**2.5. Multiple linear regressions**

Multiple linear regressions (MLR) relate a dependent variable, *y,* with two or more independents variables, *x1, x2, x3,…, xn,* by means of an equation expressed as:

$$y = a\_1\mathbf{x}\_1 + a\_2\mathbf{x}\_2 + a\_3\mathbf{x}\_3 + \dots + a\_n\mathbf{x}\_n \tag{3}$$

Comparison Between Equations Obtained by Means of Multiple

Linear Regression and Genetic Programming to Approach Measured Climatic Data in a River 249

*rs* is the average solar radiation each ten minutes interval, at instant t, in W/m2

The objective function value using equation 5 was 0.8724.

regression models in comparison with measured data.

instant t+160, in W/m2

and finally,

t+160, in m/s.

to June, 1998

0

5

10

15

**Tw ºC**

20

25

*rn* corresponds to the average net radiation each ten minutes interval, corresponding to

*vv* represents the average wind speed each ten minutes interval, corresponding to instant

Figure 4 represents both measured and calculated water temperature variation versus time using both equations (4) and (5). Measured and calculated water temperature values also appear in Figure 5 with equations (4) and (5) in comparison with the identity function.

Figure 4 indicate similar results for both genetic programming and multiple linear

In Figures 5 the measured data were compared against the identity function and the best

**Figure 4.** Time variation of measured and calculated water temperature data, Ribarroja Station. January

**0 50000 100000 150000 200000 250000 300000 t (min)**

Measured Calculated GP Calculated MLR

correlation between these values was found using genetic programming (r=0.9697).

Coefficients *a1,a2,a3,…an*, are weighting factors which allow one to see the relative importance of each variable *xi* as y is approached. Indirectly the coefficients can indicate if there is a strong correlation or lack of correlation between *xi* and *y*.

This method is often applied for several hydrology problems such as: forecasting equations for standardized runoff in a region of a country with standardized teleconnection indices, when El Niño or La Niña phenomenon occur [43] (González et al., 2000), or as an auxiliary method in estimating intensity-duration-frequency curves. In this research, regressions were made using the Microsoft Excel data analysis tool.
