fit_sa: Stochastic Approximation fitting of MRFs on 2d lattices
In mrf2d: Markov Random Field Models for Image Analysis

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Estimates the parameters of a MRF by successively sampling from a parameter configuration and updating it by comparing the sufficient statistics of the sampled field and the observed field.

This method aims to find the parameter value where the gradient of the likelihood function is equal to zero.

fit_sa(
  Z,
  mrfi,
  family = "onepar",
  gamma_seq,
  init = 0,
  cycles = 5,
  refresh_each = length(gamma_seq) + 1,
  refresh_cycles = 60,
  verbose = interactive()
)

`Z`	A `matrix` object containing the observed MRF. `NA` values can be used to create a subregion of the lattice for non-rectangular data.
`mrfi`	A `mrfi` object representing the interaction structure.
`family`	The family of parameter restrictions to potentials. Families are: `'onepar'`, `'oneeach'`, `'absdif'`, `'dif'` or `'free'`. See `mrf2d-familiy`.
`gamma_seq`	A `numeric` vector with the sequence of constants used in each step γ_t.
`init`	The initial value to be used in the optimization. It can be: A valid `array` of parameter values according to `family`. `0`. If set to `0` an array with '0“ in all entries is created.
`cycles`	The number of updates to be done (for each each pixel).
`refresh_each`	An integer with the number of iterations taken before a complete refresh (restart from a random state). This prevents the sample from being stuck in a mode for too long. Defaults to `length(gamma_seq) + 1` (no refresh happens).
`refresh_cycles`	An integer indicating how many Gibbs Sampler cycles are performed when a refresh happens. Larger is usually better, but slower.
`verbose`	`logical` indicating whether the iteration number is printed during execution.

The stochastic approximation method consists of, given an observed field Z, and a starting parameters configuration θ_0, successively sample a field Z_t from the current parameter configuration and estimate the direction of the gradient of the likelihood function by comparing the sufficient statistics in the current sample and the observed field.

The solution is updated by moving in the estimated direction with a predefined step size γ_t, a new field Z_{t+1} is sampled using the new parameter configuration and Z_t as an initial value, and the process is repeated.

θ_{t+1} = θ_t - γ_t(T(Z_t) - T(Z)),

where T(Z) is the sufficient statistics for the reference field, T(Z_t) is the sufficient statistics for a field sampled from θ_t.

gamma_seq is normalized internally by diving values by length(Z), so the choice of the sequence is invariant to the lattice dimensions. Typically, a sequence like seq(from = 1, to = 0, length.out = 1000) should be used for defining a sequence with 1000 steps. Some tuning of this sequence is required.

A mrfout object with the following elements:

theta: The estimated array of potentials.
mrfi: The interaction structure considered.
family: The parameter restriction family considered.
method: The estimation method ("Stochastic Approximation").
metrics: A data.frame containing the the euclidean distance between the sufficient statics computed for Z and the current sample.

Stochastic Approximation is called "Controllable Simulated Annealing" in some references.

Examples where Stochastic Approximation is used with MRFs are \insertCitegimel_samrf2d, \insertCiteCRmrf2d.

Victor Freguglia

\insertRef

wiki_samrf2d

\insertAllCited

A paper with detailed description of the package can be found at doi: 10.18637/jss.v101.i08.

set.seed(2)
fit1 <- fit_sa(Z_potts, mrfi(1), family = "oneeach", gamma_seq = seq(1, 0, length.out = 50))
# Estimated parameters
fit1$theta
# A visualization of estimated gradient norm over iterations.
plot(fit1$metrics, type = "l")

fit_sa(Z_potts, mrfi(1), family = "oneeach", gamma_seq = seq(1, 0, length.out = 50))