Reading and notation guide

A formula is a compact record of operations. First identify the inputs, then the operation, then what its output measures. Each chapter follows that order. The original four-person example remains one connected calculation; smaller examples are explicitly synthetic and set aside that fitted model temporarily. A successful arithmetic check does not validate the archive’s identities, attribution, or social claims.

In either notebook, run the setup and then the cells in reading order. A kernel is the running language process. Restarting it and running every cell checks that no hidden earlier calculation is needed. An assertion is a check that stops execution when a stated identity or reference answer fails. Edit the declared inputs to experiment; when an input changes, a fixed reference answer can fail deliberately while an identity should still hold. The Python and OCaml versions compute independently and need no private archive.

Read a symbol as a named object

A scalar is one number, a vector is an ordered list, and a matrix is a rectangular table. Here an observation vector is a row. A direction used for projection is a column. A shape such as $4\times3$ means four rows and three columns. The real numbers, written $\mathbb R$ , include negative values, zero, and fractions; $\mathbb R^{4\times3}$ means a table of that shape with real entries.

An index selects an entry: $v_j$ is coordinate $j$ of row $v$ , and $W_{j\ell}$ is row $j$ , column $\ell$ of matrix $W$ . The index is not a multiplier. The notation $\sum_{j=1}^{3}v_j$ means $v_1+v_2+v_3$ . Adjacent scalar symbols mean multiplication; adjacent matrices mean the row-by-column product taught below. Parentheses group operations, so work inside them first. The symbol $\in$ means membership, and $|\mathcal A|$ counts the members of a set $\mathcal A$ .

A square $a^2$ means $a$ multiplied by itself. For nonnegative $a$ , $\sqrt a$ is the nonnegative number whose square is $a$ . Single bars $|a|$ give a scalar’s absolute value, removing its sign; double bars $\lVert v\rVert$ give a vector’s length. A bar above a symbol denotes an average, and a hat denotes a reconstruction or estimate. The superscripts $\mathsf T$ and $\top$ both transpose a matrix, exchanging rows and columns. The symbol $I$ denotes a square identity matrix, with diagonal ones and other entries zero. A superscript $-1$ on a square matrix means an inverse when it exists. The symbol $\approx$ warns of rounded or approximate equality.

Keep the measuring objects distinct

This table is a reading map. The chapters introduce each object again with its dimensions before using it in a calculation.

Object	Symbol and meaning
Article row	$z_a$ : article $a$ measured in standardized news coordinates.
Comparison within a cell	$q_g$ : unique-article background; $m_{pg}$ : person $p$ ’s mean in cell $g$ .
Person before and after normalization	$r_p$ : balanced contrast; $b_p$ : its unit direction.
Population reference	$\bar b$ : average fitted person profile; $h_p=b_p-\bar b$ : centered row.
Learned people directions	$W$ : loadings mapping news coordinates to retained people coordinates $u_p$ .
Dimension counts	$d$ : embedding width; $K$ : news width; $r$ : retained people width. The scalar $r$ differs from row $r_p$ .
Sample counts	$n$ : fitted people; $N$ : reviewed training articles in the proposed concept model.
Concept model	$A$ : slopes; $\beta$ : intercepts; $s$ : predicted score row.
Local reuse of a letter	$Q$ is an SVD direction matrix; $Q_p$ is a scalar residual statistic; $Q_{\mathrm{align}}$ is an alignment matrix.

Normalization makes one row have length one. Standardization divides each coordinate by its reference standard deviation. Centering subtracts a reference mean. These answer different questions and cannot be substituted for each other. A positive coordinate names one pole of an axis; its sign does not express approval, goodness, truth, or confidence.

Across volumes, equivalent roles sometimes use different symbols. The History Math companion’s unified notation uses $L$ for the retained people dimension count, $W$ for the direction matrix, $S$ for the score table, $\Theta$ for concept coefficients, and $C$ for the concept count. This established Anthropology edition writes those roles as $r$ , the same $W$ , rows $u_p$ stacked together, $A$ , and $m$ , respectively. History counts people with $m$ and writes their covariance as $\Gamma_B$ ; this book uses $n$ and $C_P$ for those roles. Do not substitute letters without also matching their shapes and centering conventions. Here $C_P$ remains the people covariance, not the concept count. Here $Q$ names the SVD’s right-direction matrix, while $Q_p$ is a person’s scalar discarded squared length. These distinct objects never share a formula merely because their letters resemble one another.

The final notation table collects shapes. The glossary defines terms in words, and the subject index links to worked explanations. The Eigen Times companion sometimes writes observations as columns; transpose its complete formula when translating to this book’s row convention.