Estimation of State Variables and Rate Parameters Estimation of State Variables and Rate Parameters Overview 5.1 UF UF-2015

Outline: Estimation Overview Why? Why? What? What? How? How? Summary Recommendations Summary Recommendations

Why?

Why Estimate State Variables and Rate Parameters? Science Science –Understand ecological systems –Learn stuff –E.g, spotted owls Management/Conservation Management/Conservation –Apply decision-theoretic approaches –Make smart decisions –E.g., spotted owls

Why Monitor? Surveillance Monitoring & Science “Biology, with its vast informational detail and complexity, is a ‘high-information’ field, where years and decades can easily be wasted on the usual type of ‘low- information’ observations and experiments if one does not think carefully in advance about what the most important and conclusive experiments would be.” (Platt 1964)

Low-information Observations: Surveillance Monitoring Monitoring designed without guiding hypotheses about system behavior Monitoring designed without guiding hypotheses about system behavior Scientific approach: retrospective observational Scientific approach: retrospective observational Objective: Objective: –is population going up or down? –to learn inductively about system dynamics by observing time series of system state variables

Opinions About Retrospective Story-telling Claims: Claims: –(1) It is easy to view a time series of abundance estimates and build a story about the stochastic process that generated it. –(2) It is foolish to place much confidence in such a story. Phaedrus’ Law: Phaedrus’ Law: –“The number of rational hypotheses that can explain any given phenomenon is infinite.” (Pirsig 1974, Zen and the Art of Motorcycle Maintenance)

Retrospective Story-telling Homer: Not a bear in sight. The Bear Patrol must be working like a charm. Lisa: That’s specious reasoning, Dad. Homer: Thank you, dear.

High-information Science: Confront Predictions with Data Deduce predictions from hypotheses Deduce predictions from hypotheses Observe system dynamics via monitoring Observe system dynamics via monitoring Confrontation: Predictions vs. Observations Confrontation: Predictions vs. Observations –Compare estimated rate parameters to predictions (single-hypothesis) –Use correspondence between estimates and predictions to discriminate among hypotheses (multiple-hypothesis)

Observing System Dynamics: Study Designs System manipulation System manipulation –Manipulative experimentation (randomization, replication, controls) –Impact study (lacks randomization and perhaps replication, but includes time-space controls) No manipulation - observational study No manipulation - observational study –Prospective (confrontation with predictions from a priori hypotheses) –Retrospective (a posteriori story-telling)

Use of Estimates in Science Strength of inference: Strength of inference: –Manipulative experiment > Impact study > Observational study Strength of inference for observational studies: Strength of inference for observational studies: –Prospective (a priori hypotheses) > Retrospective (a posteriori stories) Prospective approach permits more focused designs, e.g., barred owls Prospective approach permits more focused designs, e.g., barred owls

Management/Conservation Key Elements Objective function Objective function Available management actions Available management actions Model set based on hypotheses Model set based on hypotheses Monitor/estimate state of system Monitor/estimate state of system Update model probabilities Update model probabilities Derive optimal management action Derive optimal management action Implement optimal management action Implement optimal management action

Management/Conservation Key Elements Objective function Objective function Available management actions Available management actions Model set based on hypotheses Model set based on hypotheses Monitor/estimate state of system Monitor/estimate state of system Update model probabilities (learn) Update model probabilities (learn) Derive optimal management action Derive optimal management action Implement optimal management action Implement optimal management action

Why Monitor? Role of monitoring in management - estimate system state for: Role of monitoring in management - estimate system state for: –Assessing success of management relative to objectives –Discrimination among competing models –State-specific decisions

Role of Monitoring: Assess System Performance Monitoring of objective-related variables permits performance assessment Monitoring of objective-related variables permits performance assessment Conservation setting: objectives may be functions of the system state variable(s) Conservation setting: objectives may be functions of the system state variable(s) Exploitation setting: objectives may include functions of other variables (e.g., accumulated harvest) estimated from the monitoring program Exploitation setting: objectives may include functions of other variables (e.g., accumulated harvest) estimated from the monitoring program

Role of Monitoring: Learning about System Dynamics Estimates of state variables are compared against model predictions Estimates of state variables are compared against model predictions Updating parameter estimates for members of model set (e.g., revised estimates of distribution of harvest rates resulting from different hunting regulations) Updating parameter estimates for members of model set (e.g., revised estimates of distribution of harvest rates resulting from different hunting regulations) Updating of model weights (learning) Updating of model weights (learning)

Bayes’ Formula p t+1 (model i | data t+1 ) = p t (model i ) P(data t+1 | model i)   p t (model j ) P(data t+1 | model j) j

Role of Monitoring: Identify System State Use in state-dependent management decisions Use in state-dependent management decisions Optimal decision = f(system state) Optimal decision = f(system state) Example: optimal harvest decisions may depend on population size Example: optimal harvest decisions may depend on population size

State-dependent Decision Matrix Ponds Mallards Johnson et al. 1997

What?

Low-information Observation: Trend Detection Trend detection: typically provides a single estimate of trend for a period of years Trend detection: typically provides a single estimate of trend for a period of years Not very useful for environmental hypotheses because environmental covariates either: Not very useful for environmental hypotheses because environmental covariates either: –vary year to year, so single trend estimate not useful –vary monotonically, so trend either does or does not follow prediction

Trend Estimation and Management No estimates of state for: No estimates of state for: – State-dependent decisions –Comparison with model-based predictions Trend most likely to be useful when management involves a single action: Trend most likely to be useful when management involves a single action: –Trends can be compared before and after action –Trends can be compared for locations that are and are not exposed to the management action

The Role of Significant Trends in Management Optimal state-dependent decisions would seldom be “wait until decline, then act” Optimal state-dependent decisions would seldom be “wait until decline, then act” –E.g., WNS Do not place management decisions in a hypothesis-testing framework Do not place management decisions in a hypothesis-testing framework –Make decisions under uncertainty –Imminent extinction –Small sample sizes

What State Variable to Monitor Population – single species Population – single species –State variable: abundance –Vital rates: P(survival, reproduction, movement) Community – multiple species Community – multiple species –State variable: Species richness –Vital rates: rates of extinction and colonization Patch – single species Patch – single species –State variable: Proportion patches occupied –Vital rates: P(patch extinction/colonization)

What State Variables (& Vital Rates) to Monitor? Depends On: Monitoring objectives Monitoring objectives –Science: what hypotheses are to be addressed? –Management/conservation: what are the objectives and available actions? Geographic and temporal scale Geographic and temporal scale Effort available for monitoring Effort available for monitoring –Required effort: species richness, patch occupancy < abundance

How?

Monitoring Based on a Count Ungulates seen while walking a line transect Ungulates seen while walking a line transect Tigers detected with camera-traps Tigers detected with camera-traps Birds heard at point count Birds heard at point count Small mammals captured on trapping grid Small mammals captured on trapping grid Bobwhite quail harvested during hunting season Bobwhite quail harvested during hunting season Kangaroos observed while flying aerial transect Kangaroos observed while flying aerial transect

Low-information Observations: Surveillance Monitoring Use counts as index to abundance Use counts as index to abundance –Confound changes in abundance and detection probability Use convenience sampling Use convenience sampling –Need probability sampling to draw inferences about area of interest Sampling may lack contrasts to distinguish among competing hypotheses Sampling may lack contrasts to distinguish among competing hypotheses –E.g., only ‘good’ habitat

Detection Probability Detection Detection –Less than 1 –Counts are unknown fraction of total animals –Proper inference requires information on detection probability

Detectability: Conceptual Basis N = abundance N = abundance C = count statistic C = count statistic p = detection probability; P(member of N appears in C) p = detection probability; P(member of N appears in C)

Detectability: Inference Inferences about N require inferences about p Inferences about N require inferences about p

Simple Indices Assume Equal Detectability N i = abundance for time/place i N i = abundance for time/place i p i = detection probability for i p i = detection probability for i C i = count statistic for i C i = count statistic for i

Indices: Dealing with Variation in Detectability Standardization (variation sources that we can identify and control) Standardization (variation sources that we can identify and control) –e.g., survey duration Covariates (variation sources that we can identify and measure) Covariates (variation sources that we can identify and measure) –e.g., observer Prayer (variation sources that we cannot identify, control or measure) Prayer (variation sources that we cannot identify, control or measure) –e.g., individual heterogeneity

Covariates Underestimated X Abundanc e 0 N True p < 1

Geographic Variation Geographic variation Geographic variation –Frequently counts/observations cannot be conducted over entire area of interest –Proper inference requires a spatial sampling design that: »Permits inference about entire area, based on a sample, and/or »Provides good opportunity for discriminating among competing hypotheses

Spatial Sampling Designs Simple random sampling Simple random sampling Stratified random sampling Stratified random sampling Systematic sampling Systematic sampling Cluster sampling Cluster sampling Double sampling Double sampling Adaptive sampling Adaptive sampling Dual-frame sampling Dual-frame sampling

Spatial Sampling N = abundance N = abundance C = count statistic C = count statistic α = proportion of total area of interest that is sampled (area from which C is obtained) α = proportion of total area of interest that is sampled (area from which C is obtained) If detection probability p=1, then: If detection probability p=1, then:

Spatial Sampling: Inference When p=1 α known: α known: α estimated: α estimated:

Canonical Estimator: Dealing w/ Detectability & Spatial Sampling α known: α known: α estimated: α estimated:

Variance of Canonical Estimator Var (C) = f(α, true spatial variance of animals over sample units)

Variance of Canonical Estimator

Summary Recommendations

Recommendations: Why Monitor? Monitoring is most useful when integrated into efforts to do science or management Monitoring is most useful when integrated into efforts to do science or management Role of monitoring in science Role of monitoring in science –Comparison of data with model predictions is used to discriminate among competing models Role of monitoring in management - determine system state for: Role of monitoring in management - determine system state for: –State-specific decisions –Assessing success of management relative to objectives –Discrimination among competing models

Recommendations: What to Monitor? Decision should be based on objectives (e.g., scientific or management context) Decision should be based on objectives (e.g., scientific or management context) Decision should consider required scale and effort Decision should consider required scale and effort Reasonable state variables Reasonable state variables –Species richness –Patch occupancy –Abundance

Recommendations: How to Monitor? Detectability Detectability –Counts represent some unknown fraction of animals in sampled area –Proper inference requires information on detection probability Geographic variation Geographic variation –Frequently counts/observations cannot be conducted over entire area of interest –Proper inference requires a spatial sampling design that: »permits inference about entire area, based on a sample, or »provides good opportunity for discriminating among competing hypotheses