12mo roadmap · 12 milestones
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.
Outcome: Math → AI systems → distributed/data → quant tools. Active only: solve, derive, implement, simulate — never aesthetic consumption.
How to work this roadmap
Treat each milestone as a claim that must be supported by evidence. Before reading, write what you think the mechanism does and where it will fail. After studying the linked concepts, run the drills without copying an answer, preserve the output, and revise the explanation. Move forward when you can connect the milestone goal to a working implementation, benchmark, architecture decision, or reviewable design artifact.
The 12mo horizon is a sequencing aid, not a completion badge. Spend more time where your prediction and the observed behavior disagree. Keep a short decision log containing the mechanism selected, alternatives rejected, expected failure mode, measurement used, and remaining uncertainty. Review that log with FSRS prompts so the roadmap produces durable system judgment rather than a temporary tour of terminology.
At the end, explain Math → AI systems → distributed/data → quant tools. Active only: solve, derive, implement, simulate — never aesthetic consumption. from first principles to a reader outside the domain. A strong explanation should survive follow-up questions about correctness, cost, latency, resource use, security, recovery, and operational visibility. If it cannot, return to the milestone that contains the missing mechanism and build a smaller falsifiable example.
Milestones
Milestone 1
Week 1 — Probability gym
Conditional probability, Bayes, distributions — plus simulate random processes.
Concepts
- ML Math Foundations
- Combinatorics
- Numerical Stability
- Queueing Theory
- Curse of Dimensionality
- Probability Fundamentals
- Classical Distributions
- Random Walks & Markov Chains
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
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
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
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
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
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
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
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
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
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
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
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.