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 2     p x x p x x      ik jk  2 ik jk 2 d d     x x   x x   k 1 ik jk k 1 ik jk • Position of points given by classical scaling

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)

Legend Plot of the aircraft data Distance measure: Clark Axes type: none r Shape parameter 1 Inter-point distances interpretable g h c q u v f i p m j d t k n e s w ab

Adding variable information Take approach in Gower and Harding (1988) Can calculate position of a new observation ( m 1 , 0, 0, 0). • – gives position of the value m 1 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 n  m 2 w d ( x , ) i i  i 1

Legend Prediction map - SPR Distance measure: Clark Axes type: none r Shape parameter 1 Inter-point distances interpretable g Prediction map: SPR 9 8 h q c u v 7 f i p m j d t 6 k n e s w 5 4 ab 3 2 1 0.5

Legend Prediction map - PLF Distance measure: Clark Axes type: none r Shape parameter 1 Inter-point distances interpretable g Prediction map: PLF 0.2 h q c u v f i p m j d t k n e s w 0.1 ab 0.01 0.001 1e-04

Normal predictive trajectories + Gower and Hand (1996), Gower and Ngounet (2003)

Legend Predictive biplot Distance measure: Clark Axes type: normal r Shape parameter 1 Inter-point 0.01 distances interpretable g 1 0.1 5 SPR 2 3 RGF PLF 4.5 3 SLF h 4 q 2 c u v f i 5 p m j d 6 t 1 4 k n e s w 0.5 3.5 ab

Legend Prediction map - SPR Distance measure: Canberra Axes type: none Shape parameter r 1 Inter-point distances interpretable g Prediction map: SPR 9 8 h d c e i 7 f q p tu v m 6 j s n k w 5 ab 4 3 2 1 0.5

Adding an arc

Adding an arc

Adding an arc

Adding an arc

Adding an arc

Map and trajectory Legend Distance measure: Canberra Axes type: normal Shape parameter r 1 Inter-point distances interpretable g SPR 1.605 2.054 2.168 2.183 2.426 2.467 h 2.607 d 2.898 c 3.618 e 3.88 i f 4.567 q 4.588 p tu v 5.855 m j s 6.081 n k w 6.502 ab Prediction map: SPR 9 8 7 6 5 4 3 2 1

Legend Distance measure: Canberra Axes type: normal Shape parameter 1 r Inter-point distances interpretable g PLF 0.106 0.117 0.138 0.143 0.15 0.155 0.177 h d c 0.166 0.172 e i f q p tu v Prediction map: PLF m 0.2 j s n k w ab 0.1 0.01 0.001 1e-04

Legend Trajectories only Distance measure: Canberra Axes type: normal Shape parameter r 1 Inter-point distances interpretable g 0.106 0.117 0.138 0.143 0.15 4.99 0.155 4.92 1.605 4.87 2.4 2.5 4.72 1.1 1.82.3 2.0542.168 SPR 2.183 2.64 4.65 RGF 0.177 2.426 h 2.7 4.53 2.467 1 PLF 2.607 d 2.8 4.484.5 2.898 0.166 c 0.172 3.618 SLF 0.9 e 3.88 i f 4.32 4.11 4.2 4.567 q 3.97 4.588 p 3.84 tu v m 5.855 3.82 j s 2.9 6.081 n 3.75 k w ab

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.