Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2021 | OriginalPaper | Chapter

11. Projection Pursuit

Author : Abdelwaheb Hannachi

Published in: Patterns Identification and Data Mining in Weather and Climate

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter describes a method that attempts to identify, sequentially, ‘interesting’ patterns by seeking directions in the data state space that optimises a specific projection index. A number of projection indexes, including indexes measuring non-normality such as skewness, are discussed and application to climate data presented.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Factor Analysis
next chapter Independent Component Analysis
Footnotes
1
Who coined it projection pursuit. In projection pursuit, the term “projection” refers to the fact that the data X is first projected onto the direction a, i.e. a T X, whereas “pursuit” refers to the optimisation used to find the correct direction.
 
2
In projection pursuit the term “projection” refers to the fact that the data X is first projected onto the direction a, i.e. a T X, whereas “pursuit” refers to the optimisation used to find the correct direction.
 
3
In fact, it is called negentropy.
 
4
Following Rényi (1961; 1970 p. 592)’s introduction of the concept of order-α entropy; the differential entropy, e.g., \(- \int f \log f\), is of order 1, whereas index (11) is of order 2.
 
5
If the projected data is centred and scaled to have zero mean and unit variance, then κ 31 = μ 31, κ 13 = μ 13, κ 22 = μ 22 − 1, κ 40 = μ 40 − 3, κ 04 = μ 04 − 3, κ 12 = μ 12, κ 21 = μ 21, κ 03 = μ 03, and κ 30 = μ 30, where the μs refer to the centred cumulants.
 
6
These are orthogonal polynomials over [−1,  1], satisfying P 0(y) = 1, P 1(y) = y, and for k ≥ 2, kP k(y) − (2k − 1)yP k−1(y) + (k − 1)P k−2(y) = 0. They are orthogonal with respect to the normal density function ϕ(x).
 
7
For example, Fisher’s linear discrimination function when there are only two groups.
 
8
If W k is the total within-groups sums-of-squares obtained using k-means, with k clusters, then it is acceptable to take k + 1 clusters if \((n-k+1) \left ( -1 + W_k/W_{k+1} \right ) > 0\).
 
9
Note that when a 1 = (1,  0, …0), …a m = (0, …,  0,  1) and α k = 1, k = 1, …m, then model (11.40) is known as an additive model.
 
10
When p = 2, the solution can be easily obtained exactly.
 
Literature
Bock H-H (1986) Multidimensional scaling in the framework of cluster analysis. In: Degens P, Hermes H-J, Opitz O (eds) Studien Zur Klasszfikation. INDEKS-Verlag, Frankfurt, pp 247–258
Bock H-H (1987) On the interface between cluster analysis, principal component analysis, and multidimensional scaling. In: Bozdogan H, Kupta AK (eds) Multivariate statistical modelling and data analysis. Reidel, Boston
Bolton RJ, Krzanowski WJ (2003) Projection pursuit clutering for exploratory data analysis. J Comput Graph Statist 12:121–142CrossRef
Chan JCL, Shi J-E (1997) Application of projection-pursuit principal component analysis method to climate studies. Int J Climatol 17(1):103–113CrossRef
Christiansen B (2009) Is the atmosphere interesting? A projection pursuit study of the circulation in the northern hemisphere winter. J Climate 22:1239–1254CrossRef
Cook D, Buja A, Cabrera J (1993) Projection pursuit indices based on expansions with orthonormal functions. J Comput Graph Statist 2:225–250CrossRef
Cover TM, Thomas JA (1991) Elements of information theory. Wiley Series in Telecommunication. Wiley, New York
Diaconis P, Freedman D (1984) Asymptotics of graphical projection pursuit. Ann Statist 12:793–815CrossRef
Eslava G, Marriott FHC (1994) Some criteria for projection pursuit. Stat Comput 4:13–20CrossRef
Franzke C, Majda AJ, Branstator G (2007) The origin of nonlinear signatures of planetary wave dynamics: Mean phase space tendencies and contributions from non-Gaussianity. J Atmos Sci 64:3987–4003CrossRef
Friedman JH, Tukey JW (1974) A projection pursuit algorithm for exploratory data analysis. IEEE Trans Comput C23:881–890CrossRef
Friedman JH, Stuetzle W, Schroeder A (1984) Projection pursuit density estimation. J Am Statist Assoc 79:599–608CrossRef
Friedman JH (1987) Exploratory projection pursuit. J Am. Statist Assoc 82:249–266CrossRef
Gordon AD (1981) Classification: methods for the exploratory analysis of multivariate data. Chapman and Hall, London
Hall, P (1989) On polynomial-based projection indices for exploratory projection pursuit. Ann Statist 17:589–605CrossRef
Hamming RW (1980) Coding and information theory. Prentice-Hall, Englewood Cliffs, New Jersey
Hannachi A, Turner GA (2013b) 20th century intraseasonal Asian monsoon dynamics viewed from isomap. Nonlin Process Geophys 20:725–741CrossRef
Hannachi A, Stephenson DB, Sperber KR (2003) Probability-based methods for quantifying nonlinearity in the ENSO. Climate Dynamics 20:241–256CrossRef
Hartigan JA (1975) Clutering algorithms. Wiley, New York
Heiser WJ, Groenen PJF (1997) Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima. Psychometrika 62:63–83CrossRef
Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Statist Soc B 52:105–124
Huber PJ (1985) Projection pursuit. Ann Statist 13:435–475
Hyvärinen A (1998) New approximations of differential entropy for independent component analysis and projection. In: Jordan MA, Kearns MJ, Solla SA (eds) Advances in neural information processing systems, vol 10. MIT Press, Cambridge, MA, pp 273–279
Jones MC (1983) The projection pursuit algorithm for exploratory data analysis. Ph.D. Thesis, University of Bath
Jones MC, Sibson R (1987) What is projection pursuit? J R Statist Soc A 150:1–36CrossRef
Kendall MG, Stuart A (1977) The advanced Theory of Statistics. Volume 1: distribution theory, 4th edn. Griffin, London
Kruskal JB (1969) Toward a practical method which helps uncover the structure of a set of multivariate observations by finding the linear transformation which optimizes a new ‘index of condensation’. In: Milton RC, Nelder JA (eds) Statistical computation, New York
Kruskal JB (1972) Linear transformations of multivariate data to reveal clustering. In: Multidimensional scaling: theory and application in the behavioural sciences, I, theory. Seminra Press, New York
Kwon S (1999) Clutering in multivariate data: visualization, case and variable reduction. Ph.D. Thesis, Iowa State University
Mardia KV (1980) Tests of univariate and multivariate normality. In: Krishnaiah PR (ed) Handbook of statistics 1: Analysis of variance. North-Holland Publishing, pp 279–320
McEliece RJ (1977) The theory of information and coding. Addison-Wesley, Reading, MA
Monahan AH, DelSole T (2009) Information theoretic measures of dependence, compactness, and non-Gaussianity for multivariate probability distributions. Nonlin Process Geophys 16:57–64CrossRef
Monahan AH, Fyfe CJ (2007) Comment on the shortcomings of nonlinear principal component analysis in identifying circulation regimes. J Climate 20:374–377CrossRef
Monahan AH (2001) Nonlinear principal component analysis: tropical Indo–Pacific sea surface temperature and sea level pressure. J Climate 14:219–233CrossRef
Nason G (1992) Design and choice of projection indices. Ph.D. Thesis, The University of Bath
Nason G (1995) Three-dimensional projection pursuit. Appl Statist 44:411–430CrossRef
Pasmanter RA, Selten MF (2010) Decomposing data sets into skewness modes. Physica D 239:1503–1508CrossRef
Posse C (1995) Tools for two-dimensional exploratory projection pursuit. J Comput Graph Statist 4:83–100
Rényi A (1961) On measures of entropy and information. In: Neyman J (ed) Proceedings of the Fourth Bekeley symposium on mathematical statistics and probability, vol I. The University of California Press, Berkeley, pp 547–561
Rennert KJ, Wallace MJ (2009) Cross-frequency coupling, skewness and blocking in the Northern Hemisphere winter circulation. J Climate 22:5650–5666CrossRef
Ross SM (1998) A first course in probability, 5th edn. Prentice-Hall, New Jersey
Sammon JW Jr (1969) A nonlinear mapping for data structure analysis. IEEE Trans Comput C-18:401–409CrossRef
Scott DW (1992) Multivariate density estimation: theory, practice, and vizualization. Wiley, New YorkCrossRef
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656CrossRef
Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London
Sura P, Hannachi A (2015) Perspectives of non-Gaussianity in atmospheric synoptic and low-frequency variability. J Cliamte 28:5091–5114CrossRef
Tukey PA, Tukey JW (1981) Preparation, prechosen sequences of views. In: Barnett V (ed) Interpreting multivariate data. Wiley, Chichester, pp 189–213
Vasicek O (1976) A test for normality based on sample entropy. J R Statist Soc B 38:54–59
Metadata
Title
Projection Pursuit
Author
Abdelwaheb Hannachi
Copyright Year
2021
Book
Patterns Identification and Data Mining in Weather and Climate

Print ISBN: 978-3-030-67072-6

Electronic ISBN: 978-3-030-67073-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-67073-3_11

Premium Partner

    Image Credits