1 Introduction
An important problem of optimization analysis surges when it is desired to guess the parameters , which minimize or maximize a function such as
(1) 
vertical bars mean given that and in this paper curly braces indicate sets. Eq. (1) is called objective function (obf) in optimization analysis (Rivett, 1973, pg. 5) where it represents a process (not necessarily including random components) whose parameters have to be determined for the process to operateat an optimum; econometric systems, water reservoirs and electric networks are some examples. Eq. (1) is also called regression function (rgf
) in regression analysis
(Reg, 2018) where a set of parameter values must be determined from a set of observables . In regression analysis are independent variates (called explanatory variables or predictors) which are usually assumed to be uncertaintyfree and and their associated observed dependent variates (response variables) are . Analytical solutions are possible, and many are well known (see for example (Montgomery et al., 2015)), when involves a linear combination of . In the case of not linearly independent pairs (Seber and Wild, 1989; Schittkowski, 2002)in spite of some empirical approaches subdividing data into subsets with quasi linear regression intervals
(Spearman, 1908; Kärber, 1931; Thompson, 1947; Miller and Urich, 2001; Miller and Ullich, 2004; Oosterbaan, 2017)to deremine subset confidence intervals to estimate uncertainties, it remains true that:
“In general, there is no closedform expression for the bestfitting parameters, as there is in linear regression. Usually numerical optimization algorithms are applied to determine the bestfitting parameters. Again in contrast to linear regression, there may be many local minima of the function to be optimized and even the global minimum may produce a biased estimate. In practice, estimated values of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares ” Non (2018).
The problem of local minima is inherent to the function fitted and cannot be avoided. Yet, efficient minimization algorithms, which start searching from a set of user provided parameters in case of many “well behaved functions” converge towards the global optimum if is within a certain boundary of the global optimum. There is, however, no analytical solution to the problem of knowing the width the boundary of guaranteed convergence to the global optimum (minimum or maximum) for a given rgf.
The gradient () represents the slope of function’s graph, that is points in the direction of the greatest rate of increase or decrease of a function (Apostol, 1969, Ch. 8). If an optimization process fitting the rgf reaches an optimum, , the following implication
(2) 
is true. It is usually assumed that the rgf exactly describes the system whose optimum set of values is being determined, and that differences between model and data (called residuals) result from lack of accuracy in measuring a given which depends on .
Yet, data from the physical world always contains uncertainty which does not result from measurement error. The sources are many:

Observation instrument limitations make the measurements fuzzy. this is the case of the point dispersion function of optical instruments, which is dependent on the wavelength of light used to observe (Scalettar et al., 1996).

Impossibility of accurately measuring something, a classical example deals with the velocity and position of a particle (Heisenberg, 1927).

Uncertainty is essential to life, otherwise any noxious factor would affect equally a whole species population making its extinction likelier, thus any parameter measured on living beings is significantly variable, uncertain, fuzzy (Conrad, 1977).

In high energy physics the existence of a particle was evidenced by an energy peak which had to be differentiated from background noise (The CMS Collaboration, 2015).
In all these instances observations randomly vary, not due to experimental “error” but due to the stochastic nature of the process under study by it self. This is most likely the case for processes such as the ones described with the Hill [Eq. (17)] or Boltzmann [Eq. (45)] equations, when these equations are used mechanistically, not just as curve fitting processes. The intermolecular reaction parameters of the Hill equation are scholastically independent, the maximum effect of speed of catalysis () does not depend on the affinity constant of the reactants () and none of them depends on the molecularity of the reaction (number of molecules of one kind reacting with a molecule of another kind, ) Ariëns et al. (1964); Segel (1975); Érdi and Tóth (1989). A similar reasoning can be used in connection with the Boltzmann equation [Eq. (45)] where and are mechanistically independent. In both cases rgf parameters are causally independent. Briefly unther these conditions we have
Condition 1.
where indicates stochastic independence, the concept of causal independence between variables or factors (Hogg and Craig, 1978; Spohn, 1980; Ghorbal and Schürman, 2002), a notion which has also recently been used in fields such as quantum thermodynamics (abd M. P. Muller and Pastena, 2015; Lostaglio et al., 2015; Goold et al., 2016), is opposite to meaning stochastic dependence. We may write a function representing these conditions based on Eq. (2)
(3) 
, is a Minkowski sum Weisstein (2018) of two independents sets such as
(4) 
as iluistrated in figure 1, Eq. (4) is a bijection for which the following holds:
(5) 
In Eq. (3),
is a random variable with a location parameter
with variability depending on a parameter . This suits “pathological distribution”, such as the Cauchy probability density function (pdf), whose statistical central moments are undefined (have no meaning) and thus neither its population mean nor its variance are defined
(Cramér, 1991; Pitman, 1993; Wolfram, 2003), for which , the median and its width factor. When dealing with other probability functions having defined central momentsdd, Eq. (
3) may be rewritten making where is the mean and is the variance. Since a gradient is a linear combination of all the function’s first derivatives (Apostol, 1969), at an optimum :(6) 
In tems of finite diferences
(7) 
and thus
(8) 
which shows that it is possible to predict small changes in parameter from small changes of ref when the regression function is at an optimum, but fluctuates stochastically, if the partial derivative of the regression function about its parameters known.
If, fluctuates stochastilly about , the fluctuations are the residuals, :
(9) 
This paper proposes that the higher a partial derivative respect to in Eq. (6) is, the more it contributes to rgf changes. Combining Eqs. (8) and (9) as
(10) 
a set of parameter fluctuation estimates.
In this work the set is used to determine how much of the empirical rgf uncertainty at an optimum is contributed by each parameter, and methods are presented to enable statistical comparisons between those parameters determined under different experimental conditions.
2 Methods
2.1 Monte Carlo random variable simulation.
To test the goodness of fitting curves to data, random data with known statistical properties were generated using Monte Carlo simulation (Dahlquist and Björk, 1974). For this purpose sets of pairs were generated as
(11) 
where, as said, is the variance and is the Cauchy pdf scale factor. Thus for population having defined nean and variance:
(12) 
When needed, Gaussian pseudorandom vaiables were generated using the Box and Muller (1958) algorithm as modified by Press et al. (1992). Fundamental to all Monte Carlo simulations (Dahlquist and Björk, 1974) is a good uniform (pseudo) random (PRNG) number generator. Data for all numerical simulations carried out in this work were produced using random numbers (
) with continuous rectangular (uniform) distribution in the closed interval [0,1] or
. All were generated using the 2002/2/10 initializationimproved 623dimensionally equidistributed uniform pseudo random number generator MT19937 algorithm (Matsumoto and Nishimura, 1998; Panneton et al., 2006). The generator has passed the stringent DIEHARD statistical tests (Marsaglia et al., 2003; Bellamy, 2013). It uses 624 words of state per generator and is comparable in speed to other generators. It has a Mersenne prime period of ().The MT19937 requires an initial starting value called seed. The seed used was a 64bit unsigned integer obtained using the /dev/random Linux PRNG , which saves environmental noise from device drivers and other sources into an entropy pool. Device /dev/random gets temporarily blocked, and stops producing random bytes, when the entropy of the device gets low, and commences producing output again when it recovers to safe levels. No such delays were perceived during this work. Using /dev/random seed makes exceedingly unlikely () that the same sequence, , of is used twice. Calculations were programmed in C++ using g++ version 5.4.0 20160609 with C++14 standards, umder Linux Mint version 18.2 running on an Apple MacBook Air computer with 8 GB RAM, Intel ® CoreTM i74650U CPU @ 1.70 GHz × 4 with a 500 GB disk.
2.2 Statistical procedures.
2.2.1 Fitting functions to data.
Functions were adjusted to data using a simplex minimization (Nelder and Mead, 1965). The simplex procedure was designed to minimize diferences between empirical data assumed to obey a function such as , where is a set of observables, and a model function . In this work the simplex was designed to minimize
(13) 
instead of the least squares procedure (Montgomery et al., 2015)
. Least squares give too much weight to outliers which may be due to random data variability (Such as in the Cauchy distribution case, see
3.3.2), but could stem from gross deviation from a prescribed experimental procedure or to error in calculating or recording numerical values.2.2.2 Details of simplex optimizations.
Simplex parameter initialization.
Simplex algorithm requires initial parameter values, , and an initial increment value, . is the initial fraction to change the parameters which is subsequently modified by the algorithm as optimization continues (Nelder and Mead, 1965).
Criteria to stop optimizations.
Optimization continued until one of the following conditions was fulfilled:
Condition 2.
Or
Simulation of residuals.
The simplex was implemented to provide a set , used to calculate the uncertainties of estimated as described in Section 3.
2.2.3 On statistical procedures utilized.
Gaussianity of data was tested witth the JarqueBera test, which allso provides data on skewness and kurtosis of data
(Bera and Jarque, 1981) and with the ShapiroWilks test (Shapiro and Wilk, 1965). Unless otherwise is indicated, data are presented as medians and their 95% confidence intervals (95% CI) calculated using nonparametric Moses (Hollander and Wolfe, 1973) statistics. Other data are presented as medians and their 95% confidence interval calculated with the procedure of Hodges and Lehmann (Hollander and Wolfe, 1973). Statistical significance of differences was decided with Mann–Whitney (Wilcoxon) test. Multiple comparisons were done with the nonparametric KruskallWallis analysis of variance. See Hollander and Wolfe (1973) for all not specified details of nonparametric methods used. Statistical differences between samples were considered significant when the probability that they stem from chance was () (Ioannidis, 2005; Benjamin et al., 2018).3 Results and discussion.
3.1 A challenging data set obtained with a procedure commonly used in cell biology.
Fraction  y_{0}  y_{m}  K_{m}  n 
(%)  (%)  (mg/mL)  
FI 



FII  ()  () 


FIII  
FIV  
FV  ()  ()  () 
Parameters of the modified Hill Equation (18): , offset parameter; , maximum effect; , concentration producing half maximum effect and , is called Hill coefficient or molecularity in some pharmacology and enzymology work (Segel, 1975). The simplex algorithm was initialized with the same set of values for the five fractions: ; was set as 0.1. All data presented as medians and their 95% CI between parenteses. Confidence intervals calculated with the Hodges and Lehman (Hodges, Jr. and Lehmann, 1963) procedure based on data determined as indicated in relation with Eq. (10). Sizes of were: FI, : FII : FIII, ; FIV, and FV, . Differences in were due to sample sizes and data peculiarities due to which Eq. (52) produced undefined values called NaN (Not a Number, such as or ) in C++ or in values such as or called inf in C++. NaN and inf results were eliminated from the calculations. Other details in the text of the communication.
The data shown in Figure 2 (taken from (QuintanaHernández, 2017)) are effects of several fractions (FI – FV) isolated from P. constellatum (Savigny, 1846) which were able to kill 4T1 breast cancer cells. Apparent effects calculated with Eq. (53) are presented in the ordinate, versus fraction concentration indicated in the abscissa (in mg/mL).
There are several oddities in the data, the effects at some concentrations are very disperse (s indicated by the brackets representing 95% CI), and, notably at low concentrations, they indicate negative percentages of death. At the lowest concentrations even median values are slightly, but significantly, bellow zero; this could be expected if the background correction [Eqs. (50) and (51)] actually overcorrects absorbance data. The large variability is most likely a result of subtractive cancellation, combined with the quotient represented by Eq. (52) which are prone to produce variance in [Eq. (52)] and [Eq. (53)].
Gaussianity tests suggests that all data set in the figures were leptokurtic and skewed. The JarqueBera test (Bera and Jarque, 1981) (based on data skewedness and kurtosis) indicated that the probability of data sets in the figures are Gaussian is . The ShaarepiroWilk test (Shapiro and Wilk, 1965) indicated the same low probability of Gaussianity ().
3.1.1 An example using a modified Hill equation.
Experimental method used in (QuintanaHernández, 2017) to estimate cell mortality shown in Figures 2 and 3 (Mosmann, 1983; Denizot and Lang, 1986; Swinehart, 1962) have been cited at least 47278 times in the literature (July 22, 2017, source: https://scholar.google.com
). This indicates that, in spite of its odd management of uncertainty, the method is believed useful by a substantial number of researchers. Notwithstanding the oddities of data in Figure
2, there are several features evidenced by the medians: all five fractions increased cell mortality as concentration raised, and in all cases there is a sigmoid aspect of the dose–effect semilogarithmic plots.The first example using the theory expressed by Eqs. (6) – (10) is fitting data to a modified Hill (1910, 1913) equation. When the Hill equation is plotted as effect versus concentration’s logarithm, a sigmoid curve is produced. In its classical form, the Hill equation is used in enzyme kinetics and in pharmacology to represent the interaction of one or more molecules of subrate with the catalytic site of an enzyme, or of a drug’s molecule with its receptor site; it derives from the mass action law (Rang, 2009):
(15) 
When the effect of a drug or the rate of enzyme cathalysis (), depend linearly on molecules of D binding receptor R, the followng holds (Rang, 2009):
(16) 
Under these conditions is called the molecularity of the reaction. Also, is used in situations where properties of the enzyme or drug receptor are modified during the interaction, the, so called, cooperative schemes, where is plainly named Hill coefficient (Monod et al., 1965; Segel, 1975).
The Hill equation (Hill, 1913) in its original form is
(17) 
which does not include a term for “offset,” occurring when .
Since data in the figures seems to include an overcorrection for the basal absorbance, this modified Hill equation will be used. as a particular case, in our analysis:
(18) 
its first derivatives on are given in Section B.1 as Eqs. (31) – (34).
Figure (3) shows the results of adjusting Eq. (18) to the data of QuintanaHernández (2017). In all cases the simplex optimization started from the same set of values: and . Since Eqs. (50) – (52) produce 24000 points per concentration, the number of pairs in each fraction’s regression analysis ranged 120000 – 144000 in the plots shown in Fig. 3. Interestingly, the curves in Fig. 3 follow, rather closely, the median percent of dead cells at each concentration in all the plots. This is particularly clear for FI and FV. The parameter values describing the curves are in Table 3.1. The curves in Figure 3, and the sets of data in Table 3.1 “look good” but uncertainty estimator for the parameters are necessary to properly state which fraction effect differs from which fractions, specially if the outliers sugested by the 95% CI and “skewness” analysis are considered.
Medians and their 95% CIs of QuintanaHernández’s compounds’ (QuintanaHernández, 2017) sets [Eq. (10)] are presented in Table 3.1. All sets used to guess residuals in Table 3.1 were tested for Gaussianty with the ShapiroWilk and JarqueBera methods, and both procedures predicted a probability that any of the sets is Gaussian.
Fraction  Loops  Parameter  Sk  Ku  Range 
FI  
y_{0}  
y_{m}  ,  
K_{m}  
n  
FII  
y_{0}  
y_{m}  
K_{m}  
n  
FIII 

y_{0}  
y_{m}  ()  
K_{m}  
n  
FIV  ^{*}  
y_{0}  
y_{m}  ()  
K_{m}  
n  
FV  
y_{0}  
y_{m}  ()  
K_{m}  
n 
Parameters of the modified Hill Equation (18): , offset parameter; , maximum effect; , concentration producing half maximum effect, , is called Hill constant; Ku, kurtosis and Sk, skewness. Range, skewness and kurtosis have the usual statistical meanings. Loop, indicates the number of times a parameter was changed during the simplex optimization (Nelder and Mead, 1965). For fractions I, II, III and V optimización stopped when when Condition 2 was fulfilled. In case of FIV the optimization was topped fulfilling Condition 3 after 3 h attempting unsuccessfully to fulfill Condition 2. See the text for further discussion.
3.1.2 An insight on the complexity of the data used for this example
Table 3.1.1 presents data on the number of iterations required by the simplex algorithm to converge to the optimum reaching Condition 2. The only exception is data for FIV, which after 1024007 (number labeled with an asterisk in the table) loops, was still unable to reach Condition 2 and after h of iterations (in the author’s computer) the process was stopped after reaching Condition 3. The table also presents some statistical properties of the sets used to calculate the uncertainty of the parameters characterizing curves fitted to data in Figure 3.
The data in the table indicates that in all cases presented sets are highly leptokurtic and very skewed, and in some cases (as indicated by the ranges presented at the leftmost column of the table), very wide ranges indicated that extreme values were observed. These extreme values are in all likelihood due to subtractive cancellation in Eqs. (50) and (51) combined with division by very small numbers in Eq. (52) and by the nature of the distribution of ratios per se (see Section 3.2.1). Interestingly, most data points seem closely packed around the median value, since the 95% CI of the medians are narrow.
r  Simulated  Predicted  Range  Loops  Sk  Kr 
3 y_{0}  
y_{m}  ()  
K_{m}  
n  
10 y_{0}  
y_{m}  ()  
K_{m}  
n  
100 y_{0}  
y_{m}  ()  
K_{m}  
n  
2000 y_{0}  ()  
y_{m}  ()  
K_{m}  
n  2 
The concentrations required by Eq. (18) were defined as: , , , , , and ; is the number of values used to calculate medians, 95% confidence in terval and ranges. Parameters and heading have same meaning as used in Table 3.1.1; indicates the number of random Cauchy values of type [Eq. (23)] which were simulated for each concentration. Please notice that the units of are irrelevan as long as they are equal to the units of [D]. See the text for further discussion.
3.2 Monte Carlo simulation of data described by Hill’s equation modified as in Eq. (18).
3.2.1 Fitting Cauchydistributed data to the modified Hill equation.
The analysis in Section 3.1.2 suggests that using the first derivatives of the regression function may produce confidence limits for stochastically independent parameters obtained from nonlinear regressions. To simulate this kind of data with Monte Carlo methods we must consider the statistical properties of a quotient of two Gaussian random variables having and variance , ,distributed following the Cauchy distribution (also called Lorentz, CauchyLorentz or Breit–Wigner distribution) (Cramér, 1991; Pitman, 1993; Wolfram, 2003) which has a pdf
(19) 
where
. The probability distribution function (
PDF) is(20) 
which is symmetric about , the median and mode of the distribution. The maximum value or amplitude of the Cauchy pdf is , located at , is called the scale factor. Using Eq. (20), it is easy to calculate the probability , thus is the % CI of , the 69% CI (like the CI in Gaussian statistics) is . The broadness the Cauchy distribution “shoulders” becomes quite evident when a 95% CI is calculated, for Caucy variables, wider than for Gaussian variables.
All statistical central moments, mean, variance, kurtosis and skewedness, of the Cauchy distribution are undefined: they lack any meaning. The Cauchy distribution is considered an example of a “pathological” distribution function. Thus even when populations and are Gaussian,, population [see Eqs. (50 – 52)], their quotient, will be “pathologically” distributed and its mean and variance will be undefined. Sample values will be symmetrically distributed about , but empirical sample means () will be increasingly variable as the number observations increases, due to increased likelihood of encountering sample points with a large absolute value (“outliers”). A similar situation applies to empirical sample variance, . Neither nor provide any information on the pdf. The distribution of will be equal to outlying observations distribution; i.e., is just an estimator of any single outlying observation from the sample. Similarly, calculating will result in values which grow larger as more observations are collected (Rothenberg et al., 1964; Fama and Roll, 1968; Lohninger, 2012).
Lemma 1.
[Wilks (1962), pg. 156] If is a random variable having a PDF then the random variable has the rectangular distribution .
Proof.
This follows at once from the fact that the PDF of is
(21) 
which is the pdf of the rectangular distribution . ∎
Wilks’ Wilks (1962) , is denoted in this paper. Lemma 1 enables to simulate random variables distributed as using uniform random variables and the following expression
(22) 
Figure 4 presents a plot of a Cauchy probability density function (pdf) [Eq. (19)] calculated with and (Panel 4A), and a Cauchy probability distribution function (PDF) [Eq. (20)] also calculated with and (Panel 4B). Also in Figure 4 (Panel 4C) is a selection of empirical distribution functions (Shorack and Wellner, 1986; Pitman, 1993) determined for the sets of killed cells fractions observed with FIII (QuintanaHernández, 2017); this set was representative of other observed with the remaining fractions in Figures 2 and 3. Sets and [Eqs. (50) and (51)] were found not to be Gaussian using the JarqueBera (Bera and Jarque, 1981) and ShapiroWilks (Shapiro and Wilk, 1965) test, this only means that in addition to their most likely “pathological” distribution the precise nature of this distribution remains unknown. Yet, Figure 4C suggests that the empirical PDF of data in Figures 2 and 3 resemble the Cauchy PDF in Figure 4B.
To test the procedure discussed in this paper Cauchydistributed data sets were generated using Monte Carlo simulation combining Eqs. (18) and (22) as
(23) 
Eq, (23) was used to generaate data sets, processed as QuintanaHernandez’ data QuintanaHernández (2017)(Section 3.1.1).
Some results of the fits of Eq. (18) to Cauchy data appear in Table 3.1.2 together with their apparent sample skewness (Pearson, 1894; Pearsons, 1963) calculated as
(24) 
and their apparent sample kurtosis calculated as
(25) 
where is the apparent sample mean. The definition represented by Eq. (25) is presented here since there are controversies and discrepancies in the definition and interpretation of “kurtosis” and “excess kurtosis” or “Pearson’s kurtosis”, in the literature (Pearson, 1905; Faleschini, 1948; Pearsons, 1963; Proschan, 1965; Darlington, 1970; Ali, 1974; Johnson et al., 1980; Moors, 1986; Ruppert, 1987; Westfall, 2014), kurtosis is used here in sense of Moors (1986):
“High kurtosis, therefore, may arise in two situations: (a) concentration of probability mass near (corresponding to a peaked unimodal distribution) and (b) concentration of probability mass in the tails of the distribution.”
In spite of their undefined central moments, both conditions occur in Cauchydistributed variables if mean is replaced by median in the preceding Moors’ quote (see Figure 5). Estimated and were incompatible to the ones expected for Gaussian variables (), this is not surprising since the data subject of the simplex optimization were generated for Cauchydistributed random variables. The values of indicate that the parameter estimates are asymmetrically distributed about the mean (median?), and increases with sample size. Thus the parameter estimates are not exactly Cauchydistributed either (Cramér, 1991; Pitman, 1993). Data in Tables 3.1.2 through 3.2.2 are very leptokurtic, clustered about the median, but extreme values are observed as indicated by the parameters’ ranges (DeCarlo, 1997). Yet, in spite of the wide ranges, 95% CIs are relatively narrow suggesting strong clustering of data arround the median. Raising the initial did not improve consistently the final estimate of , and worsened the estimations of the other parameters too.
Table 3.1.2 shows that the parameters used to simulate data distributed as Cauchy are well predicted if a simplex optimization is used. At least if the simplex is initiated with a “reasonable” set of parameters .
Table 3.2.1 presents data generated with Eq. (23) using a parameter set , and the simplex optimization started from as in Table 3.1.2. As seen in the Table 3.2.1, were well estimated. Yet, in all instances presented in Table 3.2.1.
To check if starting the simplex from higher values of improves estimate, in Table 3.2.2 are results calculated for exactly the same Monte Carlo data used in Table 3.2.1, but starting the simplex optimization with . As indicated in Section B.1 and shown in Figure 6, raising the initial value of did not improve the estimation of the parameter. More surprising, rising the initial value of n worsened the estimates of all the other parameters characterizing Eq. (18). Taken togheter the results in tables 3.2.1 and 3.2.2, suggests that is the most difficult parameter to estimate in Equation (18). The difficulty to estimate correctly agrees with the discussion on in Section B.1.
In Tables 3.1.2 – 3.2.2, specially in optimizations with larger , some sets of Monte Carlo simulated data did not reach Condition 2, and the optimization stopped on Condition 3 (numbers with asterisks), this appears once in each Table 3.1.2 – 3.2.2, but each of these data sets could have been replaced by results obtained for other pseudorandom sets of data, generated under similar conditions. where Condition 2 was indeed achieved. I.e., failure to comply with Condition 2 did not always occurred. Instances where Condition 3 stopped calculations were included in the tables to show that they do occur. Athough in Tables 3.1.2 – 3.2.2 all set of parameters are apparently leptokurtic and skewed, increased with the number of points () used for each simulated “concentration” in the optimizations, did not increase as much as but was noticeably large wit . As previously said, and depend on the 2^{nd}, 3^{rd} and 4^{th} central moments of a distribution which do not exist for the Cauchy pdf, thus sample values cannot converge towards any population value as required by sampling theory (Wilks, 1962), since population mean, variance, skewness and kurtosis do not exist for Cauchy distributed data. Sample variance and mean, grow, more or less randomly, with sample size (Section 3.2.1). This makes the mean to vary considerably even when several hundred or thousand random, Cauchy dystributed, numbers are averaged (Rothenberg et al., 1964; Fama and Roll, 1968; Lohninger, 2012). Yet, with all the uncertainties, the and estimates clearly indicate that data in tables 3.1.2 – 3.2.2 are nonGaussian.
r  Simulated  Predicted  Range  Loops  Sk  Kr 
3 y_{0}^{∙}  (5.5, 41.0)  
y_{m}  100  ()  ()  
K_{m}^{∙}  
n^{∙}  5.352  
10 y_{0}  
y_{m}  ()  
K_{m}  
n 

4.995  30.0  
100 y_{0}  5  5.9  
y_{m}  ()  
K_{m}  
n  15  
2000 y_{0}  ()  
y_{m}  ()  
K_{m}  12272  
n  15  106.3  12272 
Parameters and heading have same meaning as used in Table 3.1.1; indicates the number of random Cauchy values fitted to Eq. (18) which were exactly the same used in Table 3.1.2. Please notice that the units of are irrelevan as long as they are equal to the units of [D]. Parameter names () with a bullet () indicated that, tested with the JarqueBera test, the probability that the parameter is not Gaussian by chance is not too low (). Number of loops with an asterisk indicates the optimization stopped after fulfilling Condition 3. Other conditions as in Table 3.1.2. Please notice that in all instances. See the text of the communication for further discussion.
3.2.2 Fitting Gauss distributed data to the modified Hill equation.
To evaluate the behavior of Eq. (18) when Gaussian variables are adjusted to the equation, normal variates generated as
(26) 
and used to test the ability of the method proposed here to determine the parameters, with , in which and were the simulated effects’ sample variances and means, respectively. Tables 3.2.2 to 3.2.2 follow the same sequence of simulated sets changes and initial values as data in Tables 3.1.2 to 3.2.2, but the data adjusted to Eq. (18) was Gaussian and generated as indicated in the prior paragraph. As in the case of Cauchy distributed data in Section 3.2.1, are well estimated using the simplex minimization described. Yet, ( symbol is used to mean very significantly different) in all instances presented in Table 3.2.1, this suggests that is the most difficult parameter to estimate if the initiating value of in the simplex is very different from the value used in the Monte Carlo simulation.
(used as initiation value in Tables 3.1.2 to 3.2.2) means that the initial parameters begin increasing by 10%. In calculations, not published here, the simulations in Tables 3.1.2 to 3.2.2 were carried out setting , and yet , and the estimates of the other parameters became worse. This agrees with the discussion about Eqs. (31) to (34) in Section B.1 which holds for Gaussian random variables, since the discussion in Section B.1 is distribution independent.
r  Simulated  Predicted  Range  Loops  Sk  Kr 
3 y_{0}  5  8.64 (5.45, 41.02)  (18.1, 95.9)  31829  1.260  3.147 
y_{m}  100  93.7 ()  ()  4.072  18.0  
K_{m}  0.01  0.20 (0.06, 0.52)  (0.07, 1.07)  1.260  3.147  
n  15  11.84 (11.70, 12.17)  (11.6, 12.72)  1.260  3.147  
10 y_{0}  5  2.38 (1.22, 4.05)  (357.8, 62.1)  2176  4.995  30.0 
y_{m}  100  84.6 (, 92.0)  ()  4.89  30.79  
K_{m}  0.01  0.007 (0.023, 0.007)  (0.334, 0.363)  0.352  8.142  
n  15  9.46 (9.45, 9.48)  (9.3, 9.8)  0.352  8.142  
100 y_{0}  5  4.19 (4.50, 3.88)  (1925.1, 972.1)  1060  12.2  255.4 
y_{m}  100  97.61 (96.66, 98.39)  ()  25.6  669.7  
K_{m}  0.01  0.006 (0.003, 0.009)  (19.2, 9.8)  12.2  255.4  
n  15  14.99 (14.99, 15.00)  (4.2, 24.8)  106.3  255.4  
2000 y_{0}  5  26.5 (26.2, 26.8)  ()  2010  32.9  12272.1 
y_{m}  100  59.3 (28.4, 63.6)  ()  113.6  13261.5  
K_{m}  0.01  0.07 (0.06, 0.07)  (197.2, 1105.3) 
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