Functional Data Analysis

Transcription

1 FDA 1-1 Functional Data Analysis Michal Benko Institut für Statistik und Ökonometrie Humboldt-Universität zu Berlin

2 FDA 1-2 Outline of this talk: Introduction Turning discrete values into a functional datum Basic functional statistics Functional PCA CPCA/pCPCA Application: Volasurfaces

3 FDA 1-3 Introduction Analyzing FUNCTIONAL DATA data vectors functions functional data: x(t), t J {x(t j ), j = 1, 2,... n} x(t), t J

4 FDA 1-4 Basic Functional Statistics N Mean Function: x = 1 N x i (t) i=1 N Variance Function: var(x(t)) = 1 N (x i (t) x(t)) 2 i=1 i=1 Covariance Function: N cov(x(t), y(t)) = 1 N (x i (t) x(t)) (y i (t) y(t)) Inner Product: < x, y >= x(t)y(t)dt

5 FDA 1-5 Turning Discrete Values into a Functional Datum Measuring discrete values without errors interpolation basis (Fourier) expansions Measuring discrete values with error smoothing methods Storage of functional data set?

6 FDA 1-6 Storage of the Functional Data Storing original raw data -e.g. by interpolation Storing functional data in terms of a suitable basis

7 FDA 1-7 Basis Expansions 1. Choice of a GOOD base Fourier series {1, sin(rwt), cos(rwt), r = 1, 2,...} Polynomial base φ k (t) = (t ω) k, k = 0,... K other: Splines, Wavelets 2. Choice of dimension

8 FDA 1-8 Suppose the base functions: φ(t) = (φ 1 (t),..., φ K (t)) Observed functions can be expanded: x i (t) = K C ij φ k (t) = Ci φ(t) k=1

9 FDA 1-9 Basic statistics can be expressed: N x(t) = N 1 Ci φ(t) = C φ(t) i=1 N var(t) = N 1 φ(t) (C i C)(C i C) φ(t) i=1 N cov(s, t) = N 1 φ(s) (C i C)(C i C) φ(t) i=1 ( ) < x i (t), x j (t) > = Ci φ(s)φ(s) ds C j

10 FDA 1-10 XploRe: Solution coef=fouriertrans(tmat,n) Calculates the coefficient in applying a basis expansion by using Fourier series. In this case, it is assumed that the data unit can be expressed by a linear combination of finite terms of sine and cosine functions. Fourier1.xpl

11 FDA 1-11 Simple Functional Data Analysis x i (t), t J z i = ξ(t)x i (t) max ξ (var(z i )) in functional sence: max cov(s, t)ξ(s)ξ(t)dsdt Under constrain ξ l (t) 2 dt, ξ l (t)ξ m (t) = 0(l < m)

12 FDA 1-12 SFDA - Computational Method Discretizing the functions Discretize the observed functions to a fine grid (e.g.equally spaced) use standard multivariate PCA Earliest approach to FPCA Basis expansion W = φ φ ξ(t) = φ(t) b - basis expansion N 1 C CW b = ρb, with b W b = 1,b 1W b 2 = 0

13 FDA 1-13 Penalized Principal Component Analyses Genaralization of the (Simple) PCA Giving Smooth Conditions on PC, using Roughness Penalty Roughness Penalty e.g. {ξ }dt Condition in PCA: {ξ(t)} 2 dt = 1 {ξ(t)} 2 dt + α {ξ (t)} 2 dt = 1 ξ i (t)ξ j (t)dt = 0 ξ i (t)ξ j (t)dt + α ξ i (t)ξ j (t)dt = 0

14 FDA 1-14 XploRe Solution values,varprop,scores,harmcoef=fdapca(coef,period,lambda,npc) Carries out a penalized functional principal component analysis (PCA) based on the coefficient matrix for functional data. It is possible to choose a smoothing parameter objectively. fda1.xpl

15 FDA 1-15 Common Principal Components CPC model:ψ i = βλ i β, i = 1,..., k Ψ i = λ i1 β 1 β 1 + λ i2 β 2 β λ ip β p 1β p, i = 1,..., k Corresponds to the simultaneous decomposition Level of the similarities amoung several covariance Matrices/Functions

16 FDA 1-16 Partial Common Principal Components For fixed positive integer q < p 1: Ψ i = λ i1 β 1 β λ iq β q β q + λ i,q+1 β (i) q+1 β (i) q λ ipβ p (i) β p (i), i = 1,..., k

17 FDA 1-17 First q components are common, p q differ Corresponds to the PCA (only first q components relevant ) Vector notation: β (i) = (β 1,..., β q, β (i) q+1,..., β(i) p ) Ψ i = β (i) Λ i β (i) Estimation: Maximum Likelihood Estimation GOAL:Implement the CPC into functional data world

18 FDA 1-18 Application: Implied Volatility Modeling Financial Dynamics Black Scholes Modell Volatility Surface {ˆσ t (k, τ} T t=1 Where: k = K S t -moneyness, τ-maturity -use appropriate transformation

19 FDA 1-19 Functional Data Approach 1. Set appropriate (finite) grid for maturity τ {τ 1,..., τ N } 2. Obtain functional data σ τ (k), τ J 3. Apply PPCA, CPCA to this functional Dataset

20 FDA 1-20 Problems: Volas are Surfaces, not functions!!! 2-dimensional Smoothing / 1-dimensional Smoothing? How to choose the Grid for maturity? Smoothing Parameter Selection (Sensitivity)? Technical Questions?

21 FDA 1-21 Conclusions - Goals 1. Literature seeking 2. XploRe Implementation Basic FDA Statistics Visualisation of FPCA CPCA/FCPCA? 3. Inference for FPCA/CFPCA 4. Application: Volas/Term structure?

22 FDA 1-22 References [1] Ramsay, J.O., Silvermann, B.W Functional Data Analysis, Springer, New York. [2] Ramsay, J.O., Silvermann, B.W Applied Functional Data Analysis, Springer, New York. [3] Flury B Common Principal Components and Related Multivariate Models, Wiley, New York [4] Fengler M.R., Härdle W., Schmidt P The Analysis of Implied Volatilities,HU Berlin. [5] Fengler M.R., Härdle W., Villa C The Dynamics of Implied Volatilities,HU Berlin. [6] Yamanishi Y Studies on Principal and Regression Analyses for Functional Data, Okoyama University