Forecasting long-term precipitation for water resource management: a new multi-step data-intelligent modelling approach

Figures & data

Figure 1. Map of the study area and sites in Pakistan

Table 1. Descriptive statistics of the study sites with their climate data and climate indices over the study period (1981–2015). Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: Dera Ismail Khan. SD: standard deviation

	Climate data statistics
	Precipitation (mm)						Temperature (ºC)					Humidity (g/m^3)
Study site	Mean	Min	Max	SD	Skew	Kurt	Mean	Min	Max	SD	Skew	Kurt	Mean	Min	Max	SD	Skew	Kurt
Site 1	109.10	0.50	743.30	132.74	2.13	5.05	21.86	8.30	33.50	7.29	−0.22	−1.27	43.36	16.00	71.00	12.07	−0.06	−0.43
Site 2	45.79	1.00	409.00	52.16	2.85	12.64	23.09	9.30	81.10	8.69	1.20	7.20	45.36	19.00	69.70	10.54	−0.40	0.29
Site 3	75.66	0.20	648.60	96.12	2.50	8.42	23.75	10.50	34.90	7.16	−0.29	−1.13	46.15	15.00	77.00	13.29	−0.28	−0.49
Site 4	29.57	0.24	376.00	41.98	3.82	23.13	24.23	10.40	35.60	7.62	−0.29	−1.22	43.37	5.00	70.00	11.01	−0.36	0.28
Climate index statistics														Geographical characteristics
	Mean		Min		Max		SD			Skewness		Kurtosis		Study site		Longitude (ºE)	Latitude (ºN)	Elevation (m a.s.l.)
SOI	−1.49		−33.30		27.10		11.19			−0.13		−0.07
NINO3	25.90		23.06		29.21		1.30			−6.20		106.56
NINO3.4	27.07		24.48		29.23		1.01			−9.73		157.14		Site 1		73.08	33.73	604.00
NINO4	28.56		26.64		30.22		0.70			−14.39		210.18		Site 2		71.56	34.01	317.00
PDO	1.05		−2.33		184.00		14.68			11.77		139.17		Site 3		73.72	32.94	234.00
IOD	0.17		−0.70		1.54		0.32			0.60		1.69		Site 4		70.90	31.86	165.00
EMI	−0.08		−1.52		1.08		0.53			−0.55		−0.35
SAM	0.09		−3.25		4.92		1.10			0.08		1.41

Table 2. Selected input predictors for each site using non-dominated sorting genetic algorithm of type II (NSGA). Ratio of selected inputs and root mean square error (RMSE) are also presented

Study site	Maximum iteration	Population size	Mutation rate	No. of selected input	Name of selected input	RMSE
Site 1	10	25	0.1	2	Niño4, SAM	119.93
Site 2	10	25	0.1	7	T, H, SOI, Niño3.4, Niño4, PDO, IOD	36.38
Site 3	10	25	0.1	2	Niño4, SAM	85.89
Site 4	10	25	0.1	2	Niño4, SAM	38.24

Table 3. Schematic statistics used in the singular value decomposition (SVD) method for the decomposition of selected input predictors for each site

Study site	No. of iterations	Block size	No. of training data points	No. of testing data points
Site 1	3	2	269	148
Site 2	3	2	269	148
Site 3	3	2	269	148
Site 4	3	2	269	148

Figure 2. Detailed flow chart of the proposed multi-step non-sorting genetic algorithm of type-II (NSGA) integrated with singular value decomposition (SVD) and random forest – the NSGA-SVD-RF model

Figure 3. Scatterplot of the monthly forecasted and observed precipitation (PTCN_FOR and PTCN_OBS) (mm) in the testing phase for the proposed multi-step NSGA-SVD-RF, NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models using the coefficient of determination (r²) and a linear fit inserted in each panel for (a) Site 1: Islamabad, (b) Site 2: Peshawar, (c) Site 3: Jhelum, and (d) Site 4: D. I. Khan

Figure 4. Boxplots of the monthly forecasted error |FE| (mm) between forecasted and observed PTCN in the testing period of the proposed multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models for Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: D. I. Khan

Table 4. Training performance of the multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models between monthly forecasted and observed precipitation in terms of correlation coefficient (r) and root mean square error (RMSE, mm). Bold indicates the best model

Model	Historical lag	NSGA-SVD-RF				NSGA-SVD-KRR			NSGA-RF				NSGA-KRR
				Training period		Kernel	Training period			No. of predictor split	Training period		Kernel	Training period
		No. of trees	No. of predictor split	RMSE (mm)	r		RMSE (mm)	r	No. of trees		RMSE (mm)	r		RMSE (mm)	r
Site 1
M1	(t – 1)	10,000	100	66.40	0.937	Gaussian	121.70	0.425	10,000	100	69.36	0.924	Gaussian	121.70	0.425
M2	(t – 1) + (t – 2)	10,000	100	64.55	0.955	Gaussian	113.77	0.615	10,000	100	65.07	0.954	Gaussian	113.77	0.615
M3	(t – 1) + (t – 2) + (t – 3)	10,000	100	62.21	0.963	Gaussian	111.25	0.693	10,000	100	62.79	0.962	Gaussian	111.25	0.693
Site 2
M1	(t – 1)	10,000	100	18.44	0.951	Gaussian	15.85	0.964	10,000	100	17.90	0.961	Gaussian	15.85	0.964
M2	(t – 1) + (t – 2)	10,000	100	17.40	0.961	Gaussian	15.85	0.964	10,000	100	16.28	0.964	Gaussian	15.85	0.964
M3	(t – 1) + (t – 2) + (t – 3)	10,000	100	16.97	0.960	Gaussian	15.85	0.964	10,000	100	15.85	0.964	Gaussian	15.85	0.964
Site 3
M1	(t – 1)	10,000	100	50.36	0.929	Gaussian	46.74	0.959	10,000	100	52.28	0.918	Gaussian	46.74	0.959
M2	(t – 1) + (t – 2)	10,000	100	48.72	0.951	Gaussian	46.74	0.959	10,000	100	49.39	0.948	Gaussian	46.74	0.959
M3	(t – 1) + (t – 2) + (t – 3)	10,000	100	46.19	0.961	Gaussian	46.74	0.959	10,000	100	46.74	0.959	Gaussian	46.74	0.959
Site 4
M1	(t – 1)	10,000	100	16.02	0.935	Gaussian	14.48	0.966	10,000	100	16.52	0.925	Gaussian	14.48	0.966
M2	(t – 1) + (t – 2)	10,000	100	15.31	0.956	Gaussian	14.48	0.966	10,000	100	15.57	0.953	Gaussian	14.48	0.966
M3	(t – 1) + (t – 2) + (t – 3)	10,000	100	14.35	0.968	Gaussian	14.48	0.966	10,000	100	14.48	0.966	Gaussian	14.48	0.966

Table 5. Testing performance of the NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models measured by root mean square error (RMSE, mm), mean absolute error (MAE, mm), correlation coefficient (r) between monthly forecasted and observed precipitation. Bold indicates the best model

Model	Lags	NSGA-SVD-RF			NSGA-SVD-KRR			NSGA-RF				NSGA-KRR
			RMSE (mm)	MAE (mm)	r	RMSE (mm)	MAE (mm)	r	RMSE (mm)	MAE (mm)	r	RMSE (mm)	MAE (mm)	r
Site 1
M1	(t – 1),		67.20	49.35	0.900	112.04	84.38	0.525	68.39	50.29	0.894	112.04	84.38	0.525
M2	(t – 1), (t – 2)		64.35	47.26	0.933	104.31	78.49	0.671	64.92	47.71	0.931	104.30	78.49	0.671
M3	(t – 1), (t – 2), (t – 3)		60.26	44.17	0.947	100.56	75.39	0.741	60.74	44.28	0.946	100.55	75.39	0.741
Site 2
M1	(t – 1),		25.97	17.68	0.946	47.06	32.30	0.685	27.26	17.64	0.962	45.11	30.51	0.770
M2	(t – 1), (t – 2)		24.57	16.10	0.956	43.11	28.96	0.794	25.42	16.39	0.969	41.05	27.63	0.854
M3	(t – 1), (t – 2), (t – 3)		23.52	14.92	0.962	40.97	27.25	0.838	24.60	15.45	0.968	34.52	23.46	0.823
Site 3
M1	(t – 1),		42.76	31.55	0.917	73.89	54.18	0.497	43.24	31.92	0.915	73.89	54.18	0.497
M2	(t – 1), (t – 2)		42.17	30.22	0.932	68.66	50.05	0.625	42.44	30.40	0.931	68.65	50.05	0.625
M3	(t – 1), (t – 2), (t – 3)		38.15	27.12	0.953	63.89	46.85	0.722	38.56	27.36	0.950	63.88	46.85	0.722
Site 4
M1	(t – 1),		28.24	18.12	0.882	43.61	29.44	0.539	28.34	18.27	0.881	43.62	29.44	0.539
M2	(t – 1), (t – 2)		26.52	17.54	0.932	40.57	27.57	0.692	26.80	17.76	0.932	40.14	27.33	0.702
M3	(t – 1), (t – 2), (t – 3)		25.46	16.47	0.941	38.97	26.25	0.769	25.68	16.67	0.938	38.97	26.25	0.769

Figure 5. Empirical cumulative distribution function (ecdf) of the monthly forecasted error |FE| (mm) between forecasted and observed PTCN generated by the proposed multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models for Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: D. I. Khan

Table 6. Performance of the proposed multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models using the Willmott index (WI), the Nash-Sutcliffe efficiency (E_NS) and the Legates-McCabe index (LM) in the testing period. Bold indicates the best model

Model	Lag	NSGA-SVD-RF			NSGA-SVD-KRR			NSGA-RF			NSGA-KRR
		WI	ENS	LM	WI	ENS	LM	WI	ENS	LM	WI	ENS	LM
Site 1
M1	(t – 1)	0.667	0.720	0.494	0.196	0.223	0.136	0.657	0.710	0.485	0.196	0.223	0.136
M2	(t – 1), (t – 2)	0.683	0.743	0.512	0.268	0.324	0.190	0.677	0.738	0.508	0.268	0.324	0.190
M3	(t – 1), (t – 2), (t – 3)	0.712	0.774	0.541	0.294	0.371	0.216	0.710	0.770	0.540	0.294	0.371	0.216
Site 2
M1	(t – 1)	0.754	0.811	0.579	0.312	0.380	0.230	0.722	0.792	0.580	0.348	0.430	0.273
M2	(t – 1), (t – 2)	0.771	0.829	0.611	0.385	0.473	0.300	0.749	0.817	0.604	0.424	0.522	0.332
M3	(t – 1), (t – 2), (t – 3)	0.790	0.843	0.637	0.431	0.524	0.337	0.765	0.828	0.624	0.621	0.662	0.429
Site 3
M1	(t – 1)	0.665	0.735	0.488	0.185	0.209	0.121	0.661	0.729	0.482	0.185	0.209	0.121
M2	(t – 1), (t – 2)	0.662	0.738	0.502	0.249	0.306	0.175	0.660	0.735	0.499	0.249	0.306	0.175
M3	(t – 1), (t – 2), (t – 3)	0.709	0.781	0.544	0.302	0.385	0.212	0.705	0.776	0.540	0.302	0.385	0.212
Site 4
M1	(t – 1)	0.520	0.667	0.436	0.103	0.206	0.084	0.520	0.665	0.432	0.103	0.206	0.084
M2	(t – 1), (t – 2)	0.554	0.708	0.456	0.179	0.317	0.145	0.550	0.704	0.451	0.187	0.332	0.153
M3	(t – 1), (t – 2), (t – 3)	0.581	0.731	0.486	0.212	0.370	0.180	0.574	0.727	0.479	0.212	0.370	0.180

Figure 6. Time series of the monthly forecasted and observed PTCN in the testing period using the proposed multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models for Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: D. I. Khan

Figure 7. Polar plots showing the average monthly values of the monthly forecasted error |FE| (mm) between forecasted and observed PTCN in the testing period using the proposed multi-step NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models for Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: D. I. Khan

Table 7. Geographical comparison of the accuracy of the NSGA-SVD-RF model vs the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models in terms of relative root mean squared error (RRMSE, %) computed within the test sites. Bold indicates the best model

Model	Lag	NSGA-SVD-RF	NSGA-SVD-KRR	NSGA-RF	NSGA-KRR
		RRMSE (%)	RRMSE (%)	RRMSE (%)	RRMSE (%)
Site 1
M1	(t – 1)	58.41	97.38	59.45	97.38
M2	(t – 1), (t – 2)	56.43	91.47	56.93	91.47
M3	(t – 1), (t – 2), (t – 3)	53.31	88.96	53.73	88.95
Site 2
M1	(t – 1)	47.02	85.22	49.36	81.68
M2	(t – 1), (t – 2)	45.04	79.03	46.60	75.25
M3	(t – 1), (t – 2), (t – 3)	43.43	75.66	45.43	63.75
Site 3
M1	(t – 1)	58.27	100.69	58.92	100.69
M2	(t – 1), (t – 2)	58.28	94.88	58.65	94.87
M3	(t – 1), (t – 2), (t – 3)	53.58	89.74	54.17	89.73
Site 4
M1	(t – 1)	81.24	125.45	81.51	125.45
M2	(t – 1), (t – 2)	76.45	116.94	77.05	115.70
M3	(t – 1), (t – 2), (t – 3)	74.08	113.38	74.70	113.38

Figure 8. Taylor diagram showing the correlation coefficient between the monthly forecasted and observed PTCN (mm) and standard deviation for the proposed multi-step NSGA-SVD-RF model in comparison with the NSGA-SVD-KRR, NSGA-RF and NSGA-KRR models for Site 1: Islamabad, Site 2: Peshawar, Site 3: Jhelum, and Site 4: D. I. Khan