The main ideas behind the classical multivariate logistic regression model make sense when translated to the functional setting, where the explanatory variable $X$ is a function and the response $Y$ is binary. However, some important technical issues appear (or are aggravated with respect to those of the multivariate case) due to the functional nature of the explanatory variable. First, the mere definition of the model can be questioned: while most approaches so far proposed rely on the $L_2$-based model, we suggest an alternative (in some sense, more general) approach, based on the theory of Reproducing Kernel Hilbert Spaces (RKHS). The validity conditions of such RKHS-based model, as well as its relation with the $L_2$-based one are investigated and made explicit in two formal results. Some relevant particular cases are considered as well. Second we show that, under very general conditions, the maximum likelihood (ML) of the logistic model parameters fail to exist in the functional case. Third, on a more positive side, we suggest an RKHS-based restricted version of the ML estimator. This is a methodological paper, aimed at a better understanding of the functional logistic model, rather than focusing on numerical and practical issues.
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Decisions and the underlying rules are indispensable for driving process execution during runtime, i.e., for routing process instances at alternative branches based on the values of process data. Decision rules can comprise unary data conditions, e.g., age > 40, binary data conditions where the relation between two or more variables is relevant, e.g. temperature1 < temperature2, and more complex conditions that refer to, for example, parts of a medical image. Decision discovery aims at automatically deriving decision rules from process event logs. Existing approaches focus on the discovery of unary, or in some instances binary data conditions. The discovered decision rules are usually evaluated using accuracy, but not with regards to their semantics and meaningfulness, although this is crucial for validation and the subsequent implementation/adaptation of the decision rules. Hence, this paper compares three decision mining approaches, i.e., two existing ones and one newly described approach, with respect to the meaningfulness of their results. For comparison, we use one synthetic data set for a realistic manufacturing case and the two real-world BPIC 2017/2020 logs. The discovered rules are discussed with regards to their semantics and meaningfulness.
In mathematical finance, Levy processes are widely used for their ability to model both continuous variation and abrupt, discontinuous jumps. These jumps are practically relevant, so reliable inference on the feature that controls jump frequencies and magnitudes, namely, the Levy density, is of critical importance. A specific obstacle to carrying out model-based (e.g., Bayesian) inference in such problems is that, for general Levy processes, the likelihood is intractable. To overcome this obstacle, here we adopt a Gibbs posterior framework that updates a prior distribution using a suitable loss function instead of a likelihood. We establish asymptotic posterior concentration rates for the proposed Gibbs posterior. In particular, in the most interesting and practically relevant case, we give conditions under which the Gibbs posterior concentrates at (nearly) the minimax optimal rate, adaptive to the unknown smoothness of the true Levy density.
Dynamic treatment regimes (DTRs) consist of a sequence of decision rules, one per stage of intervention, that finds effective treatments for individual patients according to patient information history. DTRs can be estimated from models which include the interaction between treatment and a small number of covariates which are often chosen a priori. However, with increasingly large and complex data being collected, it is difficult to know which prognostic factors might be relevant in the treatment rule. Therefore, a more data-driven approach of selecting these covariates might improve the estimated decision rules and simplify models to make them easier to interpret. We propose a variable selection method for DTR estimation using penalized dynamic weighted least squares. Our method has the strong heredity property, that is, an interaction term can be included in the model only if the corresponding main terms have also been selected. Through simulations, we show our method has both the double robustness property and the oracle property, and the newly proposed methods compare favorably with other variable selection approaches.
Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Veneri [1981], we prove several facts about sets of terms having a given intersection type. One of our results is that every strongly normalizing term M admits a *uniqueness typing*, which is a pair $(\Gamma,A)$ such that 1) $\Gamma \vdash M : A$ 2) $\Gamma \vdash N : A \Longrightarrow M =_{\beta\eta} N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules. In the second part of the paper, we prove that the set of closed terms having a given intersection type is separable, and, if infinite, forms an adequate numeral system.
Competing Risks Regression for Clustered Data via the Marginal Additive Subdistribution Hazard Model
A population-averaged additive subdistribution hazard model is proposed to assess the marginal effects of covariates on the cumulative incidence function to analyze correlated failure time data subject to competing risks. This approach extends the population-averaged additive hazard model by accommodating potentially dependent censoring due to competing events other than the event of interest. Assuming an independent working correlation structure, an estimating equations approach is considered to estimate the regression coefficients and a sandwich variance estimator is proposed. The sandwich variance estimator accounts for both the correlations between failure times as well as the those between the censoring times, and is robust to misspecification of the unknown dependency structure within each cluster. We further develop goodness-of-fit tests to assess the adequacy of the additive structure of the subdistribution hazard for each covariate, as well as for the overall model. Simulation studies are carried out to investigate the performance of the proposed methods in finite samples; and we illustrate our methods by analyzing the STrategies to Reduce Injuries and Develop confidence in Elders (STRIDE) study.
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.
Correlated data are ubiquitous in today's data-driven society. A fundamental task in analyzing these data is to understand, characterize and utilize the correlations in them in order to conduct valid inference. Yet explicit regression analysis of correlations has been so far limited to longitudinal data, a special form of correlated data, while implicit analysis via mixed-effects models lacks generality as a full inferential tool. This paper proposes a novel regression approach for modelling the correlation structure, leveraging a new generalized z-transformation. This transformation maps correlation matrices that are constrained to be positive definite to vectors with un-restricted support, and is order-invariant. Building on these two properties, we develop a regression model to relate the transformed parameters to any covariates. We show that coupled with a mean and a variance regression model, the use of maximum likelihood leads to asymptotically normal parameter estimates, and crucially enables statistical inference for all the parameters. The performance of our framework is demonstrated in extensive simulation. More importantly, we illustrate the use of our model with the analysis of the classroom data, a highly unbalanced multilevel clustered data with within-class and within-school correlations, and the analysis of the malaria immune response data in Benin, a longitudinal data with time-dependent covariates in addition to time. Our analyses reveal new insights not previously known.
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.
We study the matrix completion problem that leverages hierarchical similarity graphs as side information in the context of recommender systems. Under a hierarchical stochastic block model that well respects practically-relevant social graphs and a low-rank rating matrix model, we characterize the exact information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) by proving sharp upper and lower bounds on the sample complexity. In the achievability proof, we demonstrate that probability of error of the maximum likelihood estimator vanishes for sufficiently large number of users and items, if all sufficient conditions are satisfied. On the other hand, the converse (impossibility) proof is based on the genie-aided maximum likelihood estimator. Under each necessary condition, we present examples of a genie-aided estimator to prove that the probability of error does not vanish for sufficiently large number of users and items. One important consequence of this result is that exploiting the hierarchical structure of social graphs yields a substantial gain in sample complexity relative to the one that simply identifies different groups without resorting to the relational structure across them. More specifically, we analyze the optimal sample complexity and identify different regimes whose characteristics rely on quality metrics of side information of the hierarchical similarity graph. Finally, we present simulation results to corroborate our theoretical findings and show that the characterized information-theoretic limit can be asymptotically achieved.
Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions.
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