**4. Conclusions**

250 Genetic Programming – New Approaches and Successful Applications

**Figure 5.** Comparison between genetic programming and multiple linear regression models against

Water temperature approach with multiple linear regression and genetic programming

5 7 9 11 13 15 17 19 21 23 **Tw oC**

Measured vs GP Measured vs MLR Identity

Equations (4) and (5) were applied to measured data from 1999 at the Ribarroja Station in order to arrive at the average water temperature. Measured water temperature data and the

According to Figure 6 the differences between measured and calculated water temperature shown were up to 5.5 °C (underestimation) and about 0.5 °C (overestimation) while differences with equation 5 reported an underestimation near to 4.5°C and the overestimation of almost 2°C, so the range of variation in water temperature reported by

In order to get better results in future works must be analyzed the data standardization as a preprocessing to get new mathematical linear and nonlinear models [44], The variables could be standardized by subtracting the mean and dividing by the standard deviation:

*Z*

*w w w T T T*

(6)

**r measuerd vs MLR=0.9672 r measuerd vsGP=0.9697**

measured data and the identity function. Ribarroja Station. January to June, 1998

obtained residuals using both models are shown in Figure 6.

*Tw* variable before standardization, with physical dimensions

algorithm from January to June 1999

**Tw ºC**

both equations is almost the same.

*Z* standardized variable, dimensionless

where:

Water temperature adjustment curves, in a gauged station on the Ebro River in Spain, were obtained by means of two procedures: a genetic programming algorithm (equation 4) and a multiple linear regression (equation 5), using data from 1998. The multiple linear regression method yielded a function containing the five considered variables (solar radiation, net radiation, wind speed, air temperature and relative humidity) with each variable weighted. The genetic programming algorithm yielded a function where water temperature was obtained only as a function of air temperature and relative humidity. The others variables were eliminated by the evolution algorithm due to the lack of correlation between water temperature and the remaining variables although solar radiation is implied inside the air temperature term.

Comparing measured data with calculated data, for the year 1998, led to only minor errors in estimating the average water temperature using the genetic programming algorithm. When equations (4) and (5) were applied to another year, 1999, minor mean quadratic error in estimating water temperature was obtained using the multiple linear regression equation (5). The mean quadratic error associated with the multiple linear regression equation (5) for 1999 was 1.375 ºC; whereas with the genetic programming equation (4) was 2.248 ºC. This error can be considered acceptable if one takes in account the average temperature from January to June 1998 was 12.54 ºC, whereas the average temperature in 1999 for the same period was 11.62 ºC. The residuals obtained with equations (4) and (5) using data for the year 1999 had average values of 1.04 ºC and 0.43 ºC, respectively and with this criteria, multiple linear regression model can be considered better than the GP. However, reviewing the standard deviations, both models had almost the same value (1.09 ºC and 1.08 ºC, respectively).

Comparison Between Equations Obtained by Means of Multiple

Linear Regression and Genetic Programming to Approach Measured Climatic Data in a River 253

Authors gratefully acknowledge the financial support under the project PAPIIT no.

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The described procedures are then useful because equations similar to (4) or (5) can estimate important water quality characteristics, such as water temperature, using previously measured climatic data, predicted climatic data, and hydrological parameters for a given time period.

Engineer's criteria and common sense must be considered before to apply any model to simulate physical variables.

Some standardization procedures to the involved data are suggested in order to improve the results from new models that can be obtained.

The methods here applied are undoubtedly useful in several areas of knowledge, and can led us to new approaches to physical phenomena by considering measured field data.

Future work is focuses on the use of NARMAX (Non-linear Autorregressive Moving Average with eXogenous inputs) model combined with genetic programming in order to model the water temperature providing more accurate equations.
