Franco Villoria, Maria

Ignaccolo, Rosaria

English

FRANCO VILLORIA, Maria

Institutional Research Information System University of Turin

 Uncertainty evaluation for functional kriging:an application to the Canadian temperature data setRosaria Ignaccolo*University of Torino, Torino, Italy - rosaria.ignaccolo@unito.itMaria Franco-VilloriaUniversity of Torino, Torino, Italy - maria.francovilloria@unito.itAbstractUncertainty evaluation for functional spatial prediction remains an open issue. Prediction of a curve atan unmonitored location can be obtained using a functional kriging with external drift model that takesinto account the effect of exogenous variables (either scalar or functional). To evaluate the uncertainty ofa predicted curve, a semi-parametric bootstrap for spatially correlated data is adapted to functional data.Confidence bands are obtained by ordering the bootstrapped predicted curves in two different ways accordingto band depth and L2 distance. The proposed approach is illustrated on a well known data set of Canadiantemperature.Keywords: bootstrap; P-spline; band depth; prediction bands.1. IntroductionThe use of geostatistical techniques for functional data was first addressed in the seminal paper by Goulard& Voltz (1993), but it has received increasing interest over the last few years, motivated by the availability ofcomplex data sets, where data are collected over space and e.g. time or depth, giving rise to data that can beconsidered as spatially dependent curves (Horva´th & Kokoszka (2012)). With the aim of predicting a wholecurve at an unmonitored location, several authors have extended ordinary kriging to the case of functionaldata, where the mean function is assumed to be constant (see e.g. Delicado et al. (2010), Giraldo et al.(2011), Nerini et al. (2010)). A slightly more complex alternative that allows the mean function to dependon longitude and latitude was considered by Caballero et al. (2013) and Menafoglio et al. (2013). Further,Ignaccolo et al. (2014) proposed the so called functional kriging with external drift, that allows the meanfunction to depend on scalar and/or functional exogenous variables.When predicting a curve, it is important to provide as well a measure of its uncertainty; however, literatureconcerning prediction uncertainty in the case of functional data is still very limited. An interesting approachto this matter, given the lack of distribution functions, is to use resampling methods. Ferraty et al. (2010)proposed the use of ‘wild bootstrapping’ for functional regression but with a scalar response, where thebootstrap samples are obtained multiplying the scalar errors by iid random variables constrained to satisfycertain properties. We propose a semi-parametric bootstrap approach for spatially correlated functional datathat allows to obtain uncertainty bands for a predicted curve, in the case of kriging with external drift withfunctional response. We evaluate the performance of the proposed methodology on a well known data set.2. Functional kriging with external drift (FKED)Let Υs = {Ys(t); t ∈ T} be a functional random variable observed at location s ∈ D ⊆ Rd, whose realizationis a function of t ∈ T , where T is a compact subset of R. Assume that we observe a sample of curves Υsi ,for si ∈ D, i = 1, . . . , n, that take values in a separable Hilbert space of square integrable functions. The set{Υs, s ∈ D} constitutes a functional random field or a spatial functional process (Delicado et al. (2010)), thatcan be non-stationary and whose elements are supposed to follow the model Υs = µs + s. The term µs isinterpreted as a drift describing a spatial trend while s represents a residual random field that is zero-mean,second-order stationary and isotropic. At the generic site si, i = 1, . . . , n, and at point t, the model can berewritten as a functional concurrent linear model Ysi(t) = µsi(t) + si(t) with the driftµsi(t) = α(t) +∑pγp(t)Cp,i +∑qβq(t)Xq,i(t) (1)Proceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2713                               1 / 6 where α(t) is a functional intercept, Cp,i is the p-th scalar (constant-in-time) covariate at site si, Xq,i is theq-th functional covariate at site si, γp(t) and βq(t) are the covariate coefficients and si(t) represents the resid-ual spatial functional process {s(t), t ∈ T, s ∈ D} at the site si. Once the functional regression model (1) hasbeen fitted by means of a GAM representation (for details see Ignaccolo et al. (2014)), the functional residualsesi(t) = Ysi(t)−µˆsi(t) can be used to predict the residual curve at an unmonitored site s0 via ordinary krigingfor functional data (Giraldo et al. (2011)), according to which eˆs0(t) =∑ni=1 λiesi(t), with kriging coefficientsλi ∈ R. More complex alternatives, where the kriging coefficients are not constant are available (Ignaccolo etal. (2014)). The prediction at the unmonitored site s0 is obtained by adding up, as in the classical regressionkriging, the two terms, i.e. Yˆs0(t) = µˆs0(t) + eˆs0(t), where µˆs0(t) = αˆ(t) +∑p γˆp(t)Cp,0 +∑q βˆq(t)Xq,0(t)depends on the covariate values Cp,0 and Xq,0(·) at the site s0.3. Uncertainty evaluationThe following bootstrapping algorithm is an extension of the work by Iranpanah et al. (2011) to the func-tional workframe. It allows to obtain uncertainty bands for a predicted curve Yˆs0(t) at an unmonitored site s0.Algorithm1. Estimate the drift µs following Model (1) and subtract it from the observed data to obtain the functionalresiduals si(t) = Ysi(t)− µˆsi(t).2. Estimate the residual covariance matrix Σ by means of the trace-semivariogram (Giraldo et al. (2011)),estimated as:υˆ(h) =12 |N(h)|∑i,j∈N(h)∫T(si(t)− sj (t))2dtwhere N(h) = {(si, sj) : ‖si − sj‖ = h}. Once estimated, the empirical trace-semivariogram providesa cloud of points (hg, υˆ(hg)), g = 1, . . . , G to which a parametric model (e.g. exponential, spherical,Mate´rn) can be fitted as in classical geostatistics. Decompose the estimated covariance matrix using itsCholesky decomposition as Σˆ = LˆLˆT and transform the functional residuals to make them (spatiallyuncorrelated):ζn×M = (ζ(s1), . . . , ζ(sn))′= Lˆ−1n×n (Yn×M − µˆn×M ) . (2)3. Generate B bootstrap samples ζ∗(s1), . . . , ζ∗(sn) from ζ(s1), . . . , ζ(sn) using the smoothed bootstrapas suggested in Cuevas et al. (2006); here Fn, the empirical distribution function of {ζ(s1), . . . , ζ(sn)},is replaced by a smooth version Fˆn to avoid appearance of repeated measures. In practice a booststrapsample ζ∗(s1), . . . , ζ∗(sn) from Fˆn can be obtained asζ∗si(tj) = ζ0si(tj) + Zi(tj), j = 1, . . . ,M,where ζ0(si) is drawn from Fn, (Z(t1), . . . , Z(tM )) ∼ MVN(0M , κΣζ), Σζ is the covariance matrix of{ζ(t)} and κ is a smoothing parameter.4. The final bootstrap sample Y ∗s1 , . . . , Y∗sB is obtained using an inverse transform:Y ∗si(t) = µˆsi(t) + Lˆζ∗si(t).The bootstrap samples are then fed into the FKED method to obtain B prediction curves at the unmonitoredlocation s0. The 2.5th and 97.5th quantiles of the distribution of these B curves will determine the upperand lower limits of the uncertainty band for the predicted curve Yˆs0(t). To do so, the B curves need to beordered. There is not a unique way of ordering functional data; here we consider two ordering techniques:band depth (Lopez-Pintado & Romo (2009)) and L2 distance between curves.The band in R2 delimited by k = 2 curves is defined as:B(yi1 , yi2) = {(t, y(t)) : t ∈ T,minr=1,2yir (t) ≤ y(t) ≤ maxr=1,2yir (t)} .Proceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2714                               2 / 6 The sample band depth (BD) of a curve y(t) can be calculated asBDn,2(y) =(n2)−1 ∑1≤i1<i2≤nI {G(y) ⊆ B(yi1 , yi2)} .i.e. the proportion of bands delimited by 2 curves containing the whole graph G(y) = {(t, y(t)) : t ∈ I} ofy(t). The bigger the band depth value, the more central the curve is. Band depth can be calculated forbands delimited by more than two curves, but the order induced when k > 2 remains stable (Lopez-Pintado& Romo (2009)), and using k = 2 is computationally more efficient. To avoid problems such as ties andcrossing over of the curves delimiting the band, that may happen for k = 2, a modified version is available;the modified band depth (MBD) can be estimated as:MBDn,2(y) =(n2)−1 ∑1≤i1<i2≤nλ ({t ∈ T : minr=i1,i2yr(t) ≤ y(t) ≤ maxr=i1,i2yr(t)})λ (T )where λ is the Lebesgue measure on T, with the difference that the MBD takes into account whether aportion of the curve is in the band (for further details see Lopez-Pintado & Romo (2009)).Alternatively, the curves can be ordered based on their L2 distance from the originally predicted curve Yˆs0(t)(Cuevas et al. (2006)), where distance between two curves x and y is calculated as:||x− y|| =(∫T(x(t)− y(t))2 dt)1/2. (3)The lower/upper limits of a 95% confidence band (based on band depth) are obtained by taking the pointwise(w.r.t. t) minimum/maximum of the 95% deepest curves (i.e. those closest to the center of the distribution).On the other hand, the 95% confidence ball (based on L2 distance) is made of the 95% curves closest to theoriginal FKED predicted curve.4. Canadian temperature data setWe illustrate our proposal for uncertainty evaluation on a well known data set, the Canadian temperaturedata, that has been widely used in the functional data literature (see, for example, Ramsay & Silverman(2006), Giraldo et al. (2010), Menafoglio et al. (2013), Scheipl et al. (2013)). The data set consists of dailyannual mean temperature collected at 35 meteorological stations in Canada’s Maritimes Provinces over theperiod 1960-1994 (Figure 1). For further details on the dataset, the reader is referred to one of the referencesabove. Data were converted to functional observations through smoothing by using penalized cubic B-splineswith 120 basis functions and penalty parameter equal to zero. These values were chosen using functionalcross-validation. We have chosen five locations at random and used them as validation stations. These canbe seen in red in Figure 1(a). The FKED model, with longitude and latitude as covariates, was then fittedto the remaining 30 stations and predicted temperature curves were obtained for the 5 validation stations.For each of these sites, a bootstrap sample of predicted curves of size B = 1000 was obtained following thealgorithm illustrated in Section 3. Band depth was calculated using the modified version MBD. The resultingprediction balls/bands are shown in Figure 2. The uncertainty bands are fairly narrow, as expected whenobserving the small variability between curves in Figure 1(b). Overall, the two uncertainty measures seemto agree well, although in some cases the confidence ball appears to be slightly narrower than the confidenceband. The bands appear to be wider for Station 1 than for the remaining stations, specially in winter,suggesting greater uncertainty in that station. This may be due to the fact that this is an inland station,while the other four validation stations are closer to the coast.5. Conclusions and future workSpatial functional analysis provides an interesting alternative to spatio-temporal modelling, allowing to pre-dict a whole curve taking into account exogenous covariates and the underlying spatial structure. Here, wepropose a bootstrap approach to estimate uncertainty bands for a predicted curve has been proposed, andits validity has been illustrated in a well known data set. We are currently running a simulation study toProceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2715                               3 / 6 evaluate the performance of the method proposed. Numerical results will be presented at the conference.ReferencesCaballero, W., Giraldo, R. & Mateu, J. (2013). A universal kriging approach for spatial functional data.Stochastic Environmental Research and Risk Assessment, 27(7), 1553–1563.Cuevas, A., Febrero, M. & Fraiman, R. (2006). On the use of the bootstrap for estimating functions withfunctional data. Comput. Statist. Data Anal., 51, 1063–1074.Delicado, P., Giraldo, R., Comas, C. & Mateu, J. (2010). Statistics for spatial functional data: some recentcontributions. Environmentrics, 21, 224–239.Ferraty, F., Van Keilegom, I. & Vieu, P. (2010) On the validity of the bootstrap in non-parametric functionalregression. Scand. J. Statist., 37, 286-306.Giraldo, R., Delicado, P. & Mateu, J. (2011). Ordinary kriging for function-valued spatial data. Environ-mental and Ecological Statistics, 18(3), 411–426.Goulard, M. & Voltz, M. (1993). Geostatistical interpolation of curves: A case study in soil science. In A.Soares (Ed.), Geostatistics Troia 92, Volume 2, pp. 805–816, Kluwer Academic, Dordrecht.Horva´th, L. & Kokoszka, P. (2012). Inference for functional data with applications. Springer, New York.Ignaccolo, R., Mateu, J. & Giraldo, R. (2014) Kriging with external drift for functional data for air qualitymonitoring. Stochastic Environmental Research and Risk Assessment, 28(5), 1171-1186.Iranpanah, N., Mohammadzadeh, M. & Taylor, C. C. (2011). A comparison of block and semi-parametricbootstrap methods for variance estimation in spatial statistics. Comput. Statist. Data Anal., 55, 578–587.Lopez-Pintado, S. & Romo, J. (2009). On the concept of depth for Functional Data. J. Amer. Statist.Assoc., 104(486), 718–734.Menafoglio, A., Secchi, P. & Dalla Rosa, M. (2013) A Universal Kriging predictor for spatially dependentfunctional data of a Hilbert Space. Electronic Journal of Statistics, 7, 2209–2240.Nerini, D., Monestiez, P. & Mante´, C. (2010). Cokriging for spatial functional data. J. Multivariate Anal.,101, 409-418.Ramsay, J. & Silverman, B. W. (2006) Functional Data Analysis. Springer, New York.Scheipl, F., Staicu, A. M. & Greven, S. (2013) Functional additive mixed models. arXiv:stat/1207.5947v5Proceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2716                               4 / 6 Figure 1: (a) Locations of the 35 meteorological stations in Canada’s Maritimes Provinces area(validation stations numbered in red). (b) Temperature curves (raw data)Proceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2717                               5 / 6 Figure 2: Original data (black dots), FKED predicted curve (red line), 95% prediction ball (pink)based on L2 distance and 95% prediction band (blue line) based on MBD for validation stationsProceedings of the 60th ISI World Statistics Congress, 26-31 July 2015, Rio de Janeiro, Brazil p.2718Powered by TCPDF (www.tcpdf.org)                               6 / 6

Uncertainty evaluation for functional kriging: an application to the Canadian temperature data set

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Uncertainty evaluation for functional kriging: an application to the Canadian temperature data set

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