6 Multivariate distributions
using namespace density;
gives access to a variety of multivariate normal distributions:
- Multivariate normal distributions specified via a covariance matrix (structured or unstructured).
- Autoregressive (AR) processes.
- Gaussian Markov random fields (GMRF) defined on regular grids or defined via a (sparse) precision matrix.
- Separable covariance functions, i.e. time-space separability.
These seemingly unrelated concepts are all implemented via the notion
distribution, which explains why they are placed in the same
namespace. You can combine two
distributions, and this lets you
build up complex multivariate distributions using extremely compact
notation. Due to the flexibility of the approach it is more abstract
than other parts of TMB, but here it will be explained from
scratch. Before looking at the different categories of multivariate
distributions we note the following which is of practical importance:
- All members in the
densitynamespace return the negative log density, opposed to the univariate densities in R style distributions.
6.1 Multivariate normal distributions
Consider a zero-mean multivariate normal distribution with covariance matrix Sigma (symmetric positive definite), that we want to evaluate at x:
The negative log-normal density is evaluated as follows:
In the last line
MVNORM(Sigma) should be interpreted as a
multivariate density, which via the last parenthesis
x. A less compact way of expressing this is
MVNORM_t<Type> your_dmnorm(Sigma); res = your_dmnorm(x);
your_dmnorm is a variable that holds the “density.”
Note, that the latter way (using the
MVNORM_t) is more efficient
when you need to evaluate the density more than once, i.e. for different values of
Sigma can be parameterized in different ways. Due to the symmetry of
Sigma there are at most n(n+1)/2 free parameters (n variances
and n(n-1)/2 correlation parameters). If you want to estimate all of
these freely (modulo the positive definite constraint) you can use
UNSTRUCTURED_CORR() to specify the correlation matrix, and
VECSCALE() to specify variances.
UNSTRUCTURED_CORR() takes as
input a vector a dummy parameters that internally is used to build the
correlation matrix via its cholesky factor.
using namespace density; int n = 10; vector<Type> unconstrained_params(n*(n-1)/2); // Dummy parameterization of correlation matrix vector<Type> sds(n); // Standard deviations VECSCALE(UNSTRUCTURED_CORR(unconstrained_params),sds)(x);res =
If all elements of
dummy_params are estimated we are in effect
estimating a full correlation matrix without any constraints on its
elements (except for the mandatory positive definiteness). The actual
value of the correlation matrix, but not the full covariance matrix,
can easily be assessed using the
6.2 Autoregressive processes
Consider a stationary univariate Gaussian AR1 process x(t),t=0,…,n-1. The stationary distribution is choosen so that:
- x(t) has mean 0 and variance 1 (for all t).
The multivariate density of the vector x can be evaluated as follows
Due to the assumed stationarity the correlation parameter must satisfy:
- Stationarity constraint: -1 < rho < 1
Note that cor[x(t),x(t-1)] = rho.
SCALE() function can be used to set the standard deviation.
2.1; // standard deviation of x Type sigma = res = SCALE(AR1(rho),sigma)(x);
Now, var[x(t)] = sigma^2. Because all elements of
x are scaled by
the same constant we use SCALE rather than VECSCALE.
188.8.131.52 Multivariate AR1 processes
This is the first real illustration of how distributions can be used as building blocks to obtain more complex distributions. Consider the p dimensional AR1 process
int n = 10; // Number of time steps int p=3; // dim(x) array<Type> x(p,n); // Evaluation point
The columns in
x refer to the different time points. We then
evaluate the (negative log) joint density of the time series.
MVNORM_t<Type> your_dmnorm(Sigma); // Density of x(t) // Correlation parameter Type phi; res = AR1(phi,your_dmnorm)(x);
Note the following:
- We have introduced an intermediate variable
your_dmnorm, which holds the p-dim density marginal density of x(t). This is a zero-mean normal density with covariance matrix
- All p univarite time series have the same serial correlation phi.
- The multivariate process x(t) is stationary in the same sense as the univariate AR1 process described above.
184.108.40.206 Higher order AR processes
There also exists
ARk_t of arbitrary autoregressive order.
6.3 Gaussian Markov random fields (GMRF)
GMRF may be defined in two ways:
- Via a (sparse) precision matrix Q.
- Via a d-dimensional lattice.
For further details please see
GMRF_t. Under 1) a sparse Q
corresponding to a Matern covariance function can be obtained via the
6.4 Separable construction of covariance (precision) matrices
A typical use of separability is to create space-time models with a
sparse precision matrix. Details are given in
we give a simple example.
Assume that we study a quantity
x that changes both in space and
time. For simplicity we consider only a one-dimensional space. We
discretize space and time using equidistant grids, and assume that the
distance between grid points is 1 in both dimensions. We then define
AR1(rho_s) process in space and one in time
separable assumption is that two points
x2, separated in
space by a distance
ds and in time by a distance
correlation given by
This is implemented as
Note that the arguments to
SEPARABLE() are given in the opposite order
to the dimensions of