PROCESOS DE LA EVALUACIÓN INTERNA DEL DESEMPEÑO DOCENTE

above error to solve for the coefficients, subject to the constraint that the coefficients need to be greater than 0. After calculating the coefficients, we would report the maximum relative error to the selection algorithm in the GA, and based on the error, the best individual of each tournament is selected.

aforemen-tioned goal by introducing a new adaptive fitness measurement used for the selection of individuals during the tournaments. The new adaptive fitness measurement is given by the following:

ε_adapt =w_aε_rlse+w_bε_rlae (4.3)

Here, the weights w_a and w_b are given by:

w_b = g_no−1 wf

w_a= 1−w_b (4.4)

Where g_no denotes the current generation number during the GA iteration, and w_f denotes the weight factor, a constant which controls how fast we switch emphasis from mean error to maximum error during the execution of the algorithm. For a largew_f, the transition of focus between mean to maximum error occurs more slowly over the generations. By reporting an adaptive measure of the fitness, using both the mean and maximum error as components of the overall fitness values, the GA should in theory filter out individuals that exhibit a low fitness in mean and/or maximum errors, and retain individuals which have relatively low maximum and mean errors.

By altering the fitness measurement to incorporate both mean and maximum errors, there exists three different permutations of which coefficients to use for the posynomial models, and which fitness measurement to report given two different convex optimization algorithms for the coefficients.

1. In the first case, we can use both a QP which minimizes mean error, and an LP which minimizes the maximum error to find the coefficients for the calcu-lation of ε_rlse and ε_rlae respectively. We can then calculate the adaptive error based on the two different values of error obtained above, and obtain ε_{adapt qlp}. One pitfall for this method would be to assume that the error reported to the GA corresponds to one specific model, whereas it corresponds to two different models with two different coefficients for the same exponents in the genotype.

Therefore, although we choose an error metric based on both, we decided to choose the coefficients determined by the QP as the model to adopt after the final iteration of the GA.

2. Rather than use two different optimization algorithms to obtain the values of the coefficients, we only use a QP aimed at minimizing the mean errorε_rlse, and then calculate both the maximum and mean errors produced by the resulting choice of coefficients. The error ε_{adapt qp} is obtained by adaptively adding the mean and maximum errors obtained from using the coefficients determined by the QP, and is then reported to the GA for individual selection.

3. Finally, we can use an LP which minimizes the maximum error, ε_rlae, to obtain the coefficients for the posynomial phenotype, and then use the coefficients to calculate our measures for mean and maximum error. The error measurement ε_{adapt lp} encompasses both the mean and maximum errors calculated as a result of the LP optimized coefficients, and is consequently reported to the GA so that it can perform individual selection accordingly.

Now that we have obtained several methods for determining the coefficients and measuring the fitness of the generated posynomial models, our next step is to evaluate the performance of these different methods, and determine which of them is best in terms of meeting some designer specifications on mean and maximum error.

Chapter 5

Experiments and Results

In light of the theoretical bounds we posed on the error obtained from posynomial modeling, and the various combinations of error metrics we can use as a measurement of fitness, the current chapter discusses the experimental setup we used to test the theories we postulated, and to compare the performance of the different types of models. We performed three collections of experiments, each concerned with tackling a specific question posited earlier. The first, Experiments A, compare the different types of error metrics we used as fitness measurements for propagating individuals in the GA. The comparison is performed across 4 of the output parameters we had previously performed case studies on in Chapter 2. As for the second collection of experiments, Experiments B, they collect data from the five output parameters g_m, g_ds, r₀, C_gs and I which have been modeled using a GA which optimizes for ε_rlae. The data is then plotted using the visualization tool described in section 2.2.

Finally, Experiments C generate posynomial models using a GA which optimizes for ε_rlae. They gather results corresponding to all the output parameters across input parameters A and B, for n and p-doped MOSFETs. They then compare the values of ε_rlae obtained from the GA with those obtained from Boyd’s monomial fitting.

5.1 Experimental Setup

For all collections of experiments, A through C, we used a particular setting of exper-imental constants, e.g. the number of generations, in order to facilitate the smooth execution of each run of the GA. Unless explicitly stated, the use of the constants described below was maintained across all of the experiments. For each set of exper-iments, we performed 5 different GA runs for each of the target output parameters.

The GA iterated for 500 generations per run, with a population of 50 individuals.

It randomly initialized the seed population. The choice variables were therefore ran-domly chosen so that the number of posynomial terms retained an average of 3 terms, and a maximum of 5 terms. The exponents were allowed to assume any real value between [−5 7].

Experimental Constant Value

N_pop 50

s 6

Runs 5

generations 500

minExponent -5

maxExponent 7

maxT erms 5

avgT erms 3

p_crossover 0.5

λ initial 2

λ rate Halved every 35 generations

p_{zero term} 0.7

pnon zero term 0.3

w_f 250

Table 5.1: Values of experimental constants used for the GA runs

Depending on the optimization applied to calculate the coefficients, and the error metric used to calculate the fitness measurement, the weight factor w_f = 250 was used to adaptively change the dependency of the fitness on mean or maximum error.

Individual selection occurred within tournament sizes of 6 individuals, after which crossover was applied with probabilityp_crossover = 0.5, and mutation of each exponent was performed with probability p_{zero term} = 0.7 for terms with zero coefficients, and

pnon zero term= 0.3 for terms with non-zero coefficients. Finally, the mutation variance λ was initialized to 2, and was halved every 35 generations. Table 5.1 summarizes the experimental constants we used for the experiments.

In the case of the benchmark, monomial fitting, Table 5.2 shows the experimental constants used when running the bisection algorithm.

Experimental Constant

Value

µ 1

Lower Bound 0 Upper Bound 5 Threshold 0.001

Table 5.2: Values of experimental constants for monomial fitting using bisection

5.2 Experiments A: Error Metrics for Max-Mean