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())

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 and Seongil Jo

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))

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.

Usage

estimate(object, ...)

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

Arguments

Value

Author(s)

Seongil Jo

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

Usage

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

Arguments

Value

Author(s)

Seongil Jo

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)

Draw a Heat Map for Point Estimate of Covariance Matrix

Description

Provides a heat map for posterior mean estimate of sparse covariance matrix

Usage

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

Arguments

Value

Author(s)

Seongil Jo

See Also

estimate

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)
plot(est)

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)

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())

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 and Seongil Jo

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))

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, ...)

Arguments

Value

Note

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

Author(s)

Seongil Jo

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))))
summary(fout, cols = c(1, 3, 4), rows = c(1, 3, 4))
summary(fout, cols = 1, rows = 1:p)

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)