Variable selection in nonparametric functional concurrent regression

We develop a new method for variable selection in nonparametric functional concurrent regression. The commonly used linear model (FLCM) is far too restrictive assuming linearity of the covariate effects, which not necessarily true many real-world applications. (NPFCM), on other hand, much more flexible and can capture complex dynamic relationships present between response covariates. extend classically methods, e.g., group LASSO, SCAD MCP, to perform NPFCM. show via numerical simulations that proposed with non-convex penalties identify predictors minimal false-positive rate negligible false-negative rate. also provides better out-of-sample prediction accuracy compared FLCM presence nonlinear effects predictors. method's application demonstrated by identifying influential predictor variables two real data studies: dietary calcium absorption study, some bike-sharing data. Cet article propose une nouvelle méthode de sélection dans un contexte modèles concurrents fonctionnels non paramétriques. Comme l'hypothèse linéarité des effets covariables qu'impose modèle fonctionnel linéaire usuel soit trop pour que ce dernier valide et utile applications réelles, les auteurs travail ont opté paramétriques (NPFCM). Ces sont bien plus souples capables capturer relations dynamiques complexes liant réponse ses covariables. Ils ont, donc, repris méthodes classiques, telles: MCP groupes étendre aux type Des numérqiues montrent clairement la proposée avec pénalités convexes peut identifier véritables prédicteurs taux faux positifs négatifs négligeable. Aussi, en présence d'effets linéaires fonctionnels, elle fournit meilleure précision prédiction hors échantillon comparativement à celle d'un concurent linéaire. En guise d'illustration pratique, été appliquée deux jeux données, l'un portant sur l'absorption alimentaire l'autre système partage vélos. Function-on-function regression refers class models both covariates being functions over continuous domain, such as time. particular function-on-function regression, where value at current time point depends only values specific point. most (Ramsay & Silverman, 2005), assumes relationship modelled using smooth univariate functions. These time-varying effect response. Estimation inference closely related varying coefficient (Hastie Tibshirani, 1993) have been widely studied literature. Although useful finds longitudinal analysis, it (NPFCM) overcomes this limitation specifying general modelling purposes. Multiple methods exist literature estimation NPFCM smoothing splines (Kim al. 2018a), Gaussian process (Shi 2005) local kernel techniques (Jiang Wang, 2011); see Maity (2017) references therein review various regarding With advent modern high-throughput technologies, structures recently become high-dimensional nature, has motivated researchers underlying set Several analysis scalar-on-function (Gertheiss 2013; Fan 2015) function-on-scalar (Chen 2016; Parodi Reimherr, 2018) models. However, relatively sparse. Goldsmith Schwartz variational Bayes approach. Recently, Ghosal (2020) developed extending penalized methods. these are FLCM, might actual In article, we consider an additive 2018b), extension generalized (FGAM) (McLean 2014) responses. Scheipl (2016) extended non-Gaussian responses, allowing nested or crossed random effects. NPFCM, like LASSO (Tibshirani, 1996), (Fan Li, 2001) (Zhang, 2010). To best our knowledge, ours first work follow approach problem reduces (Yuan Lin, 2006) its natural extension, namely problem. Through simulations, illustrate (FPR) (FNR). shown produce satisfactory performance even when sparsely observed contaminated measurement error. Comparisons provided potential gain terms predictive demonstrate study (Davis, 2002) bikeshare (Fanaee-T Gama, number daily bike rentals Washington, D.C., while taking into account diurnal meteorological covariates, temperature, humidity wind speed. employed relevant factors simultaneously estimating their rest organized following way. framework, up Section 2. 3, simulation studies evaluate empirical method. 4, presented. 5, deliberate contribution technical aspects future extensions based research. Remark 1.For construction B X j , k ( x ) (B-spline basis), use equally spaced knots range { i t } = 1 n . tuning parameters Kj Lj (the basis each direction) penalty parameter ? selected BIC (EBIC) Chen, 2008) corresponding likelihood. make no distributional assumption, performance, illustrated studies. Some information criterion data-driven be used, depending same K, L 1, … p, standard assumption computational tractability all less level smoothness variables. For ?, 4 3 recommended original authors grpreg package (Breheny Huang, R (R Core Team, implementation As case irregular sparse grid possibly This situation arises frequently there few subject-specific visits. Here {(Yi(tij), tij), 2, mi}, given {(U1(t1ij), t1ij), m1i}, {(U2(t2ij), t2ij), m2i},…,{(Up(tpij), tpij), mpi}. denote Uk(tkij)s, (k p) Uijk, represent error, i.e., Uijk Xk(tkij) + eijk n, mki errors assumed white noises zero mean variance ? 2 assume that, although individual observations mi small, ? m dense [0, T] 2018a; 2020). Under set-up, eigenvalues eigenfunctions curves estimated principal component (FPCA) (Yao 2005). scores through conditional expectation, finally estimates put together Karhunen–Loeve expansion get trajectory ^ · Xik(·) ? ? s S ? ? ), chosen percent explained (PVE) criterion. Hence, treat Y p performing selection. One problems likely arise irregularly B-splines any support, particularly if image Xj(t) ?. overcome limitation, employ point-wise centring scaling Xj(t), Kim (2018a). particular, transform ? ? deviation Xj(t). transformed interpreted how amount away from t. normalization, all, brings comparable scale. Second, importantly, since bounded (at least stochastically), facilitates ). practice, estimate FPCA step denoised Our shows good above strategy; henceforth, throughout pre-process (denoise standardize) before applying overall procedure presented algorithm Algorithm 1. section, investigate simulations. performance. compare (Ghosal designs: Scenario B: similar generation set-up (7) / 100 F exp 5 sin ? 16 Fj{x, t} 0 6, 20. scenario, highly unlike scenario A, specifically sample size 200 scenario. C: additional 50 40. F1(· ·), F2(· ·) F3(· 50. design kept exactly A. scenarios, generate 500 replicated datasets specified assess A: apply (PVE 99%) discussed 2.3 obtain standardized then note three criteria 2.2. repeat across Monte Carlo replications percentage predictor, reflecting true-positive (TPR) FPR percentages reported Table observe penalization (group MCP) pick out X1(·), X2(·), X3(·) 100% replications, illustrating TPR exhibits small false around 0.2% method, FPRs sizes words, they able replications. average 3.002–3.006 different sizes. successfully primary goal benefit comparing want what gained All them FLCM. results displayed 3. X2(·) (X3(·)) captured 69% (73.2%) penalty, 58.4% (64.6%) 50.8% (60%) MCP. quite large (30%–50%) FNR considerably high (29%, 21%, 16%) comparison, captures penalties. higher (87%–99%). SCAD- MCP-based exhibit (<1%). suitable predictors, remains along FPR. provide comparison intuitively should thus may 4. notice median R2 approximately 60% whereas 99% produces superior illustrates > n. found 100%. SCAD, 0.39%, 0% 0.008%, respectively. section having gives minuscule FNR, therefore article. added pseudo-variables illustration Pseudo-variables (Miller, 2002; Wu 2007) tune procedures. Next, capital casual rentals. Davis (2002). Y(t), intake X1(t), body mass index (BMI) X2(t) surface area (BSA) X3(t) 188 patients points 35 64 years age (t). patient, repeated measurements varies Figure displays patients' absorption, intake, BSA BMI function ages. primarily interested finding influence Y(t). expect associated prior FPCA-based pre-processing 95%) trajectories further add 15 simulating ? i.i.d. (j 18) b cos ???? Finally, Y(t) 18 standardized). whole times (B 100). X1(t) significant iterations. including randomly generated completely ignored So does job discarding pseudo-variables. select effect. purpose, without adding (adjusted main t) obtained 2020), comparison. (standardized) linear, matches findings fixed moderate (age), (calcium intake) increases (or than t), F(x, decreasing regions, indicating increases, decrease A observation was made (2020), one needs careful interpreting near boundaries due lack observations. comprehensive understanding association intake. 2.To happens correlation truly pseudo-variables, considered alternative Zj(t) were conditionally Z1(t) normal distribution ? Z introduce among constants a, taken scalar samples applied identical result demonstrates distinguish irrelevant ones them. Capital Fanaee-T Gama (2014). initially system contain temperature (temp), feels-like (atemp), (hum) speed (windspeed) hourly basis. Casual defined cyclists membership program. Since dynamics weekends weekdays, (2018b), restrict attention Saturdays. counts (hourly) available during period January 2011 31 December 2012, total 105 Saturdays barring exceptions (8 missing). It plausible real-time rentals, selecting well discrete skewed, log transformation (Y(·) log{Y(·) 1}). existing generating again done order X2(t), X3(t), X4(t) 2.3, 20 several 6 Also reported, FMCP 2020) pre-whitening (PW NPW, respectively). selects time, correlated, ideal picking consistently. FPR, discard uses (atemp) pre-whitening, four pre-whitening. case, positives calculated 4.85. idea, covariance matrix becomes mis-specified. (with pre-whitening) facilitate Nevertheless, performs irrespective accuracy. (no added). (in scale) reported. maximum mid-range (near mean) midday, condition biking. cold warm temperatures, decreases, natural. impact uniform day, expected weather conditions D.C. There negative humidity, decreases increasing humidity. Jim (2018b). lower (lower time), midday positive. (higher mean), drop effect, humid days fall would naturally conditions. speed, except regions bikers face difficulties. (plausibly hottest day), noticeable dip, attributed difficulty cycling amidst high-velocity winds. match those Gebhart Noland (2014), who multiple evident nonlinear. efficiency do biased lead inferior metric, mean-square error test data, MSPE ? training (80%) testing (20%) iterations Boxplots 5. smaller demonstrating advantage possible driving accurately providing 3.The pre-whitened FMCP, unable produced non-pre-whitened version used. unified framework identifies (SCAD accurate classical known oracle property 2001; Zhang, 2010) under regularity representation Equation (3) could establish properties regularized estimators Fj(· space spanned tensor product splines. way Chen (2016), established context readily time-varying. far, almost entire 2017; focused limitations forecasting outside (De Boor, 1978) will observation. Note aspect unique general, spline-based extrapolation beyond troublesome care (e.g., Eilers Marx, followed suggested (2018a) Alternatively, suggestions FGAM 2014), transformations ? I < ? ? h (where cdf ht user-chosen bandwidth) X(t) so cover 1] uniformly. Another working strategy, often practitioners, include “ghost” aim extrapolating (Eilers 2010; Kosinka 2014; Corlay, 2016) supplementing infinite support proposal Schumaker (2007). independent structure accuracy, objective take temporal dependence within similarly (2020). mis-specified, counterproductive, analysis. An challenge handle complexity arising initial step. plenty research remain explored work. interest small-sample measure uncertainty. perturbation-based bootstrap (Das 2019) interesting explore situations. distribution-free inference, split conformal bands (Lei 2015). historical lagged Extending classes diverse areas thank editor, associate editor anonymous referees valuable input suggestions, greatly helped improve Illustrations R, analyzed, Github https://github.com/rahulfrodo/NPFCM_Selection.