ppi_param_tools: PPI Parameter Tools

Description

The default parameterisation of the PPI model is a symmetric covariance-like matrix $A_L$, a location-like vector $b_L$ and a set of Dirichlet exponents $\beta$. For p components, $A_L$ has p-1 rows, $b_L$ is a vector with p-1 elements and $\beta$ is a vector with p elements. For score matching estimation this form of the parameters must be converted into a single parameter vector using ppi_paramvec(). ppi_paramvec() also includes easy methods to set parameters to NA for estimation with ppi() (in ppi() the NA-valued elements are estimated and all other elements are fixed). The reverse of ppi_paramvec() is ppi_parammats(). An alternative parametrisation of the PPI model uses a single p by p matrix $A^*$ instead of $A_L$ and $b_L$, and for identifiability $A^*$ is such that $1^T A^* 1 = 0$ where $1=(1,1,...,1)$ and $0=(0,0,..., 0)$ scealy2023scscorematchingad. Convert between parametrisations using ppi_toAstar() and ppi_fromAstar().

Usage

ppi_paramvec(
  p = NULL,
  AL = NULL,
  bL = NULL,
  Astar = NULL,
  beta = NULL,
  betaL = NULL,
  betap = NULL
)
ppi_parammats(paramvec)
ppi_toAstar(AL, bL)
ppi_fromAstar(Astar)

Value

ppi_paramvec(): a vector of length $p + (p-1)(2 + (p-1)/2)$.

ppi_parammats(): A named list of $A_L$, $b_L$, and $\beta$.

ppi_toAstar(): The matrix $A^*$.

ppi_fromAstar(): A list of the matrix $A_L$, the vector $b_L$ and a discarded constant.

Arguments

p: The number of components. If NULL then p will be inferred from other inputs.
AL: Either NULL, a p-1 x p-1 symmetric matrix, a number, or "diag". If NULL then all $A_L$ elements will be set to NA. If a single number, then $A_L$ will be fixed as a matrix of the given value. If "diag" then the non-diagonal elements of $A_L$ will be fixed to 0, and the diagonal will be NA.
bL: Either NULL, a number, or a vector of length p-1. If NULL, then all elements of $b_L$ will be set to NA. If a single number, then $b_L$ will be fixed at the supplied value.
Astar: The $A^*$ matrix (a p by p symmetric matrix)
beta: Either NULL, a number, or a vector of length p. If NULL then all elements of $\beta$ will be set to NA. If a single number then the $\beta$ elements will be fixed at the given number.
betaL: Either NULL, a number, or a vector of length p-1. If NULL then the 1...(p-1)th $\beta$ elements will be set to NA. If a single number then the 1...(p-1)th $\beta$ elements will be fixed at the given number.
betap: Either NULL or a number. If NULL then the pth element of $\beta$ will be set to NA, and ppi() will estimate it. If a number, then the pth element of $\beta$ will be fixed at the given value.
paramvec: A PPI parameter vector, typically created by ppi_paramvec() or as an output of ppi().

PPI Model

The PPI model density is proportional to $$\exp(z_L^TA_Lz_L + b_L^Tz_L)\prod_{i=1}^p z_i^{\beta_i},$$ where $p$ is the dimension of a compositional measurement $z$, and $z_L$ is $z$ without the final ($p$th) component. $A_L$ is a $p-1 \times p-1$ symmetric matrix that controls the covariance between components. $b_L$ is a $p-1$ vector that controls the location within the simplex. The $i$th component $\beta_i$ of $\beta$ controls the concentration of density when $z_i$ is close to zero: when $\beta_i \geq 0$ there is no concentration and $\beta_i$ is hard to identify; when $\beta_i < 0$ then the probability density of the PPI model increases unboundedly as $z_i$ approaches zero, with the increasing occurring more rapidly and sharply the closer $\beta_i$ is to $-1$.

Details

ppi_paramvec() returns a vector starting with the diagonal elements of $A_L$, then the off-diagonal elements extracted by upper.tri() (which extracts elements of $A_L$ along each row, left to right, then top to bottom), then $b_L$, then $\beta$.

The Astar parametrisation rewrites the PPI density as proportional to $$\exp(z^TA^*z)\prod_{i=1}^p z_i^{\beta_i},$$ where $A^*$ (Astar) is a $p$ by $p$ matrix. Because $z$ lies in the simplex (in particular $\sum z_i = 1$), the density is the same regardless of the value of $1^T A^* 1$=sum(Astar), where $1$ is the vector of ones. Thus $A_L$ and $b_L$ specify $A^*$ up to an additive factor. In the conversion ppi_toAstar(), $A^*$ is returned such that $1^T A^* 1 = 0$. NULL values or NA elements are not allowed for ppi_toAstar() and ppi_fromAstar().

Examples

Run this code

ppi_paramvec(AL = "diag", bL = 0, betap = -0.5, p = 3)
vec <- ppi_paramvec(AL = rsymmetricmatrix(2), beta = c(-0.8, -0.7, 0))
ppi_parammats(vec)
Astar <- rWishart(1, 6, diag(3))[,,1]
ppi_fromAstar(Astar)
ppi_toAstar(ppi_fromAstar(Astar)$AL, ppi_fromAstar(Astar)$bL)

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