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Data Science for Finance
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Theory
»
Introduction
About this work
Parts: the business process
Modules: the teaching process
Channels: the learning process
Audience and prerequisites
About quantitative finance: P and...
Differences between P and Q
Commonalities between P and Q
Notation
Key notation tenets
Operators and special functions
Calculus
Probability and general distribut...
Summary statistical features
Distributions
Stochastic processes
Time conventions and counting ind...
Risk drivers
Invariants
Performance
Asset classes
Market estimation and modeling
Investor preferences/profile
Acronyms
Bibliography
I. The “Checklist”
Executive summary
[ 1 ]
Risk drivers identification
[ 1.1 ]
Equities
[ 1.2 ]
Currencies
[ 1.2.1 ]
Exchange rates
[ 1.2.2 ]
Contracts
[ 1.3 ]
Fixed-income
[ 1.3.1 ]
Zero-coupon bond
[ 1.3.2 ]
Rolling value
[ 1.3.3 ]
Yield to maturity
[ 1.3.4 ]
Alternative representations
[ 1.3.5 ]
Parsimonious representations
[ 1.3.6 ]
Spreads
[ 1.4 ]
Derivatives
[ 1.4.1 ]
Call option
[ 1.4.2 ]
Rolling value
[ 1.4.3 ]
Implied volatility
[ 1.4.4 ]
Alternative representations
[ 1.4.5 ]
Parsimonious representations
[ 1.4.6 ]
Pure volatility products
[ 1.5 ]
Credit
[ 1.5.1 ]
Obligor-level risk drivers
[ 1.5.2 ]
Aggregate risk drivers
[ 1.5.3 ]
Structural models
[ 1.5.4 ]
Merton’s structural credit mode...
[ 1.6 ]
Insurance
[ 1.7 ]
Operations
[ 1.8 ]
High frequency
[ 1.8.1 ]
Market microstructure
[ 1.8.2 ]
Activity time
[ 1.8.3 ]
Time-changed variables
[ 1.9 ]
Strategies
[ 1.10 ]
The final output
[ 1.11 ]
Points of interest, pitfalls, pra...
[ 1.11.1 ]
Spurious heteroscedasticity
[ 1.11.2 ]
Spot, par, yield, forward, zero, ...
[ 2 ]
Quest for invariance
[ 2.1 ]
Efficiency: random walk
[ 2.1.1 ]
Continuous-state random walk
[ 2.1.2 ]
Discrete-state random walk
[ 2.1.3 ]
Flexible combinations
[ 2.2 ]
Mean-reversion (continuous): ARMA
[ 2.2.1 ]
AR(1) process
[ 2.2.2 ]
AR(p) process
[ 2.2.3 ]
MA(q) process
[ 2.2.4 ]
ARMA(p,q) process
[ 2.2.5 ]
ARIMA(p,d,q) process
[ 2.3 ]
Mean-reversion (discrete): Markov...
[ 2.3.1 ]
Time-homogeneous Markov chains
[ 2.3.2 ]
Time-inhomogeneous Markov chains
[ 2.3.3 ]
Invariant and next-step function
[ 2.3.4 ]
Applications to credit risk
[ 2.3.5 ]
Connection with structural model
[ 2.4 ]
Long memory: fractional integrati...
[ 2.5 ]
Seasonality
[ 2.6 ]
Volatility clustering
[ 2.6.1 ]
GARCH
[ 2.6.2 ]
Extensions of GARCH
[ 2.6.3 ]
Stochastic volatility
[ 2.7 ]
Multivariate quest
[ 2.7.1 ]
Vector autoregression
[ 2.7.2 ]
Alternative models
[ 2.8 ]
Cointegration
[ 2.8.1 ]
Modeling
[ 2.8.2 ]
Detection
[ 2.8.3 ]
Fit
[ 2.9 ]
Relationships among processes
[ 2.9.1 ]
Inversion of first-degree lag pol...
[ 2.9.2 ]
ARMA(p,q) processes as products o...
[ 2.9.3 ]
Relationships between ARMA, MA an...
[ 2.9.4 ]
AR(p) as VAR(1)
[ 2.9.5 ]
Univariate processes as VAR(1)
[ 2.9.6 ]
Relationships between VARMA, VMA ...
[ 2.9.7 ]
VAR(p) as VAR(1)
[ 2.9.8 ]
Multivariate processes as VAR(1)
[ 2.9.9 ]
Stationary as MA(∞): Wold repre...
[ 2.10 ]
Toward machine learning
[ 2.10.1 ]
Exogenous quest
[ 2.10.2 ]
Hidden quest
[ 2.11 ]
The final output
[ 2.11.1 ]
Model for groups
[ 2.11.2 ]
Fit and invariants extraction for...
[ 2.11.3 ]
Marginal test for invariance
[ 2.11.4 ]
Green light toward joint estimati...
[ 2.12 ]
Points of interest, pitfalls, pra...
[ 2.12.1 ]
Model-free non-linear invariance ...
[ 2.12.2 ]
Returns are not invariants
[ 2.12.3 ]
Sampling step size
[ 3 ]
Estimation
[ 3.1 ]
Setting the flexible probabilitie...
[ 3.1.1 ]
Exponential decay and time condit...
[ 3.1.2 ]
Kernels and state conditioning
[ 3.1.3 ]
Joint state and time conditioning
[ 3.1.4 ]
Statistical power of flexible pro...
[ 3.2 ]
Historical
[ 3.2.1 ]
From historical distribution to f...
[ 3.2.2 ]
Location-dispersion: HFP ellipsoi...
[ 3.2.3 ]
Kernel estimation with flexible p...
[ 3.3 ]
Maximum likelihood
[ 3.3.1 ]
From maximum likelihood to flexib...
[ 3.3.2 ]
Exponential family invariants
[ 3.3.3 ]
Location-dispersion: normal MLFP ...
[ 3.3.4 ]
Location-dispersion: t MLFP ellip...
[ 3.4 ]
Bayesian
[ 3.4.1 ]
Exponential family invariants
[ 3.4.2 ]
Normal invariants
[ 3.5 ]
Shrinkage
[ 3.5.1 ]
Mean shrinkage: James-Stein
[ 3.5.2 ]
Covariance shrinkage: Ledoit-Wolf
[ 3.5.3 ]
Correlation shrinkage: random mat...
[ 3.5.4 ]
Covariance shrinkage: sparse eige...
[ 3.5.5 ]
Covariance shrinkage: glasso
[ 3.6 ]
Generalized method of moments
[ 3.6.1 ]
Method of moments
[ 3.6.2 ]
Generalized method of moments - e...
[ 3.6.3 ]
Generalized method of moments - o...
[ 3.7 ]
Robustness
[ 3.7.1 ]
Local robustness
[ 3.7.2 ]
Global robustness
[ 3.8 ]
Missing data
[ 3.8.1 ]
Randomly missing data
[ 3.8.2 ]
Times series of different length
[ 3.8.3 ]
Missing series
[ 3.9 ]
(Dynamic) copula-marginal
[ 3.9.1 ]
Static copula
[ 3.9.2 ]
Dynamic copula
[ 3.10 ]
Estimation assessment
[ 3.10.1 ]
Estimators as decisions
[ 3.10.2 ]
Frequentist approach
[ 3.10.3 ]
Bayesian approach
[ 3.10.4 ]
Analytical results
[ 3.10.5 ]
Monte Carlo simulations
[ 3.10.6 ]
Cross-validation
[ 3.11 ]
Hypothesis testing
[ 3.11.1 ]
Statistics
[ 3.11.2 ]
P-value
[ 3.11.3 ]
Univariate testing: the z-statist...
[ 3.11.4 ]
Multivariate testing: the Hotelli...
[ 3.12 ]
The final output
[ 3.12.1 ]
Historical
[ 3.12.2 ]
Analytical
[ 3.12.3 ]
Copula-marginal
[ 3.12.4 ]
Estimation assessment
[ 3.13 ]
Points of interest, pitfalls, pra...
[ 3.13.1 ]
Unconditional estimation
[ 3.13.2 ]
Standardization
[ 3.13.3 ]
Non-synchronous data
[ 3.13.4 ]
High-frequency volatility/correla...
[ 3.13.5 ]
Outlier detection
[ 3.13.6 ]
Exponential moving moments and st...
[ 3.13.7 ]
Backward/forward exponential deca...
[ 3.13.8 ]
Combining estimation techniques
[ 4 ]
Projection
[ 4.1 ]
One-step historical projection
[ 4.2 ]
Analytical
[ 4.2.1 ]
Univariate analytical projection
[ 4.2.2 ]
Multivariate analytical projectio...
[ 4.3 ]
Monte Carlo
[ 4.3.1 ]
Direct Monte Carlo
[ 4.3.2 ]
Copula-marginal
[ 4.3.3 ]
General-step Monte Carlo
[ 4.4 ]
Historical
[ 4.4.1 ]
Historical bootstrapping
[ 4.4.2 ]
Hybrid Monte Carlo-historical
[ 4.5 ]
Application: multivariate Markov ...
[ 4.5.1 ]
Univariate Markov chain
[ 4.5.2 ]
Connection with structural credit...
[ 4.5.3 ]
Multivariate Markov chains
[ 4.5.4 ]
Connection with structural credit...
[ 4.6 ]
Square-root rule and generalizati...
[ 4.6.1 ]
Thin-tailed random walk
[ 4.6.2 ]
Thick-tailed random walk
[ 4.6.3 ]
Multivariate random walk
[ 4.6.4 ]
General processes
[ 4.7 ]
The final output
[ 4.7.1 ]
Analytical
[ 4.7.2 ]
Scenarios
[ 4.8 ]
Points of interest, pitfalls, pra...
[ 4.8.1 ]
Consecutive (non-)overlapping seq...
[ 4.8.2 ]
Semi-analytical random walk proje...
[ 4.8.3 ]
GARCH generalized next-step
[ 4.8.4 ]
Projection step at a later stage ...
[ 4.8.5 ]
Pitfalls for square-root rule
[ 4.8.6 ]
Thick tails from thin tails
[ 4.8.7 ]
Martingales
[ 4.8.8 ]
Linear versus compounded returns
[ 4.8.9 ]
FFT for moving averages
[ 4.8.10 ]
Scenario projection enhancements ...
[ 4.8.11 ]
Jumps versus diffusion
[ 5 ]
Pricing at the horizon
[ 5.1 ]
Exact repricing
[ 5.1.1 ]
Equities
[ 5.1.2 ]
Currencies
[ 5.1.3 ]
Fixed-income
[ 5.1.4 ]
Derivatives
[ 5.1.5 ]
Credit
[ 5.1.6 ]
High frequency
[ 5.1.7 ]
Strategies
[ 5.2 ]
Carry
[ 5.2.1 ]
Equities
[ 5.2.2 ]
Currencies
[ 5.2.3 ]
Fixed-income
[ 5.2.4 ]
Derivatives
[ 5.2.5 ]
Other asset classes
[ 5.3 ]
Taylor approximations
[ 5.3.1 ]
Equities
[ 5.3.2 ]
Fixed-income
[ 5.3.3 ]
Derivatives
[ 5.3.4 ]
Other asset classes
[ 5.4 ]
Hybrid Taylor/repricing approxima...
[ 5.4.1 ]
Global quadratic approximation
[ 5.4.2 ]
Interpolated residual
[ 5.5 ]
The final output
[ 5.5.1 ]
Analytical approach
[ 5.5.2 ]
Scenario-based approach
[ 5.6 ]
Testing the pricing function
[ 5.7 ]
Pitfalls and practical tips
[ 5.7.1 ]
Pricing at the horizon and arbitr...
[ 5.7.2 ]
Path dependence
[ 5.7.3 ]
“Pricing at the horizon”versu...
[ 5.7.4 ]
Black-Merton-Scholes is exactly c...
[ 5.7.5 ]
Greeks for intra-day updates
[ 5.7.6 ]
Greeks at the horizon
[ 5.7.7 ]
Bond carry versus accrued interes...
[ 5.7.8 ]
Option carry versus theta
[ 6 ]
Aggregation
[ 6.1 ]
Stock variables
[ 6.1.1 ]
Portfolio
[ 6.1.2 ]
Value
[ 6.1.3 ]
Exposure
[ 6.1.4 ]
Leverage
[ 6.2 ]
Liquidity P&L/risk
[ 6.3 ]
Static market/credit risk
[ 6.3.1 ]
Standardized holdings and weights
[ 6.3.2 ]
Scenario-probability distribution
[ 6.3.3 ]
Elliptical distribution
[ 6.3.4 ]
Quadratic-normal distribution
[ 6.4 ]
Dynamic market/credit risk
[ 6.5 ]
Stress-testing
[ 6.5.1 ]
Theory
[ 6.5.2 ]
Why have stress-tests
[ 6.5.3 ]
Panic copula
[ 6.5.4 ]
Extreme copula
[ 6.6 ]
Enterprise risk management
[ 6.6.1 ]
Portfolio: balance sheet
[ 6.6.2 ]
Performance: income statement
[ 6.6.3 ]
Operational risk
[ 6.6.4 ]
Simplified regulatory framework
[ 6.6.5 ]
Insurance
[ 6.7 ]
The final output
[ 6.7.1 ]
Analytical approach
[ 6.7.2 ]
Scenario-based approach
[ 6.8 ]
Points of interest and pitfalls
[ 6.8.1 ]
Solvency and collateral
[ 6.8.2 ]
Credit value adjustment
[ 6.8.3 ]
CreditRisk+ approximation
[ 7 ]
Ex-ante evaluation
[ 7.1 ]
Stochastic dominance
[ 7.2 ]
Satisfaction
[ 7.3 ]
Mean-variance
[ 7.3.1 ]
Mean
[ 7.3.2 ]
Variance
[ 7.3.3 ]
Standard deviation
[ 7.3.4 ]
Mean-variance trade-off
[ 7.3.5 ]
A strange success story
[ 7.4 ]
Expected utility and certainty-eq...
[ 7.4.1 ]
Common utility functions
[ 7.4.2 ]
Computation
[ 7.5 ]
Quantile (value at risk)
[ 7.5.1 ]
Computation
[ 7.5.2 ]
Cornish-Fisher approximation
[ 7.5.3 ]
Extreme value theory
[ 7.6 ]
Coherent indices of satisfaction
[ 7.6.1 ]
Definition
[ 7.6.2 ]
Examples
[ 7.6.3 ]
Computation
[ 7.7 ]
Spectral indices of satisfaction
[ 7.7.1 ]
Definition
[ 7.7.2 ]
General computation
[ 7.7.3 ]
Quantile (VaR) computations
[ 7.7.4 ]
Expected shortfall computations
[ 7.8 ]
Distortion measures
[ 7.8.1 ]
Definition
[ 7.8.2 ]
Equivalence with spectral measure...
[ 7.8.3 ]
Examples
[ 7.9 ]
Non-dimensional ratios
[ 7.9.1 ]
Information ratio
[ 7.9.2 ]
Downside ratios
[ 7.10 ]
Additional indices
[ 7.10.1 ]
Beta
[ 7.10.2 ]
Correlation
[ 7.10.3 ]
Buhlmann expectation
[ 7.10.4 ]
Esscher expectation
[ 7.11 ]
Enterprise risk management
[ 7.12 ]
The final output
[ 7.13 ]
Pitfalls, points of interest and ...
[ 7.13.1 ]
The Arrow-Pratt approximation of ...
[ 7.13.2 ]
Utility versus quantile
[ 7.13.3 ]
Utility versus spectrum functions
[ 7.13.4 ]
Spectral/distortion versus cohere...
[ 7.13.5 ]
The Buhlmann and Esscher expectat...
[ 7.13.6 ]
Indices of satisfaction under nor...
[ 8 ]
Ex-ante attribution
[ 8a ]
Ex-ante attribution: performance
[ 8a.1 ]
Bottom-up exposures
[ 8a.1.1 ]
Pricing factors
[ 8a.1.2 ]
Style factors/smart beta
[ 8a.2 ]
Top-down exposures: factors on de...
[ 8a.2.1 ]
Analytical computation
[ 8a.2.2 ]
Cardinality constraints
[ 8a.3 ]
Relationship between bottom-up an...
[ 8a.3.1 ]
Subportfolios
[ 8a.3.2 ]
Style factors/smart beta
[ 8a.4 ]
Joint distribution
[ 8a.4.1 ]
Elliptical distribution
[ 8a.4.2 ]
Scenario-probability distribution
[ 8a.5 ]
Application: hedging
[ 8a.6 ]
The final output
[ 8a.7 ]
Pitfalls and practical tips
[ 8a.7.1 ]
Estimation versus attribution
[ 8a.7.2 ]
The ex-ante attribution is not a ...
[ 8b ]
Ex-ante attribution: risk
[ 8b.1 ]
General criteria
[ 8b.1.1 ]
Isolated/“first in”proportion...
[ 8b.1.2 ]
“Last in”proportional attribu...
[ 8b.1.3 ]
Sequential attribution
[ 8b.1.4 ]
Shapley attribution
[ 8b.2 ]
Homogenous measures and Euler dec...
[ 8b.2.1 ]
Standard deviation
[ 8b.2.2 ]
Variance
[ 8b.2.3 ]
Certainty-equivalent
[ 8b.2.4 ]
Distortion/spectral measures
[ 8b.2.5 ]
Economic capital
[ 8b.3 ]
Esscher expectation
[ 8b.4 ]
Minimum-torsion bets attribution ...
[ 8b.4.1 ]
Effective number of bets
[ 8b.4.2 ]
Minimum-torsion bets
[ 8b.4.3 ]
Traditional marginal contribution...
[ 8b.5 ]
The final output
[ 9 ]
Construction
[ 9a ]
Construction: portfolio optimizat...
[ 9a.1 ]
A compromise: two-step mean-varia...
[ 9a.2 ]
Analytical solutions of the mean-...
[ 9a.3 ]
The final output
[ 9b ]
Construction: cross-sectional str...
[ 9b.1 ]
Simplistic portfolio construction
[ 9b.1.1 ]
Backtesting
[ 9b.2 ]
Advanced portfolio construction
[ 9b.2.1 ]
Signal-induced factor
[ 9b.2.2 ]
Flexible factor
[ 9b.2.3 ]
Backtesting
[ 9b.3 ]
Relationship with FLAM and APT
[ 9b.3.1 ]
Signal-induced moments of the P&L
[ 9b.3.2 ]
Fundamental law of active managem...
[ 9b.3.3 ]
APT assumption
[ 9b.4 ]
Multiple portfolios
[ 9b.4.1 ]
Characteristic portfolios
[ 9b.4.2 ]
Signal-induced factors
[ 9b.4.3 ]
Flexible factors
[ 9b.4.4 ]
Relationship with FLAM and APT
[ 9b.5 ]
Points of interest, pitfalls, pra...
[ 9b.5.1 ]
Machine learning
[ 9b.5.2 ]
Generalized fundamental law of ac...
[ 9c ]
Construction: time series strateg...
[ 9c.1 ]
The market
[ 9c.1.1 ]
Risky investment
[ 9c.1.2 ]
Low-risk investment
[ 9c.1.3 ]
Strategies
[ 9c.2 ]
Expected utility maximization
[ 9c.2.1 ]
The objective
[ 9c.2.2 ]
Optimization
[ 9c.3 ]
Option based portfolio insurance
[ 9c.3.1 ]
Payoff design
[ 9c.3.2 ]
Partial differential equation
[ 9c.3.3 ]
Budget
[ 9c.3.4 ]
Policy
[ 9c.3.5 ]
A unified approach
[ 9c.4 ]
Rolling horizon heuristics
[ 9c.4.1 ]
Constant proportion portfolio ins...
[ 9c.4.2 ]
Drawdown control
[ 9c.5 ]
Signal induced strategy
[ 9c.6 ]
Convexity analysis
[ 9d ]
Construction: estimation and mode...
[ 9d.1 ]
Allocation (=output) uncertainty:...
[ 10 ]
Execution
[ 10.1 ]
Market impact modeling
[ 10.1.1 ]
Liquidity curve
[ 10.1.2 ]
Exogenous impact
[ 10.1.3 ]
Endogenous impact
[ 10.2 ]
Order scheduling
[ 10.2.1 ]
Trading P&L decomposition
[ 10.2.2 ]
Model P&L
[ 10.2.3 ]
Moments of model P&L
[ 10.2.4 ]
Model P&L optimization
[ 10.2.5 ]
Quasi-optimal P&L distribution
[ 10.3 ]
Order placement
[ 10.3.1 ]
Step 1: order scheduling
[ 10.3.2 ]
Step 2: order placement
[ 10.4 ]
The final output
[ 10.5 ]
Points of interest, pitfalls, pra...
[ 10.5.1 ]
Mean-variance optimization in com...
[ 10.5.2 ]
Price manipulation
[ 10.5.3 ]
Testing
II. Factor models and learning
[ 11 ]
Executive summary
[ 12 ]
Non-parametric linear factor mode...
[ 12.1 ]
Theory
[ 12.1.1 ]
Dominant-residual models
[ 12.1.2 ]
Systematic-idiosyncratic models
[ 12.2 ]
Estimation
[ 12.2.1 ]
Time series models
[ 12.2.2 ]
Data fit
[ 13 ]
Non-parametric regression LFM
[ 13.1 ]
Theory
[ 13.1.1 ]
Solution: factor loadings
[ 13.1.2 ]
Recovery and fit
[ 13.1.3 ]
Residuals features
[ 13.1.4 ]
Natural scatter specification
[ 13.2 ]
Estimation
[ 14 ]
Non-parametric principal-componen...
[ 14.1 ]
Theory
[ 14.1.1 ]
Principal component analysis
[ 14.1.2 ]
Solution: factor loadings and fac...
[ 14.1.3 ]
Recovery and fit
[ 14.1.4 ]
Residuals features
[ 14.1.5 ]
Natural scatter specification
[ 14.2 ]
Estimation
[ 14.3 ]
Case study: swap market
[ 14.3.1 ]
Finite set of times to maturity
[ 14.3.2 ]
The continuum limit
[ 15 ]
Non-parametric systematic-idiosyn...
[ 15.1 ]
Theory
[ 15.1.1 ]
Implications on parameters
[ 15.1.2 ]
Implications on covariance
[ 15.1.3 ]
Loadings unearthing
[ 15.1.4 ]
Approximate factor extraction
[ 15.2 ]
Estimation
[ 16 ]
Non-parametric cross-sectional LF...
[ 16.1 ]
Theory
[ 16.1.1 ]
Solution: factor-construction mat...
[ 16.1.2 ]
Recovery and fit
[ 16.1.3 ]
Residuals features
[ 16.1.4 ]
Natural scatter specification
[ 16.1.5 ]
Systematic-idiosycratic assumptio...
[ 16.2 ]
Estimation
[ 16.2.1 ]
Factor construction
[ 16.2.2 ]
Problems with pure cross-sectiona...
[ 16.2.3 ]
Hybrid cross-sectional estimation
[ 17 ]
Non-parametric LFM’s: points of...
[ 17.1 ]
Principal factors are not princip...
[ 17.2 ]
Equivalent linear formulation for...
[ 17.3 ]
Linear approximation revisited
[ 17.4 ]
Performance of regression versus ...
[ 17.4.1 ]
Regression revisited
[ 17.4.2 ]
Principal component revisited
[ 17.4.3 ]
The comparison
[ 17.5 ]
Symmetric regression
[ 17.5.1 ]
Univariate symmetric regression
[ 17.5.2 ]
Multivariate symmetric regression
[ 17.6 ]
Identification issues
[ 17.6.1 ]
Systematic-idiosyncratic LFM
[ 17.6.2 ]
Principal component LFM
[ 17.7 ]
Regularized dominant-residual mod...
[ 17.7.1 ]
Regularized regression
[ 17.7.2 ]
Regularized principal component a...
[ 17.7.3 ]
Regularized cross-sectional model...
[ 17.7.4 ]
Fully specified models
[ 17.8 ]
Truncation
[ 18 ]
Non-parametric LFM’s: pitfalls
[ 18.1 ]
LFM’s are not a regression on p...
[ 18.2 ]
LFM’s are not about returns(2-3...
[ 18.3 ]
LFM’s are not about stocks(4) [...
[ 18.4 ]
LFM’s “factors”are not “f...
[ 18.5 ]
LFM’s are not systematic-idiosy...
[ 18.6 ]
LFM’s are not horizon-independe...
[ 18.7 ]
LFM’s are not a dimension reduc...
[ 18.8 ]
LFM’s are not APT and CAPM(10-1...
[ 18.9 ]
LFM’s do not include factor ana...
[ 18.10 ]
LFM’s do not extract premia-gen...
[ 18.11 ]
LFM’s are not always necessary(...
[ 19 ]
Machine learning foundations
[ 19.1 ]
Key ideas from non-parametric lin...
[ 19.1.1 ]
Supervised linear predictor
[ 19.1.2 ]
Unsupervised linear autoencoders
[ 19.1.3 ]
Latent variable linear model
[ 19.1.4 ]
Conditionally uncorrelated linear...
[ 19.2 ]
Key concepts for machine learning
[ 19.2.1 ]
Supervised versus unsupervised le...
[ 19.2.2 ]
Regression, classification, clust...
[ 19.2.3 ]
Point versus probabilistic predic...
[ 19.2.4 ]
Discriminant versus generative mo...
[ 19.2.5 ]
Linear models
[ 19.3 ]
Supervised regression
[ 19.3.1 ]
(Optimal, point) mean regression
[ 19.3.2 ]
(Optimal, point) quantile regress...
[ 19.3.3 ]
(Optimal point) analysis of varia...
[ 19.3.4 ]
(Point) linear regression
[ 19.3.5 ]
Closed-form discriminant regressi...
[ 19.3.6 ]
Modular discriminant regression
[ 19.4 ]
Supervised classification
[ 19.4.1 ]
Optimal (point and probabilistic)...
[ 19.4.2 ]
Receiver operating characteristic
[ 19.4.3 ]
(Point) linear classifiers
[ 19.4.4 ]
Discriminant classification
[ 19.4.5 ]
Generative classification
[ 19.4.6 ]
Multiple classes
[ 19.5 ]
Unsupervised autoencoders
[ 19.5.1 ]
Principal component linear factor...
[ 19.5.2 ]
Cross-sectional linear factor mod...
[ 19.5.3 ]
k-means clustering
[ 19.5.4 ]
Independent component analysis
[ 19.6 ]
Latent variable models
[ 19.6.1 ]
Probabilistic factor analysis
[ 19.6.2 ]
Latent mixture models
[ 19.6.3 ]
Missing observations
[ 19.6.4 ]
Deep learning
[ 19.7 ]
Probabilistic graphical models
[ 19.7.1 ]
Naive Bayes models
[ 19.7.2 ]
Graphs
[ 19.7.3 ]
Markov random fields
[ 19.7.4 ]
Bayes networks
[ 20 ]
Estimation and assessment
[ 20.1 ]
Bias versus variance
[ 20.2 ]
Background
[ 20.2.1 ]
Panels
[ 20.2.2 ]
Estimation
[ 20.2.3 ]
Assumptions
[ 20.2.4 ]
Testing
[ 20.3 ]
Standard estimators
[ 20.3.1 ]
Point prediction: historical with...
[ 20.3.2 ]
Probabilistic prediction: maximum...
[ 20.4 ]
Cross-validation
[ 20.4.1 ]
Training and validation sets
[ 20.4.2 ]
Joint distribution for cross vali...
[ 20.4.3 ]
Estimation improvement
[ 20.5 ]
Point prediction assessment
[ 20.5.1 ]
Specific terminology
[ 20.5.2 ]
Frequentist approach
[ 20.5.3 ]
Bayesian approach
[ 20.6 ]
Probabilistic prediction assessme...
[ 20.6.1 ]
Specific terminology
[ 20.6.2 ]
Frequentist approach
[ 20.6.3 ]
Bayesian approach
[ 21 ]
Bias-variance enhancements
[ 21.1 ]
Feature engineering
[ 21.1.1 ]
Interaction
[ 21.1.2 ]
Encoding
[ 21.1.3 ]
Trees
[ 21.1.4 ]
Segmentation
[ 21.1.5 ]
Feature basis
[ 21.1.6 ]
Exponential family
[ 21.2 ]
Gradient boosting
[ 21.3 ]
Regularization
[ 21.3.1 ]
Ridge, lasso, elastic nets
[ 21.3.2 ]
Glasso
[ 21.3.3 ]
Bayesian prior
[ 21.4 ]
Ensemble learning
[ 21.4.1 ]
Bagging
[ 21.4.2 ]
Flexible probabilities as random-...
[ 21.4.3 ]
Flexible probabilities through co...
[ 21.4.4 ]
Ensemble weighting
[ 22 ]
Deep learning
[ 22.1 ]
Point prediction
[ 22.2 ]
Probabilistic prediction
[ 23 ]
Dynamic models
[ 23.1 ]
Wiener-Kolmogorov filtering
[ 23.1.1 ]
The solution in population
[ 23.1.2 ]
The solution in sample
[ 23.2 ]
Dynamic principal component
[ 23.2.1 ]
The solution in population
[ 23.2.2 ]
Computational issue
[ 23.2.3 ]
The solution in sample
[ 23.3 ]
Linear state space models
[ 23.3.1 ]
Model
[ 23.3.2 ]
Static case
[ 23.3.3 ]
Dynamic case
[ 23.4 ]
Hidden Markov models
[ 23.4.1 ]
Static case
[ 23.4.2 ]
Dynamic case
[ 24 ]
Application: market prediction re...
[ 24.1 ]
Time series models
[ 24.2 ]
Maximum likelihood
[ 24.2.1 ]
The model
[ 24.2.2 ]
Normal assumption
[ 24.2.3 ]
Student t assumption
[ 24.3 ]
Bayesian
[ 24.3.1 ]
Normal conditional likelihood
[ 24.3.2 ]
Normal-inverse-Wishart prior dist...
[ 24.3.3 ]
Normal-inverse-Wishart posterior ...
[ 24.3.4 ]
Student t predictive distribution
[ 24.3.5 ]
Classical equivalent
[ 24.3.6 ]
Uncertainty
[ 24.4 ]
Regularization
[ 24.4.1 ]
Step-wise regression selection
[ 24.4.2 ]
Lasso regression
[ 24.4.3 ]
Ridge regression
[ 24.5 ]
Mixed approach
[ 25 ]
Application: credit default class...
[ 25.1 ]
Background
[ 25.2 ]
Fit and assessment
[ 25.3 ]
Logistic regression
[ 25.4 ]
Interactions
[ 25.5 ]
Encoding
[ 25.6 ]
Regularization
[ 25.7 ]
Trees
[ 25.8 ]
Gradient boosting
[ 25.9 ]
Cross-validation
[ 26 ]
Application: clustering
[ 26.1 ]
k-means clustering
[ 26.2 ]
Shrinkage
III. Valuation
[ 27 ]
Executive summary
[ 28 ]
Background definitions
[ 28.1 ]
Valuation foundations
[ 28.1.1 ]
Instruments
[ 28.1.2 ]
Value
[ 28.1.3 ]
Cash-flows
[ 28.1.4 ]
Re-invested cash-flows
[ 28.1.5 ]
Cash-flow adjusted value
[ 28.1.6 ]
Profit-and-loss (P&L)
[ 28.1.7 ]
Payoff
[ 28.2 ]
Points of interest and pitfalls
[ 28.2.1 ]
Value versus price
[ 28.2.2 ]
Multi-currency conversions
[ 28.2.3 ]
Actual versus simple P&L
[ 29 ]
Linear pricing theory
[ 29a ]
Linear pricing theory: core
[ 29a.1 ]
Fundamental axioms
[ 29a.1.1 ]
Law of one price
[ 29a.1.2 ]
Linearity
[ 29a.1.3 ]
No arbitrage
[ 29a.2 ]
Stochastic discount factor
[ 29a.2.1 ]
Identification
[ 29a.2.2 ]
Misidentification
[ 29a.3 ]
Fundamental theorem of asset pric...
[ 29a.4 ]
Risk-neutral pricing
[ 29a.4.1 ]
General case
[ 29a.4.2 ]
No rebalancing limit: forward mea...
[ 29a.4.3 ]
Continuous rebalancing limit
[ 29a.5 ]
Capital asset pricing model frame...
[ 29a.5.1 ]
Maximum Sharpe ratio portfolio
[ 29a.5.2 ]
Security market line
[ 29a.5.3 ]
Alternative derivation: linear fa...
[ 29a.6 ]
Covariance principle
[ 29a.7 ]
Point of interest and pitfalls
[ 29a.7.1 ]
Relationships among fundamental a...
[ 29b ]
Linear pricing theory: further as...
[ 29b.1 ]
Completeness
[ 29b.1.1 ]
Definition
[ 29b.1.2 ]
Pricing
[ 29b.1.3 ]
Arrow-Debreu securities
[ 29b.1.4 ]
Stochastic discount factor
[ 29b.2 ]
Equilibrium: pure capital asset p...
[ 29b.3 ]
Equilibrium: Buhlmann pricing equ...
[ 29b.4 ]
Arbitrage pricing theory
[ 29b.4.1 ]
Standard derivation: linear facto...
[ 29b.4.2 ]
Alternative derivation: linear fa...
[ 29b.5 ]
Intertemporal consistency
[ 29b.5.1 ]
Continuous time variables
[ 29b.5.2 ]
Martingales
[ 29b.5.3 ]
Heuristic for stochastic discount...
[ 29b.5.4 ]
Heuristic for numeraire martingal...
[ 30 ]
Non-linear pricing theory
[ 30.1 ]
Fundamental axioms
[ 30.1.1 ]
Law of one price
[ 30.1.2 ]
Non-linearity
[ 30.1.3 ]
Arbitrage
[ 30.2 ]
Valuation as evaluation
[ 30.2.1 ]
Variance and other shift principl...
[ 30.2.2 ]
Certainty-equivalent principle
[ 30.2.3 ]
Distortion principles
[ 30.2.4 ]
Esscher principle
[ 30.3 ]
Intertemporal consistency
[ 30.3.1 ]
Continuous time variables
[ 30.3.2 ]
Non-linear "martingales"?
[ 30.4 ]
Point of interest and pitfalls
[ 30.4.1 ]
Linear (mis)uses of non-linear pr...
[ 31 ]
Valuation implementation
[ 31.1 ]
Portfolio value
[ 31.1.1 ]
Long positions
[ 31.1.2 ]
Short positions
[ 31.1.3 ]
Generic positions
[ 31.1.4 ]
Funds
[ 31.1.5 ]
Sum-of-parts
[ 31.1.6 ]
Valuation recipe
[ 31.2 ]
Equities
[ 31.2.1 ]
Discounted cash-flows
[ 31.2.2 ]
Multiples
[ 31.3 ]
Options
[ 31.3.1 ]
Bachelier
[ 31.3.2 ]
Black-Scholes
[ 31.3.3 ]
Heston
[ 31.3.4 ]
Valuation recipe
[ 31.4 ]
Fixed-income
[ 31.4.1 ]
Vasicek
[ 31.4.2 ]
Other models
[ 31.4.3 ]
Valuation recipe
[ 31.5 ]
Insurance
[ 31.5.1 ]
Life insurance
[ 31.5.2 ]
Non-life insurance
[ 31.6 ]
Real assets
IV. Performance analysis
[ 32 ]
Executive summary
[ 33 ]
Performance definitions
[ 33.1 ]
Holding P&L of a portfolio
[ 33.2 ]
Trading P&L
[ 33.2.1 ]
Single transaction
[ 33.2.2 ]
Multiple transactions in one posi...
[ 33.2.3 ]
Portfolio rebalancing
[ 33.3 ]
Implementation shortfall
[ 33.4 ]
Returns
[ 33.4.1 ]
Basic definitions
[ 33.4.2 ]
Standard linear returns and weigh...
[ 33.4.3 ]
Generalized linear returns
[ 33.4.4 ]
Generalized weights and aggregati...
[ 33.4.5 ]
Investments with capital injectio...
[ 33.4.6 ]
Log-returns
[ 33.5 ]
Excess performance
[ 33.5.1 ]
Benchmark
[ 33.5.2 ]
Excess return
[ 33.6 ]
Path analysis
[ 33.7 ]
Pitfalls and practical tips
[ 33.7.1 ]
Linear versus compounded returns
[ 34 ]
Performance attribution
V. Quant toolbox
[ 35 ]
Summary
[ 36 ]
Multivariate distributions
[ 36.1 ]
Representations of a distribution
[ 36.2 ]
Marginalization
[ 36.3 ]
Conditioning
[ 36.3.1 ]
Conditional variables
[ 36.3.2 ]
Conditional features
[ 36.3.3 ]
Deterministic versus stochastic c...
[ 36.4 ]
Elliptical distributions
[ 36.4.1 ]
Fundamental concepts
[ 36.4.2 ]
Stochastic representations
[ 36.4.3 ]
Moments and dependence
[ 36.4.4 ]
Affine equivariance
[ 36.4.5 ]
Notable elliptical distributions
[ 36.4.6 ]
Generation of elliptical scenario...
[ 36.4.7 ]
Scenario generation with dimensio...
[ 36.5 ]
Scenario-probability distribution...
[ 36.5.1 ]
Types of scenario-probability dis...
[ 36.5.2 ]
Probability density function
[ 36.5.3 ]
Transformations and generalized e...
[ 36.5.4 ]
Cumulative distribution function
[ 36.5.5 ]
Quantile
[ 36.5.6 ]
Smooth quantile
[ 36.5.7 ]
Moments and other statistical fea...
[ 36.6 ]
Exponential family distributions
[ 36.6.1 ]
Normal distribution
[ 36.6.2 ]
Scenario-probability distribution
[ 36.7 ]
Other special classes of distribu...
[ 36.7.1 ]
Stable distributions
[ 36.7.2 ]
Infinitely divisible distribution...
[ 36.8 ]
Notable specific distributions
[ 36.8.1 ]
Uniform distribution
[ 36.8.2 ]
Quadratic-normal distribution
[ 36.8.3 ]
Wishart distribution
[ 37 ]
Estimation techniques
[ 37.1 ]
Maximum likelihood
[ 37.1.1 ]
General theory
[ 37.1.2 ]
Hidden variables
[ 37.1.3 ]
Relevant cases
[ 37.1.4 ]
Numerical methods
[ 37.2 ]
Bayesian statistics
[ 37.2.1 ]
General theory
[ 37.2.2 ]
Bayesian estimation
[ 37.2.3 ]
Bayesian prediction
[ 37.2.4 ]
Analytical results
[ 37.2.5 ]
Numerical methods
[ 38 ]
Views processing
[ 38.1 ]
Minimum relative entropy
[ 38.1.1 ]
Base distribution and view variab...
[ 38.1.2 ]
Point views
[ 38.1.3 ]
Distributional views
[ 38.1.4 ]
Partial views
[ 38.1.5 ]
Partial views on generalized expe...
[ 38.1.6 ]
Sanity check
[ 38.1.7 ]
Confidence
[ 38.1.8 ]
Relationship with Bayesian updati...
[ 38.2 ]
Analytical implementation
[ 38.2.1 ]
Base distribution
[ 38.2.2 ]
Views
[ 38.2.3 ]
Updated distribution
[ 38.2.4 ]
Confidence
[ 38.2.5 ]
Relevant special cases
[ 38.3 ]
Flexible probabilities implementa...
[ 38.3.1 ]
Base distribution
[ 38.3.2 ]
Views
[ 38.3.3 ]
Updated distribution
[ 38.3.4 ]
Confidence
[ 38.4 ]
Factor-based implementations
[ 38.5 ]
Copula opinion pooling
[ 38.5.1 ]
Base distribution
[ 38.5.2 ]
Views
[ 38.5.3 ]
Updated distribution
[ 38.5.4 ]
Confidence
[ 38.5.5 ]
The algorithm
[ 38.6 ]
Generalized shrinkage
[ 38.6.1 ]
Intuition
[ 38.6.2 ]
Classical shrinkage
[ 38.6.3 ]
Bayesian updating
[ 38.6.4 ]
Minimum relative entropy
[ 38.6.5 ]
Shrinkage
[ 38.6.6 ]
Regularization
[ 39 ]
Black-Litterman
[ 39.1 ]
Equilibrium prior distribution
[ 39.1.1 ]
Prior distribution of expected re...
[ 39.1.2 ]
Prior predictive distribution
[ 39.2 ]
Active views
[ 39.2.1 ]
Setting the parameters
[ 39.2.2 ]
Sanity check
[ 39.3 ]
Posterior distribution
[ 39.3.1 ]
Posterior distribution of expecte...
[ 39.3.2 ]
Posterior predictive distribution
[ 39.4 ]
Limit cases
[ 39.4.1 ]
High confidence in prior
[ 39.4.2 ]
Low confidence in views
[ 39.4.3 ]
Full confidence in views
[ 39.5 ]
Generalizations
[ 39.5.1 ]
From linear returns to risk drive...
[ 39.5.2 ]
From stock-like to generic asset ...
[ 39.5.3 ]
From normal to non-normal markets
[ 39.5.4 ]
From linear equality views to par...
[ 40 ]
Geometry of random variables
[ 40.1 ]
Length, distance and angle
[ 40.2 ]
Multivariate case: covariance pro...
[ 40.3 ]
An alternative: expectation produ...
[ 40.4 ]
Orthogonal projection
[ 40.5 ]
The r-squared
[ 40.5.1 ]
Definition
[ 40.5.2 ]
Relation to distance
[ 40.6 ]
Visualization
[ 40.7 ]
Geometry of portfolios
[ 40.7.1 ]
Length, angle and distance
[ 40.7.2 ]
Visualization
[ 40.7.3 ]
Orthogonality and collinearity
[ 40.8 ]
Alternative projections
[ 41 ]
Geometry of distributions
[ 41.1 ]
Distributions geometry
[ 41.1.1 ]
Fisher metric: length and volume
[ 41.1.2 ]
Flatness and geodesics
[ 41.1.3 ]
Duality: potentials and Legendre ...
[ 41.1.4 ]
Distance and divergence
[ 41.2 ]
Exponential distributions geometr...
[ 41.3 ]
Scenario-probability distribution...
[ 42 ]
Decision theory with model uncert...
[ 42.1 ]
Foundations of decision theory
[ 42.1.1 ]
Fundamental concepts
[ 42.1.2 ]
Frequentist approach
[ 42.1.3 ]
Bayesian approach
[ 42.1.4 ]
Model risk, estimation risk
[ 43 ]
Copulas
[ 43.1 ]
Univariate results
[ 43.2 ]
Definition and properties of copu...
[ 43.2.1 ]
Grades
[ 43.2.2 ]
Copula
[ 43.2.3 ]
Sklar’s theorem
[ 43.2.4 ]
Copula invariance
[ 43.3 ]
Special classes of copulas
[ 43.3.1 ]
Elliptical copulas
[ 43.3.2 ]
Archimedean copulas
[ 43.4 ]
Copula-marginal separation
[ 43.5 ]
Copula-marginal combination
[ 44 ]
Location and dispersion
[ 44.1 ]
Univariate location-dispersion
[ 44.1.1 ]
Z-score and affine equivariance
[ 44.1.2 ]
Taxonomy of location-dispersion f...
[ 44.2 ]
Multivariate location-dispersion
[ 44.2.1 ]
Tentative visualizations in low d...
[ 44.2.2 ]
Location-dispersion ellipsoid
[ 44.2.3 ]
Affine equivariance
[ 44.2.4 ]
Taxonomy of multivariate location...
[ 44.3 ]
Expectation and covariance
[ 44.3.1 ]
Definitions
[ 44.3.2 ]
Generalized affine equivariance
[ 44.3.3 ]
Connections with calculus
[ 44.3.4 ]
Connections with probability
[ 45 ]
Correlation and generalizations
[ 45.1 ]
Measures of dependence
[ 45.1.1 ]
Schweizer-Wolff measure
[ 45.1.2 ]
Mutual information
[ 45.2 ]
Measures of concordance
[ 45.2.1 ]
Kendall’s tau
[ 45.2.2 ]
Spearman’s rho
[ 45.3 ]
Correlation
[ 45.4 ]
Points of interest, pitfalls, pra...
[ 45.4.1 ]
Schweizer and Wolff measure via s...
[ 46 ]
Invariance tests
[ 46.1 ]
Simple tests
[ 46.2 ]
Refinements and pitfalls
[ 46.2.1 ]
Circle-like covariance (not data)
[ 46.2.2 ]
Stronger tests based on copulas
[ 47 ]
Stochastic processes cheat sheet
[ 47.1 ]
Main definitions
[ 47.2 ]
Relationships among processes
[ 47.2.1 ]
White noise ⇒ weak white noise
[ 47.2.2 ]
White noise ⇒ stationary
[ 47.2.3 ]
White noise ⇒ ergodic
[ 47.2.4 ]
Ergodic ⇒ stationary
[ 47.2.5 ]
Stationary ⇒ covariance-station...
[ 47.2.6 ]
Weak white noise ⇒ integrated
[ 47.2.7 ]
Weak white noise ⇒ covariance-s...
[ 47.2.8 ]
Covariance-stationary ⇒ integra...
[ 47.3 ]
Pitfalls
[ 48 ]
Continuous time processes
[ 48.1 ]
Efficiency: Lévy processes
[ 48.1.1 ]
Infinite divisibility
[ 48.1.2 ]
Continuous state: Brownian diffus...
[ 48.1.3 ]
Discrete state: Poisson jumps
[ 48.1.4 ]
Lévy-Khintchine representation
[ 48.1.5 ]
Subordination
[ 48.2 ]
Mean-reversion (continuous)
[ 48.2.1 ]
Ornstein-Uhlenbeck process
[ 48.2.2 ]
Square-root process and other gen...
[ 48.3 ]
Mean-reversion (discrete): Markov...
[ 48.3.1 ]
Time-homogeneous generator
[ 48.3.2 ]
Time-inhomogeneous generator
[ 48.3.3 ]
Projection of a Markov chain
[ 48.4 ]
Long memory: fractional Brownian ...
[ 48.4.1 ]
Fractional Brownian motion
[ 48.5 ]
Volatility clustering
[ 48.5.1 ]
Stochastic volatility
[ 48.5.2 ]
Time change
[ 48.5.3 ]
Connection between time-changed B...
[ 49 ]
VAR(1)/Multivariate Ornstein-Uhle...
[ 49.1 ]
AR(1)
[ 49.1.1 ]
Conditional distribution of AR(1)
[ 49.1.2 ]
Stationarity and unconditional di...
[ 49.2 ]
VAR(1)
[ 49.2.1 ]
Conditional distribution of VAR(1...
[ 49.2.2 ]
Stationarity and unconditional di...
[ 49.2.3 ]
Cointegrated VAR(1)
[ 49.3 ]
Ornstein-Uhlenbeck
[ 49.3.1 ]
Conditional distribution of OU
[ 49.3.2 ]
Stationarity and unconditional di...
[ 49.4 ]
Multivariate Ornstein-Uhlenbeck
[ 49.4.1 ]
Conditional distribution of MVOU
[ 49.4.2 ]
Stationarity and unconditional di...
[ 49.4.3 ]
Geometrical interpretation∗
[ 49.4.4 ]
Cointegrated Ornstein-Uhlenbeck
[ 49.5 ]
Relationship between (V)AR and (M...
[ 49.5.1 ]
MVOU is VAR(1)
[ 49.5.2 ]
VAR(1) is MVOU
[ 49.6 ]
VAR(1)/MVOU fit
[ 50 ]
Spectral analysis
[ 50.1 ]
The Fourier transform
[ 50.1.1 ]
Linear algebra perspective
[ 50.1.2 ]
Functional analysis perspective
[ 50.2 ]
Spectral analysis of time series
[ 50.2.1 ]
Spectrum
[ 50.2.2 ]
Estimation of the spectrum
[ 50.2.3 ]
Spectral representation
[ 50.2.4 ]
Filtering
[ 51 ]
Signals
[ 51.1 ]
Carry signals
[ 51.1.1 ]
Fixed-income
[ 51.1.2 ]
Foreign exchange
[ 51.2 ]
Value signals
[ 51.2.1 ]
Book
[ 51.2.2 ]
Pricing
[ 51.3 ]
Technical signals
[ 51.3.1 ]
Momentum
[ 51.3.2 ]
Filters
[ 51.3.3 ]
Cointegration
[ 51.4 ]
Microstructure signals
[ 51.4.1 ]
Trade autocorrelation
[ 51.4.2 ]
Order imbalance
[ 51.4.3 ]
Price prediction
[ 51.4.4 ]
Volume clustering
[ 51.5 ]
Fundamental and other signals
[ 51.6 ]
Signal processing
[ 51.6.1 ]
Smoothing
[ 51.6.2 ]
Scoring
[ 51.6.3 ]
Ranking
[ 52 ]
Stochastic dominance
[ 52.1 ]
Strong dominance (order zero domi...
[ 52.2 ]
Weak dominance (first order stoch...
[ 52.3 ]
Second order stochastic dominance
[ 52.4 ]
Order q stochastic dominance
[ 52.5 ]
Points of interest, pitfalls, pra...
[ 53 ]
Optimization primer
[ 53.1 ]
Convex programming
[ 53.1.1 ]
Conic programming
[ 53.1.2 ]
Second-order cone programming
[ 53.1.3 ]
Semidefinite programming
[ 53.1.4 ]
Quadratic programming
[ 53.1.5 ]
Linear programming
[ 53.2 ]
Integer ¯n-choose-k selection
[ 53.2.1 ]
Naive selection
[ 53.2.2 ]
Forward step-wise selection
[ 53.2.3 ]
Backward step-wise selection
[ 53.2.4 ]
Lasso
[ 54 ]
Useful algorithms
[ 54.1 ]
Conditional principal components
[ 54.2 ]
Factor analysis algorithms
[ 54.2.1 ]
Principal axis factorization
[ 54.2.2 ]
Maximum likelihood factorization
[ 54.3 ]
Matrix transpose-square-root
[ 54.3.1 ]
Spectrum
[ 54.3.2 ]
Riccati
[ 54.3.3 ]
LDL-Cholesky
[ 54.3.4 ]
Gram-Schmidt
[ 54.4 ]
Moment-matching scenarios
[ 54.4.1 ]
Twisting scenarios
[ 54.4.2 ]
Twisting probabilities
[ 54.5 ]
Fast Fourier transform for projec...
[ 54.5.1 ]
Approximating the pdf
[ 54.5.2 ]
Approximating the characteristic ...
[ 54.5.3 ]
Using the discrete Fourier transf...
[ 54.6 ]
Minimum-torsion optimization algo...
[ 55 ]
Matrix manipulations
[ 55.1 ]
Spectral theorem
[ 55.1.1 ]
The eigenvalue problem
[ 55.1.2 ]
Recursive solution
[ 55.1.3 ]
Matrix notation
[ 55.1.4 ]
The continuum limit
[ 55.2 ]
Matrix algebra
[ 55.2.1 ]
Key operators
[ 55.2.2 ]
Useful identities
[ 55.3 ]
Matrix calculus
[ 55.3.1 ]
First order derivatives
[ 55.3.2 ]
Second order derivatives
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