# A3 Index to mathematical notation and symbols

#### Matrices and vectors

**A** = matrix containing elements $a _ {ij} = - \frac{1}{2} d _ {ij} ^ 2 $

**B** = matrix of variables (*N* × *s*) that are linear combinations of normalised **X** variables having maximum correlation with CAP axes

**C** = matrix of CAP axes (*N* × *s*), standardised by the square root of their respective eigenvalues

**D** = matrix containing elements $d _ {ij}$ corresponding to distances or dissimilarities

**G** = Gower’s centred matrix, consisting of elements $g _ {ij} = a_ {ij} - \bar{a} _ {i \bullet} - \bar{a} _ {\bullet j} + \bar{a} _ {\bullet \bullet}$

**H** = ‘hat’ matrix = **X[X′X]$^ {-1}$X′**, used as a projection matrix for regression models

**I** = identity matrix, with 1’s along the diagonal and 0’s elsewhere

**Q** = matrix of PCO axes, standardised by the square root of their respective eigenvalues

**Q**$^0$ = matrix of PCO axes, orthonormalised to SSCP = **I** (‘sphericised’)
**U** = matrix whose columns contain the left singular vectors from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **U** is (*N* × *q*)

**V** = matrix whose columns contain the right singular vectors from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **V** is (*q* × *N*)

**W** = diagonal matrix of eigenvalues from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **W** is (*q* × *q*)

**X** = matrix of predictor variables (*N* × *q*) (often a set of environmental variables)

**X**$^0$ = matrix of **X** variables, orthonormalised to SSCP = **I** (‘sphericised’)

**Y** = matrix of response variables (*N* × *p*) (often a set of species variables)

**Y**$^0$ = matrix of **Y** variables, orthonormalised to SSCP = **I** (‘sphericised’)

$\hat{ {\bf Y}}$ = **HY** = matrix of fitted values (*N* × *p*)

**y**$_ {ij} $ = vector of *p* response variables for the *j*th observation in the *i*th group

$\bar{ {\bf y}}$ = the centroid vector of *p* response variables for group *i*

**Z** = matrix of dbRDA canonical axes (*N* × *s*)

#### Letters

*a*, *b*, *c*, etc… = number of levels of factor A, B, C, etc… in an ANOVA experimental design

*AIC* = multivariate analogue to Akaike’s information criterion

*AIC*$_c$ = multivariate analogue to the small-sample-size corrected version of *AIC*

*B*$_l$ = the *l*th variable in the space of normalised **X** variables that has maximum correlation with the *l*th coordinate axis (*C*$_l$) from a CAP analysis

*BIC* = multivariate analogue to Schwarz’s ‘Bayesian information criterion’

*C*$ _l $ = the *l*th coordinate axis scores from a CAP analysis

*d*$ _ {ij} $ = distance or dissimilarity between sample *i* and sample *j*
*df* = degrees of freedom

*F* = pseudo-*F* statistic for testing hypotheses in PERMANOVA or DISTLM

*i* = index used for samples (i.e., *i* = 1, …, *N*) or index used for groups (*i* = 1, …, *a*)

*j* = second index used for samples (i.e., *j* = 1, …, *N*) or index used for replicates within a group (*j* = 1,…, *n*)

*k* = index used for variables (i.e., *k* = 1, …, *p* or else *k* = 1, …, *q*)

*l* = index used for canonical axes or eigenvalues for either dbRDA or CAP (i.e., *l* = 1, …, *s*) or either the abbreviation for ‘log-likelihood’ or the ‘length’ of a vector (depending on context).

*m* = number of PCO axes chosen as a subset for analysis by CAP
*MC* = Monte Carlo

*MS* = mean square

*N* = total number of samples

*n* = number of samples (replicates) within a group or cell in an experimental design

*P* = *P*-value associated with the test of a null hypothesis
*p* = number of multivariate response variables in matrix **Y**
*q* = total number of predictor variables in matrix **X**

*r* = Pearson correlation coefficient

*R* = the ANOSIM *R* statistic (see Clarke 1993)

*R*$^2$ = proportion of explained variation from a model

*s* = number of canonical eigenvalues and associated canonical axes obtained from either a dbRDA *or* a CAP analysis
*SS* = sum of squares

*SSCP* = sum of squares and cross products

*SVD* = singular value decomposition

*t* = pseudo-*t* statistic = $\sqrt{}$pseudo- *F*

*tr* = ‘trace’ of a matrix = the sum of the diagonal elements

*X*$ _ k $ = the *k*th predictor variable

*Y*$ _ k $ = the *k*th response variable

*z*$ _ {ij} $ = distance to group centroid for the *j*th replicate within the *i*th group.

#### Greek symbols and matrices

$ \alpha$ = significance level chosen for a test (usually $\alpha$ = 0.05).

$ \delta _ l ^ 2$ = the *l*th eigenvalue from a CAP analysis, a squared canonical correlation

$ \Delta$ = diagonal matrix containing the square roots of the eigenvalues from a CAP analysis (a capital delta)

$ \gamma _ l ^ 2$ = the *l*th eigenvalue from a dbRDA analysis, a portion of the explained (regression) sum of squares from a dbRDA model.

$ \Gamma$ = diagonal matrix containing the square roots of the eigenvalues from a dbRDA analysis (a capital gamma)

$ \lambda _ i $ = the *i*th eigenvalue from a PCO analysis

$ \Lambda $ = diagonal matrix of eigenvalues from a PCO analysis (a capital lambda)

$ \nu$ = number of parameters in a particular model during model selection

$ \rho $= Spearman rank correlation (rho)

$ \sum $ = sum over the relevant index