Table of Contents

1  Summary of linear regression

versioninfo()

Julia Version 1.1.0
Commit 80516ca202 (2019-01-21 21:24 UTC)
Platform Info:
  OS: macOS (x86_64-apple-darwin14.5.0)
  CPU: Intel(R) Core(TM) i7-6920HQ CPU @ 2.90GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.1 (ORCJIT, skylake)
Environment:
  JULIA_EDITOR = code

Summary of linear regression

Methods for solving linear regression $\widehat \beta = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$:

Method	Flops	Remarks	Software	Stability
Sweep	$np^2 + p^3$	$(X^TX)^{-1}$ available	SAS	less stable
Cholesky	$np^2 + p^3/3$			less stable
QR by Householder	$2np^2 - (2/3)p^3$		R	stable
QR by MGS	$2np^2$	$Q_1$ available		stable
QR by SVD	$4n^2p + 8np^2 + 9p^3$	$X = UDV^T$		most stable

Remarks:

1. When $n \gg p$, sweep and Cholesky are twice faster than QR and need less space.
2. Sweep and Cholesky are based on the Gram matrix $\mathbf{X}^T \mathbf{X}$, which can be dynamically updated with incoming data. They can handle huge $n$, moderate $p$ data sets that cannot fit into memory.
3. QR methods are more stable and produce numerically more accurate solution.
4. Although sweep is slower than Cholesky, it yields standard errors and so on.
5. MGS appears slower than Householder, but it yields $\mathbf{Q}_1$.

There is simply no such thing as a universal 'gold standard' when it comes to algorithms.

using SweepOperator, BenchmarkTools, LinearAlgebra

linreg_cholesky(y::Vector, X::Matrix) = cholesky!(X'X) \ (X'y)

linreg_qr(y::Vector, X::Matrix) = X \ y

function linreg_sweep(y::Vector, X::Matrix)
    p = size(X, 2)
    xy = [X y]
    tableau = xy'xy
    sweep!(tableau, 1:p)
    return tableau[1:p, end]
end

function linreg_svd(y::Vector, X::Matrix)
    xsvd = svd(X)
    return xsvd.V * ((xsvd.U'y) ./ xsvd.S)
end

linreg_svd (generic function with 1 method)

using Random

Random.seed!(280) # seed

n, p = 10, 3
X = randn(n, p)
y = randn(n)

# check these methods give same answer
@show linreg_cholesky(y, X)
@show linreg_qr(y, X)
@show linreg_sweep(y, X)
@show linreg_svd(y, X);

linreg_cholesky(y, X) = [0.390365, 0.262759, 0.149047]
linreg_qr(y, X) = [0.390365, 0.262759, 0.149047]
linreg_sweep(y, X) = [0.390365, 0.262759, 0.149047]
linreg_svd(y, X) = [0.390365, 0.262759, 0.149047]

n, p = 1000, 300
X = randn(n, p)
y = randn(n)

@benchmark linreg_cholesky(y, X)

BenchmarkTools.Trial: 
  memory estimate:  708.31 KiB
  allocs estimate:  8
  --------------
  minimum time:     1.590 ms (0.00% GC)
  median time:      1.756 ms (0.00% GC)
  mean time:        1.821 ms (2.78% GC)
  maximum time:     39.979 ms (95.16% GC)
  --------------
  samples:          2733
  evals/sample:     1

@benchmark linreg_sweep(y, X)

BenchmarkTools.Trial: 
  memory estimate:  2.99 MiB
  allocs estimate:  7
  --------------
  minimum time:     5.200 ms (0.00% GC)
  median time:      7.225 ms (0.00% GC)
  mean time:        7.300 ms (2.58% GC)
  maximum time:     48.544 ms (82.00% GC)
  --------------
  samples:          684
  evals/sample:     1

@benchmark linreg_qr(y, X)

BenchmarkTools.Trial: 
  memory estimate:  4.04 MiB
  allocs estimate:  2444
  --------------
  minimum time:     10.474 ms (0.00% GC)
  median time:      11.760 ms (0.00% GC)
  mean time:        12.302 ms (3.93% GC)
  maximum time:     52.806 ms (76.80% GC)
  --------------
  samples:          406
  evals/sample:     1

@benchmark linreg_svd(y, X)

BenchmarkTools.Trial: 
  memory estimate:  8.06 MiB
  allocs estimate:  16
  --------------
  minimum time:     31.354 ms (0.00% GC)
  median time:      32.280 ms (0.00% GC)
  mean time:        33.278 ms (3.04% GC)
  maximum time:     84.534 ms (59.10% GC)
  --------------
  samples:          151
  evals/sample:     1