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Terms, Statistics, and Performance Measures for Maximum Likelihood Logistic Regression Models Estimating Hydrological Drought Probabilities in the Delaware River Basin (2020)

October 8, 2021

Tables are presented listing parameters and fit statistics for 25,453 maximum likelihood logistic regression (MLLR) models describing hydrological drought probabilities at 324 gaged locations on rivers and streams in the Delaware River Basin (DRB). Data from previous months are used to estimate chance of hydrological drought during future summer months. Models containing 1 explanatory variable use monthly mean daily streamflow data (DV) to provide hydrological drought streamflow probabilities for July, August, and September as functions of monthly mean DV from the previous 11 months. Outcomes are estimated 1 to 12 months ahead of their occurrence. Models containing 2 explanatory variables use monthly mean daily streamflow data (DV) and monthly mean precipitation data (P) to provide hydrological drought streamflow probabilities for July, August, and September as functions of monthly mean DV and monthly mean P from the previous October, November, December, January, and February. Outcomes are estimated 5 to 12 months ahead of their occurrence. Models containing 3 explanatory variables use monthly mean daily streamflow data (DV), monthly mean precipitation data (P), and monthly mean maximum daily air temperature (T) to provide hydrological drought streamflow probabilities for July, August, and September as functions of monthly mean DV, monthly mean P, and monthly mean maximum T from the previous October, November, December, January, and February. Outcomes are estimated 5 to 12 months ahead of their occurrence. Models containing 4 explanatory variables use monthly mean daily streamflow data (DV), monthly mean precipitation data (P), monthly mean maximum daily air temperature (T), and monthly mean potential evapotranspiration data (PET) to provide hydrological drought streamflow probabilities for July, August, and September as functions of monthly mean DV, monthly mean P, monthly mean maximum T, and monthly mean PET from the previous October, November, December, January, and February. Outcomes are estimated 5 to 12 months ahead of their occurrence. Explanatory variable selections for multi-parameter models were optimized using random forest statistical methods.

Selected single-parameter and multi-parameter models are provided. Overall correct classification rates tend to improve and models become more complex as the number of model explanatory variables increases from 1 to 4. Parameters for models with 1 explanatory variable are listed in the table labeled: “DRB-1_Variable_Equations.” Parameters for models with 2 explanatory variable are listed in the table labeled: “DRB-2_Variable_Equations.” Parameters for models with 3 explanatory variable are listed in the table labeled: “DRB-3_Variable_Equations.” Parameters for models with 4 explanatory variable are listed in the table labeled: “DRB-4_Variable_Equations.”

Parameters describing models containing 1 explanatory variable may be used to populate drought probability equations as follows: p =1/[1 + e-(β0 + β1 • DV)] where: e is the base of the natural logarithm, β0 is an intercept parameter, β1 is a slope parameter, DV is a factor variable describing monthly mean daily streamflow (ft3/s).

Parameters describing models containing 2 explanatory variables may be used to populate drought probability equations as follows: p =1/[1 + e-(β0+ β1• DV + β2 • P)] where: e is the base of the natural logarithm, β0 is an intercept parameter, β1 is a slope parameter, β2 is a slope parameter, DV is a factor variable describing monthly mean daily streamflow (ft3/s), P is a factor variable describing monthly mean precipitation (in/day).

Parameters describing models containing 3 explanatory variables may be used to populate drought probability equations as follows: p =1/[1 + e-(β0+ β1• DV + β2• P + β3 • T)] where: e is the base of the natural logarithm, β0 is an intercept parameter, β1 is a slope parameter, β2 is a slope parameter, β3 is a slope parameter DV is a factor variable describing monthly mean daily streamflow (ft3/s), P is a factor variable describing monthly mean precipitation (in/day), T is a factor variable describing monthly mean maximum daily air temperature (degrees F).

Parameters describing models containing 4 explanatory variables may be used to populate drought probability equations as follows: p =1/[1 + e-(β0 + β1• DV + β2• P + β3 • T + β4• PET)] where: e is the base of the natural logarithm, β0 is an intercept parameter, β1 is a slope parameter, β2 is a slope parameter, β3 is a slope parameter, β4 is a slope parameter, DV is a factor variable describing monthly mean daily streamflow (ft3/s), P is a factor variable describing monthly mean precipitation (in/day), T is a factor variable describing monthly mean maximum daily air temperature (degrees F), PET is a factor variable describing monthly mean potential evapotranspiration (in/day).

DV data span the period of record at each gage, ranging from July 1, 1899 through July 31, 2018. P, T, and PET data span the period associated with each gage beginning July 1, 1981 and ending July 31, 2018.

Equation goodness of fit parameters document model strength, identifying the utility of each relation. Receiver Operating Characteristic (ROC) AUC values, scaled from 0 to 1, identify each model’s overall correct classification rate and are listed in the table labeled: “DRB-AUC_TABLE.”

MLLR modeling of drought streamflow probabilities exploits the explanatory power of temporally linked water flows. Models with strong correct classification rates are provided for streams throughout the Delaware River Basin. Hydrological drought MLLR probability estimates inform understanding of drought streamflow conditions, provide warning of future drought conditions, and aid water management decision making.

More details of methods used may be found in: Austin, S.H., and Nelms, D.L., 2017, Modeling summer month hydrological drought probabilities in the United States using antecedent flow conditions: Journal of the American Water Resources Association, v. 53, p. 1133–1146, accessed November, 15, 2018, at https://doi.org/10.1111/1752- 1688.12562.