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Beyond changing basis and reducing dimension, the note asks what else the eigenbasis buys you — total variance surviving the rotation untouched, a gap between in-sample and out-of-sample variance explained signalling a shifted variance structure, and each loading reading as a linear regression coefficient.
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The note treats constant happiness as a loop rather than a state — happiness and sadness define each other, so sadness is the price already agreed to, arriving on a geometric schedule that makes both long sad and long happy stretches unlikely.
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Maximizing the sample variance after coordinate projection, subject to unit length, turns the first-order condition into an eigenvector equation — so the principal components are the top eigenvectors of the sample variance-covariance matrix, obtainable directly from the right singular matrix of the demeaned data.
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This note aims to answer, but is not limited to, these questions: (i) what operations interact with a limit order book? (ii) what data structures are involved to build a limit order book? (iii) what are market makers? and what are the design principles of limit order book?
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The note explains how to read a market as a two-sided auction — why price sticks in some ranges (balance) and runs through others (imbalance) — and then introduces the concrete tools (order book/DOM, volume profile, footprint chart, delta/CVD) and the entry rules (ride imbalance, fade exhaustion, read absorption vs. initiation) to act on that reading.
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The companion to the linear-algebra view: here OLS is a random object, checked against unbiasedness, consistency, and efficiency, pushed through the central limit theorem to asymptotic normality, and closed with the degrees-of-freedom-corrected variance estimator that makes the t-statistic usable.
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The note treats OLS as pure linear algebra — no randomness anywhere — deriving the closed form from the first-order condition, pinning existence and uniqueness on linear independence of the regressor columns, and then reading off three fitting properties that follow from residual orthogonality alone.
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This theorem explicitly shows the superiority of using the red-black tree as a data storage container, beating the benchmark complexity of \(O(n)\).