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20240925115408-2bcbc100d132c1729a2f1d6c6a7d85c8330b1f83 + 20240925115408 + + JOSS Admin + admin@theoj.org + + The Open Journal + + + + + Journal of Open Source Software + JOSS + 2475-9066 + + 10.21105/joss + https://joss.theoj.org + + + + + 09 + 2024 + + + 9 + + 101 + + + + LocalCop: An R package for local likelihood inference +for conditional copulas + + + + Elif Fidan + Acar + https://orcid.org/0000-0003-2908-7691 + + + Martin + Lysy + https://orcid.org/0000-0001-9974-1121 + + + Alan + Kuchinsky + + + + 09 + 25 + 2024 + + + 6744 + + + 10.21105/joss.06744 + + + http://creativecommons.org/licenses/by/4.0/ + http://creativecommons.org/licenses/by/4.0/ + http://creativecommons.org/licenses/by/4.0/ + + + + Software archive + 10.5281/zenodo.13801203 + + + GitHub review issue + https://github.com/openjournals/joss-reviews/issues/6744 + + + + 10.21105/joss.06744 + https://joss.theoj.org/papers/10.21105/joss.06744 + + + https://joss.theoj.org/papers/10.21105/joss.06744.pdf + + + + + + Generalized additive models: An introduction +with R + Wood + 10.1201/9781315370279 + 2017 + Wood, S. N. (2017). Generalized +additive models: An introduction with R (2nd ed.). Chapman; Hall/CRC. +doi:10.1201/9781315370279 + + + TMB: Automatic differentiation and Laplace +approximation + Kristensen + Journal of Statistical +Software + 5 + 70 + 10.18637/jss.v070.i05 + 2016 + Kristensen, K., Nielsen, A., Berg, C. +W., Skaug, H., & Bell, B. M. (2016). TMB: Automatic differentiation +and Laplace approximation. Journal of Statistical Software, 70(5), 1–21. +doi:10.18637/jss.v070.i05 + + + Dynamic vine copula models for multivariate +time series data + Acar + Econometrics and Statistics + 12 + 10.1016/j.ecosta.2019.03.002 + 2019 + Acar, E. F., Czado, C., & Lysy, +M. (2019). Dynamic vine copula models for multivariate time series data. +Econometrics and Statistics, 12, 181–197. +doi:10.1016/j.ecosta.2019.03.002 + + + Dependence calibration in conditional +copulas: A nonparametric approach + Acar + Biometrics + 2 + 67 + 10.1111/j.1541-0420.2010.01472.x + 2011 + Acar, E. F., Craiu, R. V., & Yao, +F. (2011). Dependence calibration in conditional copulas: A +nonparametric approach. Biometrics, 67(2), 445–453. +doi:10.1111/j.1541-0420.2010.01472.x + + + Statistical testing of covariate effects in +conditional copula models + Acar + Electronic Journal of +Statistics + 7 + 10.1214/13-EJS866 + 2013 + Acar, E. F., Craiu, R. V., & Yao, +F. (2013). Statistical testing of covariate effects in conditional +copula models. Electronic Journal of Statistics, 7, 2822–2850. +doi:10.1214/13-EJS866 + + + LocalCop: Local likelihood inference for +conditional copula models + Acar + 10.32614/CRAN.package.LocalCop + 2024 + Acar, E. F., & Lysy, M. (2024). +LocalCop: Local likelihood inference for conditional copula models. +doi:10.32614/CRAN.package.LocalCop + + + Copula: Multivariate dependence with +copulas + Hofert + 10.32614/CRAN.package.copula + 2023 + Hofert, M., Kojadinovic, I., Mächler, +M., & Yan, J. (2023). Copula: Multivariate dependence with copulas. +doi:10.32614/CRAN.package.copula + + + Enjoy the joy of copulas: With a package +copula + Yan + Journal of Statistical +Software + 4 + 21 + 10.18637/jss.v021.i04 + 2007 + Yan, J. (2007). Enjoy the joy of +copulas: With a package copula. Journal of Statistical Software, 21(4), +1–21. doi:10.18637/jss.v021.i04 + + + Modeling multivariate distributions with +continuous margins using the copula R package + Kojadinovic + Journal of Statistical +Software + 9 + 34 + 10.18637/jss.v034.i09 + 2010 + Kojadinovic, I., & Yan, J. +(2010). Modeling multivariate distributions with continuous margins +using the copula R package. Journal of Statistical Software, 34(9), +1–20. doi:10.18637/jss.v034.i09 + + + Nested Archimedean copulas meet R: The +nacopula package + Hofert + Journal of Statistical +Software + 9 + 39 + 10.18637/jss.v039.i09 + 2011 + Hofert, M., & Mächler, M. (2011). +Nested Archimedean copulas meet R: The nacopula package. Journal of +Statistical Software, 39(9), 1–20. +doi:10.18637/jss.v039.i09 + + + VineCopula: Statistical inference of vine +copulas + Nagler + 10.32614/CRAN.package.VineCopula + 2023 + Nagler, T., Schepsmeier, U., Stoeber, +J., Brechmann, E. C., Graeler, B., & Erhardt, T. (2023). VineCopula: +Statistical inference of vine copulas. +doi:10.32614/CRAN.package.VineCopula + + + gamCopula: Generalized additive models for +bivariate conditional dependence structures and vine +copulas + Nagler + 10.32614/CRAN.package.gamCopula + 2020 + Nagler, T., & Vatter, T. (2020). +gamCopula: Generalized additive models for bivariate conditional +dependence structures and vine copulas. +doi:10.32614/CRAN.package.gamCopula + + + CondCopulas: Estimation and inference for +conditional copula models + Derumigny + 10.32614/CRAN.package.CondCopulas + 2023 + Derumigny, A. (2023). 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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +6744 +10.21105/joss.06744 + +LocalCop: An R package for local likelihood inference for +conditional copulas + + + +https://orcid.org/0000-0003-2908-7691 + +Acar +Elif Fidan + + + + + +https://orcid.org/0000-0001-9974-1121 + +Lysy +Martin + + + + + +Kuchinsky +Alan + + + + + +University of Guelph + + + + +Hospital for Sick Children + + + + +University of Waterloo + + + + +University of Manitoba + + + + +9 +9 +2024 + +9 +101 +6744 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2022 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +R +C++ +copula +local likelihood +covariate effect + + + + + + Summary +

Conditional copulas models allow the dependence structure between + multiple response variables to be modelled as a function of + covariates. LocalCop + (Acar + & Lysy, 2024) is an + R/C++package + for computationally efficient semiparametric conditional copula + modelling using a local likelihood inference framework developed in + Acar, Craiu, & Yao + (2011), + Acar, Craiu, & Yao + (2013) and + Acar, Czado, & Lysy + (2019).

+
+ + Statement of Need +

There are well-developed + Rpackages + such as copula + (Hofert, + Kojadinovic, Mächler, & Yan, 2023; + Hofert + & Mächler, 2011; + Kojadinovic + & Yan, 2010; + Yan, + 2007) and VineCopula + (Nagler + et al., 2023) for fitting copulas in various multivariate data + settings. However, these software focus exclusively on unconditional + dependence modelling and do not accommodate covariate information.

+

Aside from LocalCop, + Rpackages for + fitting conditional copulas are gamCopula + (Nagler + & Vatter, 2020) and CondCopulas + (Derumigny, + 2023). gamCopula estimates the covariate-dependent + copula parameter using spline smoothing. While this typically has + lower variance than the local likelihood estimate provided by + LocalCop, it also tends to have lower accuracy + (Acar + et al., 2019). CondCopulas estimates the copula + parameter using a semi-parametric maximum-likelihood method based on a + kernel-weighted conditional concordance metric. LocalCop + also uses kernel weighting, but it uses the full likelihood + information of a given copula family rather than just that contained + in the concordance metric, and is therefore more statistically + efficient.

+

Local likelihood methods typically involve solving a large number + of low-dimensional optimization problems and thus can be + computationally intensive. To address this issue, + LocalCop implements the local likelihood function in + C++, using + the + R/C++package + TMB + (Kristensen, + Nielsen, Berg, Skaug, & Bell, 2016) to efficiently obtain + the associated score function using automatic differentiation. Thus, + LocalCop is able to solve each optimization problem very + quickly using gradient-based algorithms. It also provides a means of + easily parallelizing the optimization across multiple cores, rendering + LocalCop competitive in terms of speed with other + available software for conditional copula estimation.

+
+ + Background +

For any bivariate response vector + + (Y1,Y2), + the conditional joint distribution given a covariate + + + X + is given by + + + FX(y1,y2x)=CX(F1X(y1x),F2X(y2x)x), + where + + F1X(y1x) + and + + F2X(y2x) + are the conditional marginal distributions of + + + Y1 + and + + Y2 + given + + X, + and + + CX(u,vx) + is a conditional copula function. That is, for given + + + X=x, + the function + + CX(u,vx) + is a bivariate CDF with uniform margins.

+

The focus of LocalCop is on estimating the conditional + copula function, which is modelled semi-parametrically as + + + CX(u,vx)=𝒞(u,vθ(x),ν), + where + + 𝒞(u,vθ,ν) + is a parametric copula family, the copula dependence parameter + + + θΘ + is an arbitrary function of + + X, + and + + νΥ + is an additional copula parameter present in some models. Since most + parametric copula families have a restricted range + + + Θ, + we describe the data generating model (DGM) in terms of the + calibration function + + η(x), + such that + + θ(x)=g1(η(x)), + where + + g1:Θ + an inverse-link function which ensures that the copula parameter has + the correct range. The choice of + + g1(η) + is not unique and depends on the copula family.

+

Local likelihood estimation of the conditional copula parameter + + + θ(x) + uses Taylor expansions to approximate the calibration function + + + η(x) + at an observed covariate value + + X=x + near a fixed point + + X=x0, + i.e., + + η(x)η(x0)+η(1)(x0)(xx0)++η(p)(x0)p!(xx0)p. + One then estimates + + βk=η(k)(x0)/k! + for + + k=0,,p + using a kernel-weighted local likelihood function + + + (𝛃)=i=1nlog{c(ui,vig1(𝐱iT𝛃),ν)}Kh(xix0h), + where + + (ui,vi,xi) + is the data for observation + + i, + + + 𝐱i=(1,xix0,(xix0)2,,(xix0)p), + + + 𝛃=(β0,β1,,βp), + and + + Kh(z) + is a kernel function with bandwidth parameter + + 0]]> + h>0. + Having maximized + + (𝛃) + in Equation 3, one + estimates + + η(x0) + by + + η̂(x0)=β̂0. + Usually, a linear fit with + + p=1 + suffices to obtain a good estimate in practice.

+
+ + Usage +

LocalCop is available on + CRAN + and + GitHub. + The two main package functions are:

+ + +

CondiCopLocFit(): For estimating the + calibration function at a sequence of values + + + 𝐱0=(x01,,x0m).

+
+ +

CondiCopSelect(): For selecting a copula + family and bandwidth parameter using leave-one-out + cross-validation (LOO-CV) with subsampling as described in Acar et + al. + (2019).

+
+
+

In the following example, we illustrate the model selection/tuning + and fitting steps for data generated from a Clayton copula with + conditional Kendall + + τ + displayed in + [fig:copcomp]. + The CV metric for each combination of family and bandwidth are + displayed in + [fig:select1-plot].

+ library(LocalCop) # local likelihood estimation +library(VineCopula) # simulate copula data + set.seed(2024) + +# simulation setting +family <- 3 # Clayton Copula +n_obs <- 300 # number of observations +eta_fun <- function(x) { # calibration function + sin(5*pi*x) + cos(8*pi*x^2) +} + # simulate covariate values +x <- sort(runif(n_obs)) + +# simulate response data +eta_true <- eta_fun(x) # calibration parameter eta(x) +par_true <- BiCopEta2Par(family = family, # copula parameter theta(x) + eta = eta_true) +udata <- VineCopula::BiCopSim(n_obs, family = family, par = par_true) + # model selection and tuning +bandset <- c(.02, .05, .1, .2) # set of bandwidth parameters +famset <- c(1, 2, 3, 4, 5) # set of copula families +kernel <- KernGaus # kernel function +degree <- 1 # degree of local polynomial +n_loo <- 100 # number of LOO-CV observations + # (can be much smaller than n_obs) + # calculate cv for each combination of family and bandwidth +cvselect <- CondiCopSelect(u1= udata[,1], u2 = udata[,2], + x = x, xind = n_loo, + kernel = kernel, degree = degree, + family = famset, band = bandset) + +

Cross-validation metric for each combination of family + and bandwidth.

+ +
+ # extract the selected family and bandwidth from cvselect +cv_res <- cvselect$cv +i_opt <- which.max(cv_res$cv) +fam_opt <- cv_res[i_opt,]$family +band_opt <- cv_res[i_opt,]$band + +# calculate eta(x) on a grid of values +x0 <- seq(0, 1, by = 0.01) +copfit <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2], + x = x, x0 = x0, + kernel = kernel, degree = degree, + family = fam_opt, band = band_opt) +# convert eta to Kendall tau +tau_loc <- BiCopEta2Tau(copfit$eta, family= fam_opt) + # simulate covariate values +x <- sort(runif(n_obs)) + +# simulate response data +eta_true <- eta_fun(x) # calibration parameter eta(x) +par_true <- BiCopEta2Par(family = family, # copula parameter theta(x) + eta = eta_true) +udata <- VineCopula::BiCopSim(n_obs, family = family, par = par_true) + # model selection and tuning +bandset <- c(.02, .05, .1, .2) # set of bandwidth parameters +famset <- c(1, 2, 3, 4, 5) # set of copula families +kernel <- KernGaus # kernel function +degree <- 1 # degree of local polynomial +n_loo <- 100 # number of LOO-CV observations + # (can be much smaller than n_obs) + # calculate cv for each combination of family and bandwidth +cvselect <- CondiCopSelect(u1= udata[,1], u2 = udata[,2], + x = x, xind = n_loo, + kernel = kernel, degree = degree, + family = famset, band = bandset) + # extract the selected family and bandwidth from cvselect +cv_res <- cvselect$cv +i_opt <- which.max(cv_res$cv) +fam_opt <- cv_res[i_opt,]$family +band_opt <- cv_res[i_opt,]$band + +# calculate eta(x) on a grid of values +x0 <- seq(0, 1, by = 0.01) +copfit <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2], + x = x, x0 = x0, + kernel = kernel, degree = degree, + family = fam_opt, band = band_opt) +# convert eta to Kendall tau +tau_loc <- BiCopEta2Tau(copfit$eta, family= fam_opt) + +

True vs estimated conditional Kendall + + + τ + using various methods.

+ +
+

In + [fig:copcomp], + we compare the true conditional Kendall + + τ + to estimates using each of the three conditional copula fitting + packages LocalCop, gamCopula, and + CondCopulas, for sample sizes + + + n=300 + and + + n=1000. + In gamCopula, selection of the copula family smoothing + splines is done using the generalized CV framework provided by the + Rpackage + mgcv + (Wood, + 2017). In CondCopulas, selection of the bandwidth + parameter is done using LOO-CV. In this particular example, the sample + size of + + n=300 + is not large enough for gamCopula to pick a sufficiently + flexible spline basis, and CondCopulas picks a large + bandwidth which oversmooths the data. For the larger sample size + + + n=1000, + the three methods exhibit similar accuracy.

+
+ + Acknowledgements +

We acknowledge funding support from the Natural Sciences and + Engineering Research Council of Canada Discovery Grants + RGPIN-2020-06753 (Acar) and RGPIN-2020-04364 (Lysy).

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