Water temperature modelling: comparison between the generalized additive model, logistic, residuals regression and linear regression models

Figures & data

Figure 1. Sainte-Marguerite River drainage basin with locations of hydrometric station, weather station and thermographs (S1, S2, S3).

Table 1. Number of concomitant records by year for the three variables: water temperature, air temperature and flow.

Year	Start of period (dd/mm)	End of period (dd/mm)	No. of observations
2005	23/06	25/07	33
2006	0	0	0
2007	04/07	01/10	89
2008	0	0	0
2009	28/07	08/09	43
2010	24/05	16/10	146
2011	01/05 27/07	17/07 31/10	175
2012	20/06	31/10	134
2013	12/06	31/10	141
2014	16/05 06/09	11/06 31/10	83
2015	01/05	19/07	80

Figure 2. (a) Average daily water and air temperatures from 2005 to 2015; and (b) average daily flows from 2005 to 2015.

Table 2. Significance of effects.

Smoothing functions	Estimated degrees of freedom	Fisher test	P-value
S (air temperature)	3.6	242.9
S (flow)	8.0	43.4	2.2 × 10^−16
S (day of the year)	8.4	85.44
R^2_adj		0.94
Deviance explained		94.6%

Figure 3. Estimated smoothing functions representing the contributions of predictors: (a) air temperature; (b) flow; and (c) day of year.

Figure 4. Normal Q-Q plot of GAM residuals.

Figure 5. Normal distribution of GAM residuals.

Figure 6. Residuals versus estimated water temperatures.

Figure 7. Relationship between daily mean water temperature and mean air temperature and a fitted logistic function.

Figure 8. Mean daily water temperature, annual component estimated by sinusoidal function and the residual component.

Table 3. Results of the cross-validation of water temperature (°C) modelling expressed by RMSE, bias error and NSC by the GAM, logistic, residuals regression and linear regression approaches.

Year	Model
	GAM			Multiple regression			Residuals regression			Logistic
	RMSE(°C)	Bias(°C)	NSC	RMSE(°C)	Bias(°C)	NSC	RMSE(°C)	Bias(°C)	NSC	RMSE(°C)	Bias(°C)	NSC
2005	1.72	−0.95	0.39	1.87	−1.2	0.27	2.31	−1.59	−0.1	1.97	−1.24	0.19
2007	1.2	0.24	0.87	1.42	−0.33	0.82	1.64	0.7	0.77	1.8	−1.1	0.71
2009	1.12	0.29	0.82	1.29	0.04	0.76	1.69	1.42	0.58	1.8	−0.12	0.52
2010	1.83	−0.68	0.86	1.74	−0.68	0.87	2.32	−1.12	0.77	2.07	−1.43	0.82
2011	1.54	0.29	0.93	2.26	0.33	0.85	2.47	1.15	0.83	3.37	1.68	0.67
2012	1.27	−0.17	0.95	1.63	0	0.92	1.27	−0.71	0.95	1.82	−0.7	0.9
2013	1.12	0.39	0.94	1.59	0.29	0.89	1.53	0.44	0.9	1.76	−0.23	0.86
2014	1.48	−0.1	0.97	2.07	0.12	0.93	2.51	−0.04	0.9	3.68	0.62	0.79
2015	1.38	−0.22	0.93	2.01	0.44	0.85	1.68	0.4	0.9	3.28	1.6	0.61
Global	1.44	−0.04	0.94	1.83	−0.02	0.9	2	0.11	0.88	2.56	−0.01	0.8
Range	1.12–1.83	−0.95 to 0.39	0.39–0.97	1.29–2.26	−1.2 to 0.44	0.27–0.93	1.27–2.51	−1.59 to 1.42	−0.1 to 0.95	1.76–3.68	−1.43 to 1.68	0.19–0.86

Figure 9. Scatterplot of observed versus predicted values for (a) the GAM model; (b) the logistic model; (c) the residuals regression model; and (d) the linear regression model for the season of May–October (2005–2015).