CHAPTER 2 - Computational Modeling

Model

is a simplified representation of a real-world system, built to study, predict, or analyze behavior. *Not exact copies* but abstractions → focus only on key features relevant to the problem.

Model

helps us: • Understand complex systems • Predict future events • Test different scenarios • Reduce the cost of experiments • Save time • Improve decision-making • Minimize risks associated with real-world experimentation

Model

They • Describe how a system works. • Explain relationships among variables. • Predict future behavior. • Analyze different situations. • Test "what-if" scenarios safely. • Support scientific and engineering decisions.

Modeling

is the application of methods to analyze complex, real-world problems in order to make *predictions* about what might happen with various actions

Variables

These are the fundamental *"parts" or "actors"* within the model, representing physical things, concepts, or processes. These are elements that change. Examples: Population size, Temperature.

Parameters

Refers to *constants* that define system properties. Examples: Growth rate, Infection rate.

Equations/Rules

*Relationships* that describe how variables change.

computational model

is a simplified representation of a real-world object, process, or system that is created *using mathematical relationships and computational methods.* Instead of experimenting directly on real systems, scientists use models to understand behavior, predict outcomes, and evaluate different scenarios

Simple, accurate, reliable, easy to understand, based on valid assumptions, flexible

6 characteristics of a goof computational model (SAREBF)

Deterministic, probabilistic or stochastic

2 types of computational models

Deterministic model

type of computational model. always produces the same output when given the same input.

Characteristics

• No randomness

• Predictable results

• Exact mathematical relationships

• Easier to analyze

Examples: Simple interest computation, Projectile motion, Population growth using fixed rates, Newton's laws of motion

Deterministic model

type of computational model.

Advantages

• Easy to understand

• Easy to verify

• Produces consistent results

Limitations

• Cannot represent uncertainty

• Less realistic for many natural systems

Probabilistic/Stochastic Model

type of computational model. It includes randomness or probability. The same input may produce different outputs because uncertain events are involved.

Characteristics

• Uses probability

• Includes random variables

• Models uncertainty

• Produces multiple possible outcomes

Examples: Weather forecasting, Stock market prediction, Disease transmission, Traffic congestion, Insurance risk analysis

Probabilistic/Stochastic Model

type of computational model.

Advantages

•More realistic

•Represents uncertainty

•Useful for complex systems

Limitations

•More difficult to analyze

•Results vary from one simulation to another

Static, dynamic

subcategories of the types of computational models

Static

In a ______ model, we do not consider time so that the model is comparable to a snapshot or a map. For example, a model of the weight of a salamander as being proportional to the cube of its length has variables for weight and length but not for time.

Dynamic

By contrast, in a _______ model, time changes, so that such a model is comparable to an animated cartoon or a movie. For example, the number of salamanders in an area undergoing development changes with time— hence, a model of such a population is dynamic.

Algebraic

Type of Equation used in a static model

Differential

Type of Equation used in a dynamic model

Continuous, discrete

2 subtypes of the subcategory Dynamic

Continuous

When time changes continuously and smoothly, the model is __________.

Discrete

If time changes in incremental steps, the model is ________. Such model is analogous to a movie. A sequence of frames moves so quickly that the viewer perceives motion

Continuous

However, in a live play, the action is __________. Just as a discrete sequence of movie frames represents the continuous motion of actors, we often develop discrete computer models of continuous situations

modeling process

is cyclic and closely parallels the scientific method and the software life cycle for the development of a major software project. It is iterative— models are built, tested, refined, and reapplied

Analyze, formulate, solve, verify, report, maintain

6 general steps of the mdoeling process (AFSVR)

Analyze the problem

step in the modeling process. Identify and clearly define the problem you want to address or the question you want to answer. Define the objectives, research questions, and boundaries.

Formulate a model

step in the modeling process. In this stage, we design the model, forming an abstraction of the system we are modeling. Build assumptions, identify variables and relationships • Example: Conceptual Framework

Gather data

substep of step 2. We collect relevant data to gain information about the system's behavior.

Simplifying assumptions

substep of step 2. In formulating a model, we should attempt to be as simple as reasonably possible. Thus, we frequently decide to simplify some of the factors and to ignore other factors that do not seem as important.

Determine variables and units

substep of step 2. To simplify the model, we may decide to neglect some variables (such as air resistance), treat certain variables as constants, or aggregate several variables into one. While deciding on the variables, we must also establish their units, such as days as the unit for time

independent variable

is the variable on which others depend. In many applications, time is an example of this. The model will try to explain the dependent variables.

Establish relationships

substep of step 2. If possible, we should draw a diagram of the model, breaking it into sub-models and indicating relationships among variables. To simplify the model, we may assume that some of the relationships are simpler than they really are.

Determine equations and functions

substep of step 2. While establishing relationships between variables, we determine equations and functions for these variables. For example, we might decide that two variables are proportional to each other, or we might establish that a known scientific formula or equation applies to the model. Many computational science models involve differential equations, or equations involving a derivative

Solve the model

step in the modeling process. This stage implements the model. It is important not to jump to this step before thoroughly understanding the problem and designing the model. Otherwise, we might waste much time, which can be most frustrating. Some of the techniques and tools that the solution might employ are algebra, calculus, graphs, computer programs, and computer packages.

Solve the model

step in the modeling process. Our solution might produce an exact answer or might simulate the situation. If the model is too complex to solve, we must return to Step 2 to make additional simplifying assumptions or to Step 1 to reformulate the problem.

Verify and interpret the solution

Once we have a solution, we should carefully examine the results to make sure that they make sense (verification) and that the solution solves the original problem (validation) and is usable. The process of verification determines if the solution works correctly, while the process of validation establishes if the system satisfies the problem's requirements

Verification, validation

____________ concerns "solving the problem right," and __________ concerns "solving the right problem."

Report on the model

step in the modeling process. Reporting on a model is important for its utility. Perhaps the scientific report will be written for colleagues at a laboratory or will be presented at a scientific conference.

Maintain the model

step in the modeling process. As the model's solution is used, it may be necessary or desirable to make corrections, improvements, or enhancements

Python

is a high-level, general-purpose programming language known for its simplicity and readability. Open-source and cross platform Large ecosystem of scientific libraries

Numpy, scipy, matplotlib and seaborn, simpy

5 popular libraries in python for computational science

NumPy

is a fundamental Python library for numerical and scientific computing. It provides support for working with large, multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays efficiently.

SciPy

It builds on NumPy (which provides fast array operations) and adds a wide collection of algorithms and functions for more advanced tasks in computational science, engineering, and applied mathematics. It used for scientific computations

Matplotlib

A foundational Python plotting library for creating static, animated, and interactive plots. It uses, 2D plotting (line & scatter plots, histogram, bar charts, etc.)

Seaborn

A high-level visualization library built on top of Matplotlib. It focuses on statistical graphics and makes plots prettier and simpler with less code. It used for scientific computations. Uses specialized plots (distribution plots, heatmaps, regression & pair plots, etc)

SimPy

a library for modeling and simulating real-world systems where events happen at discrete points in time. Uses Python generators (yield) to model processes. Widely used for queueing systems, logistics, resource allocation, and operations research.

Model

key term, review. A simplified representation of a real-world system.

Computational Model

key term, review. A model implemented using mathematical and computational techniques.

Deterministic model

key term, review. Produces the same result for the same input.

Stochastic Model

key term, review. Includes randomness and probability.

Verification

key term, review. Ensures the model is built correctly.

Validation

key term, review. Ensures the correct model is being used.

Simulation

key term, review. Running a model to study system behavior under different conditions

Computational model

key takeaway, review. simplifies complex real-world systems to support analysis and decision-making.

Model

key takeaway, review. help explain system behavior, predict outcomes, and evaluate alternative scenarios without conducting costly or risky physical experiments.

Deterministic model

key takeaway, review. produce consistent results for the same inputs, while stochastic models incorporate randomness to represent uncertainty.

Computational modeling

key takeaway, review. is a foundational skill in computational science and is widely applied in engineering, healthcare, environmental science, business, and scientific research