Predictive nonlinear biplots: maps and trajectories Karen Vines - - PowerPoint PPT Presentation

Predictive nonlinear biplots: maps and trajectories

Karen Vines Department of Mathematics and Statistics The Open University

Nonlinear biplots

• Simultaneous representation of observations and

variables

• Dissimilarities between observations calculated using

an Euclidean embeddable distance function.

• e.g. Clark’s distance, square root of Canberra distance
• Position of points given by classical scaling





          

p k jk ik jk ik

x x x x d

1 2 2





  

p k jk ik jk ik

x x x x d

1 2

Aircraft data

Data originally from Stanley and Miller (1979)

• 21 fighter aircraft
• 4 variables:

– SPR (specific power) – RGF (flight range factor) – PLF (payload, as fraction of gross weight) – SLF (sustained load factor)

Plot of the aircraft data

ab c d e f g h i j k m n p q r s t u v w

Legend

Distance measure: Clark Axes type: none Shape parameter 1 Inter-point distances interpretable

Adding variable information

Take approach in Gower and Harding (1988)

• Can calculate position of a new observation (m1, 0, 0, 0).

– gives position of the value m1 for the first variable – joining a series of these points gives an axis for the first variable (SPR)

• But axis lives in high-dimensional space
• Interpolation or prediction matters. Focus here on prediction.

Prediction in biplots

For a given point, a, on the biplot, find the value of m which corresponds where the axis is closest. Use least squares... ... equivalent to finding m which minimises

) , (

2 1

m

i n i i

x d w





ab c d e f g h i j k m n p q r s t u v w

Legend

Distance measure: Clark Axes type: none Shape parameter 1 Inter-point distances interpretable 0.5 1 2 3 4 5 6 7 8 9

Prediction map: SPR

Prediction map - SPR

ab c d e f g h i j k m n p q r s t u v w

Legend

Distance measure: Clark Axes type: none Shape parameter 1 Inter-point distances interpretable 1e-04 0.001 0.01 0.1 0.2

Prediction map: PLF

Prediction map - PLF

Normal predictive trajectories

Gower and Hand (1996), Gower and Ngounet (2003)

+

ab c d e f g h i j k m n p q r s t u v w 1 2 3 4 5 6 3.5 4 4.5 5 0.01 0.1 0.5 1 2 3

Legend

Distance measure: Clark Axes type: normal Shape parameter 1 Inter-point distances interpretable SPR RGF PLF SLF

Predictive biplot

ab c d e f g h i j k m n p q r s tu v w

Legend

Distance measure: Canberra Axes type: none Shape parameter 1 Inter-point distances interpretable 0.5 1 2 3 4 5 6 7 8 9

Prediction map: SPR

Prediction map - SPR

Adding an arc

Adding an arc

Adding an arc

Adding an arc

Adding an arc

ab c d e f g h i j k m n p q r s tu v w 1.605 2.054 2.168 2.183 2.426 2.467 2.607 2.898 3.618 3.88 4.567 4.588 5.855 6.081 6.502

Legend

Distance measure: Canberra Axes type: normal Shape parameter 1 Inter-point distances interpretable SPR 1 2 3 4 5 6 7 8 9

Prediction map: SPR

Map and trajectory

ab c d e f g h i j k m n p q r s tu v w 0.106 0.117 0.138 0.143 0.15 0.155 0.166 0.172 0.177

Legend

Distance measure: Canberra Axes type: normal Shape parameter 1 Inter-point distances interpretable PLF 1e-04 0.001 0.01 0.1 0.2

Prediction map: PLF

ab c d e f g h i j k m n p q r s tu v w 1.605 2.0542.168 2.183 2.426 2.467 2.607 2.898 3.618 3.88 4.567 4.588 5.855 6.081 3.75 3.82 3.84 3.97 4.11 4.2 4.32 4.484.5 4.53 4.65 4.72 4.87 4.92 4.99 0.106 0.117 0.138 0.143 0.15 0.155 0.166 0.172 0.177 0.9 1 1.1 1.82.3 2.4 2.5 2.64 2.7 2.8 2.9

Legend

Distance measure: Canberra Axes type: normal Shape parameter 1 Inter-point distances interpretable SPR RGF PLF SLF

Trajectories only

Performance of trajectories

• Aircraft ‘f’

Variable Trajectory Map Observed SPR 1.605 1.605 1.294 RGF 3.75 3.75 3.75 PLF 0.166 0.166 0.150 SLF 1.00 1.00 0.90

Further work

• Decision rule for when two or more predicted values are indicated.
• Calculation of prediction discontinuities.
• Investigation of when non data values are predicted for non-

smooth distance functions.