TCA: Transelliptical Component Analysis

In smart: Sparse Multivariate Analysis via Rank Transformation

Description

A function to conduct Transelliptical Component Analysis

Usage

TCA(x, K, para, method = "kendall", algorithm = "tp", max.iter = 200,
    verbose = TRUE, eps.conv = 0.001)

Arguments

x	The n by d data matrix or d by d covariance matrix from the input
K	Number of components
para	A vector of length K, indicating the number of sparse loadings.
method	Method to be used to estimating the correlation matrix with 5 options: pearson, ns, npn, spearman and kendall. kendall as default.
algorithm	Algorithm to be used to obtain sparse loadings with 3 options: sp, spca and pmd. tp as default.
max.iter	Maximum number of iterations.
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.
eps.conv	Convergence criterion.

Details

PCA and Sparse PCA is sensitive to modeling assumption, outliers, missing values data dependency. We propose an alternative way using rank-based methods including ns, npn, spearman and kendall to approximate the correlation matrix. Details are refered to Han,F. and Liu,H. (2012). Three sparse PCA algorithms are used: truncated power (Yuan, X. and Zhang, T. (2011)), spca(Zou,H., Hastie, T., and Tibshirani, R. (2006)) and pmd (Witten, D., Tibshirani, R., and Hastie, T. (2009)).

Value

cov.input	An indicator of the sample covariance.
loadings	The loadings of the sparse PCs.
pev	An indicator of the sample covariance.
PC	Projected PCs. existing if cov.input=TRUE.
method	The method used in estimating the correlation matrix.
algorithm	The algorithm used in obtaining the sparse loadings.
K	The number of components.

Author(s)

Fang Han, Han Liu
Maintainer: Fang Han<[e-mail]>

References

1. Witten, D., Tibshirani, R., and Hastie, T., A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics
2. Yuan, X. and Zhang, T. (2011). Truncated power method for sparse eigenvalue problems. Techinal Report, Rutgers, 2011. 3. Zou, H., Hastie, T. and Tibshirani, R. Sparse principal component analysis. JCGS, 2006.

Examples

x=matrix(rnorm(20000),100)
fit=TCA(x,K=6, para=c(10,10,10,5,5,5))
fit
plot(fit)

Example output

Conducting nonparanormal (npn) transformation via kendall....done.
Conducting TP....123456789101112
1234567
12345678910111213
1234567
123456
123456789101112
done
Method: Kendall 
Algorithm: Truncated Power Algorithm 
Input: The Data Matrix
6 sparse PCs 
Proportion of variance explained by first k components: 0.01258167 0.02364348 0.03502151 0.04281665 0.05193145 0.05914359 
Number of nonzero loadings: 10 10 10 5 5 5 
Sparse Loadings 
              PC1        PC2        PC3        PC4        PC5       PC6
  [1,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [2,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.4074194
  [3,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [4,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [5,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [6,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [7,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [8,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
  [9,]  0.0000000  0.0000000 -0.3277447  0.0000000  0.0000000 0.0000000
 [10,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [11,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [12,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [13,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [14,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [15,]  0.0000000  0.0000000 -0.3547273  0.0000000  0.0000000 0.0000000
 [16,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [17,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [18,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [19,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [20,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [21,]  0.0000000  0.0000000  0.2733553  0.0000000  0.0000000 0.0000000
 [22,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [23,] -0.3330370  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [24,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [25,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [26,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [27,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [28,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [29,]  0.0000000  0.3367315  0.0000000  0.0000000  0.0000000 0.0000000
 [30,]  0.3628552  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [31,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [32,]  0.0000000  0.0000000 -0.2677701  0.0000000  0.0000000 0.0000000
 [33,]  0.0000000  0.0000000  0.0000000 -0.4268640  0.0000000 0.0000000
 [34,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [35,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [36,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [37,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [38,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [39,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [40,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [41,]  0.3576961  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [42,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [43,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [44,]  0.0000000 -0.3298032  0.0000000  0.0000000  0.0000000 0.0000000
 [45,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [46,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [47,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [48,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [49,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [50,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [51,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [52,]  0.2165555  0.2388028  0.0000000  0.0000000  0.0000000 0.0000000
 [53,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [54,]  0.0000000  0.0000000  0.0000000 -0.5315897  0.0000000 0.0000000
 [55,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [56,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [57,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [58,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [59,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [60,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [61,]  0.2905540  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [62,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [63,]  0.3008650  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [64,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [65,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [66,]  0.1948889  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [67,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [68,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [69,]  0.0000000  0.0000000  0.0000000 -0.4225678  0.0000000 0.0000000
 [70,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [71,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.3817567
 [72,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [73,]  0.2753218  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [74,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [75,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [76,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [77,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [78,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [79,]  0.0000000  0.0000000  0.0000000  0.0000000 -0.4778063 0.0000000
 [80,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.5726021
 [81,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [82,]  0.0000000  0.0000000  0.3180228  0.0000000  0.0000000 0.0000000
 [83,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [84,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [85,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [86,]  0.0000000  0.0000000  0.2271923  0.0000000  0.0000000 0.0000000
 [87,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [88,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [89,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [90,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [91,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [92,]  0.0000000  0.0000000  0.0000000  0.0000000 -0.4403842 0.0000000
 [93,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [94,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [95,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [96,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [97,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [98,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
 [99,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[100,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[101,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[102,]  0.0000000  0.2309130  0.0000000  0.0000000  0.0000000 0.0000000
[103,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[104,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[105,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[106,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[107,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[108,]  0.0000000  0.0000000  0.0000000  0.3828609  0.0000000 0.0000000
[109,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[110,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[111,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[112,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[113,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[114,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[115,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[116,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[117,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[118,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[119,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[120,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[121,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[122,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.4038692
[123,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[124,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[125,]  0.0000000  0.0000000  0.0000000  0.0000000 -0.4304809 0.0000000
[126,]  0.0000000  0.2305774  0.0000000  0.0000000  0.0000000 0.0000000
[127,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[128,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[129,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[130,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[131,]  0.0000000 -0.3046789  0.0000000  0.0000000  0.0000000 0.0000000
[132,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[133,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[134,]  0.0000000  0.0000000  0.0000000  0.0000000 -0.3593956 0.0000000
[135,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[136,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[137,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[138,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[139,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[140,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[141,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[142,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[143,]  0.0000000  0.0000000  0.3858369  0.0000000  0.0000000 0.0000000
[144,]  0.4150092  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[145,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[146,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[147,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[148,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[149,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[150,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[151,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[152,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[153,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[154,]  0.0000000  0.0000000  0.0000000 -0.4583160  0.0000000 0.0000000
[155,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[156,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[157,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[158,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[159,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.4441708
[160,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[161,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[162,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[163,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[164,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[165,]  0.0000000  0.0000000 -0.3199298  0.0000000  0.0000000 0.0000000
[166,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[167,]  0.3487433  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[168,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[169,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[170,]  0.0000000  0.0000000 -0.3394379  0.0000000  0.0000000 0.0000000
[171,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[172,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[173,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[174,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[175,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[176,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[177,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[178,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[179,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[180,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[181,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[182,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[183,]  0.0000000  0.3802193  0.0000000  0.0000000  0.0000000 0.0000000
[184,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[185,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[186,]  0.0000000 -0.2757548  0.0000000  0.0000000  0.0000000 0.0000000
[187,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[188,]  0.0000000 -0.3597277  0.0000000  0.0000000  0.0000000 0.0000000
[189,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[190,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[191,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[192,]  0.0000000  0.0000000  0.0000000  0.0000000  0.5131120 0.0000000
[193,]  0.0000000  0.0000000 -0.3180093  0.0000000  0.0000000 0.0000000
[194,]  0.0000000 -0.4141105  0.0000000  0.0000000  0.0000000 0.0000000
[195,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[196,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[197,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[198,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[199,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000
[200,]  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 0.0000000

smart documentation built on May 29, 2017, 8:58 p.m.