README.md
In zhoucx1119/LogConcaveDESM: Log-Concave Density Estimation Using the Score Matching Loss Function
Log-Concave Density Estimation Using the Score Matching Loss Function
LogConcaveDESM is a package to compute and visualize the (penalized)
log-concave score matching density estimate. It also contains functions
to plot the first and second derivatives of the log-density estimate.
Furthermore, functions to compute various distances between two
probability distributions are provided to assess the quality of the
density estimate.
Installation
The development version of the LogConcaveDESM can be installed from
the GitHub using devtools:
# install.packages("devtools")
devtools::install_github("zhoucx1119/LogConcaveDESM")
Load the Package
library(LogConcaveDESM, warn.conflicts = FALSE)
library(patchwork)
library(ggplot2)
Computing and Visualizing Unpenalized Log-concave Score Matching Density Estimate over a Bounded Domain
We show how to use functions in LogConcaveDESM to compute and
visualize the log-concave score matching density estimate over a bounded
domain.
We first simulate data from a truncated normal distribution over the
interval [−5,5].
set.seed(2021)
N <- 200
parent_mean <- 0
parent_sd <- 1
domain <- c(-5, 5)
t_normal <- truncated_normal(parent_mean, parent_sd, domain)
data <- t_normal$sampling(N)
We plot the histogram of the data below.
binwidth <- 2 * stats::IQR(data) / N ** (1/3)
ggplot() + geom_histogram(
data = data.frame(x = data),
aes(x = x, y = ..density..),
binwidth = binwidth,
color = "darkblue",
fill = "lightblue") +
coord_cartesian(xlim = domain, ylim = c(-0.01, 0.5)) +
theme_bw()
We assume that the second derivative of the logarithm of the log-concave
score matching density estimate is a piecewise linear function with
knots at the data points. With no penalty, the resulting optimal second
derivative values at data points can be computed as below:
result <- lcd_scorematching(
data = data,
domain = domain,
penalty_param = 0,
verbose = TRUE
)
#> -----------------------------------------------------------------
#> OSQP v0.6.0 - Operator Splitting QP Solver
#> (c) Bartolomeo Stellato, Goran Banjac
#> University of Oxford - Stanford University 2019
#> -----------------------------------------------------------------
#> problem: variables n = 202, constraints m = 203
#> nnz(P) + nnz(A) = 20504
#> settings: linear system solver = qdldl,
#> eps_abs = 1.0e-05, eps_rel = 1.0e-05,
#> eps_prim_inf = 1.0e-04, eps_dual_inf = 1.0e-04,
#> rho = 1.00e-01 (adaptive),
#> sigma = 1.00e-06, alpha = 1.60, max_iter = 10000
#> check_termination: on (interval 25),
#> scaling: on, scaled_termination: off
#> warm start: on, polish: on, time_limit: off
#>
#> iter objective pri res dua res rho time
#> 1 -4.3429e+00 1.35e+01 4.63e-03 1.00e-01 2.43e-03s
#> 200 -6.7500e+01 2.79e+00 1.88e-03 1.98e-02 1.06e-02s
#> 400 -7.1064e+01 4.32e+00 4.59e-04 3.81e-03 1.85e-02s
#> 600 -6.5517e+01 1.60e+00 1.22e-04 3.81e-03 2.57e-02s
#> 800 -6.7135e+01 4.42e-01 3.10e-05 3.81e-03 3.31e-02s
#> 1000 -6.6824e+01 1.01e-01 6.51e-06 3.81e-03 4.00e-02s
#> plsh -6.6874e+01 4.27e-10 3.14e-09 -------- 4.16e-02s
#>
#> status: solved
#> solution polish: successful
#> number of iterations: 1000
#> optimal objective: -66.8737
#> run time: 4.16e-02s
#> optimal rho estimate: 1.93e-03
#> The status of solving the constrained quadratic optimization problem is: optimal.
Then, the second derivative of the logarithm of the log-concave score
matching density estimate at arbitrary points can be computed using the
evaluate_logdensity_deriv2 function, and can be visualized using the
plot_logdensity_deriv2 function.
logden_deriv2_vals <- evaluate_logdensity_deriv2(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_deriv2_vals
#> newx_sorted logderiv2_vals
#> 1 -5 -1.488096e-23
#> 2 -4 -7.214839e+01
#> 3 -3 -1.442968e+02
#> 4 -2 2.368584e-13
#> 5 -1 2.606746e-12
#> 6 0 -1.251196e-10
#> 7 1 1.092312e-10
#> 8 2 -5.711015e-11
#> 9 3 -7.400681e-12
#> 10 4 -3.700341e-12
#> 11 5 0.000000e+00
plot_deriv2 <- plot_logdensity_deriv2(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_deriv2
Subsequently, the first derivative of the logarithm of the log-concave
score matching density estimate at arbitrary points can be computed
using the evaluate_logdensity_deriv1 function, and can be visualized
using the plot_logdensity_deriv1 function.
logden_deriv1_vals <- evaluate_logdensity_deriv1(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_deriv1_vals
#> newx_sorted logderiv1_vals
#> 1 -5 294.125904
#> 2 -4 258.051708
#> 3 -3 149.829119
#> 4 -2 13.667716
#> 5 -1 9.167263
#> 6 0 5.598630
#> 7 1 -13.979790
#> 8 2 -33.035110
#> 9 3 -33.035110
#> 10 4 -33.035110
#> 11 5 -33.035110
plot_deriv1 <- plot_logdensity_deriv1(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_deriv1
Then, the (un-normalized) log-density estimate at arbitrary point can be
computed using the evaluate_logdensity function, and can be visualized
using the plot_logdensity function.
logden_vals <- evaluate_logdensity(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_vals
#> newx_sorted logdensity_vals
#> 1 -5 0.0000
#> 2 -4 282.1012
#> 3 -3 492.0540
#> 4 -2 562.3994
#> 5 -1 574.0298
#> 6 0 580.8876
#> 7 1 575.0199
#> 8 2 553.7940
#> 9 3 520.7589
#> 10 4 487.7238
#> 11 5 454.6887
plot_logden <- plot_logdensity(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_logden
Finally, to evaluate and visualize the density estimate itself, we use
evaluate_density and plot_density functions, respectively.
den_vals <- evaluate_density(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
den_vals
#> newx_sorted density_vals
#> 1 -5 5.718881e-253
#> 2 -4 1.871947e-130
#> 3 -3 2.842021e-39
#> 4 -2 1.009840e-08
#> 5 -1 1.135730e-03
#> 6 0 1.080366e+00
#> 7 1 3.056907e-03
#> 8 2 1.849261e-12
#> 9 3 8.318257e-27
#> 10 4 3.741679e-41
#> 11 5 1.683064e-55
plot_den <- plot_density(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE
)
plot_den
Computing and Visualizing Penalized Log-concave Score Matching Density Estimate over a Bounded Domain
As we can see above, the un-penalized log-concave score matching density
estimate is too concentrated at the place where data are abundant. To
remedy this, we consider penalized log-concave score matching density
estimate, where the optimal penalty parameter can be chosen using the
cv_optimal_density_estimate function.
lambda_cand <- exp(seq(-5, 1, by = 0.5))
opt_den <- cv_optimal_density_estimate(
data = data,
domain = domain,
penalty_param_candidates = lambda_cand,
fold_number = 5
)
#> Penalty parameter value: 0.00673794699908547
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0111089965382423
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0183156388887342
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0301973834223185
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0497870683678639
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0820849986238988
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.135335283236613
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.22313016014843
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.367879441171442
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.606530659712633
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 1
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 1.64872127070013
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 2.71828182845905
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Optimal penality parameter is 0.22313016014843.
#> The status of solving the constrained quadratic optimization problem is: optimal.
The second and first derivatives of the logarithm of the optimal density
estimate, the logarithm of the un-normalized optimal density estimate,
and the optimal density estimate itself are plotted as below.
plot_ld2 <- plot_logdensity_deriv2(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_ld1 <- plot_logdensity_deriv1(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_ld <- plot_logdensity(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_den <- plot_density(
scorematching_logconcave = opt_den,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE
)
plot_ld2 + plot_ld1 + plot_ld + plot_den
We can also use the function plot_mle_scorematching to view the
log-concave maximum likelihood and score matching density estimates
together with the histogram.
plot_mle_scorematching(
scorematching_logconcave = opt_den,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE)
We could assess the quality of this density estimate using various
metrics between two probability distributions, for example, the
Kullback-Leibler divergence, the Hyvarinen divergence, the L1
distance, and the Hellinger distance.
kl <- kl_div(
true_density = t_normal,
density_estimate = opt_den)
hyva <- hyvarinen_div(
true_density = t_normal,
density_estimate = opt_den)
l1 <- L1_dist(
true_density = t_normal,
density_estimate = opt_den)
he <- hellinger_dist(
true_density = t_normal,
density_estimate = opt_den)
metric_table <- cbind(
Metrics = c("Kullback-Leibler Divergence", "Hyvarinen divergence",
"L1 Distance", "Hellinger Distance"),
Values = round(c(kl, hyva, l1, he), 5)
)
knitr::kable(metric_table)
| Metrics | Values |
|:----------------------------|:--------|
| Kullback-Leibler Divergence | 0.09961 |
| Hyvarinen divergence | 0.10237 |
| L1 Distance | 0.29509 |
| Hellinger Distance | 0.1774 |
Computing and Visualizing Penalized Log-concave Score Matching Density Estimate over Other Types of Domain
Besides the bounded domain demonstrated above, LogConcaveDESM can also
compute and visualize log-concave score matching density estimate over
other types of domain, including the entire real line, an interval of
the form (−∞,b) for some b \< ∞, and an interval of the form (a,∞)
for some a > − ∞. Functions used are the same as those used in the
bounded interval case. Demonstrations are omitted.
zhoucx1119/LogConcaveDESM documentation built on Aug. 28, 2024, 3:25 p.m.
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Log-Concave Density Estimation Using the Score Matching Loss Function
LogConcaveDESM is a package to compute and visualize the (penalized)
log-concave score matching density estimate. It also contains functions
to plot the first and second derivatives of the log-density estimate.
Furthermore, functions to compute various distances between two
probability distributions are provided to assess the quality of the
density estimate.
Installation
The development version of the LogConcaveDESM can be installed from
the GitHub using devtools:
# install.packages("devtools")
devtools::install_github("zhoucx1119/LogConcaveDESM")
Load the Package
library(LogConcaveDESM, warn.conflicts = FALSE)
library(patchwork)
library(ggplot2)
Computing and Visualizing Unpenalized Log-concave Score Matching Density Estimate over a Bounded Domain
We show how to use functions in LogConcaveDESM to compute and
visualize the log-concave score matching density estimate over a bounded
domain.
We first simulate data from a truncated normal distribution over the interval [−5,5].
set.seed(2021)
N <- 200
parent_mean <- 0
parent_sd <- 1
domain <- c(-5, 5)
t_normal <- truncated_normal(parent_mean, parent_sd, domain)
data <- t_normal$sampling(N)
We plot the histogram of the data below.
binwidth <- 2 * stats::IQR(data) / N ** (1/3)
ggplot() + geom_histogram(
data = data.frame(x = data),
aes(x = x, y = ..density..),
binwidth = binwidth,
color = "darkblue",
fill = "lightblue") +
coord_cartesian(xlim = domain, ylim = c(-0.01, 0.5)) +
theme_bw()
We assume that the second derivative of the logarithm of the log-concave score matching density estimate is a piecewise linear function with knots at the data points. With no penalty, the resulting optimal second derivative values at data points can be computed as below:
result <- lcd_scorematching(
data = data,
domain = domain,
penalty_param = 0,
verbose = TRUE
)
#> -----------------------------------------------------------------
#> OSQP v0.6.0 - Operator Splitting QP Solver
#> (c) Bartolomeo Stellato, Goran Banjac
#> University of Oxford - Stanford University 2019
#> -----------------------------------------------------------------
#> problem: variables n = 202, constraints m = 203
#> nnz(P) + nnz(A) = 20504
#> settings: linear system solver = qdldl,
#> eps_abs = 1.0e-05, eps_rel = 1.0e-05,
#> eps_prim_inf = 1.0e-04, eps_dual_inf = 1.0e-04,
#> rho = 1.00e-01 (adaptive),
#> sigma = 1.00e-06, alpha = 1.60, max_iter = 10000
#> check_termination: on (interval 25),
#> scaling: on, scaled_termination: off
#> warm start: on, polish: on, time_limit: off
#>
#> iter objective pri res dua res rho time
#> 1 -4.3429e+00 1.35e+01 4.63e-03 1.00e-01 2.43e-03s
#> 200 -6.7500e+01 2.79e+00 1.88e-03 1.98e-02 1.06e-02s
#> 400 -7.1064e+01 4.32e+00 4.59e-04 3.81e-03 1.85e-02s
#> 600 -6.5517e+01 1.60e+00 1.22e-04 3.81e-03 2.57e-02s
#> 800 -6.7135e+01 4.42e-01 3.10e-05 3.81e-03 3.31e-02s
#> 1000 -6.6824e+01 1.01e-01 6.51e-06 3.81e-03 4.00e-02s
#> plsh -6.6874e+01 4.27e-10 3.14e-09 -------- 4.16e-02s
#>
#> status: solved
#> solution polish: successful
#> number of iterations: 1000
#> optimal objective: -66.8737
#> run time: 4.16e-02s
#> optimal rho estimate: 1.93e-03
#> The status of solving the constrained quadratic optimization problem is: optimal.
Then, the second derivative of the logarithm of the log-concave score
matching density estimate at arbitrary points can be computed using the
evaluate_logdensity_deriv2 function, and can be visualized using the
plot_logdensity_deriv2 function.
logden_deriv2_vals <- evaluate_logdensity_deriv2(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_deriv2_vals
#> newx_sorted logderiv2_vals
#> 1 -5 -1.488096e-23
#> 2 -4 -7.214839e+01
#> 3 -3 -1.442968e+02
#> 4 -2 2.368584e-13
#> 5 -1 2.606746e-12
#> 6 0 -1.251196e-10
#> 7 1 1.092312e-10
#> 8 2 -5.711015e-11
#> 9 3 -7.400681e-12
#> 10 4 -3.700341e-12
#> 11 5 0.000000e+00
plot_deriv2 <- plot_logdensity_deriv2(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_deriv2
Subsequently, the first derivative of the logarithm of the log-concave
score matching density estimate at arbitrary points can be computed
using the evaluate_logdensity_deriv1 function, and can be visualized
using the plot_logdensity_deriv1 function.
logden_deriv1_vals <- evaluate_logdensity_deriv1(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_deriv1_vals
#> newx_sorted logderiv1_vals
#> 1 -5 294.125904
#> 2 -4 258.051708
#> 3 -3 149.829119
#> 4 -2 13.667716
#> 5 -1 9.167263
#> 6 0 5.598630
#> 7 1 -13.979790
#> 8 2 -33.035110
#> 9 3 -33.035110
#> 10 4 -33.035110
#> 11 5 -33.035110
plot_deriv1 <- plot_logdensity_deriv1(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_deriv1
Then, the (un-normalized) log-density estimate at arbitrary point can be
computed using the evaluate_logdensity function, and can be visualized
using the plot_logdensity function.
logden_vals <- evaluate_logdensity(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
logden_vals
#> newx_sorted logdensity_vals
#> 1 -5 0.0000
#> 2 -4 282.1012
#> 3 -3 492.0540
#> 4 -2 562.3994
#> 5 -1 574.0298
#> 6 0 580.8876
#> 7 1 575.0199
#> 8 2 553.7940
#> 9 3 520.7589
#> 10 4 487.7238
#> 11 5 454.6887
plot_logden <- plot_logdensity(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500)
plot_logden
Finally, to evaluate and visualize the density estimate itself, we use
evaluate_density and plot_density functions, respectively.
den_vals <- evaluate_density(
scorematching_logconcave = result,
newx = seq(domain[1], domain[2], length.out = 11)
)
den_vals
#> newx_sorted density_vals
#> 1 -5 5.718881e-253
#> 2 -4 1.871947e-130
#> 3 -3 2.842021e-39
#> 4 -2 1.009840e-08
#> 5 -1 1.135730e-03
#> 6 0 1.080366e+00
#> 7 1 3.056907e-03
#> 8 2 1.849261e-12
#> 9 3 8.318257e-27
#> 10 4 3.741679e-41
#> 11 5 1.683064e-55
plot_den <- plot_density(
scorematching_logconcave = result,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE
)
plot_den
Computing and Visualizing Penalized Log-concave Score Matching Density Estimate over a Bounded Domain
As we can see above, the un-penalized log-concave score matching density
estimate is too concentrated at the place where data are abundant. To
remedy this, we consider penalized log-concave score matching density
estimate, where the optimal penalty parameter can be chosen using the
cv_optimal_density_estimate function.
lambda_cand <- exp(seq(-5, 1, by = 0.5))
opt_den <- cv_optimal_density_estimate(
data = data,
domain = domain,
penalty_param_candidates = lambda_cand,
fold_number = 5
)
#> Penalty parameter value: 0.00673794699908547
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0111089965382423
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0183156388887342
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0301973834223185
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0497870683678639
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.0820849986238988
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.135335283236613
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.22313016014843
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.367879441171442
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 0.606530659712633
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 1
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 1.64872127070013
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Penalty parameter value: 2.71828182845905
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> The status of solving the constrained quadratic optimization problem is: optimal.
#> Optimal penality parameter is 0.22313016014843.
#> The status of solving the constrained quadratic optimization problem is: optimal.
The second and first derivatives of the logarithm of the optimal density estimate, the logarithm of the un-normalized optimal density estimate, and the optimal density estimate itself are plotted as below.
plot_ld2 <- plot_logdensity_deriv2(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_ld1 <- plot_logdensity_deriv1(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_ld <- plot_logdensity(
scorematching_logconcave = opt_den,
plot_domain = domain
)
plot_den <- plot_density(
scorematching_logconcave = opt_den,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE
)
plot_ld2 + plot_ld1 + plot_ld + plot_den
We can also use the function plot_mle_scorematching to view the
log-concave maximum likelihood and score matching density estimates
together with the histogram.
plot_mle_scorematching(
scorematching_logconcave = opt_den,
plot_domain = domain,
plot_points_cnt = 500,
plot_hist = TRUE)
We could assess the quality of this density estimate using various metrics between two probability distributions, for example, the Kullback-Leibler divergence, the Hyvarinen divergence, the L1 distance, and the Hellinger distance.
kl <- kl_div(
true_density = t_normal,
density_estimate = opt_den)
hyva <- hyvarinen_div(
true_density = t_normal,
density_estimate = opt_den)
l1 <- L1_dist(
true_density = t_normal,
density_estimate = opt_den)
he <- hellinger_dist(
true_density = t_normal,
density_estimate = opt_den)
metric_table <- cbind(
Metrics = c("Kullback-Leibler Divergence", "Hyvarinen divergence",
"L1 Distance", "Hellinger Distance"),
Values = round(c(kl, hyva, l1, he), 5)
)
knitr::kable(metric_table)
| Metrics | Values | |:----------------------------|:--------| | Kullback-Leibler Divergence | 0.09961 | | Hyvarinen divergence | 0.10237 | | L1 Distance | 0.29509 | | Hellinger Distance | 0.1774 |
Computing and Visualizing Penalized Log-concave Score Matching Density Estimate over Other Types of Domain
Besides the bounded domain demonstrated above, LogConcaveDESM can also
compute and visualize log-concave score matching density estimate over
other types of domain, including the entire real line, an interval of
the form (−∞,b) for some b \< ∞, and an interval of the form (a,∞)
for some a > − ∞. Functions used are the same as those used in the
bounded interval case. Demonstrations are omitted.
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