, Xi’an, People’s Republic of China 3 Department of Instrumentation Science and Technology, Hunan University, Changsha, People’s Republic of China

E-mail: [email protected] and [email protected]

Received 11 June 2013, accepted for publication 6 August 2013 Published 11 September 2013 Online at stacks.iop.org/PM/34/1239 Abstract The Cole function is widely used in bioimpedance spectroscopy (BIS) applications. Fitting the measured BIS data onto the model and then extracting the Cole parameters (R0, R∞, α and τ ) is a common practice. Accurate extraction of the Cole parameters from the measured BIS data has great significance for evaluating the physiological or pathological status of biological tissue. The traditional least-squares (LS)-based curve fitting method for Cole parameter extraction is often sensitive to noise or outliers and becomes non-robust. This paper proposes an improved Cole parameter extraction based on the least absolute deviation (LAD) method. Comprehensive simulation experiments are carried out and the performances of the LAD method are compared with those of the LS method under the conditions of outliers, random noises and both disturbances. The proposed LAD method exhibits much better robustness under all circumstances, which demonstrates that the LAD method is deserving as an improved alternative to the LS method for Cole parameter extraction for its robustness to outliers and noises. Keywords: bioimpedance spectroscopy, Cole function, curve fitting, parameter extraction, least absolute deviation (Some figures may appear in colour only in the online journal)

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1. Introduction Bioimpedance spectroscopy (BIS) is a kind of biological tissue monitoring technology based on multi-frequency, complex impedance measurements, which can reflect the physiological and pathological status of biological tissue (Grimnes and Martinsen 2008). As a non-invasive detection technique, BIS technology has been regarded as one of the most potential methods for early disease diagnosis and is increasingly widely studied in monitoring tissue ischemia (Mellert et al 2011) and muscle and cardiovascular activity (Hornero et al 2013), diagnosing mammary cancer (Czerniec et al 2011), assessing human hydration status (Utter et al 2012), assessing fluid overload for hemodialysis patients (Hur et al 2013), etc. The BIS of a biological structure is normally obtained by measuring the real/resistive (R) and imaginary/reactive (X) impedance components, or the modulus (Z) and phase angle (θ ) (Dantchev and Al-Hatib 1999). The measured BIS data are then used for evaluating the physiological or pathological status, and the Cole function has been widely adopted for this procedure. The Cole function, given by KS Cole in 1940 (Cole 1940), is an empirically derived equation representing the tissue impedance within one dispersion in the form R0 − R∞ , (1) Z(ω) = R∞ + 1 + (jωτ )α where Z(ω) denotes the complex impedance at angular frequency ω and ω = 2π f ( f = frequency), R0 is the resistance at zero frequency, R∞ is the resistance at infinite frequency, α is an empirical exponent (dimensionless) with values between 0 and 1 as a measure of the position of the center of the circle below the horizontal axis and τ is a characteristic time constant corresponding to a characteristic frequency fC (the frequency at which the reactance is maximum) (Bernt et al 2011), ω 1 fC = = . (2) 2π 2π τ The Cole parameters (R0, R∞, α and τ ) are therefore the base of the BIS data analysis, and fitting the complex BIS measurements data onto the Cole equation (1) and then extracting the Cole parameters has become a common practice in BIS applications (Ayllon et al 2009). For example, in the most spread and known body composition assessment (BCA), fitting the BIS data to the Cole equation to obtain the two Cole parameters, R0 and R∞, is a key process in obtaining the body fluid distribution (Buendia et al 2011b). In fact, Cole parameters extracted from the obtained BIS data have been used as the major indicators of the physiological and pathological status in BIS applications such as gastric cancer detection (Keshtkar et al 2012), hepatic tumor diagnosis (Laufer et al 2010), breast cancer discrimination (Kim et al 2007), hemodialysis monitoring (Jaffrin et al 1997), hypoxic ischemic encephalopathy identification (Seoane et al 2005, 2012), etc. Hence, accurate estimation of the Cole parameters is an essential task for the BIS applications. A key procedure for Cole parameter extraction is to fit a semi-circular arc in the complex plane called the Cole plot, in which the resistive part R (on the horizontal axis) is plotted against the conjugate part of the reactance X (on the vertical axis). A typical Cole plot is shown in figure 1, where the fitted semi-circle travels through the original measured data (hollow points) according to certain rules from the right side to the left side along the locus as the frequency f increases (Cole 1940). It is noteworthy that the Cole plot is just an impedance plot, without frequency information on it. In figure 1, R0 and R∞ are the intersections of the arc and the horizontal (real) axis, and the semi-circle has an approximated radius (R0–R∞)/2. α is a measure of the position of the center of the circle below the horizontal axis. τ corresponds to the inverse of the angular frequency ω at the impedance with the highest reactance (Ward et al 2006).

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Figure 1. Cole plot is the plot of negative tissue reactance of –X against tissue resistance R at all frequencies, showing the form of the semi-circle (frequency increases from right to left along the locus). Cole parameters include R0, R∞, α and τ . R0 is the resistance at zero frequency and R∞ the resistance at infinite frequency. α is an empirical exponent with values between 0 and 1 as a measure of the position of the center of the circle below the horizontal axis. τ is a characteristic time constant corresponding to a characteristic frequency f C (the frequency at which the reactance is maximum), where f C = 1/(2π τ ). ei indicates a radial error between a measured data point Pi(xi, yi) and the fitted arc in the radial direction. xi and yi denote the real part and imaginary part, respectively, of the complex impedance Zi.

The Cole parameters are usually evaluated through nonlinear curve fitting, and the leastsquares (LS) fitting technique or one of its modifications is used for this purpose. Generally there are three approaches to Cole parameter fitting: (a) without requiring direct measurement of the real and imaginary impedance parts (Elwakil and Maundy 2010, Freeborn et al 2012); (b) using only the magnitude (modulus) of the measured impedances (Ward et al 2006, Buendia et al 2011a, Bernt et al 2011); (c) using both real and imaginary parts simultaneously (Macdonald 1992, Kun et al 1999). Approaches (a) and (b) have been proposed in recent years for some specific applications such as BCA (Buendia et al 2011a), and may simplify the hardware design of future impedance measurement devices (Buendia et al 2011b). But for most impedance instruments and in broader applications, complex impedance measurements have been widely implemented, so approach (c) is still the most common practice in BIS data analysis, in which the complex nonlinear LS fitting was introduced to the field by Macdonald et al (Macdonald and Garber 1977, Macdonald 1992). Later, Kun et al (1999) proposed an iterative LS fitting algorithm for Cole parameter extraction, in which a semi-circle with known circle core (x0, y0) and radius r0 is fitted using BIS data, and then the Cole parameters can be calculated through geometrical graph analysis. All of these LS-based fitting methods mentioned above perform well if data errors are preferably normally or near-normally distributed and not infected with big outliers (Dasgupta and Mishra 2004). However, the LS fitting may be sensitive to outliers and become nonrobust when data errors are prominently non-normally distributed or contain sizeable outliers (Chen et al 2008, Youshen and Kamel 2008). Unfortunately, the BIS data errors are difficult to guarantee as being normally distributed because the number of BIS data, namely the number of frequency points in BIS measurements, is relatively small (usually from several to tens). There is a literature which reported that the Cole parameter estimation based on LS

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fitting is sensitive to the accuracy of measurements from a collected dataset (Freeborn et al 2011). The least absolute deviation (LAD) method, which is also known as the L1 method and has an equally long history as the LS method (Chen et al 2008), has been shown to be a robust alternative to the LS method in small samples for its robustness to outliers (Siemsen and Bollen 2007). In this paper, we propose an improved curve fitting method based on the LAD method to extract the Cole parameters. Comprehensive simulation experiments are carried out and the performances of the LAD-based curve fitting are compared with the LS-based curve fitting under the conditions of outliers, random noises and both disturbances. Moreover, the essential differences between the LAD and LS methods are discussed. 2. Method 2.1. LS estimation According to the Cole plot as shown in figure 1, the radial error ei is defined by the distance between the measured BIS data point Pi (xi, yi) and the fitted semi-circle in the radial direction. The LS error function is the sum of squared ei at various frequencies (where i = 1 . . . , m; m is the number of measurement frequencies): F (x0 , y0 , r0 ) =

m 

e2i =

i=1

m   

2 (xi − x0 )2 + (yi − y0 )2 − r0 .

(3)

i=1

The LS-based optimal parameter set (x0, y0, r0) is determined by minimizing the squared error function in equation (3). The method used to calculate parameters (x0, y0, r0) in this paper is the iterative LS algorithm reported by Kun et al (1999) in the literature. 2.2. LAD estimation As a widely known alternative to the LS method, the LAD method, also known as the L1 norm problem, is a mathematical optimization method that attempts to minimize the sum of absolute errors (Chen et al 2008). The LAD error function for the Cole fitting is the sum of absolute values of ei and can be expressed as m  m   (xi − x0 )2 + (yi − y0 )2 − r0 . |ei | = (4) f (x0 , y0 , r0 ) = i=1

i=1

The LAD-based optimal Cole parameter set (x0, y0, r0) could be obtained by minimizing the LAD error function f (x0, y0, r0). Unlike the LS method which has calculation convenience because minimizing the Euclidean norm (L2 norm) is amenable to calculus methods, the LAD method has had mathematical difficulty in working with absolute value functions for lack of amenability to calculus methods for a long time (Dasgupta and Mishra 2004). But in recent years, this computational difficulty has been entirely overcome by the availability of computing power and the effectiveness of linear programming (Chen et al 2008). In this paper, determining the minimum value of the LAD error function f (x0, y0, r0) in equation (4) is essentially an unconstrained multivariable nonlinear optimization problem. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is the most popular quasi-Newton method for solving the unconstrained nonlinear optimization problems (Head and Zerner 1985). There is a free pBFGS C++ program package which uses the BFGS quasi-Newton method (www.loshchilov.com/pbfgs.html). Also, in the Matlab Optimization Toolbox, the fminunc function uses BFGS with cubic line search when the problem size is set to ‘medium

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Table 1. The three datasets with outliers or/and noises used for simulation.

Datasets

Data characteristic description

D1: outliers

Added 30% radial deviation at two odd frequency points f 13, f 25, and the rest 30 points remain standard data Added 0 to ± 10% radial random noises at the 16 even frequency points: f 2, f 4, . . . , f 32; the rest half remain standard data Added 30% radial deviation at two odd frequency points f 13, f 25, and added 0 to ± 10% radial random noises at the 16 even frequency points: f 2, f 4, . . . , f 32; the rest 14 points remain standard data

D2: random noises D3: D1 and D2

scale’ (Mathworks Documentation Center 2013). Both the pBFGS and the fminunc function could be adopted to realize the LAD estimation. And as iterative approaches, both algorithms need proper iterative initial values to start computing. 3. Experimental design 3.1. Iterative initial values Iterative initial value is an important factor for iterator-based algorithms, which often have different outputs at various initial values. In order to objectively compare the performance of the LS method and the LAD method, we set the same iterative initial values of the target parameters (x0, y0, r0): ⎧ m 1 ⎪ ⎪ ⎪ x0(0) = xi ⎪ ⎪ m i=1 ⎪ ⎪ ⎨ m . (5) 1 (0) ⎪ y = yi ⎪ 0 ⎪ m i=1 ⎪ ⎪ ⎪ ⎪ ⎩ (0) r0 = [max(xi ) − min(xi )] /2 Apparently, the three formulas in equation (5) are roughly estimated values of (x0, y0, r0) from the BIS dataset Pi(xi, yi) (i = 1, . . . , m). xi means the real part of the impedance Zi, while yi is the imaginary part. 3.2. Simulated datasets The BIS data used for simulation are generated from the Cole function as standard data, and added noises. The standard BIS data are generated according to equation (1) with the parameters R0 = 150 , R∞ = 50 , α = 0.8, τ = 3.0 × 10−6, which represent a set of typical characteristic parameters of muscle impedance (Rigaud et al 1995). The 32 frequency points ( f1 , f2 , . . . , f32 ) range from 1 kHz to 1 MHz with a homogeneously logarithmical distribution. The characteristic frequency f C = 53.0516 kHz according to equation (2). All these standard data are exactly on the Cole locus with circle core x0 = 100, y0 = −16.2460 and r0 = 52.5731. The simulation datasets include three subsets D1, D2 and D3 as listed in table 1, which are added outliers, random noises and both disturbances, respectively. Let (xsi, ysi) be the standard BIS data generated from the Cole function. The dataset D1 with outliers is calculated by  xi = xsi + 0.3 × r0 × cos(θi ) (i = 13, 25), (6) yi = ysi − 0.3 × r0 × sin(θi )

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(a)

(b)

Figure 2. Fitting result comparison between the LS and LAD methods for the dataset D1 (outliers).

Table 2. Results comparison between the LS and LAD fitting methods for the dataset D1.

Fitting results Fitting method Std data LS fitting LAD fitting

R0 150.0000 150.1104 150.0000

α

R∞ eR0 (%) 0.0736 0.0000

50.0000 49.6020 50.0000

eR∞ (%) 0.7959 0.0000

0.8000 0.8237 0.8000

fC eα (%) 2.9593 0.0000

53.0516 47.0124 53.0516

e fC (%) 11.3836 0.0000

where θ i denotes the angle between the radial direction at frequency f i and the horizontal axis, and can be obtained by

xsi − x0 . (7) θi = arccos r0 The dataset D2 with random noises added is calculated according to the equation  xi = xsi + 0.1 × rand × r0 × cos(θi ) (i = 2, 4, . . . , 32), yi = ysi + 0.1 × rand × r0 × sin(θi )

(8)

where rand denotes a series of random decimals with continuous uniform distributions on the interval (−1, 1). 4. Simulation results In this section, the performance of the LAD-based curve fitting is compared with the iterative LS-based fitting. The target Cole parameters are reported as (R0, R∞, α and f C) since the fourth parameter τ is converted into the characteristic frequency f C according to equation (2). 4.1. Results under the dataset D1 For the dataset D1, 30% radial deviations are added at two odd frequency points on the basis of standard Cole function data. The fitting results based on the LS and LAD methods for the dataset D1 are shown in figures 2(a) and (b), respectively, in which the red dashed line denotes the standard Cole plot, while the blue solid line is the fitted Cole plot (the same in the similar figures below). The extracted Cole parameters (R0, R∞, α and f C) and their relative errors (eR0 , eR∞ , eα and e fC ) are listed in table 2.

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(a)

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(b)

Figure 3. Fitting comparison between the LS and LAD methods for the dataset D2 (random noises).

The fitting results indicate that the LS method is sensitive to outliers, while the results of the LAD method completely coincide with the standard Cole plot, which validates the fact that the LAD method is robust to outliers. 4.2. Results under the dataset D2 For the dataset D2, 0 to ± 10% radial random noises are added at the 16 even frequency points on the basis of standard data. Because the random noises in the dataset D2 are generated differently each time, the data deviation distribution will vary and the corresponding fitting experiment result will fluctuate to a certain extent. To measure and compare the overall performance of the LS versus LAD curve fitting methods more comprehensively, 50 times fitting experiments were implemented. The fitted Cole plots based on the LS and LAD methods in one time are shown in figures 3(a) and (b), respectively, and the total relative errors of the 50 times extracted Cole parameters (R0, R∞, α and f C) are shown in figures 4(a), (b), (c) and (d), respectively. The statistic results of the 50 times experiments, as well as their relative errors (eR0 , eR∞ , eα and e fC ), are listed in table 3, where the results were expressed as: mean value ± standard deviation. The fitting results indicate that the LS-based fitting results are sensitive to random noises, while the LAD-based fitting results are nearly not affected. The fitting experiments under the dataset D2 prove again that the LAD method is much more robust to random noise than the LS method. 4.3. Results under the dataset D3 The dataset D3 is the combination of D1 and D2, in which 0 to ± 10% radial random noises are added at the 16 even frequency points, and 30% radial deviations are added at two odd frequency points on the basis of standard data. For the same reason, 50 times fitting experiments were implemented in order to thoroughly investigate the performance of the LS versus LAD curve fitting methods under the outliers and random noise conditions. Similarly, the LS-based and LAD-based fitted Cole plots in one experiment are shown in figures 5(a) and (b), respectively, and the total relative errors of the extracted Cole parameters (R0, R∞, α and f C) in 50 experiments are shown in figures 6(a), (b), (c) and (d), respectively. Also, the statistic results of the 50 times experiments are listed in table 4 (mean value ± standard deviation).

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Table 3. Results comparison between the LS and LAD fitting methods for the dataset D2.

Fitting results Fitting method Std data LS fitting LAD fitting

R0 150.0000 149.83 ± 1.01 150.00 ± 0.00

α

R∞ eR0 (%) 0.54 ± 0.41 0.00 ± 0.00

50.0000 50.45 ± 1.10 50.00 ± 0.00

eR∞ (%) 1.97 ± 1.29 0.00 ± 0.01

0.8000 0.80 ± 0.02 0.80 ± 0.00

fC eα (%) 1.67 ± 1.42 0.00 ± 0.00

53.0516 53.19 ± 2.31 53.05 ± 0.00

e fC (%) 3.02 ± 3.12 0.00 ± 0.01

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Fitting results Fitting method Std data LS fitting LAD fitting

R0 150.0000 150.13 ± 0.76 150.01 ± 0.09

α

R∞ eR0 (%) 0.39 ± 0.32 0.02 ± 0.05

50.0000 49.67 ± 1.37 49.89 ± 0.56

eR∞ (%) 2.30 ± 1.59 0.42 ± 1.06

0.8000 0.83 ± 0.02 0.80 ± 0.01

fC eα (%) 3.45 ± 2.31 0.21 ± 0.61

53.0516 52.73 ± 4.62 53.15 ± 0.45

e fC (%) 7.09 ± 4.99 0.33 ± 0.80

Improved Cole parameter extraction based on the LAD method

Table 4. Results comparison between the LS and LAD fitting methods for the dataset D3.

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(a)

(b)

(c)

(d) Figure 4. Fitting result comparison between the LS and LAD methods for the dataset D2 (random noises).

(a)

(b)

Figure 5. Fitting comparison between the LS and LAD methods for the dataset D3 (outliers + random noises).

The experiments under the dataset D3 show that the LS-based fitting results become worse than the results under the dataset D2, and the LAD-based fitting results appear as obvious deviations, but still much better than the LS-based fitting results. The fitting experiments demonstrate that the LAD method has much better robustness than the LS method even under poor conditions, but will be discounted if most (more than half) of the data points occur with deviations.

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(a)

(b)

(c)

(d) Figure 6. Fitting result comparison between the LS and LAD methods for the dataset D3 (outliers + random noises).

5. Discussion The method of LS (L2) is one of the oldest and most widely used statistical tools which minimizes the sum of squared residuals (Euclidean norm) (Dasgupta and Mishra 2004). But the LS estimate can be sensitive to outliers and its performance in terms of accuracy and statistical inferences may be compromised (Chen et al 2008). As a widely known alternative to the LS method, the LAD (L1) methods minimize the sum of absolute residuals (absolute norm) (Dasgupta and Mishra 2004). Unlike the LS method, the LAD method is not sensitive to outliers and produces robust estimates. The LS method has a tendency to be strongly influenced by outliers in small samples since the estimator tends to avoid large discrepancies between the predicted and the observed scores at the expense of allowing smaller discrepancies. However, the LAD principle avoids the squaring error, and is thus less inclined to avoiding large discrepancies and therefore is more robust to outliers. The main advantage of LAD over LS is its small-sample robustness to outliers (Siemsen and Bollen 2007). The essential difference between the LS and the LAD method can be partly revealed in an extreme case as shown in figures 7(a)–(d), in which the Cole plots are fitted from five points. In figures 7(a) and (b), three impedance points remain standard, while the other two outliers have added 20% radial deviations. In figure 7(a), the LS-based fitted Cole plot (the blue solid line) deviates dramatically from the original curve (the red dashed line), but in figure 7(b), the LAD-based fitted Cole plots remain nearly standard. The conditions in figures 7(c) and (d) are in reverse, where at three outliers are added 20% radial deviations, and the rest two

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(a)

(b)

(c)

(d)

Figure 7. Comparison between the LS and LAD methods in an extreme case of five points. In (a) and (b), three impedance points remain standard, while at the other two points are added 20% radial deviations. In (c) and (d), at three points are added 20% radial deviations, and the rest two impedance points remain standard. The red dashed line denotes the standard Cole plot, while the blue solid line is the fitted Cole plot.

impedance points remain standard. Under this circumstance, both the LS-based fitted and the LAD-based fitted Cole plots deviate dramatically from the original curve, but in different ways. In figure 7(c), the LS method always seeks compromised smaller discrepancies for all points, thus minimizing the sum of squared residuals. But in figure 7(d), the LAD method pursues smaller discrepancies for a majority of points, thus minimizing the sum of absolute residuals. This experiment tells us that the LAD method maintains better robustness to outliers than the LS method provided that the majority of data points remain credible, but will become less robust if the majority of data points occur with deviations. The LS method has computational convenience which has led to an increase in its popularity since the end of the 18th century, but the LAD method has faced mathematical difficulty for a long time in history (Dasgupta and Mishra 2004). Due to the developments in theoretical and computational aspects, the LAD method has become increasingly popular for its robustness, and many traditional LS-based algorithms could be improved by the LAD method in the future. 6. Conclusion In BIS applications, the Cole function is widely used for characterizing biological tissues and biochemical materials. Accurate extraction of the Cole parameters from the measured BIS data

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is of great significance in aiding researchers in evaluating the physiological or pathological status. The key step of Cole parameter extraction is to find the circle core (x0, y0, r0) by curve fitting according to the experimentally obtained BIS data. The traditional LS-based parameter extraction is sensitive to noise or outliers and becomes non-robust. As a robust alternative to the LS method, the LAD method is adopted to extract the Cole parameters. Comprehensive simulation experiments are carried out and the performances of the LAD-based curve fitting are compared with the LS-based curve fitting under the conditions of outliers, random noises and both disturbances, and the proposed LAD method exhibits a much better robustness under all circumstances. The LAD method has been proven to be an improved alternative to the LS method for Cole parameter extraction for its robustness to outliers and noises, and is especially suitable for the small-sample situation in BIS applications. Acknowledgments This study has been partly supported by several grants from the National Natural Science Foundation of China (Nos 30900317, 61001140, 61273271), a grant from the Scientific Research Plan of Education Bureau of Shaanxi Province, China (No. 12JK0527) and a grant from the China Postdoctoral Science Foundation (No. 20110491674). References Ayllon D, Seoane F and Gil-Pita R 2009 Cole equation and parameter estimation from electrical bioimpedance spectroscopy measurements—a comparative study EMBC ’09: Annu. Int. Conf. IEEE Engineering in Medicine and Biology Society pp 3779–82 Bernt J N, Christian T, Ørjan G M and Sverre G 2011 Evaluation of algorithms for calculating bioimpedance phase angle values from measured whole-body impedance modulus Physiol. Meas. 32 755 Buendia R, Gil-Pita R and Seoane F 2011a Cole parameter estimation from the modulus of the electrical bioimpedance for assessment of body composition. A full spectroscopy approach J. Electr. Bioimpedance 2 72–78 Buendia R, Gil-Pita R and Seoane F 2011b Cole parameter estimation from total right side electrical bioimpedance spectroscopy measurements—influence of the number of frequencies and the upper limit Proc. Conf. IEEE Engineering in Medicine and Biology Society pp 1843–6 Chen K, Ying Z, Zhang H and Zhao L 2008 Analysis of least absolute deviation Biometrika 95 107–22 Cole K S 1940 Permeability and impermeability of cell membranes for ions Quant. Biol. 8 110–22 Czerniec S A, Ward L C, Lee M J, Refshauge K M, Beith J and Kilbreath S L 2011 Segmental measurement of breast cancer-related arm lymphoedema using perometry and bioimpedance spectroscopy Support. Care Cancer 19 703–10 Dantchev S and Al-Hatib F 1999 Nonlinear curve fitting for bioelectrical impedance data analysis: a minimum ellipsoid volume method Physiol. Meas. 20 N1–9 Dasgupta M and Mishra S K 2004 Least absolute deviation estimation of linear econometric models: a literature review MPRA Paper 1781 (University Library of Munich, Germany) (http://mpra.ub.uni-muenchen.de/1781/) Elwakil A S and Maundy B 2010 Extracting the Cole–Cole impedance model parameters without direct impedance measurement Electron. Lett. 46 1367–8 Freeborn T J, Maundy B and Elwakil A 2011 Numerical extraction of Cole–Cole impedance parameters from step response Nonlinear Theory Appl. 2 548–61 Freeborn T J, Maundy B and Elwakil A S 2012 Least squares estimation technique of Cole–Cole parameters from step response Electron. Lett. 48 752–4 Grimnes S and Martinsen Ø G 2008 Bioimpedance and Bioelectricity Basics 2nd edn (London: Elsevier) Head J D and Zerner M C 1985 A Broyden–Fletcher–Goldfarb–Shanno optimization procedure for molecular geometries Chem. Phys. Lett. 122 264–70 Hornero G, Diaz D and Casas O 2013 Bioimpedance system for monitoring muscle and cardiovascular activity in the stump of lower-limb amputees Physiol. Meas. 34 189–201 Hur E et al 2013 Effect of fluid management guided by bioimpedance spectroscopy on cardiovascular parameters in hemodialysis patients: a randomized controlled trial Am. J. Kidney Dis. 61 957–65

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