Confidence bands are commonly obtained from linear regression, but they are rarely shown for nonlinear functions. In part this is due to mathematical complexity. A simple, intuitive and easily automated alternative is to employ interval arithmetic to approximate the confidence band. We illustrate the method by applying it to a straight line, and then apply it to the rate equation of a Michaelis-Menten enzyme and a general polynomial. In each case the approximation is generally larger (and never smaller) than the corresponding standard confidence band, but in at least some instances the upper bound of the discrepancy is about 40%

Brown, Simon

Muhamad, Noorzaid

Pedley, Kevin C.

Simcock, David C.

ResearchOnline at James Cook University

 Banat’s University of Agricultural Sciences and Veterinary Medicine from Timisoara,  Contact: web: http://dse.usab-tm.ro, e-mail: bjb@usab-tm.ro  47  CONFIDENCE BAND APPROXIMATION USING INTERVAL ARITHMETIC   Simon Brown1, Kevin C. Pedley2, Noorzaid Muhamad3, David C. Simcock2  1 School of Human Life Sciences, University of Tasmania,Launceston, Tasmania 7250, Australia, Simon.Brown@utas.edu.au 2 Institute of Food, Nutrition and Human Health, Massey University, Palmerston North, New Zealand 3 Universiti Kuala Lumpur, Royal College of Medicine Perak, 30450 Ipoh, Perak, Malaysia  Abstract. Confidence bands are commonly obtained from linear regression, but they are rarely shown for nonlinear functions. In part this is due to mathematical complexity.  A simple, intuitive and easily automated alternative is to employ interval arithmetic to approximate the confidence band.  We illustrate the method by applying it to a straight line, and then apply it to the rate equation of a Michaelis-Menten enzyme and a general polynomial.  In each case the approximation is generally larger (and never smaller) than the corresponding standard confidence band, but in at least some instances the upper bound of the discrepancy is about 40%. Keywords: confidence interval, interval arithmetic, Michaelis-Menten kinetics, polynomial.  Introduction Most experiments yield parameter estimates in the form of  <x> ± , where <x> is an average and  is the standard deviation, standard error or other measure of the error of the estimate.  While this makes the precision of the estimates clear, the significance of the error can be more difficult to assess.  For example, a function y = f(x; ai), i = 1, 2, … n, might have an error associated with each of the n parameters of i, so an estimate of the total error of y (y) would be  niiiiy aaxf122);(  . (1) Considering each ai in isolation can be misleading, especially for nonlinear functions.  However, numerical differentiation is ill-posed in the sense that small variations in ai can result in large differences in the computed derivative [LU & PEREVERZEV 2006, CHENG et al. 2007] which may make the application of (1) problematic [BROWN et al. 2012].  An alternative simple and intuitively attractive approach to the problem would be to estimate (1) using interval arithmetic [Moore 1979], which relies on simple algebraic analysis, some of which might be automated [HICKEY et al. 2001, JAULIN et al. 2001], and does not requires differentiation.  Here we show that the error estimate obtained using interval arithmetic is only about 40% larger than that estimated from (1) for sufficiently large x. We summarise the basic operations of interval arithmetic which we then apply to the approximation of the confidence band of a straight line, to illustrate the approach used.  We then extend this approach to the rectangular hyperbola used in enzyme kinetics [Briggs & Haldane 1925] and in so many other contexts and to a general polynomial.  Interval arithmetic Assuming that X = <x> ± , then the effective lower and upper limits of X are xL = <x> –  and xU = <x> + , respectively, and of course xL  xU.  This interval can be written as X = [xL, xU] and a real number is a degenerate interval in which xL = xU.  If A = [aL, aU] and B = [bL, bU] are intervals defined in this way, then the following basic arithmetic operations can be defined [Moore 1979]  UULL babaBA  ,   LUUL babaBA  ,    UULUULLLUULUULLLbabababababababaAB,,,max,,,,min which reduces to  UULL babaAB ,  if 0 < aL < aU and 0 < bL < bU, and   LUUL bbaaBA 1,1, , if 0  {bL, bU}.   In addition, we make use of the width of the interval A, which is given by   LU aaAw   (2) and  2* UL aaA  , (3)  Available on-line at http://dse.usab-tm.ro  Banats Journal of Biotechnology 2012, III(5),  48  which is the midpoint of the interval A.  Theory For a straight line y = ax + c for which the coefficient estimates are  â = a ± a and ĉ = c ± c, then ŷ = y ± y where (1) yields 2222222cacayxcyay (4) where x is usually taken in relation to the midpoint of the range (xmid) [Dudewicz & Mishra 1988].  Interval analysis of the equation, based on the same coefficient estimates, implies that A = [aL, aU] = [â – a, â + a] and C = [cL, cU] = [ĉ – c, ĉ + c], and yields  UULL cxacxaCAxY  ,  (5) for x > 0, and so the limits are a linear function of x and the width (2) and midpoint (3) of the interval are   caLULUxccxaaYw 2)( (6) and    ycxaccxaaY ULULˆˆˆ21*, (7) respectively.  In this case the error is symmetrical about ŷ = Y* since the error estimated from interval arithmetic (y) is caULy xyyyy   ˆˆ . (8) Using (4) and (8) and assuming that x  0 yields an expression for the error estimated from interval arithmetic (y) in terms of the total error (y)  22222 21 ycacay xx   . (9) From (9) it is clear that (i) 222 2 yyy    for x  0 and (ii) 22 yy    when x = 0 or x → ∞.  Equation (9) can be used to estimate the discrepancy between y and y yyy 22  , (10) which has a maximum at (c/a, 1).  Equation (10) provides an upper bound on the relative difference between y and y   yyy   , since      222 yyyyyyyy   . Combining these bounds yields    01  yyy   (11) which can be improved using the empirical approximation      12yyy   (12) (Figure 1) which facilitates estimation of y from y. The rate of a reaction catalysed by a Michaelis-Menten enzyme depends on the concentration of substrate (S) ,maxSKSVvm   (13) where VmV max is the maximum rate and KmmK   is a measure of the affinity of the enzyme for the substrate.  Using (1), the confidence band of (13) is 22maxSKVvmKmVmv  (14) [Brown et al 2012].  Interval analysis yields    SKSVSKSVVLmUUmL maxmax ,  (15) which means that the bounds are not symmetrical because )()(maxmaxmax SKVVVvvvKmmVmKmVmU   and )()(maxmaxmax SKVVVvvvKmmVmKmVmL   are not equal.  If Km << Km + S and Vmax ± Vm  Vmax then  Figure 1. Confidence band estimates for a straight line (a = 1, a = 0.4, c = 0.3,  c = 0.1).  The grey region represents the confidence  Banat’s University of Agricultural Sciences and Veterinary Medicine from Timisoara,  Contact: web: http://dse.usab-tm.ro, e-mail: bjb@usab-tm.ro  49  band estimated from (4) and the dashed curves represents that estimated using (8). The dotted curves are given by (10) and (12) and the lowest curve is calculated from (4) and (8).   Figure 2.  Confidence band estimates for the rate of reaction of a Michaelis-Menten enzyme (13) assuming Vmax = 1, Vm = 0.01, Km = 0.25 and  Km = 0.05.  The grey region represents the confidence band estimated from (14) and the dashed curves represents that estimated using (15).   The dotted curves are given by (10) and (12) and the lowest curve is calculated from (14) and (16).  vvvvSKVv ULmKmVmv max (16) from which  222maxmax221 vmKmVmmKmVmvSKVSKV   (17)  and v > v since S   0.  While (17) is of a form similar to (9), there are no values of S ≥ 0 for which v = v.  However,  has a maximum at (KmVmax/Vm – Km, 1) and, for sufficiently large S, (11) also applies in this case (Figure 2).  So, as for the straight line, (10) provides an upper bound on the relative difference between y and y   yyy    and this can be improved using (12) for sufficiently large S (Figure 2) which facilitates estimation of y from y. For a general polynomial in x ≥ 0 niii xaxf0)( , each of the coefficients (ai) of which has an error estimate (i), the total error of f(x) estimated from (1) is niiif x022   (18) and the interval expression for the polynomial  is )()(1,)(1)(,)(0000xfxfxxfxxaxaFniiiniiiniiiiniiii  from which niiiULfxxfFFxf0)()(. (19) Using (18), (19) can be written in the same form as (9) and (17) 202210 1221 fniiininijjijifxx    (20) from which it is clear that 22 yy    when x = 0 or x → ∞.  Discussion We have outlined a simple, intuitive and easily automated means of estimating the confidence band of nonlinear functions.  While ξy > εy in each of the three examples considered here, as is apparent from (9), (17) and (20), the discrepancy () is not constant.  However, for a straight line or the Michaelis-Menten equation  is no more than 1 and provides an approximation of the relative difference between ξy and εy when scaled appropriately (12) as is apparent from Figures 1 and 2. We have not defined an apparent upper bound in the case of a general polynomial, but we conjecture that  is of a similar magnitude (11).   In those cases where several coefficients or parameters are estimated, it can be misleading to consider one of them while neglecting the others [BROWN ET AL 2012].  The approach outlined here provides an estimate of the confidence band using only simple  Available on-line at http://dse.usab-tm.ro  Banats Journal of Biotechnology 2012, III(5),  50  algebra.  Obviously, where the statistical significance of a difference in small it is important to carry out the full analysis, but in many cases the slight over-estimate of the confidence band given by interval arithmetic may be sufficient.  References 1. Lu, S.; Pereverzev, S.V., Numerical differentiation from a viewpoint of regularization theory, Mathematics of Computation 2006, 75, 1853-1870. 2. Cheng, J.; Jia, X.Z.; Wang, Y.B., Numerical differentiation and its applications, Inverse Problems in Science and Engineering 2007, 15, 339-357. 3. Brown, S.; Muhamad, N.; Pedley, K.C.; Simcock, D.C., A simple confidence band for the Michaelis-Menten equation, International Journal of Emerging Sciences 2012, in press,  4. Moore, R.E., Methods and applications of interval analysis, Society for Industrial and Applied Mathematics, Philadelphia, 1979. 5. Hickey, T.; Ju, Q.; van Emden, M.H., Interval arithmetic: from principles to implementation, Journal of the Association of Computing Machinery 2001, 48, 1038-1068. 6. Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, E., Applied interval analysis, Springer-Verlag, London, 2001. 7. Briggs, G.E.; Haldane, J.B.S., A note on the kinetics of enzyme action, Biochemical Journal 1925, 19, 338-339. 8. Dudewicz, E.J.; Mishra, S.N., Modern mathematical statistics, John Wiley and Sons, Inc., New York, 1988.    Received: April 28, 2012 Accepted: May 12, 2012     

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Confidence band approximation using interval arithmetic

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Confidence band approximation using interval arithmetic

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