Improved Cole parameter extraction based on the least absolute deviation method

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Physiol. Meas. 34 (2013) 1239–1252 doi:10.1088/0967-3334/34/10/1239

Improved Cole parameter extraction based on theleast absolute deviation method

Yuxiang Yang1, Wenwen Ni1, Qiang Sun2, He Wen3

and Zhaosheng Teng3

1 Department of Precision Instrumentation Engineering, Xi’an University of Technology, Xi’an,People’s Republic of China2 Department of Electronic Engineering, Xi’an University of Technology, Xi’an,People’s Republic of China3 Department of Instrumentation Science and Technology, Hunan University, Changsha,People’s Republic of China

E-mail: yyxflyinger@gmail.com and yangyuxiang@xaut.edu.cn

Received 11 June 2013, accepted for publication 6 August 2013Published 11 September 2013Online at stacks.iop.org/PM/34/1239

AbstractThe Cole function is widely used in bioimpedance spectroscopy (BIS)applications. Fitting the measured BIS data onto the model and then extractingthe Cole parameters (R0, R∞, α and τ ) is a common practice. Accurate extractionof the Cole parameters from the measured BIS data has great significance forevaluating the physiological or pathological status of biological tissue. Thetraditional least-squares (LS)-based curve fitting method for Cole parameterextraction is often sensitive to noise or outliers and becomes non-robust. Thispaper proposes an improved Cole parameter extraction based on the leastabsolute deviation (LAD) method. Comprehensive simulation experiments arecarried out and the performances of the LAD method are compared with thoseof the LS method under the conditions of outliers, random noises and bothdisturbances. The proposed LAD method exhibits much better robustness underall circumstances, which demonstrates that the LAD method is deserving asan improved alternative to the LS method for Cole parameter extraction for itsrobustness to outliers and noises.

Keywords: bioimpedance spectroscopy, Cole function, curve fitting, parameterextraction, least absolute deviation

(Some figures may appear in colour only in the online journal)

0967-3334/13/101239+14$33.00 © 2013 Institute of Physics and Engineering in Medicine Printed in the UK & the USA 1239

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1. Introduction

Bioimpedance spectroscopy (BIS) is a kind of biological tissue monitoring technology basedon multi-frequency, complex impedance measurements, which can reflect the physiologicaland pathological status of biological tissue (Grimnes and Martinsen 2008). As a non-invasivedetection technique, BIS technology has been regarded as one of the most potential methodsfor 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), diagnosingmammary 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 thephysiological or pathological status, and the Cole function has been widely adopted for thisprocedure. The Cole function, given by KS Cole in 1940 (Cole 1940), is an empirically derivedequation representing the tissue impedance within one dispersion in the form

Z(ω) = R∞ + R0 − R∞1 + (jωτ )α

, (1)

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 theposition of the center of the circle below the horizontal axis and τ is a characteristic timeconstant corresponding to a characteristic frequency fC (the frequency at which the reactanceis maximum) (Bernt et al 2011),

fC = ω

2π= 1

2πτ. (2)

The Cole parameters (R0, R∞, α and τ ) are therefore the base of the BIS data analysis, andfitting the complex BIS measurements data onto the Cole equation (1) and then extracting theCole parameters has become a common practice in BIS applications (Ayllon et al 2009). Forexample, in the most spread and known body composition assessment (BCA), fitting the BISdata to the Cole equation to obtain the two Cole parameters, R0 and R∞, is a key process inobtaining the body fluid distribution (Buendia et al 2011b). In fact, Cole parameters extractedfrom the obtained BIS data have been used as the major indicators of the physiological andpathological 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 essentialtask for the BIS applications.

A key procedure for Cole parameter extraction is to fit a semi-circular arc in the complexplane called the Cole plot, in which the resistive part R (on the horizontal axis) is plottedagainst the conjugate part of the reactance X (on the vertical axis). A typical Cole plot isshown 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 asthe frequency f increases (Cole 1940). It is noteworthy that the Cole plot is just an impedanceplot, without frequency information on it. In figure 1, R0 and R∞ are the intersections of the arcand the horizontal (real) axis, and the semi-circle has an approximated radius (R0–R∞)/2. α isa measure of the position of the center of the circle below the horizontal axis. τ correspondsto the inverse of the angular frequency ω at the impedance with the highest reactance (Wardet al 2006).

Improved Cole parameter extraction based on the LAD method 1241

Figure 1. Cole plot is the plot of negative tissue reactance of –X against tissue resistance R atall frequencies, showing the form of the semi-circle (frequency increases from right to left alongthe locus). Cole parameters include R0, R∞, α and τ . R0 is the resistance at zero frequency andR∞ the resistance at infinite frequency. α is an empirical exponent with values between 0 and 1 asa measure of the position of the center of the circle below the horizontal axis. τ is a characteristictime constant corresponding to a characteristic frequency f C (the frequency at which the reactanceis maximum), where f C = 1/(2πτ ). ei indicates a radial error between a measured data pointPi(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 least-squares (LS) fitting technique or one of its modifications is used for this purpose. Generallythere are three approaches to Cole parameter fitting:

(a) without requiring direct measurement of the real and imaginary impedance parts (Elwakiland 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 applicationssuch as BCA (Buendia et al 2011a), and may simplify the hardware design of future impedancemeasurement devices (Buendia et al 2011b). But for most impedance instruments and inbroader applications, complex impedance measurements have been widely implemented, soapproach (c) is still the most common practice in BIS data analysis, in which the complexnonlinear LS fitting was introduced to the field by Macdonald et al (Macdonald and Garber1977, Macdonald 1992). Later, Kun et al (1999) proposed an iterative LS fitting algorithm forCole parameter extraction, in which a semi-circle with known circle core (x0, y0) and radiusr0 is fitted using BIS data, and then the Cole parameters can be calculated through geometricalgraph analysis.

All of these LS-based fitting methods mentioned above perform well if data errors arepreferably normally or near-normally distributed and not infected with big outliers (Dasguptaand Mishra 2004). However, the LS fitting may be sensitive to outliers and become non-robust 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 difficultto guarantee as being normally distributed because the number of BIS data, namely thenumber of frequency points in BIS measurements, is relatively small (usually from severalto tens). There is a literature which reported that the Cole parameter estimation based on LS

1242 Y Yang et al

fitting is sensitive to the accuracy of measurements from a collected dataset (Freeborn et al2011).

The least absolute deviation (LAD) method, which is also known as the L1 method andhas an equally long history as the LS method (Chen et al 2008), has been shown to be a robustalternative to the LS method in small samples for its robustness to outliers (Siemsen and Bollen2007). In this paper, we propose an improved curve fitting method based on the LAD methodto extract the Cole parameters. Comprehensive simulation experiments are carried out and theperformances of the LAD-based curve fitting are compared with the LS-based curve fittingunder the conditions of outliers, random noises and both disturbances. Moreover, the essentialdifferences 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 distancebetween 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; mis the number of measurement frequencies):

F(x0, y0, r0) =m∑

i=1

e2i =

m∑i=1

(√(xi − x0)2 + (yi − y0)2 − r0

)2. (3)

The LS-based optimal parameter set (x0, y0, r0) is determined by minimizing the squarederror function in equation (3). The method used to calculate parameters (x0, y0, r0) in this paperis 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 L1norm problem, is a mathematical optimization method that attempts to minimize the sum ofabsolute errors (Chen et al 2008). The LAD error function for the Cole fitting is the sum ofabsolute values of ei and can be expressed as

f (x0, y0, r0) =∑m

i=1|ei| =

∑m

i=1

∣∣√(xi − x0)2 + (yi − y0)2 − r0

∣∣. (4)

The LAD-based optimal Cole parameter set (x0, y0, r0) could be obtained by minimizing theLAD error function f (x0, y0, r0).

Unlike the LS method which has calculation convenience because minimizing theEuclidean norm (L2 norm) is amenable to calculus methods, the LAD method has hadmathematical difficulty in working with absolute value functions for lack of amenabilityto calculus methods for a long time (Dasgupta and Mishra 2004). But in recent years, thiscomputational difficulty has been entirely overcome by the availability of computing powerand 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-Newtonmethod for solving the unconstrained nonlinear optimization problems (Head and Zerner1985). There is a free pBFGS C++ program package which uses the BFGS quasi-Newtonmethod (www.loshchilov.com/pbfgs.html). Also, in the Matlab Optimization Toolbox, thefminunc function uses BFGS with cubic line search when the problem size is set to ‘medium

Improved Cole parameter extraction based on the LAD method 1243

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

D2: random noises 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

D3: D1 and D2 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

scale’ (Mathworks Documentation Center 2013). Both the pBFGS and the fminunc functioncould be adopted to realize the LAD estimation. And as iterative approaches, both algorithmsneed 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 havedifferent outputs at various initial values. In order to objectively compare the performanceof the LS method and the LAD method, we set the same iterative initial values of the targetparameters (x0, y0, r0):⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

x(0)

0 = 1

m

m∑i=1

xi

y(0)

0 = 1

m

m∑i=1

yi

r(0)

0 = [max(xi) − min(xi)] /2

. (5)

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, whileyi 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 theparameters R0 = 150 �, R∞ = 50 �, α = 0.8, τ = 3.0 × 10−6, which represent a set oftypical characteristic parameters of muscle impedance (Rigaud et al 1995). The 32 frequencypoints ( f1, f2, . . . , f32) range from 1 kHz to 1 MHz with a homogeneously logarithmicaldistribution. The characteristic frequency f C = 53.0516 kHz according to equation (2). Allthese standard data are exactly on the Cole locus with circle core x0 = 100, y0 = −16.2460and r0 = 52.5731.

The simulation datasets include three subsets D1, D2 and D3 as listed in table 1, whichare added outliers, random noises and both disturbances, respectively.

Let (xsi, ysi) be the standard BIS data generated from the Cole function. The dataset D1with outliers is calculated by{

xi = xsi + 0.3 × r0 × cos(θi)

yi = ysi − 0.3 × r0 × sin(θi)(i = 13, 25), (6)

1244 Y Yang et al

(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 R0 R∞ α f C

Std data 150.0000 eR0 (%) 50.0000 eR∞ (%) 0.8000 eα (%) 53.0516 e fC (%)LS fitting 150.1104 0.0736 49.6020 0.7959 0.8237 2.9593 47.0124 11.3836LAD fitting 150.0000 0.0000 50.0000 0.0000 0.8000 0.0000 53.0516 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

θi = arccos

(xsi − x0

r0

). (7)

The dataset D2 with random noises added is calculated according to the equation{xi = xsi + 0.1 × rand × r0 × cos(θi)

yi = ysi + 0.1 × rand × r0 × sin(θi)(i = 2, 4, . . . , 32), (8)

where rand denotes a series of random decimals with continuous uniform distributions on theinterval (−1, 1).

4. Simulation results

In this section, the performance of the LAD-based curve fitting is compared with the iterativeLS-based fitting. The target Cole parameters are reported as (R0, R∞, α and f C) since thefourth 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 basisof standard Cole function data. The fitting results based on the LS and LAD methods for thedataset D1 are shown in figures 2(a) and (b), respectively, in which the red dashed line denotesthe standard Cole plot, while the blue solid line is the fitted Cole plot (the same in the similarfigures 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.

Improved Cole parameter extraction based on the LAD method 1245

(a) (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 resultsof the LAD method completely coincide with the standard Cole plot, which validates the factthat 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 pointson the basis of standard data. Because the random noises in the dataset D2 are generateddifferently each time, the data deviation distribution will vary and the corresponding fittingexperiment result will fluctuate to a certain extent. To measure and compare the overallperformance of the LS versus LAD curve fitting methods more comprehensively, 50 timesfitting experiments were implemented. The fitted Cole plots based on the LS and LAD methodsin one time are shown in figures 3(a) and (b), respectively, and the total relative errors of the50 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 relativeerrors (eR0 , eR∞ , eα and e fC ), are listed in table 3, where the results were expressed as: meanvalue ± 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 thedataset D2 prove again that the LAD method is much more robust to random noise than theLS method.

4.3. Results under the dataset D3

The dataset D3 is the combination of D1 and D2, in which 0 to ± 10% radial randomnoises are added at the 16 even frequency points, and 30% radial deviations are added attwo odd frequency points on the basis of standard data. For the same reason, 50 times fittingexperiments were implemented in order to thoroughly investigate the performance of the LSversus 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 andf C) in 50 experiments are shown in figures 6(a), (b), (c) and (d), respectively. Also, the statisticresults of the 50 times experiments are listed in table 4 (mean value ± standard deviation).

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Yang

etal

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

Fitting results

Fitting method R0 R∞ α f C

Std data 150.0000 eR0 (%) 50.0000 eR∞ (%) 0.8000 eα (%) 53.0516 e fC (%)LS fitting 149.83 ± 1.01 0.54 ± 0.41 50.45 ± 1.10 1.97 ± 1.29 0.80 ± 0.02 1.67 ± 1.42 53.19 ± 2.31 3.02 ± 3.12LAD fitting 150.00 ± 0.00 0.00 ± 0.00 50.00 ± 0.00 0.00 ± 0.01 0.80 ± 0.00 0.00 ± 0.00 53.05 ± 0.00 0.00 ± 0.01

Improved

Cole

parameter

extractionbased

onthe

LA

Dm

ethod1247

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

Fitting results

Fitting method R0 R∞ α f C

Std data 150.0000 eR0 (%) 50.0000 eR∞ (%) 0.8000 eα (%) 53.0516 e fC (%)LS fitting 150.13 ± 0.76 0.39 ± 0.32 49.67 ± 1.37 2.30 ± 1.59 0.83 ± 0.02 3.45 ± 2.31 52.73 ± 4.62 7.09 ± 4.99LAD fitting 150.01 ± 0.09 0.02 ± 0.05 49.89 ± 0.56 0.42 ± 1.06 0.80 ± 0.01 0.21 ± 0.61 53.15 ± 0.45 0.33 ± 0.80

1248 Y Yang et al

(a) (b)

(c) (d)

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

(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 worsethan the results under the dataset D2, and the LAD-based fitting results appear as obviousdeviations, but still much better than the LS-based fitting results. The fitting experimentsdemonstrate that the LAD method has much better robustness than the LS method even underpoor conditions, but will be discounted if most (more than half) of the data points occur withdeviations.

Improved Cole parameter extraction based on the LAD method 1249

(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 whichminimizes the sum of squared residuals (Euclidean norm) (Dasgupta and Mishra 2004). Butthe LS estimate can be sensitive to outliers and its performance in terms of accuracy andstatistical inferences may be compromised (Chen et al 2008). As a widely known alternativeto the LS method, the LAD (L1) methods minimize the sum of absolute residuals (absolutenorm) (Dasgupta and Mishra 2004). Unlike the LS method, the LAD method is not sensitiveto outliers and produces robust estimates. The LS method has a tendency to be stronglyinfluenced by outliers in small samples since the estimator tends to avoid large discrepanciesbetween 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 avoidinglarge discrepancies and therefore is more robust to outliers. The main advantage of LAD overLS 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 anextreme 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 outliershave added 20% radial deviations. In figure 7(a), the LS-based fitted Cole plot (the blue solidline) 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

1250 Y Yang et al

(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 twoimpedance points remain standard. The red dashed line denotes the standard Cole plot, while theblue solid line is the fitted Cole plot.

impedance points remain standard. Under this circumstance, both the LS-based fitted and theLAD-based fitted Cole plots deviate dramatically from the original curve, but in differentways. In figure 7(c), the LS method always seeks compromised smaller discrepancies for allpoints, thus minimizing the sum of squared residuals. But in figure 7(d), the LAD methodpursues smaller discrepancies for a majority of points, thus minimizing the sum of absoluteresiduals. This experiment tells us that the LAD method maintains better robustness to outliersthan the LS method provided that the majority of data points remain credible, but will becomeless robust if the majority of data points occur with deviations.

The LS method has computational convenience which has led to an increase in itspopularity since the end of the 18th century, but the LAD method has faced mathematicaldifficulty for a long time in history (Dasgupta and Mishra 2004). Due to the developmentsin theoretical and computational aspects, the LAD method has become increasingly popularfor its robustness, and many traditional LS-based algorithms could be improved by the LADmethod in the future.

6. Conclusion

In BIS applications, the Cole function is widely used for characterizing biological tissues andbiochemical materials. Accurate extraction of the Cole parameters from the measured BIS data

Improved Cole parameter extraction based on the LAD method 1251

is of great significance in aiding researchers in evaluating the physiological or pathologicalstatus. The key step of Cole parameter extraction is to find the circle core (x0, y0, r0) by curvefitting according to the experimentally obtained BIS data. The traditional LS-based parameterextraction is sensitive to noise or outliers and becomes non-robust. As a robust alternative tothe LS method, the LAD method is adopted to extract the Cole parameters. Comprehensivesimulation experiments are carried out and the performances of the LAD-based curve fittingare compared with the LS-based curve fitting under the conditions of outliers, random noisesand both disturbances, and the proposed LAD method exhibits a much better robustness underall circumstances. The LAD method has been proven to be an improved alternative to the LSmethod for Cole parameter extraction for its robustness to outliers and noises, and is especiallysuitable for the small-sample situation in BIS applications.

Acknowledgments

This study has been partly supported by several grants from the National Natural ScienceFoundation of China (Nos 30900317, 61001140, 61273271), a grant from the ScientificResearch Plan of Education Bureau of Shaanxi Province, China (No. 12JK0527) and a grantfrom the China Postdoctoral Science Foundation (No. 20110491674).

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