All the direct search methods originated from the basic flexible polyhedron algorithm (Nelder and Mead, 1965; Himmelblau, 1972; Kuester and Mize, 1974). The basic idea is to evaluate the error functional for several combinations of the parameter values, discard the worst and substitute it with a better one, whose selection is the core of the search algorithm. Given np parameters, the ‘flexible polyhedron’ has np+1 vertices P P
(
1, , ,2…P Pnp, np+1)
, so that it covers a convex non-degenerate region of the parameter space (the ‘hull’). After testing the error functional with each of the np+1 combinations, a ranking is made, and the worst one is replaced with a better one.p2 p2
p1 p1
(a) (b)
FIGURE 2.17 Problems created by a ‘narrow valley’ error functional. The gradient-based method (a) makes little progress because its projection is orthogonal to the contour lines. A search method (b) reaches the mini-mum quickly once the valley direction has been detected.
The problem is to make an educated guess to select the new parameter vector, which will be referred to simply as ‘point’, in the parameter space. Figure 2.18 shows the worst point replace-ment procedure in a two-parameter space: after evaluating E P
( )
for all the vertices by model simulation, the worst is discarded and replaced by a new one. The question now is how to select a new point that will improve the search.2.3.1.1 Polyhedron Reshaping
The simplex polyhedron is called ‘flexible’, and not without reason. In fact, it is conceived to change its shape and adapt to the error surface in its search for the minimum. Once all the vertices have been tested and ranked as P
(
max, ,… Pmin)
, the worst point Pmax is excluded from the set, and the cen-troid of the remaining np points is computed as their averagePcen P
1 max
= 1
=≠
np
∑
i ii
np
(2.34)
One of the following operations is then performed, depending on the relative merit of each point, along the search direction defined by the line joining Pmax to Pcen:
Reflection: This preliminary operation defines a new point Pr along the direction P
(
max→Pcen)
as Pr=Pcen+α(
Pcen−Pmax)
α>0 (2.35) E( )
Pr is then compared with the previous values. Based on its relative merit, one of the three opera-tions of Figure 2.19 is made:Expansion: If E( )Pr <E(Pmin), the reflected point is better than the previous best P
(
min)
, so the search is continued along the same direction, and a new expanded point Pexp is determined asPexp=Pcen+γ
(
Pcen−Pmax)
γ 1 (2.36)>p2
Model simulation
E(P) evaluation
New P1 p1 P3 ≡ (p13 3, p2) → E(P3)
P2 ≡ (p12 2, p2) → E(P2)
P1 ≡ (p11 1, p2) → E(P1)
Pi ≡ (p1i, p2i)
E(P1)> E (P2)> E(P3)
FIGURE 2.18 Substitution of the worst point (vertex) of the old polyhedron (hatched triangle). In this case, the ranking resulted in P1 being the worst point, and it is replaced by the new P1. Convexity requires the poly-hedron to have np + 1 vertices.
Now, the new point Pnew is determined, depending on the comparison between Pr and Pexp, as follows:
P P P P
Pr P P
new
if
= if <
>
exp exp
exp
( ) ( )
( ) ( )
E E
E E
r r
(2.37)
Reflection: If the merit of the reflected point is intermediate between the maximum and minimum points, then the reflected point already computed with Equation 2.35 is retained as the new point, that is,
Pnew=Pr if E(Pmax)>E( )Pr >E(Pmin) (2.38) Contraction: If the expansion does not bring any improvement, that is, if E
( )
Pr >E(
Pmax)
, then the new point is selected inside the old polyhedron according toPnew=Pc=Pcen+β
(
Pcen−Pmax)
0< <β 1 (2.39) Finally, If E( )
Pc >E(
Pmax)
, then the simplex must be shrunk to sink deeper into the bottom of the error surface. This is accomplished through aReduction: All the vertices are brought closer to the best point P
(
min)
by reducing all of its sides by a fixed amountPi(new)=Pi(old)+δ
(
Pi(old)−Pmin)
0< <δ 1 i=1, , np+1 and i≠min (2.40)Pmax
E(Pr) < E(Pmin) Expansion E(Pexp) < E(Pr) ⇒ Pnew = Pexp
E(Pmax) > E(Pr) > E(Pmin) Reflection
Pnew = Pr
E(Pr) > E(Pmax) Contraction
Pnew = Pc
Pexp E(Pexp) > E(Pr) ⇒ Pnew = Pr
Pmin Pr Pcen
Basic reflection
Pmax
Pmin Pcen
Pr Pc
Pr
Pmin
Pmax
Pmin
Pcen Pr
Pcen Pmax
FIGURE 2.19 The three basic operations through which the simplex changes its shape starting with the reflected point (filled triangle). The final new point is indicated by the filled square, while a hollow circle indicates the centroid.
2.3.2 termInatIon crIterIon
To terminate the search, a test for flatness is made on the polyhedron. In fact, when all the vertices yield nearly the same functional level within a specified tolerance ε, it is assumed that the search has reached the bottom of the surface, that is, it is close to the minimum. Therefore, the terminating criterion can be stated as
ε ≥
( )
−( )
=
∑
+1 2
1 1
n E E
p
i i np
P Pcen (2.41)
It might be suggested that the search can be stopped when the error has reached a certain value.
Optimistically (or naively), one might suggest to stop the search when E P
(
min)
= 0, but this value would never be reached in practice, because there will always be a residual error due to the noise affecting either the data and/or model approximations. Setting a nonzero value for E P(
min)
would not work either, because there is no clue on how to set an appropriate value. Hence, the constraint (2.41) emerges as the only sensible stopping criterion. To elaborate further on the behaviour of the search near the minimum, consider that the data actually used for the estimation are the sum of the‘true’ model error y t( ) and the measurement noise v t( ), as shown in Figure 2.20 for the case of a single output and a single parameter. Thus, the minimum of the error functional is the combination of two error terms, due to the model mismatch and/or to the presence of measurement noise, that is,
E p
E E p
min
mod min
( )
= perfect model & no noise = & 0( )
= perfe0 σ = 0
cct model & noise & 0 model error &
Emod=0 σ≠ E p
(
min)
=(
N n s− p)
2noise Emod≠ & ≠0 E p
(
min)
=Emod+(
N n s− p)
0 σ 2
(2.42)
Figure 2.21 shows the effects of modelling and measurement errors following the scheme of Figure 2.20.
The bottom curve represents the ideal case of perfect model and noiseless measurements. In this case, the estimated value of the parameter p
(
pmin)
converges to its true value ptrue and E p1(
true)
= . 0 The second curve represents the case of perfect model and noisy measurements. In this case, the residual error is just the noise variance, that is, E p2 N s2( min)=
(
−1)
. In the third curve, there are both modelling(
Emod)
and measurement errors (σ2 s2), so the minimum error will be the sum of both factors, that is, E p3 E N s2min mod
( )
= +(
−1)
. If both errors are moderate, then the algorithm will still converge to the true parameter value pmin =ptrue, but if modelling and/or measurementModel
True model response y(t) v(t) ∼ N(0, σ2)
y(t) m(t) = y(t) + v(t)
Noise-free samples m(t) = y(t) Noisy samples m(t) = y(t) + v(t)
+
FIGURE 2.20 Modelling the measurement error as an additive noise affecting the output.
errors are large, the minimum of the error functional may be shifted, so that the estimator will no longer converge to the true value (upper curve). An example of the effect of noise on the estimation is provided in Figure 2.22, where the maximum growth rate
(
µmax)
of the Monod kinetics is esti-mated with ‘clean’ and noisy data. The static sensitivity curve is affected by noise that also shifts the minimum variance that only coincides with the true value for very low noise.2.3.2.1 Optimized Simplex Search
One of the weaknesses of the basic simplex search is that the polyhedron reshaping parameters α β γ δ, , ,
( )
are constant throughout the search. On the other hand, the search could be improved by adjusting the local expansion to reach the local minimum in the current search direction, as shown in Figure 2.23. This improvement was introduced by setting up a unidirectional search (Marsili-Libelli and Castelli, 1987; Marsili-Libelli, 1992) based on the Fibonacci interval elimina-tion (Himmelblau, 1972). This approach results in fewer, but more effective expansions, and has proved particularly efficient in coping with the ‘narrow valley’ problem.pmin ≠ ptrue
pmin = ptrue
E4 (pmin)
E3 (ptrue)
E2 (ptrue)
E1 (ptrue) (N − 1)s2
Emod + (N − 1)s2 E(p)
FIGURE 2.21 Effect of model and measurement error on the estimation accuracy, in the single parameter case.
0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.00
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
0 20 40 60 80 100 120 140 160
0 2 4 6 8 10 12
Time (h) μmax
Conc. (mg/l) E(μmax)
Substrate σ2 = 0 Substrate σ2 = 0.2485 Biomass σ2 = 0 Biomass σ2 = 0.2485
σ2 = 0.2485 σ2 = 0.0543
σ2 = 0.0
σ2 = 0.0147 σ2 = 0.1210
(a) (b)
FIGURE 2.22 An example of how noisy measurement may shift the minimum of the error functional. The noiseless and noisy measurement are shown on the left (a), while (b) depicts the changes in the error functional due to the noisy measurements. If the noise is moderate, the minimum still coincides with the true parameter value, but it is shifted away as the noise increases.