Kernel Matrix Using Guasian Kernel

Description

Many radial basis function kernels, such as the Gaussian kernel, map X into a infinte dimensional space. While the Gaussian kernel has a free parameter (bandwidth), it still follows a number of theoretical properties such as boundedness, consistence, universality, robustness etc. It is the most applicable kernel of the positive definite kernel based methods.

Usage

gkm(X)

Arguments

Details

Many radial basis function kernels, such as the Gaussian kernel, map input sapce into a infinite dimensional space. The Gaussian kernel has a a number of theoretical properties such as boundedness, consistence, universality and robustness, etc.

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md. Ashad Alam, Hui-Yi Lin, HOng-Wen Deng, Vince Calhour Yu-Ping Wang (2018), A kernel machine method for detecting higher order interactions in multimodal datasets: Application to schizophrenia, Journal of Neuroscience Methods, Vol. 309, 161-174.

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

Examples

##Dummy data:
X<-matrix(rnorm(1000),100)
gkm(X)

A helper function

Description

#An matrices dicomposition function

Usage

gm3edc(Amat, Bmat, Cmat)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

A helper function

Description

#An matrices dicomposition function

Usage

gmedc(A, B = diag(nrow(A)))

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

A helper function

Description

###An function to adjust

Usage

gmi(X, tol = sqrt(.Machine$double.eps))

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

Hampel's psi function

Description

##The ratio of the first derivative of the Hampel loss fuction to the argument. Tuning constants are fixed in different quintiles.

Usage

hadr(u)

Arguments

Value

a real value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

#See Also as gkm, hudr

A Hampel loss function

Description

#Tuning constants of the Hampel loss fuction are fixed in different quintiles of the arguments.

Usage

halfun(u)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See Also as hulfun, hadr, hudr

Objective function

Description

Objective function of Hampel's loss fucntion

Usage

halofun(x)

Arguments

Value

a real value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as hulofun

Huber's psi function

Description

The ratio of the first derivative of the Huber loss fuction to the argument. Tuning constants is fixed as a meadian vlue.

Usage

hudr(x)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as hadr

A Huber loss function

Description

Tuning constants of the Huber loss fuction are fixed in different quintiles of the arguments.

Usage

hulfun(x)

Arguments

Details

Tuning constants of the Huber fuction is fixed as a median.

Value

a real number

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as halfun

Objective function

Description

Objective function of Huber's loss fucntion

Usage

hulofun(x)

Arguments

Value

a real value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See Also as halofun, ~~~

Kernel Matrix Using Identity-by-state Kernel

Description

For GWASs, a kernel captures the pairwise similarity across a number of SNPs in each gene. Kernel projects the genotype data from original high dimensional space to a feature space. One of the more popular kernels used for genomics similarity is the identity-by-state (IBS) kernel (non- parametric function of the genotypes)

Usage

ibskm(Z)

Arguments

Details

For genome-wide association study, a kernel captures the pairwise similarity across a number of SNPs in each gene. Kernel projects the genotype data from original high dimensional space to a feature space. One popular kernel used for genomics similarity is the identity-by-state (IBS) kernel, The IBS kernel does not need any assumption on the type of genetic interactions.

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md. Ashad Alam, Hui-Yi Lin, HOng-Wen Deng, Vince Calhour Yu-Ping Wang (2018), A kernel machine method for detecting higher order interactions in multimodal datasets: Application to schizophrenia, Journal of Neuroscience Methods, Vol. 309, 161-174.

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as gkm, lkm

Examples

##Dummy data:
X <- matrix(rnorm(200),50)
ibskm(X)

Influence Funciton of Canonical Correlation Analysis

Description

##To define the robustness in statistics, different approaches have been pro- posed, for example, the minimax approach, the sensitivity curve, the influence function (IF) and the finite sample breakdown point. Due to its simplic- ity, the IF is the most useful approach in statistical machine learning

Usage

ifcca(X, Y, gamma = 1e-05, ncomps = 2, jth = 1)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as rkcca, ifrkcca

Examples

##Dummy data:

X <- matrix(rnorm(500),100); Y <- matrix(rnorm(500),100)

ifcca(X,Y,  1e-05,  2, 2)

Influence Function of Multiple Kernel Canonical Analysis

Description

## To define the robustness in statistics, different approaches have been pro- posed, for example, the minimax approach, the sensitivity curve, the influence function (IF) and the finite sample breakdown point. Due to its simplic- ity, the IF is the most useful approach in statistical machine learning.

Usage

ifmkcca(xx, yy, zz, kernel = "rbfdot", gamma = 1e-05, ncomps = 1, jth=1)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as ifcca

Examples

##Dummy data:

X <- matrix(rnorm(500),100); Y <- matrix(rnorm(500),100); Z <- matrix(rnorm(500),100)

ifmkcca(X,Y, Z, "rbfdot",  1e-05,  2, 1)

Influence Function of Robust Kernel Canonical Analysis

Description

##To define the robustness in statistics, different approaches have been pro- posed, for example, the minimax approach, the sensitivity curve, the influence function (IF) and the finite sample breakdown point. Due to its simplic- ity, the IF is the most useful approach in statistical machine learning.

Usage

ifrkcca(X, Y, lossfu = "Huber", kernel = "rbfdot", gamma = 0.00001, ncomps = 10, jth = 1)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as rkcca, ifrkcca

Examples

##Dummy data:

X <- matrix(rnorm(500),100); Y <- matrix(rnorm(500),100)

ifrkcca(X,Y, lossfu = "Huber", kernel = "rbfdot", gamma = 0.00001, ncomps = 10, jth = 2)

A helper function

Description

#A function ..............

Usage

lcv(X, Y, res)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

Kernel Matrix Using Linear Kernel

Description

The linear kernel is used by the underlying Euclidean space to define the similarity measure. Whenever the dimensionality is high, it may allow for more complexity in the function class than what we could measure and assess otherwise

Usage

lkm(X)

Arguments

Details

The linear kernel is used by the underlying Euclidean space to define the similarity measure. Whenever the dimensionality of the data is high, it may allow for more complexity in the function class than what we could measure and assess otherwise.

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md. Ashad Alam, Hui-Yi Lin, HOng-Wen Deng, Vince Calhour Yu-Ping Wang (2018), A kernel machine method for detecting higher order interactions in multimodal datasets: Application to schizophrenia, Journal of Neuroscience Methods, Vol. 309, 161-174.

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

Md Ashad Alam, Vince D. Calhoun and Yu-Ping Wang (2018), Identifying outliers using multiple kernel canonical correlation analysis with application to imaging genetics, Computational Statistics and Data Analysis, Vol. 125, 70- 85

See Also

See also as gkm, ibskm

Examples

##Dummy data:

X <- matrix(rnorm(500),100)
lkm(X)

Bandwidth of the Gaussian kernel

Description

A median of the pairwise distance of the data

Usage

mdbw(X)

Arguments

Details

While the Gaussian kernel has a free parameter (bandwidth), it still follows a number of theoretical properties such as boundedness, consistenc, universality, robustness, etc. It is the most applicable one. In a Gaussian RBF kernel, we need to select an appropriate a bandwidth. It is well known that the parameter has a strong influence on the result of kernel methods. For the Gaussian kernel, we can use the median of the pairwise distance as a bandwidth.

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md. Ashad Alam, Hui-Yi Lin, HOng-Wen Deng, Vince Calhour Yu-Ping Wang (2018), A kernel machine method for detecting higher order interactions in multimodal datasets: Application to schizophrenia, Journal of Neuroscience Methods, Vol. 309, 161-174.

Md. Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

Md. Ashad Alam and Kenji Fukumizu (2015), Higher-order regularized kernel canonical correlation analysis, International Journal of Pattern Recognition and Artificial Intelligence, Vol. 29(4) 1551005.

Arthu Gretton, Kenji. Fukumizu, C. H. Teo, L. Song, B. Scholkopf and A. Smola (2008), A Kernel statistical test of independence, in Advances in Neural Information Processing Systems, Vol. 20 585–592.

See Also

See also as lkm, gkm

Examples

##Dummy data:

X <- matrix(rnorm(1000),100)

mdbw(X)

A helper function

Description

# A function

Usage

medc(A, fn = sqrt)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

A helper function

Description

## A function

Usage

mvnod(n = 1, mu, Sigma, tol = 1e-06, empirical = FALSE, EISPACK = FALSE)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

A helper function

Description

A function

Usage

ranuf(p)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

Robust kernel canonical correlation analysis

Description

#A robust correlation

Usage

rkcca(X, Y, lossfu = "Huber", kernel = "rbfdot", gamma = 1e-05, ncomps = 10)

Arguments

Value

An S3 object containing the following slots:

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as ifcca, rkcca, ifrkcca

Examples

##Dummy data:

X <- matrix(rnorm(1000),100); Y <- matrix(rnorm(1000),100)

rkcca(X,Y, "Huber",  "rbfdot", 1e-05,  10)

Robust kernel cross-covariance opetator

Description

# A function

Usage

rkcco(X, Y, lossfu = "Huber", kernel = "rbfdot", gamma = 1e-05)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

M. Romanazzi (1992), Influence in canonical correlation analysis, Psychometrika vol 57(2) (1992) 237-259.

See Also

See also as rkcca snpfmridata, ifrkcca

Examples

##Dummy data:

X <- matrix(rnorm(2000),200); Y <- matrix(rnorm(2000),200)

rkcco(X,Y, "Huber","rbfdot", 1e-05)

Robsut Kernel Center Matrix

Description

# A functioin

Usage

rkcm(X, lossfu = "Huber", kernel = "rbfdot")

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

Md Ashad Alam, Vince D. Calhoun and Yu-Ping Wang (2018), Identifying outliers using multiple kernel canonical correlation analysis with application to imaging genetics, Computational Statistics and Data Analysis, Vol. 125, 70- 85

See Also

See also as ifcca, rkcca, ifrkcca

Examples

##Dummy data:

X <- matrix(rnorm(2000),200); Y <- matrix(rnorm(2000),200)

rkcm(X, "Huber","rbfdot")

A helper fuction

Description

#A function to calcualte generalized logit function.

Usage

rlogit(x)

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>

An example of imaging genetics data to calcualte influential observations from two view data

Description

#A function

Usage

snpfmridata(n = 300, gamma=0.00001, ncomps = 2, jth = 1)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

Md Ashad Alam, Vince D. Calhoun and Yu-Ping Wang (2018), Identifying outliers using multiple kernel canonical correlation analysis with application to imaging genetics, Computational Statistics and Data Analysis, Vol. 125, 70- 85

See Also

See also as rkcca, ifrkcca, snpfmrimth3D

Examples

##Dummy data:

n<-100

snpfmridata(n, 0.00001,  10, jth = 1)

An example of imaging genetics and epi-genetics data to calcualte influential observations from three view data

Description

#A function

Usage

snpfmrimth3D(n = 500, gamma = 1e-05, ncomps = 1, jth=1)

Arguments

Value

Author(s)

Md Ashad Alam <malam@tulane.edu>

References

Md Ashad Alam, Kenji Fukumizu and Yu-Ping Wang (2018), Influence Function and Robust Variant of Kernel Canonical Correlation Analysis, Neurocomputing, Vol. 304 (2018) 12-29.

Md Ashad Alam, Vince D. Calhoun and Yu-Ping Wang (2018), Identifying outliers using multiple kernel canonical correlation analysis with application to imaging genetics, Computational Statistics and Data Analysis, Vol. 125, 70- 85

See Also

See also as rkcca, snpfmridata, ifrkcca

Examples

##Dummy data:

n<-100

snpfmrimth3D(n, 0.00001,  10, 1)

A helper function

Description

### A function to a measure of a system's real point computing power

Usage

 udtd(x)
 

Arguments

Author(s)

Md Ashad Alam <malam@tulane.edu>