Uncertainty and Monte Carlo
Lecture 08
September 23, 2024
Review of Last Classes
Simulation
- “Mimic” data generation given a certain model for the relevant processes;
- Often requires some type of discretization (not always!)
Simulation Workflow
Simulation
- “Mimic” data generation given a certain model for the relevant processes;
- Often requires some type of discretization (not always!)
- More complex approaches (e.g. discrete-event simulation) for specific use cases.
What We Haven’t Discussed
- Calibration: How do we select parameter values?
- Validation: Does the model adequately reproduce the system dynamics?
Dissolved Oxygen
- Dissolved oxygen (DO) is essential for water quality and aquatic life.
- Commonly regulated to keep DO above a minimum threshold.
- DO impacted by a number of factors, notably organic waste decomposition and nitrification.
- “Sag Curve”: DO reduced near a discharge until waste decomposition reduces OD and re-aeration can occur.
Questions?
Text: VSRIKRISH to 22333
Uncertainty and Systems Analysis
Systems and Uncertainty
- Deterministic models: uncertainty due to the separation between the “internals” of the system and the “external” environment.
- Stochastic models: additional uncertainty from internal stochasticity
What is Uncertainty?
Glib Answer: A lack of certainty!
More Seriously: Uncertainty refers to an inability to exactly describe current or future values or states.
Types of Uncertainty
Two (broad) types of uncertainties:
- Aleatory uncertainty, or uncertainties resulting from randomness;
- Epistemic uncertainty, or uncertainties resulting from lack of knowledge.
Risk
Systems and Risk
Designing and managing environmental systems is often about minimizing or managing risk:
- Maintaining clean air/water;
- Power grid reliabiliy standards;
- Flooding/other hazards;
- Climate change mitigation/adaptation.
What is Risk?
The Society for Risk Analysis definition:
“risk” involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environment), often focusing on negative, undesirable consequences.
So What is Risk?
Important: “Risk” is not just another words for probability, but:
- Involves uncertainty;
- Undesireable outcomes;
- Effects matter, not just the events themselves.
Components of Risk
Multiple components which contribute to risk:
- Probability of a hazard;
- Exposure to that hazard;
- Vulnerability to outcomes;
- Socioeconomic responses.
Source: Simpson et al (2021)
Systems and Risk Management
Risk management is often a key consideration in systems analysis. For example, consider regulatory standards.
- Often a tradeoff between strictness of a regulation and costs of compliance.
- Systems modeling is a key way to understand
- the impacts of changing a regulation
- the probability of failure to meet standards.
Probability
Two Definitions of Probability
We often represent uncertainties with probabilities. What is a probability?
- Long-run frequency of an event (frequentist)
- Degree of belief that a proposition is true (Bayesian)
Conditional Probabilities
We often don’t want to just know if a particular event
In other words:
We want the conditional probability of
Conditional Probabilities
We can write conditional probabilities in terms of unconditional probabilities:
Bayes’ Theorem
Conditional probabilities can be inverted according to Bayes’ Theorem:
Probability Distributions
The probability of possible values of an unknown quantity are often represented as a probability distribution.
Probability distributions associate a probability to every event under consideration (the event space) and have to follow certain rules (for example, total probability = 1).
Selecting a Distribution
The specification of distributions can strongly influence the analysis.
Selecting a Distribution
A distribution implicitly answers questions like:
- What is the most probable event? How much more likely is it than the others?
- Are larger or smaller events more, less, or equally probable?
- How probable are extreme events?
- Are different events correlated, or are they independent?
Key Features of Probability Distributions
- Mean/Mode (what events are “typical”)
- Skew (are larger or smaller events more or equally probable)
- Variance (how spread out is the distribution around the mode)
- Tail Probabilities (how probable are extreme events)
Common Distributions
- Gaussian / Normal
- Lognormal
- Binomial
- Uniform / Discrete Uniform
Probability Distribution Tails
The tails of distributions represent the probability of high-impact outcomes.
Key consideration: Small changes to these (low) probabilities can greatly influence risk.
Monte Carlo
Stochastic Simulation
Monte Carlo simulation: Propagating random samples through a model to estimate a value (usually an expectation or a quantile).
Goals of Monte Carlo
Monte Carlo is a broad method, which can be used to:
- Obtain probability distributions of outputs;
- Estimate deterministic quantities (Monte Carlo estimation).
Monte Carlo Estimation
Monte Carlo estimation involves framing the quantity of interest as a summary statistic (such as an expected value).
MC Example: Finding
Finding
MC Example: Dice
What is the probability of rolling 4 dice for a total of 19?
Can simulate dice rolls and find the frequency of 19s among the samples.
Monte Carlo Estimation
This type of estimation can be repeated with any simulation model that has a stochastic component.
For example, consider our dissolved oxygen model. Suppose that we have a probability distribution for the inflow DO.
How could we compute the probability of DO falling below the regulatory standard somewhere downstream?
Monte Carlo and Uncertainty Propagation
- Draw samples from some distribution;
- Eun them through one or more models;
- Compute the (conditional) probability of outcomes of interest (for good or bad).
Key Takeaways
Key Takeaways
- Systems analysis features many uncertainties, which may be neglected with certain methods/model setups.
- Choice of probability distribution can have large impacts on uncertainty and risk estimates: try not to use distributions just because they’re convenient.
- Monte Carlo: Estimate expected values of outcomes using simulation.
Upcoming Schedule
Next Classes
Wednesday: Monte Carlo Lab (clone before class, maybe instantiate environment too)
Monday: Why Does Monte Carlo Work?
Assessments
HW3: Released today, due 10/03 at 9pm.
Uncertainty and Monte Carlo Lecture 08 September 23, 2024