Introduction to computational social science, notation, and functions

Why we are here

• Social science
  • Anthropology
  • Economics
  • History
  • Political science
  • Psychology
  • Sociology
  • Comparative human development(?)

Computational social science

• Computational approaches to the social sciences
• Historic definition
• Modern definition
  • Intersection of computer science, math/statistics, and social science

Computational social science

Acquiring CSS skills

• Computer science
• Social science
• Math/statistics

Mathematics

• Purely abstract
• Based on axioms that are independent from the real world
• If the axioms are accepted, mathematical inferences are certain
• Language for expressing structure and relationships
• Generally proof-based

Probability

• Systematic and rigorous method for treating uncertainty
• “Mathematical models of uncertain reality”
• Derivation of “applied mathematics”

Statistics

• Practice or science of collecting and analyzing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a sample
• Making inferences from data that are not entirely certain
• Based on mathematical models, but with deviations from the model (not deterministic)

Their uses

• Mathematical models
  • Game theory
  • Formal theory
  • Much bigger in economics
  • Defining statistical models and relationships
• Probability/statistics
  • Establishing a structure for relationships between variables using data
  • Inferring relationships and assessing their validity

Goals for the camp

• Computational Mathematics and Statistics Camp
• Survey math and statistical tools that are foundational to CSS
• Review common mathematical notation
• Apply math/statistics methods
• Prepare students to pass the MACSS math/stats placement exam

Course logistics

All course materials are located on GitHub

Course staff

• Me
• Teaching assistants
  • Sanja Miklin
  • Nora Nickels

Prerequisites

• No formal prerequisites
• Most likely to succeed
  • Linear algebra
  • Calculus
  • Probability theory
  • Statistical inference
• Assume prior exposure

Alternatives to this camp

• SOSC 30100 - Mathematics for Social Sciences (aka the Hansen math camp)
• MACS 30100 - Mathematics and Statistics for Computational Social Science
• Departmental methods sequences
• Math/stats departments

Evaluation

• Pass/fail (no credit)
• Grades no longer matter (or should not)
• Learn as much material as possible
• If you truly only care about learning material, you’ll get amazing grades

Pedagogical approach

• Flipped-classroom design
• Instructional staff is to be your “sherpa” and guide you along your journey

Computational tools for the future

• Vision is open-source
• Shift away from proprietary formats (e.g. SPSS, Stata, SAS)
• Emphasis on reproducibility
  • Code
  • Results
  • Analysis
  • Publication

Programming languages

Programming languages

• Python
• R
• Things R does well
  • Statistical analysis
  • Data visualization
• Things R does not do as well
  • Speed
• Things Python does well
  • General computation
  • Speed
  • The Jupyter Notebook
• Things Python does not do as well
  • Visualizations
  • Add-on libraries

Version control

• Revisions in research
• Tracking revisions
• Multiple copies
  • analysis-1.r
  • analysis-2.r
  • analysis-3.r
• Cloud storage (e.g. Dropbox, Google Drive, Box)
• Version control software
  • Repository
  • Git

Move away from WYSIWYG

• What You See Is What You Get
• Microsoft Word or Google Docs
• Not a reproducible format

Notebooks

• Integrate code, output, and written text
• Reproducible
  • Rerun the notebook to regenerate all the output
• Good for prototyping and exploratory data analysis
• For Python - Jupyter Notebooks
• For R - R Markdown

\(\LaTeX\)

• High-quality typesetting system
• De facto standard for production of technical and scientific documentation
• Free software
• Renders documents as PDFs

• Makes typesetting easy

  $$f(x) = \frac{\exp(-\frac{(x - \mu)^2}{2\sigma^2} )}{ \sqrt{2\pi \sigma^2}}$$

  \[f(x) = \frac{\exp(-\frac{(x - \mu)^2}{2\sigma^2} )}{ \sqrt{2\pi \sigma^2}}\]

• Tables/figures/general typesetting/nice presentations - easier in \(\LaTeX\)

• Steep learning curve up front, but leads to big dividends later

Markdown

• Lightweight markup language with plain text formatting syntax
• Easy to convert to HTML, PDF, and more
• Used commonly on GitHub documentation, Jupyter Notebooks, R Markdown, and more
• Simplified syntax compared to \(\LaTeX\) - also less flexibility
• Publishing formats

How you will acquire these skills

Irony - you won’t learn any of that here. No time to teach programming AND math AND statistics in 3 weeks.

• Python
• R
• Git
• \(\LaTeX\)/Markdown/notebooks

Why is math important to social science

• Consistent language to communicate ideas in an orderly and systematic way
• Science uses highly precise language that is not easily interpretable to outsiders
• Mathematics is an effective way to describe our world
• Mathematical notation lets us convey precision and minimizes the risk of misinterpretation by other scholars

Example: Rational voter theory

• Rational voters should weigh the rewards vs. costs of voting

• Mathematical notation

  \[R = PB - C\]

  • \(R =\) the utility satisfaction of voting
  • \(P =\) the actual probability that the voter will affect the outcome with her particular vote
  • \(B =\) the perceived difference in benefits between the two candidates measured in utiles (units of utility)
  • \(C =\) the actual cost of voting in utiles (e.g. time, effort, money)
• Does the model seem reasonable?

• What implications does the model provide?

Example: Rational voter theory

• Leads to the paradox of voting
• Clearly a question worth answering in political science
• Pure mathematical model may not be fully accurate, but allows us to delve deeper into the paradox

Equations

• An equation “equates” two quantities - they are arithmetically identical
  • \(R = PB - C\)
  • \(R\) and \(PB - C\) are exactly equal to one another
• Equations do not need to be equalities
  • Greater than/less than
  • Approximately equal

Functions

• A mapping which gives a correspondence from one measure onto exactly one other for that value

• Mapping from one defined space to another, such as \(f \colon \Re \rightarrow \Re\)

  \[f(x) = x^2 - 1\]

  • Maps \(x\) to \(f(x)\) by squaring \(x\) and subtracting 1

Not a function

• All functions are relations, but not all relations are functions
• Functions have exactly one value returned by \(f(x)\) for each value of \(x\)
• Relations may have more than one value returned

Not a function

Not a function

Two major properties of functions

\[f(x) = y\]

1. Continuous

2. Invertible

   \[g^{-1}(y) = x, \text{where } g^{-1}(g(x)) = x\]

Not all functions are continuous and invertible

\[ f(x) = \left\{ \begin{array}{ll} \frac{1}{x} & \quad x \neq 0 \text{ and } x \text{ is rational}\\ 0 & \quad \text{otherwise} \end{array} \right. \]

Not all functions are continuous and invertible

Not all functions are continuous and invertible

Why this is important

• Functions must be continuous AND invertible to calculate derivatives in calculus
• Important for optimization and solving for parameter values in modeling strategies
• Non-continuous functions - cannot do much about
• Non-invertible functions
  • Can make invertible by restricting the domain

Restrict the domain

Logarithms and exponents

• Important component to many mathematical and statistical methods in social science
• Exponents - repeatedy multiply a number by itself
• Logarithms - reverse of an exponent

Common rules of exponents

• \(x^0 = 1\)
• \(x^1 = x\)
• \(\left ( \frac{x}{y} \right )^a = \left ( \frac{x^a}{y^a}\right ) = x^a y^{-a}\)
• \((x^a)^b = x^{ab}\)
• \((xy)^a = x^a y^a\)
• \(x^a \times x^b = x^{a+b}\)

Logarithms

• Class of functions
  • \(\log_{b}(x) = a \Rightarrow b^a = x\)
  • What number \(a\) solves \(b^a = x\)
• Commonly used bases

  • Base 10

    \[\log_{10}(100) = 2 \Rightarrow 10^2 = 100\]

    \[\log_{10}(0.1) = -1 \Rightarrow 10^{-1} = 0.1\]

  • Base 2

    \[\log_{2}(8) = 3 \Rightarrow 2^3 = 8\]

    \[\log_{2}(1) = 0 \Rightarrow 2^0 = 1\]

Logarithms

• Base \(e\) - Euler’s number (aka a natural logarithm)

  \[\log_{e}(e) = 1 \Rightarrow e^1 = e\]

  • Natural logarithms are incredibly useful in math
  • Often \(\log()\) is assumed to be a natural log
  • Also seen as \(\ln()\)
• Rules of logarithms
  • \(\log_b(1) = 0\)
  • \(\log(x \times y) = \log(x) + \log(y)\)
  • \(\log(\frac{x}{y}) = \log(x) - \log(y)\)
  • \(\log(x^y) = y \log(x)\)