Miscellanea of functions

Description

Utility functions for least squares estimation in large data sets.

Usage

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control(B, symmetric = TRUE, tol.values = 1e-7, tol.vectors = 1e-7,
       out.B = TRUE, method = c("eigen", "Cholesky"))
cp(X, w = NULL, row.chunk = NULL, sparse = FALSE)
is.sparse(X, sparselim = .9, camp = .05)               

Arguments

B

a squared matrix.

symmetric

logical, is B symmetric?

tol.values

tolerance to be consider eigenvalues equals to zero.

tol.vectors

tolerance to be consider eigenvectors equals to zero.

out.B

Have the matrix B to be returned?

method

the method to check for singularity. By default is "eigen", and an eigendecomposition of X'X is made. The "Cholesky" method is faster than "eigen" and does not use tolerance, but the former seems to be more stable for opportune tolerance values.

X

the model matrix.

w

a weights vector.

sparse

logical, is X sparse?

sparselim

a real in the interval [0; 1]. It indicates the minimal proportion of zeroes in the data matrix X in order to consider X as sparse

eigendec Logical. Do you want to investigate on rank of X? You may set to

row.chunk

an integer which indicates the total rows number compounding each of the first g-1 blocks. If row.chunk is not a divisor of nrow(X), the g-th block will be formed by the remaining data.

camp

the sample proportion of elements of X on which the survey will be based.

Details

Function control makes an eigendecomposition of B according established values of tolerance. Function cp makes the cross-product X'X by partitioning X in row-blocks. When an optimized BLAS, such as ATLAS, is not installed, the function represents an attempt to speed up the calculation and avoid overflows with medium-large data sets loaded in R memory. The results depending on processor type. Good results are obtained, for example, with an AMD Athlon dual core 1.5 Gb RAM by setting row.chunk to some value less than 1000. Try the example below by changing the matrix size and the value of row.chunk. If the matrix X is sparse, it will have class "dgCMatrix" (the package Matrix is required) and the cross-product will be made without partitioning. However, good performances are usually obtained with a very high zeroes proportion. Function is.sparse makes a quick sample survey on sample proportion of zeroes in X.

Value

for the function control, a list with the following elements:

XTX

the matrix product B without singularities (if there are).

rank

the rank of B

pivot

an ordered set of column indeces of B with, if the case, the last rank+1,...,p columns which indicate possible linear combinations.

for the function cp:

new.B

the matrix product X'X (weighted, if w is given).

for the function is.sparse:

sparse

a logical value which indicates if the sample proportion of zeroes is greater than sparselim, with the sample proportion as attribute.

Author(s)

Marco ENEA

See Also

eigen, chol, qr, crossprod

Examples

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#### example 1.

n <- 100000
k <- 100
x <- round(matrix(rnorm(n*k),n,k),digits=4)
y <- rnorm(n)

# if an optimized BLAS is not installed, depending on processor type, cp() may be 
# faster than crossprod() for large matrices.

system.time(a1 <- crossprod(x))
system.time(a2 <- cp(x,,row.chunk = 500))
all.equal(a1, a2)  

#### example 2.1.
n <- 100000
k <- 10
x <- matrix(rnorm(n*k),n,k)
x[,2] <- x[,1] + 2*x[,3]  # x has rank 9
y <- rnorm(n)

# estimation by least squares 
A <- function(){
  A1 <- control(crossprod(x))
  ok <- A1$pivot[1:A1$rank]
  as.vector(solve(A1$XTX,crossprod(x[,ok],y)))
}
# estimation by QR decomposition
B <- function(){
  B1 <- qr(x)
  qr.solve(x[,B1$pivot[1:B1$rank]],y)    
}  
system.time(a <- A())
system.time(b <- B())

all.equal(a,b)

###  example 2.2
x <- matrix(c(1:5, (1:5)^2), 5, 2)
x <- cbind(x, x[, 1] + 3*x[, 2])
m <- crossprod(x)
qr(m)$rank # is 2, as it should be
control(m,method="eigen")$rank # is 2, as it should be
control(m,method="Cholesky")$rank # is wrong


### example 3. 
n <- 10000
fat1 <- gl(20,500)
y <- rnorm(n)
da <- data.frame(y,fat1)
m <- model.matrix(y ~ factor(fat1),data = da)
is.sparse(m)