Introduction

Computational Science

  • often hard science (physics, chemistry, biology); engineering
  • using computers to solve scientific problems
  • models
  • simulations
  • numerical analysis

Data Science

  • can be used in both hard and soft sciences
    • soft sciences often categorized by ill-defined systems (economics, finance, sociology, but also, business/predictive analytics)
  • machine learning, natural language processing, information extraction
  • information retrieval
  • knowledge discovery in networks
  • database theory (big data)
  • high performance computing
  • MapReduce framework

What is systems modeling?

  • a system is a set of inter-related factors
  • a model is a mathematical representation of some object or process
  • modeling is the application of methods to analyze real-world problems in order to predict how the model might behave under different scenarios
  • can be physical or mathematical/computational
  • this usually entails a visualization

Types of System Models

  • deterministic
    • static
    • dynamic
      • continuous
      • discrete
  • stochastic (probabilistic)
    • static (Monte Carlo simulation)
    • dynamic
      • continuous
      • discrete (discrete-event simulation)

Characterizing the Model

  • Deterministic of stochastic?
    • Does the model contain significant randomness/uncertainty?
  • Static or dynamic?
    • Is the time dimension important?
  • Continuous or discrete
    • How does the system evolve?
    • Continuous: classical mechanics
      • state variables are infinitely updated within any finite timespan
      • commonly described by ordinary differential equations
      • E.g., \dot{x}=f(x, u, t)
    • Discrete: queuing, inventory
      • time axis is discretized
      • commonly described by difference equations
      • E.g., \frac{x_{t+1}-x_t}{\Delta t} \approx f(x_t, u_t)

Examples

Static:Birthday Problem
from scipy import misc, stats
prob = lambda n : stats.poisson.sf(0, misc.comb(n,2)/365.)
Deterministic simulation:
 Manufacturing process
Stochastic simulation:
 Realistic manufacturing process; Hardware/server time to failure
Continuous simulation:
 Rocket trajectory, robotics

Applications

  • Data Science
    • Finding cats on youtube
    • Prediction - Nate Silver’s 538 Model
    • Recommendation engines (Netflix, Amazon, JSTOR)
    • Health Care/Bioinformatics
    • Network Analysis
    • Disaster Response
  • Systems Modeling
    • Hardware simulators (embedded systems)
    • Hydrodynamic simulations (storm surges, tsunamis, floods)
    • Earthquakes
    • Flight simulators
    • Traffic simulation
    • Biology/Health Care (Disease spread, Population dynamics)
    • Ecology
    • Manufacturing processes (inventory, machine shops)
  • Visualization - D3.js

(Iterative) Modeling Process

  1. Analyze the problem (Goals and objectives)
  2. Formulate a model (Specification)
    1. diagram relationships on paper
    2. gather data
    3. determine input/output and units
    4. determine equations and functions
    5. document simplifying assumptions
    6. pseudocode
  3. Solve the model
    1. Analytic or Simulation
  4. Verify and validate then interpret the solution
    1. verification - ensure the code is faithful to the specification model (model right?)
    2. validation - ensure that simulation results are congruent with experimental results / the system being analyzed (right model?)
  5. Report on the model
    1. Describe the problem, the model design, the model solution, and results
  6. Maintenance
    1. Importance of unit testing

Why Simulate?

  • Mathematical model may not have an analytical solution
  • Physical system is not available
  • Dangerous (atomic reactor, flight simulator)
  • Prohibitively high cost
  • Time-constraints
  • Control over input variables and state variables
  • Isolation of particular controls
  • Suppression of higher-order effects (non-linearities)

Why Python

  • Alternatives
  • Python
    • Low-level enough to understand the concepts
    • High-level enough not to get in your way
    • Batteries included
    • Prominence in scientific computing
    • Free

Good Software Engineering Habits

  • Coding standards
  • Version control
  • Unit testing / Test-driven development
  • Maintain backups (rsync)

Unsolicited Advice