# miscellanea-Internal: Miscellanea of functions In speedglm: Fitting Linear and Generalized Linear Models to Large Data Sets

## Description

Utility functions for least squares estimation in large data sets.

## Usage

 ```1 2 3 4``` ```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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53``` ```#### 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) ```