Notation

Notation reference — the Y family

Symbol	Meaning	Shape	Defined
\(\mathbf{Y}\)	Raw spot yields	\(T \times N\)	Step 1
\(\widetilde{\mathbf{Y}}\)	Log yields, 12-month differenced	\((T-12) \times N\)	Steps 3–4
\(\widetilde{y}_{s,n}\)	Entry of \(\widetilde{\mathbf{Y}}\), row \(s\) col \(n\)	scalar	Steps 3–4
\(\vec{\widetilde{y}}\)	Column means of \(\widetilde{\mathbf{Y}}\)	\(N\) (vector)	Step 5
\(\bar{\widetilde{y}}_n\)	Column mean for maturity \(n\) (entry of \(\vec{\widetilde{y}}\))	scalar	Step 5
\(\widetilde{\mathbf{Y}}_c\)	\(\widetilde{\mathbf{Y}}\), column-demeaned	\((T-12) \times N\)	Step 5
\(\widetilde{y}^{\,c}_{s,n}\)	Entry of \(\widetilde{\mathbf{Y}}_c\), row \(s\) col \(n\)	scalar	Step 5
\(\mathbf{A}\)	Covariance matrix of \(\widetilde{\mathbf{Y}}_c\): \(\mathbf{A} = \frac{1}{T-13}\widetilde{\mathbf{Y}}_c^\top \widetilde{\mathbf{Y}}_c\)	\(N \times N\)	Step 6

Notation reference — z (two distinct uses)

Symbol	Meaning	Shape	Defined
\(\mathbf{z}\) (classical, general)	Principal component scores, \(\mathbf{z} = \widetilde{\mathbf{Y}}_c\mathbf{W}\) — computed directly from observed data, using all \(N\) components (no subscript = nothing dropped)	\((T-12) \times N\)	—
\(\mathbf{z}_3\) (classical, reduced)	Principal component scores, \(\mathbf{z}_3 = \widetilde{\mathbf{Y}}_c\mathbf{W}_3\) — computed from only the first 3 components	\((T-12) \times 3\)	Step 9
\(\mathbf{z}\) (probabilistic)	Latent variable in the generative model \(\mathbf{x} = \mathbf{W}\mathbf{z} + \boldsymbol{\mu} + \boldsymbol{\epsilon}\) — unobserved, inferred rather than computed	\(k\) (vector)	Probabilistic PCA

Notation reference — W (two distinct uses)

Symbol	Meaning	Shape	Defined
\(\mathbf{w}_k\)	Loading vector for component \(k\) — same object as the generic eigenvector \(\vec{v}\)	\(N\) (vector)	Step 7
\(\mathbf{W}\) (classical)	Loading matrix — eigenvectors of \(\mathbf{A}\) stacked as columns	\(N \times N\)	Step 7
\(\mathbf{W}_3\)	\(\mathbf{W}\) restricted to the first 3 columns (top 3 components)	\(N \times 3\)	Step 9
\(\mathbf{W}\) (probabilistic)	Maps latent space to observed space in \(\mathbf{x} = \mathbf{W}\mathbf{z} + \boldsymbol{\mu} + \boldsymbol{\epsilon}\)	\(d \times k\)	Probabilistic PCA

Notation reference — Bayesian parameters

Symbol	Meaning	Shape	Defined
\(\mu_i\)	Mean of component \(i\)'s score distribution (Bayesian parameter, flat prior)	scalar, per component \(i \in \{1,2,3\}\)	Step 1 (bayesian)
\(\sigma_i^2\)	Variance of component \(i\)'s score distribution (Jeffreys prior on \(\sigma_i^2\), flat prior on \(\log\sigma_i\))	scalar, per component	Step 1 (bayesian)
\(z_i\) (bayesian likelihood)	Component \(i\)'s observed PC scores from Step 9, modelled as \(z_i \sim \text{Normal}(\mu_i,\sigma_i^2)\) — a third, distinct use of \(z\) beyond the two above	\((T-12)\) (vector)	Step 2 (bayesian)