# Variance Models

## CHOLk, CHOLkC, ANTE, US

In CHOL`k`
models **S** = **LDL'** where **L** is lower
triangular with ones on the diagonal, **D** is diagonal
and ` k` is the number of non-zero off diagonals in **L**.
In
CHOL`k`C
models **S** = **LDL'** where **L** is lower
triangular with ones on the diagonal, **D** is diagonal
and ` k` is the number of non-zero sub diagonal columns in **L**.
This is somewhat similar to the factor analytic model.
In
ANTE`k`
models inverse(**S**) = **UDU**' where **U** is
upper triangular with ones on the diagonal, **D** is diagonal
and ` k` is the number of non-zero off diagonals in **U**
The CHOL`k`
and ANTE`k` models are
equivalent to the US structure, that is, the full
variance structure, when ` k`
is ` w - 1`,
Initial values for US,
CHOL
and ANTE structures are
given in the form of a US matrix which is specified lower
triangle row-wise, viz

V11

C21 V22

C31 C32 V22

that is, initial values are given in the order,
` V11 C21 V22 C31 C32 V33 `
The US model is associated with several special features of ASReml.
When used in the R structure for
multivariate data,
ASReml automatically
recognises patterns of
missing values in the responses.
When the
!GP qualifier is set, ASReml checks the matrix after every
AI update and of the matrix is not Positive Definite, backs up
and does an EM update instead.
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