Bayesian logistic regression analysis for spatial patterns of inter-seasonal drought persistence

Figures & data

Figure 1. A map of the study area and spatial distribution of meteorological stations.

Figure 2. Histograms of precipitation data and varying fitted distributions for SPI-3.

Figure 3. A q-q plot of empirical vs. theoretical distributions for SPI-3 at selected meteorological stations.

Figure 4. Temporal variations of SPI-3 at selected meteorological stations.

Table 1. Descriptive analysis of precipitation during the period 1971–2017 of selected meteorological stations.

Station	Mean	1st quartile	Median	3rd quartile	Max	Std. dev.
Balakot	132.7	46.9	101.1	187.2	914.9	121.0969
Kakul	111.44	39.38	85.6	161.22	494.6	93.17354
Murree	146.2	47.2	112.2	216.1	704.3	127.193
Muzaffarabad	124.37	43.65	97.2	176.03	721.0	113.1248
Garhi Dupatta	122.14	46.92	97.7	172.2	570.5	100.0412
Kotli	104.15	29.18	69.25	141.15	711.0	109.139

Table 2. Best-fitted probability distributions for the calculation of SPI-3 with their respective BIC values at selected meteorological stations.

Station	Distribution	BIC
Balakot	Gumbel	−1777.45
Kakul	Rayleigh	−1170.10
Murree	Gumbel	−1614.46
Muzaffarabad	Gumbel	−1540.48
Garhi Dupatta	Logistic	−1727.82
Kotli	Generalized Extreme Value	−1478.21

Table 3. Seasonal rainfall variations and climatic features of selected meteorological stations.

Stations	Longitude (N)	Latitude (E)	Elevation (m)	Climate	Seasonal rainfall mm/annum
					Winter	Spring	Summer	Autumn
Balakot	72̊ 21′	34̊ 33′	995	Humid subtropical	308.99	363.17	708.66	211.30
Kakul	73̊ 15′	34̊ 11′	1308	Humid subtropical	228.21	320.10	604.03	184.99
Murree	73̊ 39′	33̊ 91′	2291	Subtropical highland	345.82	363.72	799.49	245.11
Muzaffarabad	73̊ 29′	34̊ 22′	702	Humid subtropical	296.82	333.17	673.58	188.82
Garhi Dupatta	73̊ 37′	34̊ 13′	814	Humid subtropical	324.88	360.30	588.79	191.75
Kotli	73̊ 54′	33̊ 31′	614	Humid subtropical	216.54	240.46	637.16	155.69

Figure 5. Temporal behaviour of $\mathit{SPI}\le 0$ selected meteorological stations.

Figure 6. Seasonal drought counts (total number of moths with $\mathit{SPI}\le 0$) at selected stations.

Figure 7. Average number of months with $\mathit{SPI}\le 0$ per year at each selected station.

Figure 8. Seasonal drought frequency as a percentage of total years (1070–2016) at each station.

Figure 9. Drought persistence percent probability for all seasons in the selected stations.

Table 4. Regional average of seasonal drought frequency as measured by percent probability.

Stations	Winter	Spring	Summer	Autumn	Average
Balakot	87.0	39.9	34.1	43.5	51.1
Kakul	99.3	73.2	45.7	52.9	67.8
Murree	84.1	50.7	26.1	40.6	50.4
Muzaffarabad	83.3	42.8	26.8	45.7	49.6
Garhi Dupatta	93.5	64.5	63.0	73.9	73.7
Kotli	87.7	54.3	28.3	39.1	52.4
Average	89.1	54.2	37.3	49.3

Table 5. Regional average of seasonal drought persistence as measured by percent probability.

Stations	Autumn to winter	Winter to spring	Spring to summer	Summer to autumn	Average
Balakot	40.0	92.7	36.2	5.0	43.5
Kakul	52.6	100.0	68.3	17.8	59.7
Murree	35.3	85.7	44.4	3.6	42.3
Muzaffarabad	41.7	84.7	37.8	1.6	41.5
Garhi Dupatta	72.9	96.6	56.3	52.9	69.7
Kotli	35.5	90.7	46.2	3.7	44.0
Average	46.3	91.7	48.2	14.1

Figure 10. Posterior trace plots of the logistic regression coefficient associated with predictor (SPI) for all seasons at Balakot station.

Figure 11. Posterior predictive checks for seasonal drought occurrence at Balakot station.

Figure 12. Histograms with the distribution of posterior values combining all chains by estimated parameters (β) for all seasons at Balakot station.

Figure 13. Seasonal drought probability distributions as a function of the previous season’s SPI for Balakot station.

Figure 14. Odds ratio for various seasons in the selected stations. The bold form stations are significant at 1%.

Table 6. Estimated coefficients, odds ratios (OR), standard errors (SE) of estimates, and 99% credible intervals (CI) of odds ratios obtained from Bayesian logistic regression models for inter-seasonal combinations at all the selected stations.

					99% CI (OR)
Stations	Season	Coefficient	OR	SE	Lower	Upper
Balakot	Winter-to-Spring	−0.1	0.87	0.2	0.47	1.56
	Spring-to-summer	0.3	1.35	0.3	0.68	2.76
	Summer-to-autumn	1.9*	6.78	0.3	3.29	15.51
	Autumn-to-winter	0.5	1.7	0.2	0.98	3.20
Kakul	Winter-to-spring	0	1.00	0.3	0.48	2.11
	Spring-to-summer	0.3	1.33	0.3	0.69	2.56
	Summer-to-autumn	1.9*	6.94	0.3	3.23	18.48
	Autumn-to-winter	0.2	1.16	0.4	0.42	3.28
Murree	Winter-to-spring	0.0	0.97	0.2	0.56	1.65
	Spring-to-summer	0.3	1.41	0.3	0.75	2.93
	Summer-to-autumn	2.2*	8.61	0.4	3.61	23.37
	Autumn-to-winter	0.8*	2.24	0.2	1.24	4.32
Muzaffarabad	Winter-to-spring	−0.1	0.88	0.2	0.52	1.54
	Spring-to-summer	0.2	1.23	0.3	0.63	2.50
	Summer-to-autumn	1.9*	7.01	0.3	3.39	16.84
	Autumn-to-winter	0.7	1.93	0.2	1.09	3.56
Garhi Dupatta	Winter-to-spring	−0.2	0.8	0.2	0.44	1.43
	Spring-to-summer	0.6*	1.75	0.2	1.04	3.19
	Summer-to-autumn	0.8*	2.17	0.2	1.23	4.29
	Autumn-to-winter	0.0	1.00	0.3	0.52	2.15
Kotli	Winter-to-spring	−0.2	0.82	0.2	0.44	1.51
	Spring-to-summer	0.4	1.47	0.3	0.73	2.87
	Summer-to-autumn	1.9*	6.46	0.3	2.90	16.21
	Autumn-to-winter	0.6*	1.87	0.2	1.02	3.76