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Quantitative Finance · Glossary

What is Basis expansion?

Definition 4.7 Machine Learning for Markets · Chapter 4 — Linear and Regularised Baselines

A basis expansion replaces each feature xjx_j by functions hj1(xj),…,hjm(xj)h_{j1}(x_j),\dots,h_{jm}(x_j) (splines, indicators of bins, polynomials) and fits a linear model on them; the model is additive in the features and nonlinear in each.

model (parameter)Roos2R^2_{\mathrm{oos}}ICtt (HAC)Sharpe grossSharpe netturnover
ordinary least squares0.25%0.0605.61.851.610.83
ridge (λ=104\lambda = 10^4)0.30%0.0605.61.831.590.83
lasso (λ=10−3\lambda = 10^{-3})0.32%0.0666.72.171.910.80
elastic net0.32%0.0666.72.171.910.80
PCR (10 components)0.13%0.0434.11.230.990.77
PLS (1 component)0.25%0.0615.61.871.620.82
logistic regression0.22%0.0605.71.841.590.84
ridge on splines0.35%0.0646.21.981.690.95
boosted trees0.47%0.0767.82.201.900.97
ceiling0.71%
Table 4.1. Ten test years of the forty-characteristic panel with factor risk (500 stocks, twenty training years). Sharpe ratios of the equal-weighted top-minus-bottom decile portfolio, annualised, net of 20 basis points per unit traded; turnover is the share of the gross book traded each month. Data: ml_baselines.table.
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