| Title: | Computationally Efficient Maximum Likelihood Identification of Linear Dynamical Systems |
|---|---|
| Description: | Provides implementations of computationally efficient maximum likelihood parameter estimation algorithms for models that represent linear dynamical systems. Currently, one such algorithm is implemented for the one-dimensional cumulative structural equation model with shock-error output measurement equation and assumptions of normality and independence. The corresponding scientific paper is yet to be published, therefore the relevant reference will be provided later. |
| Authors: | Vytautas Dulskis [cre, aut], Leonidas Sakalauskas [aut] |
| Maintainer: | Vytautas Dulskis <vytautas.dulskis@gmail.com> |
| License: | GPL-2 |
| Version: | 0.2.0 |
| Built: | 2025-02-14 03:24:43 UTC |
| Source: | https://github.com/cran/EMLI |
Calculates the likelihood function value for given data and statistical measure values of the output-differenced version of the one-dimensional cumulative structural equation model with shock-error output measurement equation and assumptions of normality and independence. Suitable when there are no contradictions in statistical measure values.
calculate_likelihood(dat, params)calculate_likelihood(dat, params)
dat |
An (n + 1) x (m + 1) data frame of finite numeric elements (possibly except for row 1 columns 1 to m) containing observed input (columns 1 to m) and output (column m + 1) data of the original model. |
params |
A list consisting of 3 elements: 1) Sigma ((m + 1) x (m + 1) matrix of finite numeric elements); 2) sigma_y^2 (vector of length 1, finite numeric element); 3) mu ((m + 1) x 1 matrix of finite numeric elements). |
Calculated likelihood function value (vector of length 1, numeric element).
set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) data <- generate_data(100, L, sigma, mu) estimated_parameters <- estimate_parameters(data, 0.00001) calculate_likelihood(data, estimated_parameters)set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) data <- generate_data(100, L, sigma, mu) estimated_parameters <- estimate_parameters(data, 0.00001) calculate_likelihood(data, estimated_parameters)
Calculates maximum likelihood estimates of the statistical measures of the output-differenced version of the one-dimensional cumulative structural equation model with shock-error output measurement equation and assumptions of normality and independence.
estimate_parameters(dat, tol)estimate_parameters(dat, tol)
dat |
An (n + 1) x (m + 1) data frame of finite numeric elements (possibly except for row 1 columns 1 to m) containing observed input (columns 1 to m) and output (column m + 1) data of the original model. |
tol |
A tolerance parameter of the golden section search algorithm used for minimizing the one-dimensional likelihood function (vector of length 1, finite positive numeric element). |
A list consisting of 3 elements: 1) estimate of the covariance at lag 0 of the data that result from the output-differenced model (Sigma; (m + 1) x (m + 1) matrix of numeric elements); 2) estimate of the only non-zero element of the negative covariance at lag 1 of the data that result from the output-differenced model (sigma_y^2; vector of length 1, numeric element); 3) estimate of the mean of the data that result from the output-differenced model (mu; (m + 1) x 1 matrix of numeric elements).
set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) data <- generate_data(100, L, sigma, mu) estimate_parameters(data, 0.00001)set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) data <- generate_data(100, L, sigma, mu) estimate_parameters(data, 0.00001)
Calculates a discrepancy-function-based metric of accuracy of the statistical measure estimates for the output-differenced version of the one-dimensional cumulative structural equation model with shock-error output measurement equation and assumptions of normality and independence. Suitable when there are no contradictions in the factuals/estimates.
evaluate_estimates(f, e, n)evaluate_estimates(f, e, n)
f |
A list consisting of 3 elements: 1) the factual Sigma ((m + 1) x (m + 1) matrix of finite numeric elements); 2) the factual sigma_y^2 (vector of length 1, finite numeric element); 3) the factual mu ((m + 1) x 1 matrix of finite numeric elements). |
e |
Analogous to parameter f but with estimates instead of factuals. |
n |
The number of time moments used for obtaining parameter e (vector of length 1, finite positive integer). |
Calculated accuracy metric value (vector of length 1, numeric element). The lower the value, the better the accuracy, with 0 indicating perfect accuracy.
set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) n <- 100 data <- generate_data(n, L, sigma, mu) Sigma <- L %*% t(L) + diag(sigma[1:(m + 1), ] ^ 2) sigma_y_squared <- sigma[m + 2, ] ^ 2 Sigma[m + 1, m + 1] <- Sigma[m + 1, m + 1] + 2 * sigma_y_squared factual_parameters <- list(Sigma, sigma_y_squared, mu) estimated_parameters <- estimate_parameters(data, 0.00001) evaluate_estimates(factual_parameters, estimated_parameters, n)set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) n <- 100 data <- generate_data(n, L, sigma, mu) Sigma <- L %*% t(L) + diag(sigma[1:(m + 1), ] ^ 2) sigma_y_squared <- sigma[m + 2, ] ^ 2 Sigma[m + 1, m + 1] <- Sigma[m + 1, m + 1] + 2 * sigma_y_squared factual_parameters <- list(Sigma, sigma_y_squared, mu) estimated_parameters <- estimate_parameters(data, 0.00001) evaluate_estimates(factual_parameters, estimated_parameters, n)
Generates data according to the one-dimensional cumulative structural equation model with shock-error output measurement equation and assumptions of normality and independence with given model parameter values.
generate_data(n, L, sigma, mu)generate_data(n, L, sigma, mu)
n |
The number of time moments to generate the data for (vector of length 1, finite positive integer). |
L |
Factor loadings ((m + 1) x k matrix of finite numeric elements: the first m rows correspond to the input measurement equation; the last row corresponds to the transition equation). |
sigma |
Standard deviations of the error/noise terms ((m + 2) x 1 matrix of finite non-negative numeric elements: the first m rows correspond to the input measurement equation; the row before the last one corresponds to the transition equation; the last row corresponds to the output measurement equation). |
mu |
Intercept terms ((m + 1) x 1 matrix of finite numeric elements; the first m rows correspond to the input measurement equation; the last row corresponds to the transition equation). |
An (n + 1) x (m + 1) data frame of numeric elements (except for row 1 columns 1 to m that contain NA's) containing observed input (columns 1 to m) and output (column m + 1) data.
set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) generate_data(10, L, sigma, mu)set.seed(1) m <- 4 k <- 2 L <- matrix(runif((m + 1) * k, min = -10, max = 10), nrow = m + 1) sigma <- matrix(runif(m + 2, min = 0, max = 10), nrow = m + 2) mu <- matrix(runif(m + 1, min = -10, max = 10), nrow = m + 1) generate_data(10, L, sigma, mu)