January 2015

creativecommons

Overview



  1. Shiny: the democratisation of R
  2. Graphical network example
  3. The mplot package
    • Variable inclusion plots
    • Model stability plots
    • Bootstrapping the lasso
    • Diabetes data example
  4. Technology involved

Shiny: the democratisation of R

What is Shiny?

How did I get started?

Graphical Newtork Example

Sharing your shiny app

The mplot package

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 appear to be significant.

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

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 \(\mathcal{A}=\{\{1\},\ldots,\alpha_f\}\).
  • Coefficent vector \(\beta\).

A smörgåsbord of tuning parameters…

Information Criterion

  • Generalised IC: \(\text{GIC}(\alpha;\lambda) = -2\times \text{LogLik}(\alpha)+\lambda p_\alpha\)

With important special cases:

  • AIC: \(\lambda=2\)
  • BIC: \(\lambda = \log(n)\)
  • HQIC: \(\lambda= 2\log(\log(n))\)

Regularisation routines

  • Lasso: minimises \(-\text{LogLik}(\alpha) +\lambda\ ||\beta_\alpha||_1\)
  • Many variants of the Lasso, SCAD,…

A stability based approach

Aim

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

Method

  • interactive graphical tools
  • exhaustive searches (where feasible)
  • bootstrapping to assess selection stability

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

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")

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)

Bootstrapping the lasso

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 residual sum of squares subject to the constraint that \[\sum_j |\beta_j| \leq t.\]

"The encouraging results reported here suggest that absolute value constraints 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

## 
## 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"

Artificial example – Lasso

Artificial example – Lasso

Artificial example – Lasso

Artificial example – Lasso

Bootstrapping the lasso

bgn.art = bglmnet(lm.art, lambda = seq(0.05, 2.75, 0.1))
plot(bgn.art, which = "models")

Bootstrapping the lasso

bgn.art = bglmnet(lm.art)
plot(bgn.art, which = "variables")

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

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

Diabetes example

Variables

Variable Description
age Age
sex Gender
bmi Body mass index
map Mean arterial pressure (average blood pressure)
tc Total cholesterol (mg/dL)
ldl Low-density lipoprotein ("bad" cholesterol)
hdl High-density lipoprotein ("good" cholesterol)
tch Blood serum measurement
ltg Blood serum measurement
glu Blood serum measurement (glucose?)
y A quantitative measure of disease progression one year after baseline

Source: Efron, B., Hastie, T., Johnstone, I., Tibshirani, R., (2004). "Least angle regression. The Annals of Statistics 32 (2): 407-499. doi:10.1214/009053604000000067

Diabetes data set

Variable importance

Model stability

Adaptive fence

glmnet VIP

Technology involved

Underlying technology

R packages

  • leaps
  • bestglm
  • googleVis
  • shiny
  • parallel
  • mvoutlier

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

Additional resources

Session Info

sessionInfo()
## R version 3.1.2 (2014-10-31)
## Platform: x86_64-apple-darwin13.4.0 (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] lars_1.2         xtable_1.7-4     mplot_0.5.0      glmnet_1.9-8    
##  [5] Matrix_1.1-5     mvoutlier_2.0.6  sgeostat_1.0-25  shiny_0.11.1    
##  [9] googleVis_0.5.8  doParallel_1.0.8 iterators_1.0.7  foreach_1.4.2   
## [13] bestglm_0.34     leaps_2.9        knitr_1.9       
## 
## loaded via a namespace (and not attached):
##  [1] cluster_2.0.1         codetools_0.2-11      colorspace_1.2-6     
##  [4] DEoptimR_1.0-2        digest_0.6.8          evaluate_0.5.5       
##  [7] formatR_1.0           GGally_0.5.0          ggplot2_1.0.0        
## [10] grid_3.1.2            gtable_0.1.2          htmltools_0.2.6      
## [13] httpuv_1.3.2          lattice_0.20-30       MASS_7.3-39          
## [16] mime_0.2              munsell_0.4.2         mvtnorm_1.0-2        
## [19] pcaPP_1.9-60          pls_2.4-3             plyr_1.8.1           
## [22] proto_0.3-10          R6_2.0.1              Rcpp_0.11.5          
## [25] reshape_0.8.5         reshape2_1.4.1        RJSONIO_1.3-0        
## [28] rmarkdown_0.5.1       robCompositions_1.9.0 robustbase_0.92-3    
## [31] rrcov_1.3-8           scales_0.2.4          stats4_3.1.2         
## [34] stringr_0.6.2         tools_3.1.2           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.

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