Package 'huge'

High-Dimensional Undirected Graph Estimation

Description

A package for high-dimensional undirected graph estimation

Details

Package:	huge
Type:	Package
Version:	1.2.7
Date:	2015-09-14
License:	GPL-2
LazyLoad:	yes

The package "huge" provides 8 main functions:
(1) the data generator creates random samples from multivariate normal distributions with different graph structures. Please refer to huge.generator.
(2) the nonparanormal (npn) transformation helps relax the normality assumption. Please refer to huge.npn.
(3) The correlation thresholding graph estimation. Please refer to huge.
(4) The Meinshausen-Buhlmann graph estimation. Please refer to huge.
(5) The graphical Lasso algorithm using lossless screening rule. Please refer and huge.

**Both (4) and (5) can be further accelerated by the lossy screening rule preselecting the neighborhood of each node via thresholding sample correlation.
(6) The model selection using the stability approach to regularization selection. Please refer to huge.select.
(7) The model selection using the rotation information criterion. Please refer to huge.select.
(8) The model selection using the extended Bayesian information criterion. Please refer to huge.select.

Author(s)

Tuo Zhao, Han Liu, Haoming Jiang, Kathryn Roeder, John Lafferty, and Larry Wasserman
Maintainers: Haoming Jiang<[email protected]>;

References

1. T. Zhao and H. Liu. The huge Package for High-dimensional Undirected Graph Estimation in R. Journal of Machine Learning Research, 2012
2. H. Liu, F. Han, M. Yuan, J. Lafferty and L. Wasserman. High Dimensional Semiparametric Gaussian Copula Graphical Models. Annals of Statistics,2012
3. D. Witten and J. Friedman. New insights and faster computations for the graphical lasso. Journal of Computational and Graphical Statistics, to appear, 2011. 4. Han Liu, Kathryn Roeder and Larry Wasserman. Stability Approach to Regularization Selection (StARS) for High Dimensional Graphical Models. Advances in Neural Information Processing Systems, 2010.
5. R. Foygel and M. Drton. Extended bayesian information criteria for gaussian graphical models. Advances in Neural Information Processing Systems, 2010.
6. H. Liu, J. Lafferty and L. Wasserman. The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs. Journal of Machine Learning Research, 2009
7. J. Fan and J. Lv. Sure independence screening for ultra-high dimensional feature space (with discussion). Journal of Royal Statistical Society B, 2008.
8. O. Banerjee, L. E. Ghaoui, A. d'Aspremont: Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data. Journal of Machine Learning Research, 2008.
9. J. Friedman, T. Hastie and R. Tibshirani. Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 2008.
10. J. Friedman, T. Hastie and R. Tibshirani. Sparse inverse covariance estimation with the lasso, Biostatistics, 2007.
11. N. Meinshausen and P. Buhlmann. High-dimensional Graphs and Variable Selection with the Lasso. The Annals of Statistics, 2006.

See Also

huge.generator, huge.npn, huge, huge.plot and huge.roc

---

High-dimensional undirected graph estimation

Description

The main function for high-dimensional undirected graph estimation. Three graph estimation methods, including (1) Meinshausen-Buhlmann graph estimation (mb) (2) graphical lasso (glasso) (3) correlation thresholding graph estimation (ct) and (4) tuning-insensitive graph estimation (tiger), are available for data analysis.

Usage

huge(
  x,
  lambda = NULL,
  nlambda = NULL,
  lambda.min.ratio = NULL,
  method = "mb",
  scr = NULL,
  scr.num = NULL,
  cov.output = FALSE,
  sym = "or",
  verbose = TRUE
)

Arguments

x	There are 2 options: (1) x is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. (n is the sample size and d is the dimension).
lambda	A sequence of decreasing positive numbers to control the regularization when method = "mb", "glasso" or "tiger", or the thresholding in method = "ct". Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. When method = "mb", "glasso" or "tiger", use with care - it is better to supply a decreasing sequence values than a single (small) value.
nlambda	The number of regularization/thresholding parameters. The default value is 30 for method = "ct" and 10 for method = "mb", "glasso" or "tiger".
lambda.min.ratio	If method = "mb", "glasso" or "tiger", it is the smallest value for lambda, as a fraction of the upperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to 0. The program can automatically generate lambda as a sequence of length = nlambda starting from MAX to lambda.min.ratio*MAX in log scale. If method = "ct", it is the largest sparsity level for estimated graphs. The program can automatically generate lambda as a sequence of length = nlambda, which makes the sparsity level of the graph path increases from 0 to lambda.min.ratio evenly.The default value is 0.1 when method = "mb", "glasso" or "tiger", and 0.05 method = "ct".
method	Graph estimation methods with 4 options: "mb", "ct", "glasso" and "tiger". The default value is "mb".
scr	If scr = TRUE, the lossy screening rule is applied to preselect the neighborhood before the graph estimation. The default value is FALSE. NOT applicable when method = "ct", "mb", or "tiger".
scr.num	The neighborhood size after the lossy screening rule (the number of remaining neighbors per node). ONLY applicable when scr = TRUE. The default value is n-1. An alternative value is n/log(n). ONLY applicable when scr = TRUE and method = "mb".
cov.output	If cov.output = TRUE, the output will include a path of estimated covariance matrices. ONLY applicable when method = "glasso". Since the estimated covariance matrices are generally not sparse, please use it with care, or it may take much memory under high-dimensional setting. The default value is FALSE.
sym	Symmetrize the output graphs. If sym = "and", the edge between node i and node j is selected ONLY when both node i and node j are selected as neighbors for each other. If sym = "or", the edge is selected when either node i or node j is selected as the neighbor for each other. The default value is "or". ONLY applicable when method = "mb" or "tiger".
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Details

The graph structure is estimated by Meinshausen-Buhlmann graph estimation or the graphical lasso, and both methods can be further accelerated via the lossy screening rule by preselecting the neighborhood of each variable by correlation thresholding. We target on high-dimensional data analysis usually d >> n, and the computation is memory-optimized using the sparse matrix output. We also provide a highly computationally efficient approaches correlation thresholding graph estimation.

Value

An object with S3 class "huge" is returned:

data	The n by d data matrix or d by d sample covariance matrix from the input
cov.input	An indicator of the sample covariance.
ind.mat	The scr.num by k matrix with each column corresponding to a variable in ind.group and contains the indices of the remaining neighbors after the GSS. ONLY applicable when scr = TRUE and approx = FALSE
lambda	The sequence of regularization parameters used in mb or thresholding parameters in ct.
sym	The sym from the input. ONLY applicable when method = "mb" or "tiger".
scr	The scr from the input. ONLY applicable when method = "mb" or "glasso".
path	A list of k by k adjacency matrices of estimated graphs as a graph path corresponding to lambda.
sparsity	The sparsity levels of the graph path.
icov	A list of d by d precision matrices as an alternative graph path (numerical path) corresponding to lambda. ONLY applicable when method = "glasso" or "tiger".
cov	A list of d by d estimated covariance matrices corresponding to lambda. ONLY applicable when cov.output = TRUE and method = "glasso"
method	The method used in the graph estimation stage.
df	If method = "mb" or "tiger", it is a k by nlambda matrix. Each row contains the number of nonzero coefficients along the lasso solution path. If method = "glasso", it is a nlambda dimensional vector containing the number of nonzero coefficients along the graph path icov.
loglik	A nlambda dimensional vector containing the likelihood scores along the graph path (icov). ONLY applicable when method = "glasso". For an estimated inverse covariance Z, the program only calculates log(det(Z)) - trace(SZ) where S is the empirical covariance matrix. For the likelihood for n observations, please multiply by n/2.

Note

This function ONLY estimates the graph path. For more information about the optimal graph selection, please refer to huge.select.

See Also

huge.generator, huge.select, huge.plot, huge.roc, and huge-package.

Examples

#generate data
L = huge.generator(n = 50, d = 12, graph = "hub", g = 4)

#graph path estimation using mb
out1 = huge(L$data)
out1
plot(out1)         #Not aligned
plot(out1, align = TRUE) #Aligned
huge.plot(out1$path[[3]])

#graph path estimation using the sample covariance matrix as the input.
#out1 = huge(cor(L$data), method = "glasso")
#out1
#plot(out1)         #Not aligned
#plot(out1, align = TRUE) #Aligned
#huge.plot(out1$path[[3]])

#graph path estimation using ct
#out2 = huge(L$data,method = "ct")
#out2
#plot(out2)

#graph path estimation using glasso
#out3 = huge(L$data, method = "glasso")
#out3
#plot(out3)

#graph path estimation using tiger
#out4 = huge(L$data, method = "tiger")
#out4
#plot(out4)

---

Graph estimation via correlation thresholding (ct)

Description

See more details in huge

Usage

huge.ct(
  x,
  nlambda = NULL,
  lambda.min.ratio = NULL,
  lambda = NULL,
  verbose = TRUE
)

Arguments

x	There are 2 options: (1) x is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. (n is the sample size and d is the dimension).
nlambda	The number of regularization/thresholding parameters. The default value is 30 for method = "ct" and 10 for method = "mb", "glasso" or "tiger".
lambda.min.ratio	If method = "mb", "glasso" or "tiger", it is the smallest value for lambda, as a fraction of the upperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to 0. The program can automatically generate lambda as a sequence of length = nlambda starting from MAX to lambda.min.ratio*MAX in log scale. If method = "ct", it is the largest sparsity level for estimated graphs. The program can automatically generate lambda as a sequence of length = nlambda, which makes the sparsity level of the graph path increases from 0 to lambda.min.ratio evenly.The default value is 0.1 when method = "mb", "glasso" or "tiger", and 0.05 method = "ct".
lambda	A sequence of decreasing positive numbers to control the regularization when method = "mb", "glasso" or "tiger", or the thresholding in method = "ct". Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. When method = "mb", "glasso" or "tiger", use with care - it is better to supply a decreasing sequence values than a single (small) value.
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

See Also

huge, and huge-package.

---

Data generator

Description

Implements the data generation from multivariate normal distributions with different graph structures, including "random", "hub", "cluster", "band" and "scale-free".

Usage

huge.generator(
  n = 200,
  d = 50,
  graph = "random",
  v = NULL,
  u = NULL,
  g = NULL,
  prob = NULL,
  vis = FALSE,
  verbose = TRUE
)

Arguments

n	The number of observations (sample size). The default value is 200.
d	The number of variables (dimension). The default value is 50.
graph	The graph structure with 4 options: "random", "hub", "cluster", "band" and "scale-free".
v	The off-diagonal elements of the precision matrix, controlling the magnitude of partial correlations with u. The default value is 0.3.
u	A positive number being added to the diagonal elements of the precision matrix, to control the magnitude of partial correlations. The default value is 0.1.
g	For "cluster" or "hub" graph, g is the number of hubs or clusters in the graph. The default value is about d/20 if d >= 40 and 2 if d < 40. For "band" graph, g is the bandwidth and the default value is 1. NOT applicable to "random" graph.
prob	For "random" graph, it is the probability that a pair of nodes has an edge. The default value is 3/d. For "cluster" graph, it is the probability that a pair of nodes has an edge in each cluster. The default value is 6*g/d if d/g <= 30 and 0.3 if d/g > 30. NOT applicable to "hub" or "band" graphs.
vis	Visualize the adjacency matrix of the true graph structure, the graph pattern, the covariance matrix and the empirical covariance matrix. The default value is FALSE
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Details

Given the adjacency matrix theta, the graph patterns are generated as below:

(I) "random": Each pair of off-diagonal elements are randomly set theta[i,j]=theta[j,i]=1 for i!=j with probability prob, and 0 other wise. It results in about d*(d-1)*prob/2 edges in the graph.

(II)"hub":The row/columns are evenly partitioned into g disjoint groups. Each group is associated with a "center" row i in that group. Each pair of off-diagonal elements are set theta[i,j]=theta[j,i]=1 for i!=j if j also belongs to the same group as i and 0 otherwise. It results in d - g edges in the graph.

(III)"cluster":The row/columns are evenly partitioned into g disjoint groups. Each pair of off-diagonal elements are set theta[i,j]=theta[j,i]=1 for i!=j with the probability probif both i and j belong to the same group, and 0 other wise. It results in about g*(d/g)*(d/g-1)*prob/2 edges in the graph.

(IV)"band": The off-diagonal elements are set to be theta[i,j]=1 if 1<=|i-j|<=g and 0 other wise. It results in (2d-1-g)*g/2 edges in the graph.

(V) "scale-free": The graph is generated using B-A algorithm. The initial graph has two connected nodes and each new node is connected to only one node in the existing graph with the probability proportional to the degree of the each node in the existing graph. It results in d edges in the graph.

The adjacency matrix theta has all diagonal elements equal to 0. To obtain a positive definite precision matrix, the smallest eigenvalue of theta*v (denoted by e) is computed. Then we set the precision matrix equal to theta*v+(|e|+0.1+u)I. The covariance matrix is then computed to generate multivariate normal data.

Value

An object with S3 class "sim" is returned:

data	The n by d matrix for the generated data
sigma	The covariance matrix for the generated data
omega	The precision matrix for the generated data
sigmahat	The empirical covariance matrix for the generated data
theta	The adjacency matrix of true graph structure (in sparse matrix representation) for the generated data

See Also

huge and huge-package

Examples

## band graph with bandwidth 3
L = huge.generator(graph = "band", g = 3)
plot(L)

## random sparse graph
L = huge.generator(vis = TRUE)

## random dense graph
L = huge.generator(prob = 0.5, vis = TRUE)

## hub graph with 6 hubs
L = huge.generator(graph = "hub", g = 6, vis = TRUE)

## hub graph with 8 clusters
L = huge.generator(graph = "cluster", g = 8, vis = TRUE)

## scale-free graphs
L = huge.generator(graph="scale-free", vis = TRUE)

---

The graphical lasso (glasso) using sparse matrix output

Description

See more details in huge

Usage

huge.glasso(
  x,
  lambda = NULL,
  lambda.min.ratio = NULL,
  nlambda = NULL,
  scr = NULL,
  cov.output = FALSE,
  verbose = TRUE
)

Arguments

x	There are 2 options: (1) x is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. (n is the sample size and d is the dimension).
lambda	A sequence of decreasing positive numbers to control the regularization when method = "mb", "glasso" or "tiger", or the thresholding in method = "ct". Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. When method = "mb", "glasso" or "tiger", use with care - it is better to supply a decreasing sequence values than a single (small) value.
lambda.min.ratio	If method = "mb", "glasso" or "tiger", it is the smallest value for lambda, as a fraction of the upperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to 0. The program can automatically generate lambda as a sequence of length = nlambda starting from MAX to lambda.min.ratio*MAX in log scale. If method = "ct", it is the largest sparsity level for estimated graphs. The program can automatically generate lambda as a sequence of length = nlambda, which makes the sparsity level of the graph path increases from 0 to lambda.min.ratio evenly.The default value is 0.1 when method = "mb", "glasso" or "tiger", and 0.05 method = "ct".
nlambda	The number of regularization/thresholding parameters. The default value is 30 for method = "ct" and 10 for method = "mb", "glasso" or "tiger".
scr	If scr = TRUE, the lossy screening rule is applied to preselect the neighborhood before the graph estimation. The default value is FALSE. NOT applicable when method = "ct", "mb", or "tiger".
cov.output	If cov.output = TRUE, the output will include a path of estimated covariance matrices. ONLY applicable when method = "glasso". Since the estimated covariance matrices are generally not sparse, please use it with care, or it may take much memory under high-dimensional setting. The default value is FALSE.
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

See Also

huge, and huge-package.

---

Graph inference

Description

Implements the inference for high dimensional graphical models, including Gaussian and Nonparanormal graphical models We consider the problems of testing the presence of a single edge and the hypothesis is that the edge is absent.

Usage

huge.inference(data, T, adj, alpha = 0.05, type = "Gaussian", method = "score")

Arguments

data	The input n by d data matrix(n is the sample size and d is the dimension).
T	The estimated inverse of correlation matrix of the data.
adj	The adjacency matrix corresponding to the graph.
alpha	The significance level of hypothesis.The default value is 0.05.
type	The type of input data. There are 2 options: "Gaussian" and "Nonparanormal". The default value is "Gaussian".
method	When using nonparanormal graphical model. Test method with 2 options: "score" and "wald". The default value is "score".

Details

For Nonparanormal graphical model we provide Score test method and Wald Test. However it is really slow for inferencing on Nonparanormal model, especially for large data.

Value

An object is returned:

data	The n by d data matrix from the input.
p	The d by d p-value matrix of hypothesis.
error	The type I error of hypothesis at alpha significance level.

References

1.Q Gu, Y Cao, Y Ning, H Liu. Local and global inference for high dimensional nonparanormal graphical models.
2.J Jankova, S Van De Geer. Confidence intervals for high-dimensional inverse covariance estimation. Electronic Journal of Statistics, 2015.

See Also

huge, and huge-package.

Examples

#generate data
L = huge.generator(n = 50, d = 12, graph = "hub", g = 4)

#graph path estimation using glasso
est = huge(L$data, method = "glasso")

#inference of Gaussian graphical model at 0.05 significance level
T = tail(est$icov, 1)[[1]]
out1 = huge.inference(L$data, T, L$theta)

#inference of Nonparanormal graphical model using score test at 0.05 significance level
T = tail(est$icov, 1)[[1]]
out2 = huge.inference(L$data, T, L$theta, type = "Nonparanormal")

#inference of Nonparanormal graphical model using wald test at 0.05 significance level
T = tail(est$icov, 1)[[1]]
out3 = huge.inference(L$data, T, L$theta, type = "Nonparanormal", method = "wald")

#inference of Nonparanormal graphical model using wald test at 0.1 significance level
T = tail(est$icov, 1)[[1]]
out4 = huge.inference(L$data, T, L$theta, 0.1, type = "Nonparanormal", method = "wald")

---

Meinshausen & Buhlmann graph estimation

Description

See more details in huge

Usage

huge.mb(
  x,
  lambda = NULL,
  nlambda = NULL,
  lambda.min.ratio = NULL,
  scr = NULL,
  scr.num = NULL,
  idx.mat = NULL,
  sym = "or",
  verbose = TRUE
)

Arguments

x	There are 2 options: (1) x is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. (n is the sample size and d is the dimension).
lambda	A sequence of decreasing positive numbers to control the regularization when method = "mb", "glasso" or "tiger", or the thresholding in method = "ct". Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. When method = "mb", "glasso" or "tiger", use with care - it is better to supply a decreasing sequence values than a single (small) value.
nlambda	The number of regularization/thresholding parameters. The default value is 30 for method = "ct" and 10 for method = "mb", "glasso" or "tiger".
lambda.min.ratio	If method = "mb", "glasso" or "tiger", it is the smallest value for lambda, as a fraction of the upperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to 0. The program can automatically generate lambda as a sequence of length = nlambda starting from MAX to lambda.min.ratio*MAX in log scale. If method = "ct", it is the largest sparsity level for estimated graphs. The program can automatically generate lambda as a sequence of length = nlambda, which makes the sparsity level of the graph path increases from 0 to lambda.min.ratio evenly.The default value is 0.1 when method = "mb", "glasso" or "tiger", and 0.05 method = "ct".
scr	If scr = TRUE, the lossy screening rule is applied to preselect the neighborhood before the graph estimation. The default value is FALSE. NOT applicable when method = "ct", "mb", or "tiger".
scr.num	The neighborhood size after the lossy screening rule (the number of remaining neighbors per node). ONLY applicable when scr = TRUE. The default value is n-1. An alternative value is n/log(n). ONLY applicable when scr = TRUE and method = "mb".
idx.mat	Index matrix for screening.
sym	Symmetrize the output graphs. If sym = "and", the edge between node i and node j is selected ONLY when both node i and node j are selected as neighbors for each other. If sym = "or", the edge is selected when either node i or node j is selected as the neighbor for each other. The default value is "or". ONLY applicable when method = "mb" or "tiger".
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

See Also

huge, and huge-package.

---

Nonparanormal(npn) transformation

Description

Implements the Gausianization to help relax the assumption of normality.

Usage

huge.npn(x, npn.func = "shrinkage", npn.thresh = NULL, verbose = TRUE)

Arguments

x	The n by d data matrix representing n observations in d dimensions
npn.func	The transformation function used in the npn transformation. If npn.func = "truncation", the truncated ECDF is applied. If npn.func = "shrinkage", the shrunken ECDF is applied. The default is "shrinkage". If npn.func = "skeptic", the nonparanormal skeptic is applied.
npn.thresh	The truncation threshold used in nonparanormal transformation, ONLY applicable when npn.func = "truncation". The default value is 1/(4*(n^0.25)* sqrt(pi*log(n))).
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Details

The nonparanormal extends Gaussian graphical models to semiparametric Gaussian copula models.Motivated by sparse additive models, the nonparanormal method estimates the Gaussian copula by marginally transforming the variables using smooth functions.Computationally, the estimation of a nonparanormal transformation is very efficient and only requires one pass of the data matrix.

Value

data	A d by d nonparanormal correlation matrix if npn.func = "skeptic", and A n by d data matrix representing n observations in d transformed dimensions other wise.

See Also

huge and huge-package.

Examples

# generate nonparanormal data
L = huge.generator(graph = "cluster", g = 5)
L$data = L$data^5

# transform the data using the shrunken ECDF
Q = huge.npn(L$data)

# transform the non-Gaussian data using the truncated ECDF
Q = huge.npn(L$data, npn.func = "truncation")

# transform the non-Gaussian data using the truncated ECDF
Q = huge.npn(L$data, npn.func = "skeptic")

---

Graph visualization

Description

Implements the graph visualization using adjacency matrix. It can automatic organize 2D embedding layout.

Usage

huge.plot(
  G,
  epsflag = FALSE,
  graph.name = "default",
  cur.num = 1,
  location = NULL
)

Arguments

G	The adjacency matrix corresponding to the graph.
epsflag	If epsflag = TRUE, save the plot as an eps file in the target directory. The default value is FALSE.
graph.name	The name of the output eps files. The default value is "default".
cur.num	The number of plots saved as eps files. Only applicale when epsflag = TRUE. The default value is 1.
location	Target directory. The default value is the current working directory.

Details

The user can change cur.num to plot several figures and select the best one. The implementation is based on the popular package "igraph".

See Also

huge and huge-package.

Examples

## visualize the hub graph
L = huge.generator(graph = "hub")
huge.plot(L$theta)

## visualize the band graph
L = huge.generator(graph = "band",g=5)
huge.plot(L$theta)

## visualize the cluster graph
L = huge.generator(graph = "cluster")
huge.plot(L$theta)

## plot 5 graphs and save the plots as eps files in the tempdir()
huge.plot(L$theta, epsflag = TRUE, cur.num = 5, location = tempdir())

---

Draw ROC Curve for a graph path

Description

Draws ROC curve for a graph path according to the true graph structure.

Usage

huge.roc(path, theta, verbose = TRUE)

Arguments

path	A graph path.
theta	The true graph structure.
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Details

To avoid the horizontal oscillation, false positive rates is automatically sorted in the ascent order and true positive rates also follow the same order.

Value

An object with S3 class "roc" is returned:

F1	The F1 scores along the graph path.
tp	The true positive rates along the graph path
fp	The false positive rates along the graph paths
AUC	Area under the ROC curve

Note

For a lasso regression, the number of nonzero coefficients is at most n-1. If d>>n, even when regularization parameter is very small, the estimated graph may still be sparse. In this case, the AUC may not be a good choice to evaluate the performance.

See Also

huge and huge-package.

Examples

#generate data
L = huge.generator(d = 200, graph = "cluster", prob = 0.3)
out1 = huge(L$data)

#draw ROC curve
Z1 = huge.roc(out1$path,L$theta)

#Maximum F1 score
max(Z1$F1)

---

Model selection for high-dimensional undirected graph estimation

Description

Implements the regularization parameter selection for high dimensional undirected graph estimation. The optional approaches are rotation information criterion (ric), stability approach to regularization selection (stars) and extended Bayesian information criterion (ebic).

Usage

huge.select(
  est,
  criterion = NULL,
  ebic.gamma = 0.5,
  stars.thresh = 0.1,
  stars.subsample.ratio = NULL,
  rep.num = 20,
  verbose = TRUE
)

Arguments

est	An object with S3 class "huge".
criterion	Model selection criterion. "ric" and "stars" are available for all 3 graph estimation methods. ebic is only applicable when est$method = "glasso" in huge(). The default value is "ric".
ebic.gamma	The tuning parameter for ebic. The default value is 0.5. Only applicable when est$method = "glasso" and criterion = "ebic".
stars.thresh	The variability threshold in stars. The default value is 0.1. An alternative value is 0.05. Only applicable when criterion = "stars".
stars.subsample.ratio	The subsampling ratio. The default value is 10*sqrt(n)/n when n>144 and 0.8 when n<=144, where n is the sample size. Only applicable when criterion = "stars".
rep.num	The number of subsamplings when criterion = "stars" or rotations when criterion = "ric". The default value is 20. NOT applicable when criterion = "ebic".
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Details

Stability approach to regularization selection (stars) is a natural way to select optimal regularization parameter for all three estimation methods. It selects the optimal graph by variability of subsamplings and tends to overselect edges in Gaussian graphical models. Besides selecting the regularization parameters, stars can also provide an additional estimated graph by merging the corresponding subsampled graphs using the frequency counts. The subsampling procedure in stars may NOT be very efficient, we also provide the recent developed highly efficient, rotation information criterion approach (ric). Instead of tuning over a grid by cross-validation or subsampling, we directly estimate the optimal regularization parameter based on random Rotations. However, ric usually has very good empirical performances but suffers from underselections sometimes. Therefore, we suggest if user are sensitive of false negative rates, they should either consider increasing r.num or applying the stars to model selection. Extended Bayesian information criterion (ebic) is another competitive approach, but the ebic.gamma can only be tuned by experience.

Value

An object with S3 class "select" is returned:

refit	The optimal graph selected from the graph path
opt.icov	The optimal precision matrix from the path only applicable when method = "glasso"
opt.cov	The optimal covariance matrix from the path only applicable when method = "glasso" and est$cov is available.
merge	The graph path estimated by merging the subsampling paths. Only applicable when the input criterion = "stars".
variability	The variability along the subsampling paths. Only applicable when the input criterion = "stars".
ebic.scores	Extended BIC scores for regularization parameter selection. Only applicable when criterion = "ebic".
opt.index	The index of the selected regularization parameter. NOT applicable when the input criterion = "ric"
opt.lambda	The selected regularization/thresholding parameter.
opt.sparsity	The sparsity level of "refit".

and anything else included in the input est

Note

The model selection is NOT available when the data input is the sample covariance matrix.

See Also

huge and huge-package.

Examples

#generate data
L = huge.generator(d = 20, graph="hub")
out.mb = huge(L$data)
out.ct = huge(L$data, method = "ct")
out.glasso = huge(L$data, method = "glasso")

#model selection using ric
out.select = huge.select(out.mb)
plot(out.select)

#model selection using stars
#out.select = huge.select(out.ct, criterion = "stars", stars.thresh = 0.05,rep.num=10)
#plot(out.select)

#model selection using ebic
out.select = huge.select(out.glasso,criterion = "ebic")
plot(out.select)

---

Tuning-insensitive graph estimation

Description

See more details in huge

Usage

huge.tiger(
  x,
  lambda = NULL,
  nlambda = NULL,
  lambda.min.ratio = NULL,
  sym = "or",
  verbose = TRUE
)

Arguments

x	There are 2 options: (1) x is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. (n is the sample size and d is the dimension).
lambda	A sequence of decreasing positive numbers to control the regularization when method = "mb", "glasso" or "tiger", or the thresholding in method = "ct". Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. When method = "mb", "glasso" or "tiger", use with care - it is better to supply a decreasing sequence values than a single (small) value.
nlambda	The number of regularization/thresholding parameters. The default value is 30 for method = "ct" and 10 for method = "mb", "glasso" or "tiger".
lambda.min.ratio	If method = "mb", "glasso" or "tiger", it is the smallest value for lambda, as a fraction of the upperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to 0. The program can automatically generate lambda as a sequence of length = nlambda starting from MAX to lambda.min.ratio*MAX in log scale. If method = "ct", it is the largest sparsity level for estimated graphs. The program can automatically generate lambda as a sequence of length = nlambda, which makes the sparsity level of the graph path increases from 0 to lambda.min.ratio evenly.The default value is 0.1 when method = "mb", "glasso" or "tiger", and 0.05 method = "ct".
sym	Symmetrize the output graphs. If sym = "and", the edge between node i and node j is selected ONLY when both node i and node j are selected as neighbors for each other. If sym = "or", the edge is selected when either node i or node j is selected as the neighbor for each other. The default value is "or". ONLY applicable when method = "mb" or "tiger".
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

See Also

huge, and huge-package.

---

Plot function for S3 class "huge"

Description

Plot sparsity level information and 3 typical sparse graphs from the graph path.

Usage

## S3 method for class 'huge'
plot(x, align = FALSE, ...)

Arguments

x	An object with S3 class "huge"
align	If align = FALSE, 3 plotted graphs are aligned
...	System reserved (No specific usage)

See Also

huge

---

Plot function for S3 class "roc"

Description

Plot the ROC curve for an object with S3 class "roc".

Usage

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

Arguments

x	An object with S3 class "roc"
...	System reserved (No specific usage)

See Also

huge.roc

---

Plot function for S3 class "select"

Description

Plot the optimal graph by model selection.

Usage

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

Arguments

x	An object with S3 class "select"
...	System reserved (No specific usage)

See Also

huge.select

---

Plot function for S3 class "sim"

Description

Visualize the covariance matrix, the empirical covariance matrix, the adjacency matrix and the graph pattern of the true graph structure.

Usage

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

Arguments

x	An object with S3 class "sim"
...	System reserved (No specific usage)

See Also

huge.generator and huge

---

Print function for S3 class "huge"

Description

Print the information about the model usage, the graph path length, graph dimension, sparsity level.

Usage

## S3 method for class 'huge'
print(x, ...)

Arguments

x	An object with S3 class "huge".
...	System reserved (No specific usage)

See Also

huge

---

Print function for S3 class "roc"

Description

Print the information about true positive rates, false positive rates, the area under curve and maximum F1 score.

Usage

## S3 method for class 'roc'
print(x, ...)

Arguments

x	An object with S3 class "roc".
...	System reserved (No specific usage)

See Also

huge.roc

---

Print function for S3 class "select"

Description

Print the information about the model usage, graph dimension, model selection criterion, sparsity level of the optimal graph.

Usage

## S3 method for class 'select'
print(x, ...)

Arguments

x	An object with S3 class "select".
...	System reserved (No specific usage)

See Also

huge.select

---

Print function for S3 class "sim"

Description

Print the information about the sample size, the dimension, the pattern and sparsity of the true graph structure.

Usage

## S3 method for class 'sim'
print(x, ...)

Arguments

x	An object with S3 class "sim".
...	System reserved (No specific usage)

See Also

huge.generator

---

Stock price of S&P 500 companies from 2003 to 2008

Description

This data set consists of stock price and company information.

Usage

data(stockdata)

Format

The format is a list containing contains two matrices. 1. data - 1258x452, represents the 452 stocks' close prices for 1258 trading days. 2. info - 452x3: The 1st column: the query symbol for each company. The 2nd column: the category for each company. The 3rd column: the full name of each company.

Details

This data set can be used to perform high-dimensional graph estimation to analyze the relationships between S&P 500 companies.

Source

It was publicly available at finance.yahoo, which is now out of date

Examples

data(stockdata)
image(stockdata$data)
stockdata$info

---