SP500 dataset

Description

The S&P 500 dataset from State Street Global Advisors with the collection period from Jan 2013 to Nov 2023.

Format

list

Source

State Street Global Advisors.

Examples

data("SP500")
names(SP500)

Bayesian Estimation of a Banded Covariance Matrix

Description

Provides a post-processed posterior for Bayesian inference of a banded covariance matrix.

Usage

bandPPP(X, k, eps, prior = list(), nsample = 2000)

Arguments

Details

Lee, Lee, and Lee (2023+) proposed a two-step procedure generating samples from the post-processed posterior for Bayesian inference of a banded covariance matrix:

• Initial posterior computing step: Generate random samples from the following initial posterior obtained by using the inverse-Wishart prior IW_p(B_0, \nu_0)

  \Sigma \mid X_1, \ldots, X_n \sim IW_p(B_0 + nS_n, \nu_0 + n),

  where S_n = n^{-1}\sum_{i=1}^{n}X_iX_i^\top.

• Post-processing step: Post-process the samples generated from the initial samples

  \Sigma_{(i)} := \left\{\begin{array}{ll}B_{k}(\Sigma^{(i)}) + \left[\epsilon_n - \lambda_{\min}\{B_{k}(\Sigma^{(i)})\}\right]I_p, & \mbox{ if } \lambda_{\min}\{B_{k}(\Sigma^{(i)})\} < \epsilon_n, \\ B_{k}(\Sigma^{(i)}), & \mbox{ otherwise }, \end{array}\right.

where \Sigma^{(1)}, \ldots, \Sigma^{(N)} are the initial posterior samples, \epsilon_n is a small positive number decreasing to 0 as n \rightarrow \infty, and B_k(B) denotes the k-band operation given as

B_{k}(B) = \{b_{ij}I(|i - j| \le k)\} \mbox{ for any } B = (b_{ij}) \in R^{p \times p}.

For more details, see Lee, Lee and Lee (2023+).

Value

Author(s)

Kwangmin Lee

References

Lee, K., Lee, K., and Lee, J. (2023+), "Post-processes posteriors for banded covariances", Bayesian Analysis, DOI: 10.1214/22-BA1333.

See Also

cv.bandPPP estimate

Examples

n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X,2,0.01,nsample=100)

Bayesian Sparse Covariance Estimation

Description

Provides a Bayesian sparse and positive definite estimate of a covariance matrix via the beta-mixture shrinkage prior.

Usage

bmspcov(
  X,
  Sigma,
  prior = list(),
  nsample = list(),
  nchain = 1,
  seed = NULL,
  do.parallel = FALSE,
  show_progress = TRUE
)

Arguments

Details

Lee, Jo and Lee (2022) proposed the beta-mixture shrinkage prior for estimating a sparse and positive definite covariance matrix. The beta-mixture shrinkage prior for \Sigma = (\sigma_{jk}) is defined as

\pi(\Sigma) = \frac{\pi^{u}(\Sigma)I(\Sigma \in C_p)}{\pi^{u}(\Sigma \in C_p)}, ~ C_p = \{\mbox{ all } p \times p \mbox{ positive definite matrices }\},

where \pi^{u}(\cdot) is the unconstrained prior given by

\pi^{u}(\sigma_{jk} \mid \rho_{jk}) = N\left(\sigma_{jk} \mid 0, \frac{\rho_{jk}}{1 - \rho_{jk}}\tau_1^2\right)

\pi^{u}(\rho_{jk}) = Beta(\rho_{jk} \mid a, b), ~ \rho_{jk} = 1 - 1/(1 + \phi_{jk})

\pi^{u}(\sigma_{jj}) = Exp(\sigma_{jj} \mid \lambda).

For more details, see Lee, Jo and Lee (2022).

Value

Author(s)

Kyoungjae Lee, Seongil Jo, and Kyeongwon Lee

References

Lee, K., Jo, S., and Lee, J. (2022), "The beta-mixture shrinkage prior for sparse covariances with near-minimax posterior convergence rate", Journal of Multivariate Analysis, 192, 105067.

See Also

sbmspcov

Examples

set.seed(1)
n <- 20
p <- 5

# generate a sparse covariance matrix:
True.Sigma <- matrix(0, nrow = p, ncol = p)
diag(True.Sigma) <- 1
Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8)
nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1)
zeroIND = (1:(p*(p-1)/2))[-nonzeroIND]
Values[zeroIND] <- 0
True.Sigma[lower.tri(True.Sigma)] <- Values
True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)]
if(min(eigen(True.Sigma)$values) <= 0){
  delta <- -min(eigen(True.Sigma)$values) + 1.0e-5
  True.Sigma <- True.Sigma + delta*diag(p)
}

# generate a data
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma)

# compute sparse, positive covariance estimator:
fout <- bspcov::bmspcov(X = X, Sigma = diag(diag(cov(X))))
post.est.m <- bspcov::estimate(fout)
sqrt(mean((post.est.m - True.Sigma)^2))
sqrt(mean((cov(X) - True.Sigma)^2))

# Multiple chains example with parallel processing:
# fout_multi <- bspcov::bmspcov(X = X, Sigma = diag(diag(cov(X))), 
#                               nchain = 4, do.parallel = TRUE)
# post.est.multi <- bspcov::estimate(fout_multi)
# When do.parallel = TRUE, the function automatically sets up 
# a multisession plan with nchain workers for parallel execution.

colon dataset

Description

The colon cancer dataset, which includes gene expression values from 22 colon tumor tissues and 40 non-tumor tissues.

Format

data.frame

Source

http://genomics-pubs.princeton.edu/oncology/affydata/.

Examples

data("colon")
head(colon)

CV for Bayesian Estimation of a Banded Covariance Matrix

Description

Performs leave-one-out cross-validation (LOOCV) to calculate the predictive log-likelihood for a post-processed posterior of a banded covariance matrix and selects the optimal parameters.

Usage

cv.bandPPP(X, kvec, epsvec, prior = list(), nsample = 2000, ncores = 2)

Arguments

Details

The predictive log-likelihood for each k and \epsilon_n is estimated as follows:

\sum_{i=1}^n \log S^{-1} \sum_{s=1}^S p(X_i \mid B_k^{(\epsilon_n)}(\Sigma_{i,s})),

where X_i is the ith observation, \Sigma_{i,s} is the sth posterior sample based on (X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n), and B_k^{(\epsilon_n)} represents the banding post-processing function. For more details, see (3) in Lee, Lee and Lee (2023+).

Value

Author(s)

Kwangmin Lee

References

Lee, K., Lee, K., and Lee, J. (2023+), "Post-processes posteriors for banded covariances", Bayesian Analysis, DOI: 10.1214/22-BA1333.

Gelman, A., Hwang, J., and Vehtari, A. (2014). "Understanding predictive information criteria for Bayesian models." Statistics and computing, 24(6), 997-1016.

See Also

bandPPP

Examples

Sigma0 <- diag(1,50)
X <- mvtnorm::rmvnorm(25,sigma = Sigma0)
kvec <- 1:2
epsvec <- c(0.01,0.05)
res <- bspcov::cv.bandPPP(X,kvec,epsvec,nsample=10,ncores=4)
plot(res)

CV for Bayesian Estimation of a Sparse Covariance Matrix

Description

Performs cross-validation to estimate spectral norm error for a post-processed posterior of a sparse covariance matrix.

Usage

cv.thresPPP(
  X,
  thresvec,
  epsvec,
  prior = NULL,
  thresfun = "hard",
  nsample = 2000,
  ncores = 2
)

Arguments

Details

Given a set of train data and validation data, the spectral norm error for each \gamma and \epsilon_n is estimated as follows:

||\hat{\Sigma}(\gamma,\epsilon_n)^{(train)} - S^{(val)}||_2

where \hat{\Sigma}(\gamma,\epsilon_n)^{(train)} is the estimate for the covariance based on the train data and S^{(val)} is the sample covariance matrix based on the validation data. The spectral norm error is estimated by the 10-fold cross-validation. For more details, see the first paragraph on page 9 in Lee and Lee (2023).

Value

Author(s)

Kwangmin Lee

References

Lee, K. and Lee, J. (2023), "Post-processes posteriors for sparse covariances", Journal of Econometrics, 236(3), 105475.

See Also

thresPPP

Examples

Sigma0 <- diag(1,50)
X <- mvtnorm::rmvnorm(25,sigma = Sigma0)
thresvec <- c(0.01,0.1)
epsvec <- c(0.01,0.1)
res <- bspcov::cv.thresPPP(X,thresvec,epsvec,nsample=100)
plot(res)

Point-estimate of posterior distribution

Description

Compute the point estimate (mean) to describe posterior distribution. For multiple chains, combines all chains to compute a more robust estimate.

Usage

estimate(object, ...)

## S3 method for class 'bspcov'
estimate(object, ...)

Arguments

Value

Author(s)

Seongil Jo and Kyeongwon Lee

See Also

plot.postmean.bspcov

Examples

n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X,2,0.01,nsample=100)
est <- bspcov::estimate(res)

Plot Diagnostics of Posterior Samples and Cross-Validation

Description

Provides a trace plot of posterior samples and a plot of a learning curve for cross-validation. For multiple chains, each chain is displayed in a different color for convergence assessment.

Usage

## S3 method for class 'bspcov'
plot(x, ..., cols, rows)

Arguments

Value

Author(s)

Seongil Jo and Kyeongwon Lee

Examples

set.seed(1)
n <- 100
p <- 20

# generate a sparse covariance matrix:
True.Sigma <- matrix(0, nrow = p, ncol = p)
diag(True.Sigma) <- 1
Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8)
nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1)
zeroIND = (1:(p*(p-1)/2))[-nonzeroIND]
Values[zeroIND] <- 0
True.Sigma[lower.tri(True.Sigma)] <- Values
True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)]
if(min(eigen(True.Sigma)$values) <= 0){
  delta <- -min(eigen(True.Sigma)$values) + 1.0e-5
  True.Sigma <- True.Sigma + delta*diag(p)
}

# generate a data
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma)

# compute sparse, positive covariance estimator:
fout <- bspcov::sbmspcov(X = X, Sigma = diag(diag(cov(X))))
plot(fout, cols = c(1, 3, 4), rows = c(1, 3, 4))
plot(fout, cols = 1, rows = 1:3)

# Cross-Validation for Banded Structure
Sigma0 <- diag(1,50)
X <- mvtnorm::rmvnorm(25,sigma = Sigma0)
kvec <- 1:2
epsvec <- c(0.01,0.05)
res <- bspcov::cv.bandPPP(X,kvec,epsvec,nsample=10,ncores=4)
plot(res)

Plot method for postmean.bspcov objects

Description

Create heatmap visualization for posterior mean estimate of sparse covariance matrix. Provides flexible visualization options with customizable aesthetics and labeling.

Usage

## S3 method for class 'postmean.bspcov'
plot(
  x,
  title = NULL,
  color_limits = NULL,
  color_low = "black",
  color_high = "white",
  base_size = 14,
  legend_position = "bottom",
  x_label = "Variable",
  y_label = "Variable",
  show_values = FALSE,
  ...
)

Arguments

Value

A ggplot object showing heatmap visualization of the posterior mean covariance matrix.

Author(s)

Seongil Jo, Kyeongwon Lee

See Also

estimate, plot.bspcov, plot.quantile.bspcov

Examples

# Example with simulated data
n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::thresPPP(X, eps=0.01, thres=list(value=0.5,fun='hard'), nsample=100)
est <- bspcov::estimate(res)

# Basic plot
plot(est)

# Plot with custom color scheme
plot(est, color_low = "blue", color_high = "red")

Plot method for quantile.bspcov objects

Description

Create visualization plots for posterior quantiles of covariance matrices. Produces heatmap visualizations showing uncertainty across different quantile levels.

Usage

## S3 method for class 'quantile.bspcov'
plot(
  x,
  type = "heatmap",
  titles = NULL,
  ncol = 3,
  color_limits = NULL,
  color_low = "black",
  color_high = "white",
  base_size = 14,
  legend_position = "bottom",
  x_label = "Variable",
  y_label = "Variable",
  width = NULL,
  height = 6,
  ...
)

Arguments

Value

A ggplot object (single quantile) or patchwork object (multiple quantiles) showing heatmap visualizations.

Author(s)

Kyeongwon Lee

See Also

quantile, plot.bspcov, plot.postmean.bspcov

Examples

# Example with simulated data
n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X, 2, 0.01, nsample = 100)
quant <- quantile(res)

# Plot uncertainty visualization
plot(quant)

# Plot with custom titles and labels
plot(quant, titles = c("Lower Bound", "Median", "Upper Bound"),
     x_label = "Variable", y_label = "Variable")

Preprocess SP500 data

Description

The proc_SP500 function preprocesses the SP500 stock data by calculating monthly returns for selected sectors and generating idiosyncratic errors.

Usage

proc_SP500(SP500data, sectors)

Arguments

Details

1. Calculates monthly returns for each stock in the specified sectors

2. Estimates the number of factors via hdbinseg::get.factor.model(ic="ah")

3. Uses POET::POET() to extract factor loadings/factors and form idiosyncratic errors

Value

A list with components:

Examples

data("SP500")
set.seed(1234)
sectors <- c("Energy", "Financials", "Materials",
   "Real Estate", "Utilities", "Information Technology")

Uhat <- proc_SP500(SP500, sectors)$Uhat
PPPres <- thresPPP(Uhat, eps = 0, thres = list(value = 0.0020, fun = "hard"), nsample = 100)
postmean <- estimate(PPPres)
diag(postmean) <- 0 # hide color for diagonal
plot(postmean)

Preprocess Colon Gene Expression Data

Description

The proc_colon function preprocesses colon gene expression data by:

1. Log transforming the raw counts.

2. Performing two-sample t-tests for each gene between normal and tumor samples.

3. Selecting the top 50 genes by absolute t-statistic.

4. Returning the filtered expression matrix and sample indices/groups.

Usage

proc_colon(colon, tissues)

Arguments

Value

A list with components:

X

A numeric matrix (samples x 50 genes) of selected, log‐transformed expression values.

normal_idx

Integer indices of normal‐tissue columns in the original data.

tumor_idx

Integer indices of tumor‐tissue columns in the original data.

group

Integer vector of length ncol(colon), with 1 = normal, 2 = tumor.

Examples

data("colon")
data("tissues")
set.seed(1234)
colon_data <- proc_colon(colon, tissues)
X <- colon_data$X

foo <- bmspcov(X, Sigma = cov(X))
sigmah <- estimate(foo)

Quantiles of posterior distribution

Description

Compute quantiles to describe posterior distribution. For multiple chains, combines all chains to compute more robust quantiles.

Usage

## S3 method for class 'bspcov'
quantile(x, probs = c(0.025, 0.5, 0.975), ...)

Arguments

Value

Author(s)

Kyeongwon Lee

See Also

estimate, plot.postmean.bspcov

Examples

# Example with simulated data
n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X, 2, 0.01, nsample=100)
quant <- quantile(res)

# Get 95% credible intervals
quant <- quantile(res, probs = c(0.025, 0.975))

Save quantile plot to file

Description

Convenience function to save quantile plots with appropriate dimensions.

Usage

save_quantile_plot(x, filename, width = NULL, height = 6, ...)

Arguments

Examples

# Example with simulated data
n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X, 2, 0.01, nsample = 100)
quant <- quantile(res)

Bayesian Sparse Covariance Estimation using Sure Screening

Description

Provides a Bayesian sparse and positive definite estimate of a covariance matrix via screened beta-mixture prior.

Usage

sbmspcov(
  X,
  Sigma,
  cutoff = list(),
  prior = list(),
  nsample = list(),
  nchain = 1,
  seed = NULL,
  do.parallel = FALSE,
  show_progress = TRUE
)

Arguments

Details

Lee, Jo, Lee, and Lee (2023+) proposed the screened beta-mixture shrinkage prior for estimating a sparse and positive definite covariance matrix. The screened beta-mixture shrinkage prior for \Sigma = (\sigma_{jk}) is defined as

\pi(\Sigma) = \frac{\pi^{u}(\Sigma)I(\Sigma \in C_p)}{\pi^{u}(\Sigma \in C_p)}, ~ C_p = \{\mbox{ all } p \times p \mbox{ positive definite matrices }\},

where \pi^{u}(\cdot) is the unconstrained prior given by

\pi^{u}(\sigma_{jk} \mid \psi_{jk}) = N\left(\sigma_{jk} \mid 0, \frac{\psi_{jk}}{1 - \psi_{jk}}\tau_1^2\right), ~ \psi_{jk} = 1 - 1/(1 + \phi_{jk})

\pi^{u}(\psi_{jk}) = Beta(\psi_{jk} \mid a, b) ~ \mbox{for } (j, k) \in S_r(\hat{R})

\pi^{u}(\sigma_{jj}) = Exp(\sigma_{jj} \mid \lambda),

where S_r(\hat{R}) = \{(j,k) : 1 < j < k \le p, |\hat{\rho}_{jk}| > r\}, \hat{\rho}_{jk} are sample correlations, and r is a threshold given by user.

For more details, see Lee, Jo, Lee and Lee (2022+).

Value

Author(s)

Kyoungjae Lee, Seongil Jo, and Kyeongwon Lee

References

Lee, K., Jo, S., Lee, K., and Lee, J. (2023+), "Scalable and optimal Bayesian inference for sparse covariance matrices via screened beta-mixture prior", arXiv:2206.12773.

See Also

bmspcov

Examples

set.seed(1)
n <- 20
p <- 5

# generate a sparse covariance matrix:
True.Sigma <- matrix(0, nrow = p, ncol = p)
diag(True.Sigma) <- 1
Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8)
nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1)
zeroIND = (1:(p*(p-1)/2))[-nonzeroIND]
Values[zeroIND] <- 0
True.Sigma[lower.tri(True.Sigma)] <- Values
True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)]
if(min(eigen(True.Sigma)$values) <= 0){
  delta <- -min(eigen(True.Sigma)$values) + 1.0e-5
  True.Sigma <- True.Sigma + delta*diag(p)
}

# generate a data
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma)

# compute sparse, positive covariance estimator:
fout <- bspcov::sbmspcov(X = X, Sigma = diag(diag(cov(X))))
post.est.m <- bspcov::estimate(fout)
sqrt(mean((post.est.m - True.Sigma)^2))
sqrt(mean((cov(X) - True.Sigma)^2))

# Multiple chains example with parallel processing:
# fout_multi <- bspcov::sbmspcov(X = X, Sigma = diag(diag(cov(X))), 
#                                nchain = 4, seed = 123, do.parallel = TRUE)
# post.est.multi <- bspcov::estimate(fout_multi)
# summary(fout_multi, cols = 1, rows = 1:3)  # Shows MCMC diagnostics
# plot(fout_multi, cols = 1, rows = 1:3)     # Shows colored trace plots
# When do.parallel = TRUE, the function automatically sets up 
# a multisession plan with nchain workers for parallel execution.

Summary of Posterior Distribution

Description

Provides the summary statistics for posterior samples of covariance matrix.

Usage

## S3 method for class 'bspcov'
summary(object, cols, rows, quantiles = c(0.025, 0.25, 0.5, 0.75, 0.975), ...)

Arguments

Value

Note

If both cols and rows are vectors, they must have the same length.

Author(s)

Seongil Jo and Kyeongwon Lee

Examples

# Example with simulated data
set.seed(1)
n <- 20
p <- 5

# generate a sparse covariance matrix:
True.Sigma <- matrix(0, nrow = p, ncol = p)
diag(True.Sigma) <- 1
Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8)
nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1)
zeroIND = (1:(p*(p-1)/2))[-nonzeroIND]
Values[zeroIND] <- 0
True.Sigma[lower.tri(True.Sigma)] <- Values
True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)]
if(min(eigen(True.Sigma)$values) <= 0){
  delta <- -min(eigen(True.Sigma)$values) + 1.0e-5
  True.Sigma <- True.Sigma + delta*diag(p)
}

# generate a data
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma)

# compute sparse, positive covariance estimator:
fout <- bspcov::sbmspcov(X = X, Sigma = diag(diag(cov(X))))
summary(fout, cols = c(1, 3, 4), rows = c(1, 3, 4))
summary(fout, cols = 1, rows = 1:p)

# Custom quantiles:
summary(fout, cols = 1, rows = 1:3, quantiles = c(0.05, 0.5, 0.95))

Bayesian Estimation of a Sparse Covariance Matrix

Description

Provides a post-processed posterior (PPP) for Bayesian inference of a sparse covariance matrix.

Usage

thresPPP(X, eps, thres = list(), prior = list(), nsample = 2000)

Arguments

Details

Lee and Lee (2023) proposed a two-step procedure generating samples from the post-processed posterior for Bayesian inference of a sparse covariance matrix:

• Initial posterior computing step: Generate random samples from the following initial posterior obtained by using the inverse-Wishart prior IW_p(B_0, \nu_0)

  \Sigma \mid X_1, \ldots, X_n \sim IW_p(B_0 + nS_n, \nu_0 + n),

  where S_n = n^{-1}\sum_{i=1}^{n}X_iX_i^\top.

• Post-processing step: Post-process the samples generated from the initial samples

  \Sigma_{(i)} := \left\{\begin{array}{ll}H_{\gamma_n}(\Sigma^{(i)}) + \left[\epsilon_n - \lambda_{\min}\{H_{\gamma_n}(\Sigma^{(i)})\}\right]I_p, & \mbox{ if } \lambda_{\min}\{H_{\gamma_n}(\Sigma^{(i)})\} < \epsilon_n, \\ H_{\gamma_n}(\Sigma^{(i)}), & \mbox{ otherwise }, \end{array}\right.

where \Sigma^{(1)}, \ldots, \Sigma^{(N)} are the initial posterior samples, \epsilon_n is a positive real number, and H_{\gamma_n}(\Sigma) denotes the generalized threshodling operator given as

(H_{\gamma_n}(\Sigma))_{ij} = \left\{\begin{array}{ll}\sigma_{ij}, & \mbox{ if } i = j, \\ h_{\gamma_n}(\sigma_{ij}), & \mbox{ if } i \neq j, \end{array}\right.

where \sigma_{ij} is the (i,j) element of \Sigma and h_{\gamma_n}(\cdot) is a generalized thresholding function.

For more details, see Lee and Lee (2023).

Value

Author(s)

Kwangmin Lee

References

Lee, K. and Lee, J. (2023), "Post-processes posteriors for sparse covariances", Journal of Econometrics.

See Also

cv.thresPPP

Examples

n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::thresPPP(X, eps=0.01, thres=list(value=0.5,fun='hard'), nsample=100)
est <- bspcov::estimate(res)

tissues dataset

Description

The tissues data of colon cancer dataset, which includes gene expression values from 22 colon tumor tissues and 40 non-tumor tissues.

Format

numeric

Source

http://genomics-pubs.princeton.edu/oncology/affydata/.

Examples

data("tissues")
head(tissues)