update indices for stochastic matrices

Author: SteveBronder

src/reference-manual/transforms.qmd

@@ -682,30 +682,30 @@ constrained to have non-negative entries that sum to one.
 ### Definition of a Stochastic Matrix {-}
 
 A column stochastic matrix \(X \in \mathbb{R}^{N \times M}\) is defined such
-that each column is a simplex. For column \(j\) (where \(1 \leq j \leq M\)):
+that each column is a simplex. For column \(m\) (where \(1 \leq m \leq M\)):
 
 $$
-X_{i, j} \geq 0 \quad \text{for } 1 \leq i \leq N,
+X_{n, m} \geq 0 \quad \text{for } 1 \leq n \leq N,
 $$
 
 and
 
 $$
-\sum_{i=1}^N X_{i, j} = 1.
+\sum_{n=1}^N X_{n, m} = 1.
 $$
 
 A row stochastic matrix is any matrix whose transpose is a column stochastic matrix
 (i.e. the rows of the matrix are simplexes)
 
 
 $$
-X_{i, j} \geq 0 \quad \text{for } 1 \leq j \leq N,
+X_{n, m} \geq 0 \quad \text{for } 1 \leq n \leq N,
 $$
 
 and
 
 $$
-\sum_{j=1}^N X_{i, j} = 1.
+\sum_{m=1}^N X_{n, m} = 1.
 $$
 
 This definition ensures that each column (row) of the matrix \(X\) lies on the
@@ -719,19 +719,19 @@ as simplex, but applied to each column (row).
 
 ### Absolute Jacobian Determinant for the Inverse Transform {-}
 
-The Jacobian determinant of the inverse transform for each column \(j\) in
-the matrix is given by the product of the diagonal entries \(J_{i,i,j}\) of
+The Jacobian determinant of the inverse transform for each column \(m\) in
+the matrix is given by the product of the diagonal entries \(J_{n, m}\) of
 the lower-triangular Jacobian matrix. This determinant is calculated as:
 
 $$
-\left| \det J_j \right| = \prod_{i=1}^{N-1} \left( z_{i, j} (1 - z_{i, j}) \left( 1 - \sum_{i'=1}^{i-1} X_{i'j} \right) \right).
+\left| \det J_m \right| = \prod_{n=1}^{N-1} \left( z_{n, m} (1 - z_{n, m}) \left( 1 - \sum_{n'=1}^{n-1} X_{n', m} \right) \right).
 $$
 
 Thus, the overall Jacobian determinant for the entire `column_stochastic_matrix` and `row_stochastic_matrix`
 is the product of the determinants for each column (row):
 
 $$
-\left| \det J \right| = \prod_{j=1}^{M} \left| \det J_j \right|.
+\left| \det J \right| = \prod_{m=1}^{M} \left| \det J_m \right|.
 $$
 
 ### Transform for Stochastic Matrix {-}

src/reference-manual/types.qmd

@@ -675,7 +675,7 @@ priors for some parameters.
 
 ### Stochastic Matrices {-}
 
-A stochastic matrix is a matrix where each column, row, or both is a
+A stochastic matrix is a matrix where each column or row is a
 unit simplex, meaning that each column (row) vector has non-negative
 values that sum to 1. The following example is a \(3 \times 4\)
 column-stochastic matrix.