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 $g _ {ij} = a_ {ij} - \bar{a} _ {i.} - \bar{a} _ {j.} + \bar{a} _ {..}$
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 jth observation in the ith 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 'An 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 kth predictor variable
Y$ _ k $ = the kth response variable
z$ _ {ij} $ = distance to group centroid for the jth replicate within the ith 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 ith 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