STAT 101
M12 · L02
Module 12 — Capstone

Advanced Topics Preview

You have the foundation. Now see where statistics goes next — six frontiers that define modern statistical practice.

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STAT 101
M12 · L02
The Map

Six Frontiers

  • Survival Analysis — time-to-event data
  • Spatial Statistics — geographic dependence
  • Functional Data — curves as observations
  • High-Dimensional — p ≫ n settings
  • Causal Inference — beyond correlation
  • ML & Statistics — converging fields
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STAT 101
M12 · L02
Frontier 1

Survival Analysis

How long until a patient dies, a machine fails, a customer churns? The key challenge: censoring — the event has not occurred yet by study end. This observation is incomplete but still informative.

The Fix
Kaplan–Meier estimator + Cox proportional hazards model
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STAT 101
M12 · L02
Kaplan–Meier

Survival Curve

At each event time, multiply by the fraction still surviving. The result: a step function that drops only when events occur, ignoring the censored.

Kaplan–Meier Estimator
\hat{S}(t) = \prod_{t_i \le t} \left(1 - \frac{d_i}{n_i}\right)
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Frontier 2

Spatial Statistics

Nearby locations are more similar than distant ones — spatial autocorrelation. Ignoring it underestimates standard errors and overstates significance.

Variogram — quantifies how correlation decays with distance
Kriging — optimal spatial interpolation
Moran’s I — tests for spatial clustering
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STAT 101
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Frontier 3

Functional Data

Each observation is a curve, not a scalar. Growth curves, EEG traces, temperature profiles — treat the whole function as the data unit.

Classical
x ∈ ℝp
Functional
x(t) ∈ L²
Key Tool
Functional PCA (FPCA) decomposes curves into eigenfunctions + scalar scores
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Frontier 4

High-Dimensional Statistics

When p ≫ n, ordinary least squares fails completely. Regularization adds a penalty that shrinks or zeroes out coefficients.

Ridge (L2) — shrinks all coefficients, never zeros
LASSO (L1) — exact zeros → variable selection
Elastic Net — L1 + L2 for correlated predictors
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STAT 101
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LASSO

Sparse Estimation

Minimize residual sum of squares plus an L1 penalty. The diamond-shaped constraint induces corners — solutions land exactly at zero.

LASSO Objective
\min_{\boldsymbol{\beta}} \left\{ \tfrac{1}{n}\|\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\|_2^2 + \lambda\|\boldsymbol{\beta}\|_1 \right\}
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STAT 101
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Frontier 5

Causal Inference

The fundamental problem: you can observe Y(1) or Y(0) for any unit, never both. Only the group average is estimable.

Observed
Y(W)
Target
ATE
Methods
IV, RD, DiD
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STAT 101
M12 · L02
Frontier 6

ML & Statistics

Statistics: inference, uncertainty, hypothesis testing. Machine learning: prediction, generalization, test error. They grew apart — and are now converging.

  • Conformal prediction — valid intervals for any ML model
  • Double ML — causal inference with high-dim controls
  • Regularization — the bridge between both worlds
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STAT 101
Knowledge Check

Check whatstuck

Four questions on the frontiers this preview names — survival, spatial, high-dimensional and causal.

Question 1 of 0
Score 0/0

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STAT 101
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The Synthesis

One Community

The most interesting work sits exactly at the intersection. Modern statisticians need ML fluency; modern ML practitioners benefit from statistical discipline.

Survival — deep learning for censored data
Spatial — graph neural networks for areal data
Causal ML — heterogeneous treatment effects
Conformal — distribution-free prediction sets
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STAT 101
Summary
Recap

What you previewed

Six frontiers beyond the core: survival, spatial, functional, high-dimensional, causal, and the ML intersection. Each is a discipline worth a year of study. Pick your direction.

Final Lesson
Resources & Career →
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