Syllabus#
This is the course handout for Objective Analysis — a graduate-level introduction to the statistical and mathematical tools used in the atmospheric and oceanic sciences.
How to read these notes#
Throughout the notes, two types of highlighted boxes are used:
Figure / in-class demonstration
Green boxes mark figures, in-class demonstrations, or Python/Matlab examples to be shown during lecture. They are placeholders for visual material that accompanies the written derivations.
Example / deeper dive
Blue boxes contain worked examples, mathematical proofs, or deeper conceptual discussions that go beyond the main narrative. These are important — they are where most of the practice problems and physical intuition live.
Course Outline#
The course runs over 16 weeks, with the mid-term in Week 7 and the final in Week 16. There is no class in the final-exam week.
Weeks |
Topic |
|---|---|
— |
|
1-3 |
|
4-6 |
|
7 |
Mid-term exam |
8-11 |
|
12-15 |
|
16 |
Final exam |
Chapter 0: The Idea Behind the Course#
Chapter 0: The Idea Behind the Course — read once now, revisit after each Part
Every method in this course is the same move: optimize a functional subject to constraints. Maximizing entropy under known moments gives you the PDF a hypothesis test needs; minimizing a loss gives you the best model, because a loss function is a negative log-likelihood. The Lagrange multiplier that appears in both turns out, in Part III, to be the eigenvalue.
This chapter is the most abstract material of the semester and nothing in Weeks 1-3 depends on it. Each Part opens with a short “The pattern” box pointing back here and filling in that Part’s row.
Part I: Foundations of Statistics#
Week 1-3: Rule101 — Week 1-3: Rule 101
The origin of statistics: why the bell curve?
Mean, variance and higher moments
Basic probabilities; unions; intersections; conditional probabilities; Bayes theorem
Tying it together: Moments, PDFs, and Bayes’ Theorem
Statistical significance testing
Hypothesis testing
Monte Carlo and resampling techniques
Compositing
Other common distributions
Non-parametric tests
Part II: Regression & Autocorrelation#
Week 4-6: Regression & AR1 — Week 4-6: Regression & AR1
Linear regression: least squares, slope & intercept
Theory of correlation: Pearson’s r, Fisher-Z, Spearman’s rank
Autocorrelation & effective sample size: AR1/red noise, Leith & Bretherton
Multiple regression: generalized normal equations, overfitting
Week 7 — Mid-term exam, covering Parts I and II.
Part III: Seeking Structure in Data#
Week 8-11: Seeking Structure in Data — Week 8-11: EOFs & Clustering
Linear algebra review: inner products, covariance matrices, inverse, rank & null space
Eigenvalues and eigenvectors: diagonalizing the covariance matrix
EOFs via eigenanalysis: maximizing explained variance, orthogonality, principal components
EOFs via Singular Value Decomposition, and its equivalence to eigenanalysis
EOFs with real data: weighting, standardization, presentation in physical units
How many EOFs to retain: North et al. (1982) and degeneracy
Cluster analysis: k-means and Lloyd’s algorithm
Self-organizing maps (SOMs): training, mapping, quantization & topographic error
Part IV: Time Series Analysis#
Week 12-15: Time Series Analysis — Week 12-15: Spectral Analysis & Filtering
Harmonic analysis: Fourier sums, the Nyquist frequency and aliasing
The discrete Fourier transform as orthogonal multiple regression
The power spectrum: line vs. continuous, and the resolution–reliability trade-off
The complex Fourier transform
Significance of spectral peaks: red-noise null, the F-test, a priori vs. a posteriori
Windows and finite data: convolution, the convolution theorem, boxcar & Hanning, WOSA
Filtering: response functions, non-recursive & Lanczos, recursive & Butterworth
Cross-spectrum analysis: co-spectrum, quadrature spectrum, coherence² and phase
Mixed space-time analysis: propagating waves and phase speed
Week 16 — Final exam. No class this week.