# 12-Week Active Math Stack > 5 days/week: 90m math + 30m implementation. Weeks 1–4 probability & statistics (simulate, implement stats, debunk fake signals). Weeks 5–8 linear algebra (matmul, OLS, PCA from scratch). Weeks 9–12 quant bridge (returns, vol, Sharpe, drawdown, momentum backtest vs SPY/QQQ). One public note per week. No artifact = no learning. Canonical: https://learn.significanthobbies.com/curriculum/roadmaps/math-stack-12w 5 days/week: 90m math + 30m implementation. Weeks 1–4 probability & statistics (simulate, implement stats, debunk fake signals). Weeks 5–8 linear algebra (matmul, OLS, PCA from scratch). Weeks 9–12 quant bridge (returns, vol, Sharpe, drawdown, momentum backtest vs SPY/QQQ). One public note per week. No artifact = no learning. - Horizon: 12mo - Outcome: Math → AI systems → distributed/data → quant tools. Active only: solve, derive, implement, simulate — never aesthetic consumption. - Tracks: Mathematics Work each milestone as a claim that needs evidence. Predict the mechanism and failure mode first, study the linked concepts, complete the drills without copying an answer, and preserve the resulting code, measurements, or design. ## Milestone 1: Week 1 — Probability gym Conditional probability, Bayes, distributions — plus simulate random processes. ### Concepts - [ML Math Foundations](https://learn.significanthobbies.com/curriculum/concepts/ml-math) — Vectors, matrices, dot products, matmul, shapes. - [Combinatorics](https://learn.significanthobbies.com/curriculum/concepts/combinatorics) — Counting without enumerating — permutations, combinations, inclusion-exclusion, pigeonhole. - [Numerical Stability](https://learn.significanthobbies.com/curriculum/concepts/numerical-stability) — Floating point, catastrophic cancellation, and the log-sum-exp trick. - [Queueing Theory](https://learn.significanthobbies.com/curriculum/concepts/queueing-theory) — Little's law and the utilisation curve — why latency explodes before a system runs out of capacity. - [Curse of Dimensionality](https://learn.significanthobbies.com/curriculum/concepts/curse-of-dimensionality) — Why distances concentrate in high dimensions, and what that does to nearest-neighbour search. - [Probability Fundamentals](https://learn.significanthobbies.com/curriculum/concepts/probability-fundamentals) — Sample spaces, conditional probability, independence, and Bayes' rule for updating beliefs. - [Classical Distributions](https://learn.significanthobbies.com/curriculum/concepts/classical-distributions) — Bernoulli, Binomial, Poisson, Normal, and Exponential — when each models the world and what to expect from them. - [Random Walks & Markov Chains](https://learn.significanthobbies.com/curriculum/concepts/random-walks-markov) — Markov property, random walks, transition matrices — generative story behind market efficiency intuition. ### Build evidence - **Simulate coin flips, dice, and random walks** — Code simulators (no stats libraries): Bernoulli trials, dice sums, and a 1D random walk. Plot or print distributions and compare sample mean to theory. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 2: Week 2 — Expectation & summaries RVs, expectation, variance — implement descriptive stats from scratch. ### Concepts - [Random Variables & Distributions](https://learn.significanthobbies.com/curriculum/concepts/random-variables) — Discrete and continuous distributions, expectation, variance, and the law of large numbers. - [Descriptive Statistics](https://learn.significanthobbies.com/curriculum/concepts/descriptive-statistics) — Summarizing data: mean, median, variance, correlation, and when each summary lies. - [Covariance & Correlation](https://learn.significanthobbies.com/curriculum/concepts/covariance-correlation) — Covariance measures co-movement; correlation normalizes to [−1, 1]. Foundation for regression, PCA, and portfolio risk. ### Build evidence - **Implement mean, variance, correlation from scratch** — On a real CSV (or synthetic data), compute mean, variance, std dev, covariance, and Pearson r without pandas/numpy — loops only. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 3: Week 3 — Sampling & CI CLT, standard error, confidence intervals. ### Concepts - [Sampling & the Central Limit Theorem](https://learn.significanthobbies.com/curriculum/concepts/sampling-and-clt) — Sampling distributions, standard error of the mean, and why averages become Normal as n grows. - [Estimation & Confidence Intervals](https://learn.significanthobbies.com/curriculum/concepts/estimation-confidence) — Point estimates, standard error, confidence intervals, and what '95% confident' actually means. ### Build evidence - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 4: Week 4 — Inference & overfitting Hypothesis tests, A/B testing, regression, bias-variance — debunk a fake signal. ### Concepts - [Hypothesis Testing](https://learn.significanthobbies.com/curriculum/concepts/hypothesis-testing) — Null and alternative hypotheses, p-values, significance, power, and Type I/II errors. - [A/B Testing for Engineers](https://learn.significanthobbies.com/curriculum/concepts/ab-testing-engineering) — Sample size, statistical power, practical significance, SRM checks, and multiple-comparison traps in product experiments. - [Linear Regression](https://learn.significanthobbies.com/curriculum/concepts/regression-basics) — Fitting a line (or hyperplane) by least squares; residuals, R², and the geometry of projection. - [Bias, Variance & Overfitting](https://learn.significanthobbies.com/curriculum/concepts/bias-variance-overfitting) — Underfitting vs memorizing noise; why in-sample greatness lies. ### Build evidence - **A/B test analysis report** — Analyze a mock or real experiment: check SRM, compute CI on the lift, interpret p-value and practical significance, and write a ship/no-ship recommendation. - **Analyze a fake trading signal** — Given (or generate) a spurious 'alpha' signal on synthetic or historical noise, show why it fails: multiple testing, in-sample overfit, or non-stationarity. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 5: Week 5 — Vectors & matrices Dot products, matmul, transforms — implement multiplication. ### Concepts - [Vectors & Vector Spaces](https://learn.significanthobbies.com/curriculum/concepts/vectors-and-spaces) — Vectors as ordered lists, dot products, norms, orthogonality, and the geometric picture of n-dimensional space. - [Matrices & Linear Transformations](https://learn.significanthobbies.com/curriculum/concepts/matrices-and-transformations) — Matrix multiplication as composing linear maps: rotation, scaling, projection, and change of basis. - [Rank, Basis & Subspaces](https://learn.significanthobbies.com/curriculum/concepts/matrix-rank-basis) — Column space, rank, independence — why matrices are often low-rank in practice. ### Build evidence - **Implement matrix multiplication from scratch** — General m×n by n×p matmul with nested loops; apply to a 2D rotation + scale transform on points. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 6: Week 6 — Eigen & regression geometry Eigenvalues, least squares as projection — OLS from scratch. ### Concepts - [Eigenvalues & Matrix Decomposition](https://learn.significanthobbies.com/curriculum/concepts/eigenvalues-decomposition) — Eigenvectors as directions preserved by a transformation; eigenvalues as stretch factors; SVD as the universal factorization. - [Linear Regression](https://learn.significanthobbies.com/curriculum/concepts/regression-basics) — Fitting a line (or hyperplane) by least squares; residuals, R², and the geometry of projection. ### Build evidence - **Least-squares regression from scratch** — Fit y = Xβ via normal equations (XᵀX)β = Xᵀy using your matmul — no sklearn. Report β, residuals, R². - **Visualize linear projections** — Draw vector projection onto a line/subspace: show orthogonal residual. Connect to OLS geometry. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 7: Week 7 — PCA Covariance eigenstructure — PCA without libraries. ### Concepts - [PCA & Projection](https://learn.significanthobbies.com/curriculum/concepts/pca-projection) — Principal components as variance-maximizing orthogonal directions; projection as subspace approximation. - [Covariance & Correlation](https://learn.significanthobbies.com/curriculum/concepts/covariance-correlation) — Covariance measures co-movement; correlation normalizes to [−1, 1]. Foundation for regression, PCA, and portfolio risk. ### Build evidence - **PCA from scratch** — Center data, compute covariance matrix, eigen-decompose, project to 2D. Visualize. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 8: Week 8 — Optimization bridge Gradients, convexity, MLE — runway to ML training. ### Concepts - [Derivatives & Gradients](https://learn.significanthobbies.com/curriculum/concepts/derivatives-and-gradients) — Partial derivatives, the gradient vector, and reading a loss surface for descent direction. - [Multivariable Optimization](https://learn.significanthobbies.com/curriculum/concepts/multivariable-optimization) — Convexity, critical points, constrained optimization, and why SGD works on non-convex losses anyway. - [Maximum Likelihood Estimation](https://learn.significanthobbies.com/curriculum/concepts/maximum-likelihood) — Choosing parameters that make the observed data most probable; log-likelihood; connection to cross-entropy loss. ### Build evidence - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 9: Week 9 — Returns pipeline Ingest prices, compute daily returns — no look-ahead. ### Concepts - [Returns & Volatility](https://learn.significanthobbies.com/curriculum/concepts/returns-volatility) — Simple/log returns, realized volatility, annualization — basic quant units. ### Build evidence - **Price data → daily returns pipeline** — Ingest adjusted close prices (CSV or free API), compute simple/log daily returns, handle missing days. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 10: Week 10 — Risk metrics Volatility, Sharpe, drawdown, correlation matrix. ### Concepts - [Sharpe, Drawdown & Portfolio Risk](https://learn.significanthobbies.com/curriculum/concepts/portfolio-risk-metrics) — Sharpe, max drawdown, correlation matrices — risk-adjusted comparison. - [Stationarity & Autocorrelation](https://learn.significanthobbies.com/curriculum/concepts/stationarity-autocorrelation) — When series statistics are stable; memory in lags; noise vs signal. ### Build evidence - **Volatility, Sharpe, drawdown, correlation matrix** — On 2–5 tickers: annualized vol, Sharpe (assume rf=0 or document rate), max drawdown, pairwise correlation matrix. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 11: Week 11 — Stochastic intuition Random walks, Markov, stationarity — noise vs signal. ### Concepts - [Random Walks & Markov Chains](https://learn.significanthobbies.com/curriculum/concepts/random-walks-markov) — Markov property, random walks, transition matrices — generative story behind market efficiency intuition. - [Bayesian Inference](https://learn.significanthobbies.com/curriculum/concepts/bayesian-inference) — Priors, posteriors, credible intervals, and when Bayesian updating beats frequentist tests. ### Build evidence - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making. ## Milestone 12: Week 12 — Backtest discipline Simple momentum vs SPY/QQQ with honest limitations. ### Concepts - [Momentum Backtest Discipline](https://learn.significanthobbies.com/curriculum/concepts/momentum-backtest) — Momentum rules, SPY/QQQ benchmarks, holdout honesty — why most backtests lie. - [Information & Entropy](https://learn.significanthobbies.com/curriculum/concepts/information-entropy) — Entropy as surprise, cross-entropy as a loss, KL divergence as a distributional distance. ### Build evidence - **Simple momentum backtest vs SPY/QQQ** — 12-month momentum rule on 1–3 names; compare cumulative return to SPY or QQQ over the same window. State assumptions and survivorship bias. - **Weekly math writeup (public note)** — Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making.