 Methodology article
 Open Access
Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems
 Maria RodriguezFernandez^{1},
 Jose A Egea^{1} and
 Julio R Banga^{1}Email author
https://doi.org/10.1186/147121057483
© RodriguezFernandez et al; licensee BioMed Central Ltd. 2006
 Received: 29 May 2006
 Accepted: 02 November 2006
 Published: 02 November 2006
Abstract
Background
We consider the problem of parameter estimation (model calibration) in nonlinear dynamic models of biological systems. Due to the frequent illconditioning and multimodality of many of these problems, traditional local methods usually fail (unless initialized with very good guesses of the parameter vector). In order to surmount these difficulties, global optimization (GO) methods have been suggested as robust alternatives. Currently, deterministic GO methods can not solve problems of realistic size within this class in reasonable computation times. In contrast, certain types of stochastic GO methods have shown promising results, although the computational cost remains large. RodriguezFernandez and coworkers have presented hybrid stochasticdeterministic GO methods which could reduce computation time by one order of magnitude while guaranteeing robustness. Our goal here was to further reduce the computational effort without loosing robustness.
Results
We have developed a new procedure based on the scatter search methodology for nonlinear optimization of dynamic models of arbitrary (or even unknown) structure (i.e. blackbox models). In this contribution, we describe and apply this novel metaheuristic, inspired by recent developments in the field of operations research, to a set of complex identification problems and we make a critical comparison with respect to the previous (above mentioned) successful methods.
Conclusion
Robust and efficient methods for parameter estimation are of key importance in systems biology and related areas. The new metaheuristic presented in this paper aims to ensure the proper solution of these problems by adopting a global optimization approach, while keeping the computational effort under reasonable values. This new metaheuristic was applied to a set of three challenging parameter estimation problems of nonlinear dynamic biological systems, outperforming very significantly all the methods previously used for these benchmark problems.
Keywords
 Local Search
 Fisher Information Matrix
 Scatter Search
 Global Optimization Method
 Parameter Estimation Problem
Background
Modelling approaches are central in systems biology and provide new ways towards the analysis of omics data, ultimately leading to a greater understanding of the language of cells and organisms [1–3]. Further, these approaches provide systematic strategies for key issues in medicine [4] and the pharmaceutical and biotechnological industries. For example, modelbased approaches can provide a rational framework to guide drug development, taking into account the effects of possible new drugs on biochemical pathways and physiology [5]. A common approach to model inter and intracellular dynamic processes is by means of dynamic models, usually consisting of sets of differential equations [6].
The general area of system identification deals with the development of mathematical models of dynamic systems from input/output data [7, 8]. When building mathematical models, one starts from the definition of the purpose of the model and uses the a priori available knowledge (i.e. physical, chemical or biological laws, initial hypothesis and/or preliminary data) to choose a model framework and to propose a model structure. This model contain parameters and we need to know whether is it possible to uniquely determine their values (identifiability analysis) and if so, to estimate them with maximum precision and accuracy. This leads to a first working model that must be validated with new experiments, revealing in most cases a number of deficiencies. In this case, a new model structure and/or a new experimental design must be planned, and the process is repeated iteratively until the validation step is considered satisfactory. This iterative process (i.e. the model building cycle) should also contain other elements like optimal experimental design and model discrimination steps [9–13].
This work is focused on the key step of parameter estimation, assuming the structure of the nonlinear dynamic model as given. Parameter estimation (also known as model calibration) aims to find the parameters of the model which give the best fit to a set of experimental data. Examples of recent efforts in the particular case of biochemical pathways are the works of Sugimoto and coworkers [14], Voit and Almeida [15], RodriguezFernandez and coworkers [13] and Polisetty and coworkers [16]. The key issues considered here in this work were to ensure reliable and accurate parameter estimation, paying especial attention to the computational cost, and also to perform the identifiability analysis of the models.
Parameter estimation in nonlinear dynamic biological models
Estimating the parameters of a nonlinear dynamic model is more difficult than for the linear case, as no general analytic result exists. Biological models are often dynamic and highly nonlinear, thus, in order to find the estimates, we must resort to nonlinear optimization techniques where a measure of the distance between model predictions and experimental data (Z = $\tilde{Y}$  Y) is used as the optimality criterion to be minimized. The criterion selection will depend on the assumptions about the data disturbance and on the amount of information provided by the user. As explained in detail in the Methods section, the maximum likelihood estimator maximizes the probability of the occurrence of the observed measurements. If we make the assumption that the residuals are normally distributed and independent with the same variance σ^{2}, then the maximum likelihood criterion is equivalent to the least squares and we aim to find $\widehat{p}$ which minimizes the sum of squared residuals of all the responses. That is, the estimation criterion would be to minimize the trace of Z^{ T }Z [17]. This is subject to the dynamics of the system, plus possibly other algebraic constraints, and model parameters are also subject to upper and lower bounds. This formulation is that of a nonlinear programming problem (NLP) with differentialalgebraic (DAEs) constraints. In this work, we have followed the socalled single shooting approach [18], where an initial value problem (IVP, i.e., the systems dynamics) is solved as an inner problem of the master NLP problem. When estimating parameters of dynamical systems a number of difficulties may arise, like e.g. convergence to local solutions if standard local methods are used, very flat objective function in the neighborhood of the solution, overdetermined models, badly scaled model functions or nondifferentiable terms in the systems dynamics [18].
Due to the nonlinear and constrained nature of the systems dynamics, these problems are very often multimodal (nonconvex). Thus, traditional gradient based methods, like LevenbergMarquardt or GaussNewton, may fail to identify the global solution and may converge to a local minimum when a better solution exists just a small distance away. Moreover, in the presence of a bad fit, there is no way of knowing if it is due to a wrong model formulation, or if it is simply a consequence of local convergence. Thus, there is a distinct need for using global optimization methods which provide more guarantees of converging to the globally optimal solution, as shown in [19]. The importance of using global optimization methods for parameter estimation in systems biology has been increasingly recognized in recent years [16, 20–23]. Global optimization methods can be roughly classified as deterministic, stochastic and hybrid strategies. Deterministic methods can guarantee, under some conditions and for certain problems, the location of the global optimum solution. Nevertheless, no deterministic algorithm can solve global optimization problems of the class considered here with certainty in finite time. Actually, computational effort increases very rapidly (often exponentially) with the problem size. Although very significant advances have been recently made [24–26], these methods have a number of requirements about the dynamics of the system, and currently they do not seem to be applicable to problems with a relatively large number of parameters. Stochastic methods are based on probabilistic algorithms, and they rely on statistical arguments to prove their convergence in a weak way. However, many stochastic methods can locate the vicinity of global solutions in modest computational times [27]. Additionally, stochastic methods do not require transformation of the original problem, which can be treated as a blackbox.
In our group, and during the last decade, we have compared a number of different stochastic and deterministic global optimization methods. The overall conclusions from these studies indicate that modern evolution strategies have several key advantages over genetic algorithms and simulated annealing, namely better efficiency/robustness ratio, good scaling properties, an inherent parallel nature and an almost selftuning mechanism for the search parameters of the method. Our tests and comparisons indicate that DE [28] and SRES [29] are one of the most competitive algorithms, with the additional advantage of being able to handle arbitrary constraints if needed. The main problem presented by these methods is that they require too many evaluations of the objective function [19]. In order to surmount this difficulty, we have recently proposed a hybrid method [13] that speeds up these methodologies while retaining their robustness.
The key concept behind hybrid methods is the well known idea of synergy, that is, a mutually advantageous conjunction of distinct elements. There are several nontrivial questions associated with the actual development of such method, namely choosing which methods to combine, and how to structure such combination. Our work is then focused on selecting more efficient stochastic GO methods and designing better hybrid methods in order to improve the ratio efficiency/robustness. RodriguezFernandez and coworkers [13] combined a global and a local optimization method in a sequential, twophase hybrid method in order to speedup the stochastic global optimization methods while retaining their robustness. However, computational times were still rather significant, especially if one considers its possible application to larger scale problems.
In order to further increase computational efficiency, in the present work we present a novel metaheuristic approach based on extensions of scatter search combined with various local methods. As it will be shown below, this metaheuristic shows speeds up of between one and two orders of magnitude with respect to previous results obtained with the above mentioned methods. Moreover, this method eliminates the delicate task of deciding where to set the switching point from the global to the local method.
Global optimization: novel metaheuristic
Scatter search (SS) is a populationbased method based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, such as the surrogate constraint method. It was first introduced by Glover [30] as a heuristic for integer programming, although it has been extended for other problem classes more recently [31, 32]. Scatter search orients its explorations systematically relative to a set of reference points that typically consist of good solutions obtained by prior problem solving efforts.
The justification for choosing this algorithm as the starting framework for our metaheuristic was based on a recent review comparing a number of global optimization solvers over a set of over 1000 constrained GO problems [33], in which a solver based on scatter search proved to be the best among all the stochastic solvers, and the best among all methods for blackbox problems. Furthermore, for problems with a large number of decision variables, this solver also proved to be the most reliable.
Scatter search, when the local search is activated, can be defined as a hybrid method since it combines a global search with an intensification phase (i.e. local search). The algorithm uses different heuristics to efficiently choose suitable initial points for the local search, based on merit and distance filters as well as a memory term. This feature helps overcome the problem of switching from global to local search as explained in [13]. The user does not have to worry about stopping the global search and starting the local solver since the algorithm performs this work automatically.

A diversification generation method to generate a collection of diverse trial solutions.

An improvement method to transform a trial solution into one or more enhanced trial solutions.

A reference set update method to build and maintain a reference set consisting of the b "best" solutions found (where the value of b is typically small, e.g., no more than 20), organized to provide efficient accessing by other parts of the method. Solutions gain membership to the reference set according to their quality or their diversity.

A subset generation method to operate on the reference set, to produce several subsets of its solutions as a basis for creating combined solutions.

A solution combination method to transform a given subset of solutions produced by the subset generation method into one or more combined solution vectors.
Of the five methods in the SS methodology, only four are strictly required. The improvement method is usually needed if high quality outcomes are desired, but a scatter search procedure can be implemented without it. Differences among scatter search implementations are based in the level of sophistication in which these steps are implemented, not in the presence or absence of other steps. In the algorithm presented here, we have added some advanced features in order to improve its performance when solving parameter estimation problems:

A logarithmic distribution for generating initial trial solutions can be chosen by the user to favor their presence close to the bounds in terms of Euclidean distance, since the location of the global optimum near or even touching the bounds (i.e. being active at any of the bounds) is quite usual in parameter estimation problems.

Mechanisms to avoid flat zones (also frequent in parameter estimation problems), as well as others to avoid getting stuck in local solutions, have been added. In every iteration the algorithm analyzes if the elite solutions have very similar objective function values regardless their Euclidean distances. If the variance of these values is too low, our procedure considers that the search is located in a flat zone and resets the elite solutions to explore different (and scattered) areas. This mechanism also prevents the algorithm getting stuck in a local solution when the elite solutions have converged to that minimum.

A new solution combination method allows to explore deeper the search space. Apart from the traditional method of linear combination between solutions, another method based on building hypercubes around the solutions to generate new solutions inside them has been implemented. Now new points around elite solutions (and not only on the segments joining these elite solutions) can be generated, which favors the intensification and accelerates the convergence.

When all the combinations among elite solutions have been done, the algorithm can stop or continue by partially rebuilding the set of the elite solutions. A new strategy for rebuilding this set, based in orthogonal search directions has been implemented. Instead of simply maximizing the Euclidean distances between the new elite solutions to be generated and the remaining ones, the algorithm takes into account the directions generated by every pair of elite solutions and force the generator to build new solutions that create new search directions.

The user can choose a number of different local solvers such us SQP methods like fmincon (The MathWorks Inc.), SOLNP [34], SNOPT [35], direct methods like Nomad [36] for the cases of very noisy data, and others specifically designed for parameter estimation problems such as N2FB/DN2FB [37].
It is interesting to observe similarities and contrasts between scatter search and the original genetic algorithm proposals. Both are instances of what are sometimes called "population based" or "evolutionary" approaches. Both incorporate the idea that a key aspect of producing new elements is to generate some form of combination of existing elements. However, genetic algorithm approaches are predicated on the idea of choosing parents randomly to produce offspring, and further on introducing randomization to determine which components of the parents should be combined. By contrast, the scatter search approach does not emphasize randomization, particularly in the sense of being indifferent to choices among alternatives. Instead, the approach is designed to incorporate strategic responses, both deterministic and probabilistic, that take account of evaluations and history. Scatter search focuses on generating relevant outcomes without losing the ability to produce diverse solutions, due to the way the generation process is implemented.
A detailed description of the method is given in the Methodology section.
Confidence intervals
Determining the parameter values with the maximum likelihood of being correct is only part of the parameter estimation problem. Moreover, it is equally important to find a realistic measure of the precision of those parameters [38, 39].
It should be noted that, unlike for the linear case for which a neat theory already exists, there is no exact theory for the evaluation of confidence intervals for systems which are nonlinear in the parameters. An approximate method based on a local linearization of the output function is often used and was also adopted in this study, thus the confidence region is evaluated as a function of the parameter covariance matrix C, based on the Fisher information matrix (see details in the Methods section).
However, the confidence intervals obtained with the Fisher method are statistically optimistic due to the use of a linear approximation of the nonlinear model in the neighborhood of the best parameter estimates [40].
Alternatively, more robust techniques such as the jackknife and bootstrap methods produce parameter variances that are more realistic. As a drawback, one should mention that these methods are very computing intensive. Another way to obtain the true confidence region of the parameters in nonlinear models is by a systematic exploration of the objective functional for an extensive number of parameter combinations. This is a computing intensive task as well, because the number of evaluations increases as a power function of the number of parameters. Therefore, in this study we will make use of the method based on the FIM.
Precision of parameter estimates
Many difficulties found during parameter estimation are due to a poor identifiability of the model parameters. Parameter identifiability tests should be performed prior to the estimation process to ensure that the parameter estimation problem is wellposed [11]. The identifiability analysis investigates if the unknown parameters of the postulated model can be estimated in a unique way.
Regarding this problem, we can distinguish between structural and practical (or a posteriori) identifiability [41]. Structural identifiability is a theoretical property of the model structure depending only on the observation function and the input function. The parameters of a model are structurally globally identifiable if, under ideal conditions of noisefree observations and errorfree model structure, and independently of the particular values of the parameters, they can be uniquely estimated from the designed experiment [8].
The requirements for global structural identifiability are rather strict, since we can find realistic situations where the parameters are not identifiable according to this definition, but nevertheless they would be identifiable for a reasonably restricted set of all possible parameters. This leads to the definition of local structural identifiability, where the requirement for the parameters is to be identifiable in a ε neighborhood of a parameter set. Although necessary, structural identifiability is obviously not sufficient to guarantee successful parameter estimation from real data, and this is when the concept of practical identifiability comes into play. In contrast to the theoretical properties of structural identifiability, the practical identifiability is limited by the finite amount of data and the observational noise. Hence, in the presence of large observation errors and/or few data, no reliable estimate is possible and these parameters are called practical nonidentifiable.
Assessing a priori global identifiability is very difficult for nonlinear dynamic models, although techniques based on differential algebra have shown very promising results [42]. However, it has been argued that these techniques have somewhat limited applicability [43, 44]. These limitations, taken in conjunction with the need for practical methods, provides a key argument for emphasizing the use of practical identifiability despite its limitations derived from its local nature. The question addressed in the a posteriori or practical identifiability analysis is the following: with the available experimental data, can the parameters be uniquely estimated? Or, in other words, if a small deviation of the parameter set occurs, does this have a great impact on the quality of the fit?
The output sensitivity functions (partial derivatives of the measured states with respect to the parameters), are central to the evaluation of practical identifiability. If the sensitivity functions are linearly dependent the model is not identifiable, and sensitivity functions that are nearly linearly dependent, result in parameter estimates that are highly correlated. An easy way to study the practical identifiability of a simple model is to plot the sensitivity functions calculated for a given parameter set. However, this straightforward procedure becomes intractable when the number of measured states and parameters is of realistic size. In the Methods section, a numerical procedure to test practical identifiability based on the Fisher information matrix (FIM), as well as an approximate computation of the correlation matrix, are described.
The correlation matrix measures the interrelationship between the parameters and gives an idea of the compensation effects of changes in the parameter values on the model output. If two parameters are highly correlated, a change in the model output caused by a change in a model parameter can be (nearly) compensated by an appropriate change in the other parameter value. This prevents the parameters from being uniquely identifiable even if the model output is very sensitive to changes in the individual parameters.
In order to perform the practical identifiability analysis, prior knowledge of the model parameters is required. In an experimental situation, the parameters values will not be known a priori, and the identifiability analysis will be an important step in an iterative process involving experimental design and parameter estimation [45].
In this work, the new global optimization metaheuristic described above has been coupled with a computational procedure to check identifiability and related measures. This has resulted in an integrated environment to perform robust parameter estimation and identifiability analysis.
Results and discussion
In order to evaluate the performance and reliability of the novel metaheuristic presented here, which we will denote SSm (scatter search method), we have considered three challenging benchmark problems of increasing order of complexity. All the computations were carried out using a PC/Pentium 4 (1.80 GHz).
Isomerization of αpinene
Assuming this model to be appropriate, the initial conditions for the five species are known, and we can estimate the unknown coefficients p_{1},...,p_{5} by minimization of a cost function which is usually a weighted distance measure between the experimental values corresponding to the measured variables and the predicted values for those variables. For this problem the cost function can be formulated as:
Box and coworkers [46] tried, in a first instance, to solve this problem without analyzing the multiresponse data, finding parameter values which provided an unsatisfactory data fit. Since ignoring possible dependencies among the responses can lead to difficulties when estimating the parameters (e.g. multiple local minima, very flat objective function, etc.), Box and coworkers described a method for detecting and handling these linear relationships. They showed that there are dependencies in the data and thus only three independent linear combinations of the five responses are used in the identification improving significantly the fit of the data. This analysis of multiresponse data, although efficient, requires a considerable effort specially to uncover the dependencies causes once they have been found, and a deep understanding of the model (that can no longer be considered as a blackbox) is essential. Moreover, it becomes unaffordable when the model complexity increases.
Tjoa and Biegler [50] also considered this problem and used a robust local estimation approach to estimate the unknown parameters. They considered the entire data set in order to asses the performance of this method with dependencies in the data, finally reaching the same optimal parameters reported by Box et al. However, the initial value considered for the parameters was very close to the truly optimal solution, which explains why this local method reached the global optimum without getting trapped in a local solution. As pointed out by Averick and coworkers [51], the solution of this problem is not difficult to obtain from initial values of p which are close to the global solution, but becomes increasingly difficult to solve from more remote starting points.
In order to avoid the convergence to local solutions without a good initialization value for the parameters and/or further analysis of the multiresponse data, the use of a global optimization approach is proposed here. The lower bounds considered for the five parameters arise from physical considerations, p_{ i }≥ 0, and we took the upper bounds to be p_{ i }≤ 1, very far from the best known solution (p_{ 1 }= 5.93e  5, p_{2} = 2.96e  5, p_{3} = 2.05e  5, p_{4} = 27.5e  5, p_{5} = 4.00e  5). As initial point, we chose p_{ i }= 0.5. It should be noted that all the local solvers that we tried with this initial point failed to converge, or converged to bad local solutions.
Optimal parameters for the alpha pinene isomerization problem
Optimal parameters (J = 19.87)  

Parameter  Optimal value 
P _{1}  5.9259e5 ± 1.4391e6 
P _{2}  2.9634e5 ± 1.3039e6 
P _{3}  2.0473e5 ± 6.6657e6 
P _{4}  2.7449e4 ± 5.5314e5 
P _{5}  3.9980e5 ± 1.9514e5 
Inhibition of HIV proteinase
We considered the same problem solved by Kuzmic [52] and Mendes and Kell [53] fitting five of the rate constants to the experimental data. In this fit, a certain degree of uncertainty (± 50 %) in the value of the initial concentrations of substrate and enzyme (titration errors) was also assumed. In addition, the offset (baseline) of the fluorimeter was also considered as a degree of freedom. Given that there are five time course curves, there are a total of 20 adjustable parameters: the five rate constants, five initial concentrations of enzyme, five initial concentrations of substrate and five offset values.
By minimization of the sumofsquares function of the residuals between the measured and the simulated data, the best known solution was obtained by Mendes and Kell using simulated annealing, with a computational cost of 3 million simulations. The next best solution was obtained using a LevenbergMarquardt method in a considerable shorter computational time (4000 simulations) although this method is only guaranteed to converge to the global minimum if started in its vicinity.
Optimal parameters for the HIV proteinase inhibition problem
Results SSm  

Parameter  Parameter value (J = 0.0199)  Parameter value (J = 0.0203) 
k _{3}  6.235 ± 3.2546  5.656 ± 1.953 
k _{42}  8772 ± 46120  688.4 ± 3436 
k _{22}  473.0 ± 624.6  120.6 ± 508.1 
k _{52}  0.09726 ± 0.1288  4.615 ± 583.4 
k _{6}  0.01417 ± 0.01032  3.531 ± 455.4 
S_{0} (exp 1)  24.63 ± 0.07817  24.69 ± 0.08049 
S_{0} (exp 2)  23.32 ± 1.349  23.43 ± 0.1541 
S_{0} (exp 3)  26.93 ± 1.222  27.11 ± 0.1672 
S_{0} (exp 4)  13.34 ± 1.822  17.07 ± 1.986 
S_{0} (exp 5)  12.50 ± 1.812  14.49 ± 1.757 
E_{0} (exp 1)  0.005516 ± 0.001968  0.005397 ± 0.0009091 
E_{0} (exp 2)  0.005321 ± 0.001309  0.005199 ± 0.0005520 
E_{0} (exp 3)  0.006000 ± 0.001111  0.006000 ± 0.0005489 
E_{0} (exp 4)  0.004391 ± 0.00008686  0.004264 ± 0.00005821 
E_{0} (exp 5)  0.003981 ± 0.00008844  0.003973 ± 0.00005648 
offset (exp 1)  0.004339 ± 0.001788  0.005611 ± 0.001836 
offset (exp 2)  0.001577 ± 0.002966  0.004247 ± 0.003432 
offset (exp 3)  0.01117 ± 0.002734  0.01522 ± 0.003865 
offset (exp 4)  0.001661 ± 0.001881  0.009649 ± 0.003277 
offset (exp 5)  0.007133 ± 0.001764  0.001329 ± 0.003178 
Threestep biochemical pathway
The novel metaheuristic, SSm, presented in this work was able to improve this result in an additional order of magnitude regarding the computational time. Moreover, SSm had the additional advantage of not requiring preliminary runs, or any user inputs, for tuning the method, making it a very easy to use strategy. In short, using SSm we have reduced the computation time from two days [19] to a couple of minutes, while ensuring robustness.
Conclusion
Parameter estimation from experimental data remains a bottleneck for a major breakthrough in systems biology. Traditional global optimization methods can ensure proper solutions, but suffer from the large computational burden required for largescale model identification. In this contribution, we have presented a novel global optimization metaheuristic, SSm, which increases very significantly the efficiency of the estimation while keeping robustness. Its capabilities were tested considering three challenging benchmark problems. This new method was able to successfully find the best known solutions for these problems while reducing the computation time by several orders of magnitude with respect to previous approaches.
Methods
Problem statement
In this work, we consider deterministic, nonlinear dynamic models of biochemical systems, i.e. those described by deterministic ordinary differential equations (ODEs), or differentialalgebraic equations (DAEs). In the case of ODEs, a popular statement is the so called statespace formulation:
x(p,t) = f[x(p,t), u(t),p], x(0) = x_{0}, (8)
y(p,t) = g[x(p,t), u(p,t), p] (9)
where x is the vector of N_{ x }state variables and p the vector of N_{ p }model parameters. Note that f specify the model, u specifies the vector of inputs (i.e. for a particular experiment) and y the vector of N_{ y }measured states. An experiment is specified by the initial conditions x(0), the inputs u chosen from among some set of possible inputs U and the observations y. Note that the inputs can be time dependent.
Given a model structure and a set of experimental data, the goal of the parameter estimation problem is to calibrate the model so as to reproduce the experimental results in the best possible way. This is performed by minimizing a cost function that measures the goodness of the fit. Several estimator functions have been suggested as metrics, standing out the maximum likelihood estimator introduced by Fisher (1912), for being the one that maximizes the probability of the observed event.
Maximum likelihood estimation consists of maximizing the socalled likelihood function, J_{ ml }, which is the probability density of a model for the occurrence of the measurements for given parameters. The likelihood function depends on the probability of the measurements. Assuming these to be uncorrelated normal distributions, the loglikelihood function (which yields the same estimate that the likelihood function but is easier to handle in practice) is given as:
For given measurements ${\tilde{y}}_{i}$, the maximum likelihood estimates of the parameters are those values of p for which the likelihood function has its minimum. Moreover, if we assume the noise to be Gaussian with known of constant (homoscedastic) variance, minimizing J_{ ml }(Equation 10) is equivalent to minimizing the function:
J_{ ls }(p) = w_{ i }[${\tilde{y}}_{i}$  y_{ i }(p)]^{2} (11)
with the weights W_{ i }= $\frac{1}{{\sigma}^{2}}$. One thus obtains a weighted leastsquares estimator. If all σ_{ i }'s are equal, unweighed leastsquares should be used (w_{ i }= 1) and the noise variance do not need to be known a priori and can be estimated a posteriori from the residuals [7, 8].
Confidence intervals
In general, confidence regions can be expressed as:
The covariance matrix obtained for a linear case can be extended for nonlinear models leading an approximate covariance matrix as:
where the term $\frac{J(\widehat{p})}{N{n}_{p}}$ is an unbiased approximation of the residual variance σ^{2} and $\frac{\partial y}{\partial p}(\widehat{p})$ the sensitivity functions with respect to the parameters evaluated at $\widehat{p}$.
Under the assumption of uncorrelated measurement noise with Gaussian distribution with a mean of zero, the approximation of the covariance matrix C_{ J }given by the Equation 13 is just the inverse of the Fisher information matrix of the estimation problem defined as:
According to the CramèrRao theorem, C_{ J }($\widehat{p}$) = FIM^{1} represents the error covariance matrix of the minimum variance unbiased estimator, thus substituting C_{ J }from Equation 13 into Equation 12, yields the approximate confidence ellipsoids.
Therefore, a lower bound for the individual parameter confidence interval σ_{ i }(i = 1,...,n_{ p }) can be obtained from the diagonal of the covariance matrix as:
where ${t}_{N{n}_{p}}^{1(\alpha /2)}$ is the twotails Student's t distribution for the given confidence level α and N  n_{ p }degrees of freedom which converges to a linear distribution when the number of measurements N is high. Assuming that the measurement noise is white and uncorrelated we consider the error correlation matrix as diagonal, neglecting the offdiagonal elements of C, that is, the covariances among the parameters. When parameters are simultaneously determined they usually have a significant covariance thus the confidence intervals might be underestimated. That is why these confidence intervals obtained from the FIM can only be taken as lower bounds and never as an exact confidence region.
A posteriori local identifiability analysis
Under the assumption of uncorrelated measurement noise with Gaussian distribution with a mean of zero, the covariance matrix can be approximated by the inverse of the Fisher information matrix involving the output sensitivity functions. If the sensitivity equations shows linear dependence at the experimental data points, the FIM becomes singular and the model is not identifiable.
Useful information about the correlation between estimated parameters can be also obtained from the covariance matrix. The correlation matrix, which elements are the approximate correlation coefficients between the ith and the jth parameter, is defined by:
R_{ ij }= 1, i = j, (17)
A singular FIM indicates the presence of unidentifiable parameters, and correlations between parameters that are greater than 0.99 may lead to singular FIM.
SSm algorithm
SSm is an advanced design of the scatter search algorithm for real variables. The method uses a relatively short number, b, of elite decision vectors in a socalled reference set (refset). These elite vectors are combined in pairs to generate new ones that may enter the refset replacing existing vectors in it (i.e. the refset always maintains a fixed number of vectors). This evolutionary approach is combined with local searches from selected vectors.
In a ndimensional problem, vectors of decision variables are represented by, x ∈ ℝ^{ n } so that a particular decision variable in the population of size NP can be symbolized as ${x}_{i}^{r}$, where i = 1, 2,...,n and r = 1, 2,...,NP. The refset will have NP = b, whose default value is 10.
The main steps of the algorithm are shown below, with a diagram presented in Figure 1.
Generation of diverse vectors within the search space
The first step consists of generating a set S of m (default m = 10 · b) diverse vectors in the search space. Unlike other diversification strategies, SSm does not only generate vectors with their components uniformly distributed within the search space, but also drives the generation of values for each decision variable onto parts of the space where they have not appeared very often during the search process. For that, the method makes use of memory taking into account the number of times that every decision variable appears in different parts of the search space.
Initially, the range of every decision variable, defined by its lower and upper bounds, xl_{ i }and xu_{ i }respectively, is divided in p (default p = 4) subranges of equal size, (xu_{ i } xl_{ i })/p. Therefore, the limits that define each subrange j ∈ [1, 2,...,p] for the variable i are given by

Lower bound:

Upper bound:
Frequencies f_{ ij }are defined as the number of times that the variable i is in the sub range j along all the generated vectors, with i ∈ [1,2,...,n] and j ∈ [1,2,...,p].
To initialize all the frequencies to a value of 1, p vectors are first generated, each of them having all their variables randomly generated in the same sub range using a uniform distribution (e.g. vector 1, x^{1}, has all its variables in sub range 1, and every decision variable i is randomly generated using a uniform distribution within the bounds xl_{ i }and $x{l}_{i}+\frac{(x{u}_{i}x{l}_{i})}{p}$). This first set of vectors forms the initial matrix of diverse vectors S^{p × n} that will be extended up to a size of S^{m × n} by adding new diverse vectors.
New vectors will be generated using the following procedure:
For each new vector x^{p+t} to be generated, the probability of having its decision variable i in the sub range j is calculated as
with t ∈ [1,2,...,m  p], i ∈ [1,2,...,n] and j ∈ [1,2,...,p].
Then, a uniformly distributed random number, rnd, in the interval [0 1] is generated. The next generated vector x^{p+t} will have its ith component in the subrange j = a for the first value of a that accomplishes
Each component, ${x}_{i}^{p+t}$, will take a value randomly selected using an uniform distribution in the range [lb_{ ij }, ub_{ ij }].
Thus, for a new vector to be generated, the probability of having the variable i in the subrange j is inversely proportional to the frequency of appearance of the variables i in this subrange along the already created vectors. Therefore, the method has to "remember" and update these frequencies to enhance diversity. As new vectors x^{p+t} are generated, they are added to the matrix S in rows until it becomes mbyn dimensional.
Refset formation
When the diverse vectors have been generated a selected number of them will create the first refset, R. There are two strategies to do it.
The first one consists of evaluating the fitness f (x) (i.e. the cost function) of all diverse vectors and select the b/2 best ones in term of fitness. For example in a minimization problem, provided the diverse vectors are sorted according to their fitness (the best one first), the initial selection is R^{b/2 × n} = [x^{1}, x^{2},...,x^{b/2}] such that
f(x^{ i }) ≤ f(x^{ j }) ∀ j > i, i ∈ [1,2,...,b/2], j ∈ [2,3,...,m] (23)
The current number of vectors present in the refset is computed as h. Therefore, in this stage h = b/2 and the maximum value of h is b. We complete the refset with the remaining diverse vectors not yet included by maximizing the minimum Euclidean distance to the included vectors in the refset.
For every diverse vector not yet included in the refset, x^{ d } with d ∈ [h + 1, h + 2,...,m], Euclidean distances to all current refset vectors are computed. The minimum of these distances, d_{ min }, is stored for each vector x:
d_{ min }(x^{ d }) = min{d(x^{ d }, R} (24)
where d(x, R), represents a vector whose components are the Euclidean distances between vector x and all the vectors in the matrix R.
Then, the vector x having the highest minimum distance will join the refset, R = R ∪ x such that
d_{ min }(x) = max(d_{ min }(x^{ d })) ∀ d = h + 1, h + 2,...,m (25)
and the value of h is increased one unit since a new vector has been added to the refset. This is repeated until the refset is filled with b vectors (i.e. h = b) so that R ∈ ℝ^{b × n}.
The second strategy does not take into account the fitness of the diverse vectors. The initial refset is formed by 3 vectors: one having all the variables in their lower bounds, one having all the variables in their upper bounds and the middle point between these two vectors. This initial refset R ∈ ℝ^{3 × n} is completed using the same distance criterion described in the first strategy until it is composed of b decision vectors. Please note that the first strategy involves a higher computational cost since the fitness of all the diverse vectors has to be evaluated. However, this strategy ensures a better quality of the initial refset which can help to converge faster to the global solution.
Combination
Unlike genetic algorithms or other evolutionary strategies, scatter search does not use mutation or crossover operators among its members, but combinations among them. SSm combines all the vectors in the refset in pairs, making use of memory to avoid the combination of two vectors that have already been combined. The number of vectors created from each pair of elite vectors depends on the quality of the latter. These combinations are of the following three types, assuming x' and x" being the elite vectors to be combined and being x' superior in quality to x":

Type 1: c_{1} = x'  d

Type 2: c_{2} = x' + d

Type 3: c_{3} = x'' + d
where d = r.·(x"  x')/2
And r is a vector of dimension n with all its components being uniform random numbers in the interval [0 1].
Please note that the notation.· above indicates that the vectors are multiplied component by component, thus that is not an scalar product.
The vector has the form
If both x' and x" belong to the best b/2 elements of the refset in terms of fitness, then 4 vectors are generated: one of type 1, one of type 3 and two of type 2.
If only x' belongs to the best b/2 elements of the refset in terms of fitness, then 3 vectors are generated: one of each type.
If neither x" nor x' belong to the best b/2 elements of the refset in terms of fitness, then 2 vectors are generated: one of type 2 and another one of type 1 or 3 (randomly chosen).
This type of combination allows more diversity in the generated vectors than the linear combination used in classical implementations of scatter search. These vectors generated by combination of refset members will be named x^{ c } with c ∈ [1, 2,..., nc], and form a matrix C ∈ ℝ^{nc × n} where nc is the total number of vectors generated by combination, which is not a fixed number. It may change every iteration depending on the number of combinations made among refset members (remember that the method avoids doing combinations with pairs of vectors already combined).
Refset update
Once the combinations have been done, the new vectors generated may replace the elite vectors if the refset can increase its quality. Each new vector created by combination which is better than the worst vector in refset is compared with all the elite vectors. If new vectors outperform elite vectors in terms of fitness they replace them as long as they comply with a minimum diversity (i.e. the method avoids vector duplication in the refset by computing Euclidean distances among all vectors).
The best generated vector by combination is compared with the worst vector in refset. If the former outperforms the latter and is not included in the refset, the replacement is carried out. Otherwise, the algorithm tries to find another vector in the refset to accomplish both conditions and do the replacement.
The first candidate vector to join the refset among the nc generated vectors by combination is z such that
f(z) ≤ f(x^{ i }) ∀ i = 1,2,...,nc (27)
The possible vector to be replaced in the refset is the worst in the refset, x^{ w } such that
f(x^{ w }) ≥ f(x^{ j }) ∀ j = 1,2,...,b (28)
The replacement will be carried out if
f(z) ≤ f(x^{ w }) and z ∉ R (29)
Regardless the replacement is done or not, z is deleted from the matrix C, therefore nc is decreased in one unit. This is repeated with every generated vector by combination until no new vectors are better in quality than the current worst vector in refset.
There is an exception to these rules: if one vector has the best fitness in terms of quality found so far, it will join the refset replacing the worst vector in it or, in case that the diversity condition can not be achieved, the closest elite vector to it.
A mechanism to avoid flat zones is added to the refset update. In flat areas, many vectors with very similar (and sometimes the same) fitness can appear. To avoid including vectors from the same flat area, new vectors can only join the refset if the candidate vector has a different fitness value apart from being diverse enough. This prevents vectors in the same flat area from joining the refset at the same time.
Provided the diversity criterion is accomplished, the candidate vector z will join the refset only if
f(z) <f(x^{ i })(1  ε) ∀ x^{ i }∈ R (30)
where ε is a small value defined by the user.
Refset regeneration
When all possible new combinations have been done and none of the generated vectors can replace any of the elite vectors, the algorithm can either stop or continue by regenerating the refset. The latter strategy is used in our algorithm. The worst g elite vectors (in terms of fitness) are deleted. New diverse vectors are generated (see above) and the refset is refilled according to a diversity criterion as the one described in the refset formation.
Normally g = b/2 but in aggressive implementations it can be set to b  1 (i.e. all the vectors in the refset except the best one are deleted).
A new strategy for regenerating the refset has been implemented in SSm. Because the classical diversity criterion based on Euclidean distances described above does not ensure that the search will be performed along the different dimensions of the space. In our new strategy the vectors refilling the refset are chosen to maximize the number of relative directions defined by them and the existing vectors in the refset.
After deleting the g worst solutions the refset is (b  g) × n dimensional. Again, we compute h as the number of existing vectors in the current refset thus when the regeneration starts h = b  g.
A new matrix M containing the vectors that define the segments formed by the best vector in refset and the rest of vector in it is defined as
M^{h 1 × n} = x^{1} x^{ k } ∀ k = 2,3,...,h (31)
with x^{1} being the best element not deleted in refset in terms of fitness and x^{ k } the rest of the elements in it (note that the refset is ordered according to fitness).
For every diverse vector x^{ v } with v ∈ [1, 2,...,m] to join the refset in the regeneration phase a vector P of scalar products is also defined:
P^{ v } = (x^{1}  x^{ v })·M^{ T } (32)
where x^{1} is again the best not deleted element in refset and M^{ T } is the transpose matrix of M. For every x^{ v } the maximum value of its vector P^{ v } is computed as msp(x^{ v }).
The solution v ∈ x^{ v } will join the refset in the regeneration phase if
msp(v) = min{msp(x^{ v })} (33)
with v ∈ x^{ v }. In this stage, the value of h is increased one unit and the process continues until h = b. The application of this strategy results in a maximum diversity in search directions on the regenerated refset.
Local search – filters
Local searches are carried out from different vectors as initial points to accelerate the convergence to the minima as shown in Figure 1. The user can use a different set of local solvers (see list above) to solve their problems. When a local (maybe global) solution provided by a local search outperforms the vector used as initial point to start the local search in terms of fitness, the former replaces the latter and becomes a member to join the refset. Otherwise, the solution obtained in the local search is discarded.
To avoid doing too many local searches or start from different initial points that might provide the same local solutions, two filters are implemented in the routine. The first one is a merit filter that takes into account the fitness of the vectors so that a local search is not started from bad vectors in terms of fitness. The other filter takes into account distances from initial points to the local solutions they provide, thus it avoids starting local searches from the area of influence of already found minima.
In principle, both filters must be passed to start a local search, but depending on the characteristics of the problem, any of them (or both) can be deactivated. Furthermore, they can be relaxed if no vectors passing them are found after a number of consecutive iterations.
Stopping criterion
The stopping criterion is taken as a combination of three conditions:

maximum number of evaluations exceeded

maximum computational time exceeded

value to reach of the cost function satisfied
By default, the algorithm will stop when any of these conditions is satisfied.
Declarations
Acknowledgements
This work was supported by the European Community as part of the FP6 COSBICS Project (STREP FP6512060) and by Xunta de Galicia (PGIDIT05PXIC40201PM). Author JAE gratefully acknowledges financial support (FPU fellowship) from the Spanish Ministry of Education and Science.
Authors’ Affiliations
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