26 November 2014

creativecommons

Overview



  1. What's wrong with existing variable selection techniques?
  2. Variable inclusion plots
  3. Model stability plots
  4. Adaptive fence method
  5. The mplot package
  6. Examples with real data

What's wrong with existing variable selection techniques?

Current state of variable selection

Google

  • 2 million pages with "model selection"
  • 1/2 million pages with "variable selection"

Google Scholar

  • 620,000 articles with "model selection"
  • 170,000 articles with "variable selection"

Do we really need more?

Current approaches to variable selection

Information Criterion

  • AIC
  • BIC

Stepwise procedures

  • Mallows \(C_p\)
  • Sequential testing

Regularisation methods

  • Lasso
  • SCAD
  • LARS (stagewise)

Artificial example

Artificial example

require(mplot)
lm.art = lm(y~., data=artificialeg)
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.10 0.33 -0.31 0.76
x1 0.64 0.69 0.92 0.36
x2 0.26 0.62 0.42 0.68
x3 -0.51 1.24 -0.41 0.68
x4 -0.30 0.25 -1.18 0.24
x5 0.36 0.60 0.59 0.56
x6 -0.54 0.96 -0.56 0.58
x7 -0.43 0.63 -0.68 0.50
x8 0.15 0.62 0.24 0.81
x9 0.40 0.64 0.63 0.53

Artificial example – Stepwise

step(lm.art)
## Start:  AIC=79.3
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
## 
##        Df Sum of Sq    RSS    AIC
## - x8    1    0.2423 163.94 77.374
## - x3    1    0.6946 164.39 77.512
## - x2    1    0.7107 164.41 77.517
## - x6    1    1.3051 165.00 77.698
## - x5    1    1.4425 165.14 77.739
## - x9    1    1.6065 165.31 77.789
## - x7    1    1.8835 165.58 77.873
## - x1    1    3.4999 167.20 78.358
## - x4    1    5.7367 169.44 79.023
## <none>              163.70 79.301
## 
## Step:  AIC=77.37
## y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x9
## 
...
## 
## Call:
## lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x9, data = artificialeg)
## 
## Coefficients:
## (Intercept)           x1           x2           x3           x4  
##     -0.1143       0.8019       0.4011      -0.8083      -0.3514  
##          x5           x6           x7           x9  
##      0.4927      -0.7738      -0.5772       0.5478

Artificial example – Stepwise

Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.11 0.32 -0.36 0.72
x1 0.80 0.19 4.13 0.00
x2 0.40 0.18 2.26 0.03
x3 -0.81 0.19 -4.22 0.00
x4 -0.35 0.12 -2.94 0.01
x5 0.49 0.19 2.55 0.01
x6 -0.77 0.15 -5.19 0.00
x7 -0.58 0.15 -3.94 0.00
x9 0.55 0.19 2.90 0.01

  • We've dropped x8.
  • All the other variables are significant.
  • Job well done!

Artificial example – Stepwise

The true data generating process is:

\[y = 0.6x_8 + \varepsilon.\]

art.true = lm(y~x8,data=artificialeg)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03 0.29 0.11 0.91
x8 0.55 0.05 10.43 0.00

Artificial example – Stepwise

The true data generating process is:

\[y = 0.6x_8 + \varepsilon.\]

art.true = lm(y~x8,data=artificialeg)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03 0.29 0.11 0.91
x8 0.55 0.05 10.43 0.00 

anova(art.true,step.art)
Res.Df RSS Df Sum of Sq F Pr(>F)
1 48 200.97
2 41 163.94 7 37.03 1.32 0.2643

A short history of the lasso

  • Tibshirani (1996) suggested doing regression with an \(L_1\) norm penalty and called it the lasso (least absolute shrinkage and selection operator).

  • The lasso parameter estimates are obtained by minimising the resudual sum of squares subject to the constraint that \[\sum_j |\beta_j| \leq t.\]

"The encouraging results reported here suggest that absolute value constrants might prove to be useful in a wide variety of statistical estimation problems. Further study is needed to investigate these possibilities." - Tibshirani (1996)

  • Implemented easily and quickly through LARS (least angle regression) algorithm of Efron et al. (2004).

Artificial example – Lasso

require(lars)
x = as.matrix(subset(artificialeg, select = -y))
y = as.matrix(subset(artificialeg, select = y))
art.lars = lars(x, y)

Artificial example – Lasso

require(lars)
x = as.matrix(subset(artificialeg, select = -y))
y = as.matrix(subset(artificialeg, select = y))
art.lars = lars(x, y)

Artificial example – Lasso

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x8 
##  8 
## [1] "x8"
## [1] "x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x6 
##  6 
## [1] "x6" "x8"
## [1] "x6+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x1 
##  1 
## [1] "x1" "x6" "x8"
## [1] "x1+x6+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x4 
##  4 
## [1] "x1" "x4" "x6" "x8"
## [1] "x1+x4+x6+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x3 
##  3 
## [1] "x1" "x3" "x4" "x6" "x8"
## [1] "x1+x3+x4+x6+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x7 
##  7 
## [1] "x1" "x3" "x4" "x6" "x7" "x8"
## [1] "x1+x3+x4+x6+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x5 
##  5 
## [1] "x1" "x3" "x4" "x5" "x6" "x7" "x8"
## [1] "x1+x3+x4+x5+x6+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x2 
##  2 
## [1] "x1" "x2" "x3" "x4" "x5" "x6" "x7" "x8"
## [1] "x1+x2+x3+x4+x5+x6+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x5 
## -5 
## [1] "x1" "x2" "x3" "x4" "x6" "x7" "x8"
## [1] "x1+x2+x3+x4+x6+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x6 
## -6 
## [1] "x1" "x2" "x3" "x4" "x7" "x8"
## [1] "x1+x2+x3+x4+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x5 
##  5 
## [1] "x1" "x2" "x3" "x4" "x5" "x7" "x8"
## [1] "x1+x2+x3+x4+x5+x7+x8"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x9 
##  9 
## [1] "x1" "x2" "x3" "x4" "x5" "x7" "x8" "x9"
## [1] "x1+x2+x3+x4+x5+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x6 
##  6 
## [1] "x1" "x2" "x3" "x4" "x5" "x6" "x7" "x8" "x9"
## [1] "x1+x2+x3+x4+x5+x6+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x2 
## -2 
## [1] "x1" "x3" "x4" "x5" "x6" "x7" "x8" "x9"
## [1] "x1+x3+x4+x5+x6+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x2 
##  2 
## [1] "x1" "x2" "x3" "x4" "x5" "x6" "x7" "x8" "x9"
## [1] "x1+x2+x3+x4+x5+x6+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x3 
## -3 
## [1] "x1" "x2" "x4" "x5" "x6" "x7" "x8" "x9"
## [1] "x1+x2+x4+x5+x6+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
## x3 
##  3 
## [1] "x1" "x2" "x3" "x4" "x5" "x6" "x7" "x8" "x9"
## [1] "x1+x2+x3+x4+x5+x6+x7+x8+x9"
## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

## 
## Call:
## lars(x = x, y = y)
## R-squared: 0.751 
## Sequence of LASSO moves:
##      x8 x6 x1 x4 x3 x7 x5 x2 x5 x6 x5 x9 x6 x2 x2 x3 x3
## Var   8  6  1  4  3  7  5  2 -5 -6  5  9  6 -2  2 -3  3
## Step  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17

Artificial example – Lasso

Artificial example – Lasso

Artificial example – Lasso

Artificial example – Lasso

Artificial example – Lasso

A different approach

  • graphical tools
  • exhaustive searches
  • bootstrapping
  • assessing stability
  • interactivity

Concept of model stability independently introduced by Meinshausen and Bühlmann (2010) and Müller and Welsh (2010) for different linear regression situations.

Aim

To provide scientists/researchers with tools that give them more information about the model selection choices that they are making.

Some notation

  • Say we have a full model with an \(n\times p\) design matrix \(\mathbf{X}\).
  • Let \(\alpha\) be any subset of \(p_\alpha\) distinct elements from \(\{1,\ldots,p\}\).
  • We can define a \(n\times p_\alpha\) submodel with design matrix \(\mathbf{X}_\alpha\) subset from \(\mathbf{X}\) by the elements of \(\alpha\).
  • Denote the set of all possible models as \(\mathcal{A}=\{\{1\},\ldots,\alpha_f\}\).

Variable inclusion plots

Variable inclusion plots

Aim

To visualise inclusion probabilities as a function of the penalty multiplier \(\lambda\in [0,2\log(n)]\).

Procedure

  1. Calculate (weighted) bootstrap samples \(b=1,\ldots,B\).
  2. For each bootstrap sample, at each \(\lambda\) value, find \(\hat{\alpha}_\lambda^{(b)}\in\mathcal{A}\) as the model with smallest \(\text{GIC}(\alpha;\lambda) = -2\times \text{LogLik}(\alpha)+\lambda p_\alpha\).
  3. The inclusion probability for variable \(x_j\) is estimated as \(\frac{1}{B}\sum_{b=1}^B 1\{j\in \hat{\alpha}_\lambda^{(b)}\}\).

References

  • Müller and Welsh (2010) for linear regression models
  • Murray, Heritier, and Müller (2013) for generalised linear models

Artificial example – VIP

require(mplot)
vis.art = vis(lm.art)
plot(vis.art, which = "vip")

Model stability plots

Artificial example – Loss against size

plot(vis.art, which = "lvk")

Artificial example – Loss against size

plot(vis.art, which = "lvk", highlight = "x6")

Artificial example – Loss against size

plot(vis.art, which = "lvk", highlight = "x7")

Model stability plots

Aim

To add value to the loss against size plots by choosing a symbol size proportional to a measure of stability.

Procedure

  1. Calculate (weighted) bootstrap samples \(b=1,\ldots,B\).
  2. For each bootstrap sample, identify the best model at each dimension.
  3. Add this information to the loss against size plot using model identifiers that are proportional to the frequency with which a model was identified as being best at each model size.

References

  • Murray, Heritier, and Müller (2013) for generalised linear models

Artificial example – Model stability plot

plot(vis.art, which = "boot")

The adaptive fence

The fence

Notation

  • Let \(Q(\alpha)\) be a measure of lack of fit
  • Specifically we consider \(Q(\alpha)=-2\text{LogLik}(\alpha)\)

Main idea

The fence (Jiang et al. 2008) is based aroung the inequality: \[ Q(\alpha) \leq Q(\alpha_f) + c \]

  • A model \(\alpha\) passes the fence if the inequality holds.
  • For any \(c\geq 0\), the full model always passes the fence.
  • Among the set of models that pass the fence, model(s) with smallest dimension are preferred.

Illustration

Illustration

Problem: how to choose \(c\)?

Solution: Bootstrap over a range of values of \(c\).

Procedure

  1. For each value of \(c\):
    • Perform parametric bootstrap under \(\alpha_f\).
    • For each bootstrap sample, identify the smallest model that passes the fence, \(\hat{\alpha}(c)\). Jiang, Nguyen, and Rao (2009) suggest that if there are more than one model, choose the one with the smallest \(Q(\alpha)\).
    • Let \(p^∗(\alpha) = P^∗\{\hat{\alpha}(c) = \alpha\}\) be the empirical probability of selecting model \(\alpha\) at a given value of \(c\).
    • Calculate \(p^* = \max_{\alpha\in\mathcal{A}} p^*(\alpha)\)
  2. Plot values of \(p^*\) against \(c\) and find first peak.
  3. Use this value of \(c\) with the original data.

What does this look like?

Source: Jiang, Nguyen, and Rao (2009)

Our implementation

Core innovations

  • Enhanced the plot of \(p^*\) against \(c\) so that you can see which model is being selected most often
  • Allows for consideration of all models that pass the fence at the smallest dimension
  • Integration with the leaps and bestglm packages
  • Implemented multicore technology to speed up the embarassingly parallel calculations

Optional innovations

  • Intial stepwise procedure to restrict the values of \(c\)
  • Can force variables into the model
  • Specify the size of the largest model to be considered
  • Specify the maximum value for \(c\)

Artificial example – adaptive fence

af.art = af(lm.art, B = 100, n.c = 50, n.cores = 2)
plot(af.art)

Speed (linear models; B=50; n.c=25)

Speed (linear models; B=50; n.c=25)

The mplot package

Get it on Github 

install.packages("devtools")
require(devtools)
install_github("garthtarr/mplot",quick=TRUE)
require(mplot)

Vignettes

vignette("mplot-guide",package="mplot")
vignette("mplot-stepwise",package="mplot")

Main functions

  • af() for the adaptive fence
  • vis() for VIP and model stability plots
  • mplot() for an interactive shiny interface

Examples

Underlying technology

R packages

  • leaps
  • bestglm
  • googleVis
  • shiny
  • parallel (part of base R)

Web

Package development

  • RStudio makes setting up package structure easy
  • Version control with Github
  • Documentation with roxygen2
  • Vignettes (and these slides) with markdown
  • devtools is an essential package for building and loading R packages
  • R packages by Hadley Wickham

Future work

  • Increase speed of GLM?
  • Mixed models (speed is an issue here too)
  • Include other visualisations (suggestions?)
  • Robust methods?

Parallel processing in R

Parallel processing

Cores are easy to find:

  • NCI at ANU (and Australia wide): 3592 compute nodes each with 16 cores (57,472 cores total).
  • HPC at Wollongong: 30 compute nodes each with 8 cores (240 cores total)
  • HPC at USYD: 1500 cores (available Feb 2015).
  • Your personal computer probably has at least 4 cores.

Basic syntax

require(parallel)
ncores=1
n=10
expt = function(j) rep(j,4)
result = mclapply(1:n, FUN=expt, mc.cores=ncores)
simplify2array(result)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,]    1    2    3    4    5    6    7    8    9    10
## [2,]    1    2    3    4    5    6    7    8    9    10
## [3,]    1    2    3    4    5    6    7    8    9    10
## [4,]    1    2    3    4    5    6    7    8    9    10

Basic syntax

require(doMC,quietly=TRUE)
require(foreach)
registerDoMC(cores=ncores)
n=10
result = foreach(j = 1:n, .combine=rbind) %dopar% {
  # EXPERIMENT
  # last line is returned as a row in the result matrix
  rep(j,4)
  }
head(result)
##          [,1] [,2] [,3] [,4]
## result.1    1    1    1    1
## result.2    2    2    2    2
## result.3    3    3    3    3
## result.4    4    4    4    4
## result.5    5    5    5    5
## result.6    6    6    6    6

Session Info

sessionInfo()
## R version 3.1.1 (2014-07-10)
## Platform: x86_64-apple-darwin13.1.0 (64-bit)
## 
## locale:
## [1] en_AU.UTF-8/en_AU.UTF-8/en_AU.UTF-8/C/en_AU.UTF-8/en_AU.UTF-8
## 
## attached base packages:
## [1] parallel  stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] doMC_1.3.3      iterators_1.0.7 foreach_1.4.2   lars_1.2       
##  [5] xtable_1.7-4    mplot_0.4.7     shiny_0.10.2.1  googleVis_0.5.6
##  [9] bestglm_0.34    leaps_2.9       knitr_1.7      
## 
## loaded via a namespace (and not attached):
##  [1] codetools_0.2-9  compiler_3.1.1   digest_0.6.4     evaluate_0.5.5  
##  [5] formatR_1.0      htmltools_0.2.6  httpuv_1.3.2     mime_0.2        
##  [9] R6_2.0.1         Rcpp_0.11.3      RJSONIO_1.3-0    rmarkdown_0.3.10
## [13] stringr_0.6.2    tools_3.1.1      yaml_2.1.13

References

Efron, Bradley, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. 2004. “Least Angle Regression.” The Annals of Statistics 32 (2): 407–51. doi:10.1214/009053604000000067.

Jiang, Jiming, Thuan Nguyen, and J. Sunil Rao. 2009. “A Simplified Adaptive Fence Procedure.” Statistics & Probability Letters 79 (5): 625–29. doi:10.1016/j.spl.2008.10.014.

Jiang, Jiming, J. Sunil Rao, Zhonghua Gu, and Thuan Nguyen. 2008. “Fence Methods for Mixed Model Selection.” The Annals of Statistics 36 (4): 1669–92. doi:10.1214/07-AOS517.

Meinshausen, Nicolai, and Peter Bühlmann. 2010. “Stability Selection.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 72 (4): 417–73. doi:10.1111/j.1467-9868.2010.00740.x.

Murray, K, S Heritier, and Samuel Müller. 2013. “Graphical Tools for Model Selection in Generalized Linear Models.” Statistics in Medicine 32 (25): 4438–51. doi:10.1002/sim.5855.

Müller, Samuel, and Alan H. Welsh. 2010. “On Model Selection Curves.” International Statistical Review 78 (2): 240–56. doi:10.1111/j.1751-5823.2010.00108.x.

Tibshirani, Robert. 1996. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society: Series B (Methodological), 267–88. http://www.jstor.org/stable/10.2307/2346178.