Statistical Analysis of Aircraft Trajectories:a Functional Data Analysis Approach

Florence NICOL

ENAC-DEVI

ALLDATA 2017April 26, 2017

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Outline

Principal Component Analysis

Functional DataWhat are functional data?Functional Data Analysis

Functional Principal Component AnalysisWhat is this?Estimation Methods

Application to Aircraft TrajectoriesUnivariate FPCAMultivariate FPCA

Conclusion

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Principal Component Analysis

Outline

Principal Component Analysis

Functional Data

Functional Principal Component Analysis

Application to Aircraft Trajectories

Conclusion

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Principal Component Analysis

What is the best representation?An old problem

Figure: Fragment from the Tomb of Nebamun, Thebes, British Museum.

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Principal Component Analysis

What is the best representation?An old problem

Figure: Mug shot of American gangster Al Capone, 1931.

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Principal Component Analysis

What is the best representation?An old problem

Figure: Marie-Therese portrait, Picasso.6 / 30

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Principal Component Analysis

Main ideas

Context

• a large number of numeric variables possibly correlated,

• analyze data variability by studying the covariance structure.

What do we want to do?

• create a small number of new descriptors,

• capture the maximum amount of variation in the data.

How can we do that?

• Looking for new orthogonal directionssuch that the variance of the projecteddata is maximal.

•

Δ2

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X3

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•Δ3

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Functional Data

Outline

Principal Component Analysis

Functional DataWhat are functional data?Functional Data Analysis

Functional Principal Component Analysis

Application to Aircraft Trajectories

Conclusion

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Functional Data

What are Functional Data?

Rather than a sample of points xi , i = 1, . . . , n, we observe a sampleof entangled curves xi (t), images or functions.

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Longitude−Latitude

X(t) (Nm)

Y(t

) (N

m)

Figure: Sample of aircraft trajectories (Paris-Toulouse).9 / 30

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Functional Data

What are functional data?

Functional variable (f.v.)

X = {X (t), t ∈ J} is a functional space H-valued random variable

• a continuous stochastic process on a compact interval J,

• H is the separable Hilbert space L2(J).

Observed dataA functional dataset x1, . . . , xn is n realizations of the f.v. X (or theobservation of n f.v. X1, . . . ,Xn identically distributed as X ).

Discretized observed dataI Functional data xi (t) are observed discretely

xi (tij), i = 1, . . . , n, j = 1, . . . ,Ni .

I Often given at the same time arguments t1, . . . , tN .10 / 30

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Functional Data

Functional Data AnalysisInference about functional data

Multivariate statistical techniques are inadequate !

• They don’t take into account to the functional nature of data.

• A hard drawback: the curse of dimensionality N � n.

Extend multivariate methods to the functional case

• Functional principal component analysis (FPCA), Clustering.

• Functional linear models, Functional analysis of variance... etc

Generalization is not trivial!

• Data live in infinite dimensional spaces.

• Two types of errors:

I sampling error in random functions drawn from an underlying process,I measurement error when functions are discrete noisy sample paths.

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Functional Principal Component Analysis

Outline

Principal Component Analysis

Functional Data

Functional Principal Component AnalysisWhat is this?Estimation Methods

Application to Aircraft Trajectories

Conclusion

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Functional Principal Component Analysis

Generalization to the functional case

Multivariate PCA Functional PCAindividual vector xi ∈ Rp function xi (t) ∈ Heigenvector u eigenfunction γ

mean vector x ∈ Rp mean function µ(t) ∈ Hcovariance matrix covariance operator

inner product inner product

〈u, xi 〉 = uT xi 〈γ, xi 〉 =

∫Jγ(t)xi (t)dt

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Functional Principal Component Analysis

Generalization to the functional case

Maximization of varianceThe weight function γ1 maximizes the variance of the projected data

γ1 = argmax‖γ1‖=1

Var ( 〈γ1,X 〉 ) .

The subsequent weight functions γk can be found analogously subjectto the additional constraint (orthogonality)

〈γi , γk〉 =

∫Jγi (t)γk(t) = 0, i < k.

I γ1, γ2, . . . are called functional principal components,I orthogonality constraints ensure that γi indicates something new,I the amount of variation λi = Var ( 〈γi ,X 〉 ) will decline stepwise.

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Functional Principal Component Analysis

Estimation

Let X1, . . . ,Xn be a sample of independent functional variables.

Karhunen-Loeve representation

Xj(t) =n∑

i=1

Aij γi (t), j = 1, . . . , n.

Interpretation

• Principal components γi are modes of variation of individualtrajectories.

• Random scores Aij = 〈γi ,Xj〉 are proportionality factors: measurethe influence of the principal component γi on the shape of Xj .

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Functional Principal Component Analysis

Estimation

Reduction dimension tool: a small number L� n is needed

Xj(t) 'L∑

i=1

Aij γi (t)+µ(t).

I A small number L of components is often sufficient to account fora large part of variation.I High values of L are associated with high frequency componentswhich represent the sampling noise.

Quality of representation: % of total variation (Scree Plot)

τi =λi∑ni=1 λi

, τCL =

∑Lk=1 λk∑ni=1 λi

.

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Application to Aircraft Trajectories

Outline

Principal Component Analysis

Functional Data

Functional Principal Component Analysis

Application to Aircraft TrajectoriesUnivariate FPCAMultivariate FPCA

Conclusion

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Application to Aircraft Trajectories

Univariate FPCA: Flight LevelRoute: Paris Orly airport −→ Toulouse airport

Flight level

I Aircraft type: A319(25%), A320(41%), A321(24%), B733 (4%),B463 (2%) AT type (4%).I Aircraft trajectories measured at 4 seconds intervals.

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Application to Aircraft Trajectories

Univariate FPCA: Flight LevelEffects on the mean trajectory of adding (+) or substracting (-) PC

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8890

9294

9698

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PCA Scree Plot

Principal Components

Var

ianc

e (%

)

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Principal Components

time (sec)

PC1 88.1 %PC2 6.7 %PC3 2.6 %PC4 1.3 %

I 4 components = 98, 7% of total variation.

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Application to Aircraft Trajectories

Univariate FPCA: Flight LevelEffects on the mean trajectory of adding (+) or substracting (-) PC

Overall effect Takeoff effect

88.1% 6.7%

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6080

PC1 88.1 %

time (sec)

Z(t

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feet

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time (sec)

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time (sec)

Z(t

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feet

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++++++++ +

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PC4 1.3 %

time (sec)

Z(t

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First step effect Time shift effect

2.6% 1.3%

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Application to Aircraft Trajectories

Univariate FPCA: Flight LevelScore scatterplots by aircraft types

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Aircraft Type

A319A320A321ATB463B733

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Score Plot

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I detect outliers and clusters in the data,I interpret clusters,I explain individual behaviour relatively to modes of variation.

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Application to Aircraft Trajectories

Univariate FPCA: Flight Level

Table: Individual scores by aircraft type

PC1 PC2 PC3 PC4 OutlierAircraft type Overall Take-off First step Time shift

AT, E120, B463 + + + +A320 0 - - -B733 - - + + *A319 - + 0 0A321 + - - 0

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Application to Aircraft Trajectories

Multivariate FPCARoute: Paris Charles de Gaulle airport −→ Toulouse airport

Longitude-Latitude trajectories

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Longitude−Latitude

X(t) (Nm)

Y(t

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Route

Paris Charles de Gaullewwwwwwwwwwww�Toulouse

I Aircraft type: A319(25%), A320(41%), A321(24%), B733 (4%),B463 (2%) AT type (4%).I Aircraft trajectories measured at 4 seconds intervals.

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Application to Aircraft Trajectories

Multivariate FPCAPrincipal Components (PC): Principal components in X and Y -coordinates

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Principal Components X(t)

time (sec)

PC1 PC2 PC3 PC4

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time (sec)

Principal Component Total Var X-coord Y-coordPC1 Overall effect 58% 2% 98%PC2 Landing effect 14.7% 48% 52%PC3 Separation effect 12.9% 86% 14%PC4 Change procedure effect 6% 66% 34%

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Application to Aircraft Trajectories

Multivariate FPCAEffects on the mean trajectory of adding (+) or substracting (-) PC

Overall effect Landing effect

58% 14.7%

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PC1 58.7 %

X(t) (en Nm)

Y(t

) (e

n N

m)

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PC2 14.7 %

X(t) (en Nm)

Y(t

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m)

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PC3 12.9 %

X(t) (en Nm)

Y(t

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n N

m)

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PC4 6 %

X(t) (en Nm)

Y(t

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n N

m)

Separation effect Procedure effect

12.9% 6%

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Application to Aircraft Trajectories

Multivariate FPCAMean cluster Trajectories and the overall mean (black curve)

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Three clusters

X(t) (Nm)

Y(t

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m)

Clusters

C1C2C3

Table: k-means on scores

Aircraft type Cluster 1 Cluster 2 Cluster 3A319 15 18 0A320 14 14 1A321 25 28 0AT 2 0 8B463 10 0 2B733 22 1 2

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Application to Aircraft Trajectories

Multivariate FPCAMean cluster Trajectories (Route: Paris Orly airport � Toulouse airport)

Longitude-Latitude trajectoriesRoute

Paris Orlywwwwwwwwwwww�Toulouse

Route

Paris Orly~wwwwwwwwwwwwToulouse

I FPCA is able to separate the two clusters located at the right side:a standard approach procedure and a short one at Toulouse airport.27 / 30

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Conclusion

Outline

Principal Component Analysis

Functional Data

Functional Principal Component Analysis

Application to Aircraft Trajectories

Conclusion

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Conclusion

Conclusion and future works

A dimension reduction toolI An empirical basis function expansion.I Dimension reduction: use score vectors instead of functions xi (t).

A powerful visualization tool

I Explore the ways in which trajectories vary.I Reveal clusters and atypical trajectories

Other applications

I Generalization to 3D trajectories.I Generate samples of trajectories.I Reduce the dimension of simulated models.

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Short bibliography

P. Besse, J.O. Ramsay, Principal component analysis of sampledcurves, Psychometrika, 51 (1986), 285311.

Bookstein F.L., Morphometric tools for landmark data:geometryand biology, Cambridge: Cambridge University Press, 1991.

J. Dauxois, A. Pousse, and Y. Romain, Asymptotic theory for theprincipal component analysis of a vector random function: Someapplications to statistical inference. Journal of MultivariateAnalysis, 12 (1982), 136154.

F. Nicol, Functional Principal Component Analysis of AircraftTrajectories, ISIATM, July 810, 2013, Toulouse, France.

J.O. Ramsay, G. Hooker and S. Graves, Functional data analysiswith R and Matlab, Springer-Verlag, New York, 2009.

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