Using Maximum Likelihood Logistic Regression to Predict Drought Flow
Threshold Probabilities: A Primer
Samuel H. Austin 1 and Robert Burgholzer 2
Abstract
Introduction.
1.
Problem statement. a.
How is stream water used? i.
Withdrawals.
1.
Industrial applications.
2.
Agricultural applications.
3.
Domestic applications. ii.
Maintenance of channel morphology.
1.
Physical properties of channel function.
2.
Movement of sediments.
3.
Maintenance of stream width and depth. iii.
Maintenance of ecosystem functions.
1.
Habitat for aquatic organisms.
2.
Food source for aquatic organisms.
3.
Transportation corridors for aquatic organisms.
4.
Source water for terrestrial animals and plants. b.
Why are stream flows important during dry times? i.
To maintain anthropogenic, geomorphologic, and ecosystem functions. c.
Why make predictions of flow probabilities? i.
Predictions of flow probabilities allow us to better anticipate water availability during critical periods, facilitating improved planning and management of water resources.
2.
Purpose and scope of study. a.
Provide methods to advance the prediction of stream low-flow probabilities in drought prone months, by as much as five to eight months. b.
Accurately characterize the probabilities of stream flows exceeding specific drought flow thresholds.
Benefits of drought flow-threshold prediction.
1.
Extend the lead-time of drought response by five to eight months.
2.
Improve knowledge and prediction of low flows in drought prone months.
3.
Anticipate future low-flows as functions of change in precipitation, water withdrawals, land use management, and climate.
1 Hydrologist, USGS Virginia Water Science Center, Richmond, Virginia.
2 Environmental Specialist, Virginia Department of Environmental Quality, Richmond, Virginia.
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4.
Better manage water withdrawals and land use to maintain water availability in drought prone months.
An introduction to maximum likelihood logistic regression.
1.
Maximum likelihood logistic regressions are used to describe the chance of a particular average stream flow. a.
Specifically, they describe the chance of a particular average flow exceeding a particular flow threshold, as a function of flows from an earlier time period. b.
They answer the question: “What is the chance (likelihood) that an average flow for a particular month, or a particular day in the month, will exceed a certain flow threshold as a function of measured flows from an earlier time period?” c.
Logistic regression fits nominal Y responses to a linear model of X terms. To be more precise, it fits probabilities for the response levels using a logistic function. For two response levels, the function is: i.
P(Y = r
1
) = (1 + e –Xb ) -1 where r
1
is the first response, or equivalently: log{P(Y=r
1
)/P(Y=r
2
)} = Xb where r
1
and r
2
are the two responses d.
The fitting principal of maximum likelihood means that the βs are chosen to maximize the joint probability attributed by the model to the responses that did occur. This fitting principal is equivalent to minimizing the negative log-likelihood (–LogLikelihood) as attributed by the model:
Loss = –logLikelihood = Σ i = 1 to n –log(Prob(ith row has the y j th response))
2.
The probabilities of exceeding nominal monthly- and daily-flow threshold response values for July, August, and September are predicted using
November, December, January, and February daily-flow values as explanatory variables. These explanatory variables are surrogates for precipitation. a.
Equations describing the maximum likelihood logistic regression line. i.
Probability of “Yes.”
P[Yes] = 1/(1 + e (B0 + B1•X) ) ii.
Probability of “No.”
P[No] = 1/(1 + e -(B0 + B1•X) ) b.
Explanation of the equations and their use.
2

Predicting drought flow threshold probabilities using maximum likelihood logistic regression.
1.
Guided by the Virginia drought task force, the following return flow thresholds are used to test flow probabilities in our maximum likelihood logistic regressions: a.
5-percent b.
10-percent c.
25-percent, and d.
50-percent flow quantile for the month, for the period of record.
2.
The questions asked are: a.
What is the chance that flow exceeds the 5-percent flow quantile? b.
What is the chance that flow exceeds the 10-percent flow quantile? c.
What is the chance that flow exceeds the 25-percent flow quantile? d.
What is the chance that flow exceeds the 50-percent flow quantile?
An Example: drought flow threshold predictions for station number
02039500, the Appomattox River near Farmville, Virginia.
1.
Example maximum likelihood probability plots describing the chance that average daily flow will exceed median August period-of-record (POR) return flow as a function of February mean monthly flow, and the average of combined November, December, January, February (N-D-J-F) mean monthly flows for station number 02039500, Appomattox River near Farmville,
Virginia.
2.
Example maximum likelihood probability plots describing the chance that average monthly flow will exceed median August period-of-record (POR) return flow as a function of February mean monthly flow, and the average of combined November, December, January, February (N-D-J-F) mean monthly flows for station number 02039500, Appomattox River near Farmville,
Virginia.
Applying flow threshold predictions to statewide drought assessments.
1.
Anticipating 5 to 8 months of lead-time in likely available water supply during critical low-flow periods. a.
An example of projecting lead times needed for management decisions.
2.
Managing water withdrawals ahead of crisis points. a.
An example of a “smooth landing” when reducing water withdrawals.
3.
Education, awareness, and encouragement of voluntary changes in water use. a.
An illustration of how knowledge and understanding can help change behavior.
4.
Better management of in-stream and ecological flows. a.
An illustration of how knowledge for healthier streams and nonconsumptive uses can help change behavior.
3

Future Applications
1.
Provide predictions of flow threshold probabilities for continuous record stream gaging sites throughout Virginia, five to eight months ahead of drought prone months.
2.
Use maximum likelihood logistic regression models of flow threshold probabilities, coupled with other analytical methods such as Monte Carlo simulation methods to assist in evaluating longer-term stream flow patterns.
Summary and Conclusions.
1.
Understanding water use is essential to management of water supply, geomorphologic functions, and ecosystem functions.
2.
Stream flows are important during dry times to supply water for maintaining anthropogenic, morphologic, and ecosystem functions.
3.
Predictions of flow threshold probabilities allow us to anticipate water availability during critical periods and provide improved planning and management of water resources.
4.
Drought flow threshold prediction using maximum likelihood logistic regression improves our knowledge and prediction of low flows in drought prone months, and can extend the lead-time of drought response by five to eight months.
References
1.
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