These are lecture notes and homeworks for a course taught at the University of Rochester by Andrew White in the Chemical Engineering Department. The course is taught using Jupyter Notebooks.
View the course online: whitead.github.io/numerical_stats/
This course provides an introduction to numerical methods and engineering statistics for chemical engineers. Students learn to use computer models and statistics to understand engineering systems. The focus of numerical methods is translating engineering problems into nalgorithms and implementing them in a spreadsheet or programming language. Topics covered include basic data structures, programming flow control, plotting, function minimization, integration and differential equations. The statistics portion teaches students basic probability theory, the central limit theorem, hypothesis testing, confidence intervals, regression, model fitting and basic error analysis.
See project folder.
Unit 1 — Introduction
Lecture 1: Sample Spaces, Probability Algebra of Samples, Events
Unit 2 — Probability
Lecture 1: Combinations & Permutations, Multidimensional Sample Spaces, Random Variables, Continuous Probability Distributions
Lecture 2: Marginals, Joints, Independence of Random Variables, Table of Useful Equations
Lecture 3: Conditionals, Working with Joints/Marginals/Conditionals, Bayes' Theorem, Math Definition Independence, Compound Conditionals, Conditional Independence, Table of Useful Equations
Unit 3 — Python Basics
Lecture 1: Python Variables, String Formatting, Representing Integers
Lecture 2: Floating Point Representation, Python Booleans, Default Booleans, Floating Point Booleans, Lists, Slicing
Unit 4 — Python Basics, Expected Value
Lecture 1: List Methods, Range, Numpy Arrays, Python Tutor, For loops, Python Data Types (dictionaries, tuples, ints, floats), Function Arguments, Basic Plotting, Jupter Notebook Format
Lecture 2: Expected Values and Variance, Conditional Expectation
Unit 5 — Probability Distributions
Lecture 1: Bernoulli, Geometric, Binomial, Poisson, Exponential and Normal Distribution Equations
Lecture 2: Probability of a Sample or Interval, Prediction Interval
Unit 6 — Python Program Flow
Lecture 1: Plotting - Basics, LaTeX, Point Markers, Vertical/Horizontal Lines, Legends
Lecture 2: Break Statement, While Loops, Discrete Distribution Prediction Intervals, Scipy Stats, Working with Probability and Prediction Intervals of Normal Distribution
Lecture 3: Defining Functions, Named Arguments, Default Function Arguments, Documenting Functions,
Unit 7 — Functions and Sample Statistics
Lecture 1: Sample Statisics for 1D data: median, mean, mode, quartiles and quantiles.
Lecture 2: Presenting Results and Precision, Calculating Sample Statistics, Visualizing 1D data with histograms, Caclulating Sample Statistics with Categories, Visualizing Categorical 1D data with Boxplots and Violin Plots.
Lecture 3: Sample Statisics for 2D data: Sample Covariance, Sample Correlation.
Lecture 4: Plotting 2D data (scatter plot) and computing sample covariance/correlation
Unit 8 — Central Limit Theorem and Confidence Intervals
Lecture 1: Central Limit Theorem and Theory of Confidence Intervals
Lecture 2: Computing Confidence Intervals
Unit 9 — Linear Algebra in Python
Lecture 1: Python Tips & Tricks
Lecture 2: Matrix Algebra (linalg), Solving Systems of Equations, Eigenvector/Eigenvalue, Matrix Rank
Lecture 3: Numerical Differentiation, Numerical Integration via Trapezoidal Rule, Numerical Integration in Scipy, Anonymous Functions (lambda)
Unit 10: — Hypothesis Testing
Lecture 1: Introduction to Hypothesis Testing, the zM and Student's t-Test
Lecture 2: Non-Parametric Statistics, Reading a CSV file in Pandas, Wilcoxon Sum of Ranks, Wilcoxon Signed Rank, Poisson Test, Binomial Test
Unit 11 — Optimization
Lecture 1: Common mistakes with functions, Scope, Root Finding in 1D, Minimization in 1D, Convexity
Lecture 2: Root finding in multiple dimensions, Minimization in multiple dimensions, Bounded Optimization, Non-convex Optimization
Unit 12: — Regression
Lecture 1: Shapiro-Wilk Normality Test, Ordinary Least-Squares Linear Regression in 1- (OLS-1D) and N dimensions (OLS-ND), Standard error, Uncertainty in OLS-1D, OLS-ND, Fit coefficient hypothesis tests, Fit coefficient confidence intervals, Overview of steps to justify and perform regression (bottom of lecture)
Lecture 2: Non-linear regression and error analysis. Deconvoluting spectrum example.
Lecture 3: Regressing categorical data with discrete domains
Lecture 4: Regressing with constant uncertainty/measurement error in independent and/or dependent variables
Unit 13: — Differential Equations & Uncertainty Propagation
Lecture 1: Standard form and categorizing differential equations, Solving ODEs
Lecture 2: Error propagation through numerical derivatives, statistical fallacies
Unit 14: — Applied Python - Working with Data
Lecture 1: Dealing with duplicate, missing, NaN, non-contiguous, out of order data, Joining datasets, Using Pandas, Using Seaborn, Computing Running Means
Lecture 2: Packaging and deploying Python modules
Unit 15: — What to do now
Lecture 1: Next steps to learn more about numerical methods, statistics, and programming
Unit 16: — MATLAB
Lecture 1: An overview of MATLAB, the Jupyter Hub server and Excel
Unit 17: — User Interfaces
Lecture 1: Creating and writing animations
Lecture 2: Introduction to HTML, CSS, JS and modifying notebook style
Unit 18: — Design of Experiments
Lecture 1: Tables of experiments, vocabulary of design of experiments, ANOVA, factorial design, fractional factorial design, nuisance factors, blocking