Herramientas de diseño

3.3.1Datasets

In the last decade, regions within the UK Environment Agency (EA) have completed a number of dye tracing studies and more than one hundred different tracing studies were analysed to obtain estimates of the mean travel velocity and longitudinal dispersion. A database of travel times and dispersion was developed (Guymer 1999) comprising the tracing works cited. The database (denoted by Data I) contains 196 data samples from 27 different rivers and includes information relevant to the traces; including geographical and physical attributes of the river reaches as well as optimized Advection- Dispersion Model (ADE) and Aggregated Dead Zone Model (ADZ) travel times, velocities, ADE longitudinal dispersion coefficients and ADZ dispersive fractions.

The second dataset, Data II, contains 71 sets of measurements from 29 rivers in the United States. This dataset has previously been very well studied in the literature (Seo and Cheong 1998; Deng, Singh et al. 2001; Rowinski, Piotrowski et al. 2005; Tayfur and Singh 2005).

3.3.2Data Pre-processing

Usually data pre-processing of GNMM includes scaling the inputs and targets so that they fall within a specified range. However, since there are a total of 49 variables available in Data I, the prior objective of data pre-processing is to reduce the dimensionality of the original set of inputs by eliminating

Table 3-1: Variables in Data I and II

Data I (16 variables) Data II (8 variables) Start location: Cs (km 2 ), Ds (km) Ms (m3/s), Qs (m3/s) Independent: B (m), H (m) U (m/s), u* (m/s),α End location: Ce (km 2 ), De (km) Me (m3/s), Qe (m3/s) Reach: S, L(m), Dr (m) Dependent: B/H, U/u*, β Gauging station: Cg (km2), A(m3/s) Mg (m3/s), Qg (m3/s) I(m3/s)

redundant and/or dependant variables. This will result in a set of independent inputs that are not necessarily related to the dispersion coefficient. This subset of inputs can then be used in GNMM to determine which of these inputs are most appropriate for mapping to the output.

Among the 49 available variables in Data I, non-numerical variables such as river name, flow excedence/category, and start location grid reference are removed first of all. These variables are valuable in terms of dye tracing studies but do not provide useful information in the current context. Secondly, dependant variables are discarded. For example, start and end position of the river location elevation are discarded, while reach slope is kept; reach sinuosity is removed while reach length and straight distance are kept. After being processed, Data I contains 16 variables which belong to 4 categories in the original dataset, e.g. start/end location, reach, and gauging station. These

variables are: for the start/end location (subscripts s/e), catchment area (C), distance from injection point (D), theoretical mean flow (M), and theoretical Q95 flow (Q); for the reach in question, slope (S), reach length (L), and straight distance (Dr); for the gauging station, catchment area (Cg), average daily flow (A), daily mean flow (Mg), theoretical Q95 flow (Qg), instant flow (I). All these variables are listed in Table 3-1.

Data II contains 8 variables apart from the longitudinal dispersion coefficient. There are five independent variables: channel width (B), flow depth (H), velocity (U), shear velocity (u*) and river sinuosity (α = channel length/valley length). Dependant variables are width-to-depth ratio (B/H), relative shear velocity (U/u*) and channel shape variable (β= ln(B/H)). Data II variables are listed in Table 3-1 too. It is worth noting that dependent variables exist in Data II (B/H, U/u*), which indicates that eliminating redundant and/or dependant variables is not always necessary in GNMM. Since the aim of this step is to reduce the dimensionality of the data by eliminating redundant and/or dependant variables, obviously in Data II the dimensionality is not a main issue. On the other hand, as we will show later, Data II can be treated as an example of how GNMM handles dependent variables.

3.3.3Division into Training and Testing Data

Before the start of GA variable selection, it is necessary to divide the dataset into training and testing subsets. This is to avoid over fitting when

chromosomes are being evaluated in an MLP (Lin and Lee 1996). The division is achieved by selecting representative sets for both of the training and testing data:

(1) Among the 196 data samples contained in Data I, some contain a high percentage of missing values, or indications that the data recorded is inaccurate. In order to obtain reliable results, these data are removed. As a result, the final dataset contains 127 data samples (see Appendix A). After division, the training subset, denoted by Data It, contains 102

samples; while the testing subset, denoted by Data Is, consists of the

remaining 25.

(2) Similarly, Data II (see Appendix B) is divided into two subsets, Data IIt and

Data IIs for training and testing respectively. Data IIt contains 49 datasets

out of 71, while Data IIs consists of the remaining 22.

Normally when we have a large quantity of data we would typically use more data for training and less for testing. With small datasets we may repeat the process several times by randomly generating training and testing data to ensure that our results are reliable for the dataset. Table 3-2 and Table 3-3 show statistics of these subsets respectively. Note that in these tables, Avg,

Avgt and Avgs mean the average for the whole dataset, training and testing

Table 3-2: Data I statistics

Start Location End Location

Cs (km2) Ds (km) Ms (m3/s) Qs (m3/s) Ce (km2) De (km) Me (m3/s) Qe (m3/s) Max 3314.75 41.50 49.55 9.47 3315.25 46.50 49.55 9.47 Min 16.00 1.00 0.18 0.02 9.25 3.40 0.39 0.03 Avg 714.51 9.82 13.04 2.10 858.97 16.43 15.15 2.45 Avgt 643.05 8.49 13.57 1.97 838.79 16.27 17.10 2.61 Avgs 732.15 10.15 12.90 2.13 863.91 16.47 14.63 2.41

Reach Gauging Station

S L(m) Dr (m) Cg (km 2 ) A(m3/s) Mg (m3/s) Qg (m3/s) I(m3/s) Max 0.0244 14697.0 12133.50 3314.80 47.14 75.00 6.60 75.00 Min 0.0000 1058.00 915.57 20.00 0.44 0.48 0.06 0.50 Avg 0.0023 6037.06 4342.88 792.39 12.38 10.08 1.93 10.20 Avgt 0.0030 6775.73 4834.09 736.99 13.48 11.33 1.83 11.29 Avgs 0.0022 5856.02 4222.49 805.96 12.11 9.78 1.95 9.93

Table 3-3: Data II statistics

B (m) H (m) U (m/s) u* (m/s) B/H U/u* β α K (m2/s) Max 711.2 19.94 1.74 0.553 156.5 19.63 5.05 2.54 892.0 Min 11.9 0.22 0.03 0.002 13.8 1.29 2.62 1.08 1.9 Avg 83.0 1.70 0.54 0.088 51.7 7.62 3.79 1.39 107.7 Avgt 62.9 1.31 0.49 0.084 51.4 7.13 3.79 1.37 98.4 Avgs 127.6 2.55 0.66 0.097 52.4 8.72 3.77 1.42 128.4

When forming these two subsets, the present work follows that of Tayfur and Singh (2005), in order to compare results. However, as mentioned in Tayfur and Singh (2005): ‘In choosing the datasets for training and testing, special

attention was paid to ensure that we have representative sets so as to avoid bias in model prediction’ (p. 993).

For example, in Data II the range for the dispersion coefficient (K) varies from 1.9 to 892 m2/s, and K is greater than 100 m2/s in 21 cases, which counts for about 30% of all available measured coefficient values. The range for the width-to-depth ratio (B/H) of the datasets varies from 13.8 to 156.5 and (B/H) is greater than 50 in 26 cases (37%). After division the percentages of K> 100

m2/s and B/H > 50 are also comparable for both Data IIt and Data IIs. For

example, in Data IIs 25% of Kis greater than 100 m/s2 (this ratio is 31% in Data

IIt), also, in Data IIs 40% of B/His greater than 50 (this ratio is 31% in Data IIt).