Steric hindered frustrated Lewis pairs (FLPs) have been shown to activate hydrogen molecules, and their reactivity is strongly determined by the geometric parameters of the Lewis acid s and bases. A recent experimental study showed that ionic liquids
(ILs) could largely improve the effective configuration of FLPs. However, the detailed mechanistic profile is still unclear. Herein, we performed a molecular dynamics (MD) simulations, aimi ng to reveal the effects of ILs on the structures of FLPs, and to present a rule for selecting more efficient reaction media. For this purpose, mixture systems were adopt consisting of the ILs [Cnmim][NTf2] (n= 6, 10, 14), and the typical FLP (tBu)3P/B(C6F5)3 . Radial distribution function (RDF) results show that toluene competes with (tBu)3P to interact with B(C6F5)3 , resulting in a relatively low effective (tBu)3P/B(C6F5)3 complex. [Cnmim][NTf2] is more intended to form a solvated shell surrounding the (tBu)3P/B(C6F5)3 , which increases the amount of effective FLPs. Spatial distribution function (SDF) results show that toluene formed a continuum solvation shell, which hinders the interactions of (tBu)3P and B(C6F5)3 , while [Cnmim][NTf2] leave a relatively large empty space, which is accessible by (tBu3)P molecules, resulting in a higher probability of Lewis acids and bases interactions. Lastly, we find that the longer alkyl chain length of[Cnmim] cations, the higher probability of effective FLPs.
Accurate predictions of customers future lifetime value (LTV) given their attributes and past purchase behavior enables a more customer-centric marketing strategy. Marketers can segment customers into various buckets based on the predicted LTV and, i
n turn, customize marketing messages or advertising copies to serve customers in different segments better. Furthermore, LTV predictions can directly inform marketing budget allocations and improve real-time targeting and bidding of ad impressions. One challenge of LTV modeling is that some customers never come back, and the distribution of LTV can be heavy-tailed. The commonly used mean squared error (MSE) loss does not accommodate the significant fraction of zero value LTV from one-time purchasers and can be sensitive to extremely large LTVs from top spenders. In this article, we model the distribution of LTV given associated features as a mixture of zero point mass and lognormal distribution, which we refer to as the zero-inflated lognormal (ZILN) distribution. This modeling approach allows us to capture the churn probability and account for the heavy-tailedness nature of LTV at the same time. It also yields straightforward uncertainty quantification of the point prediction. The ZILN loss can be used in both linear models and deep neural networks (DNN). For model evaluation, we recommend the normalized Gini coefficient to quantify model discrimination and decile charts to assess model calibration. Empirically, we demonstrate the predictive performance of our proposed model on two real-world public datasets.
Dealing with biased data samples is a common task across many statistical fields. In survey sampling, bias often occurs due to unrepresentative samples. In causal studies with observational data, the treated versus untreated group assignment is often
correlated with covariates, i.e., not random. Empirical calibration is a generic weighting method that presents a unified view on correcting or reducing the data biases for the tasks mentioned above. We provide a Python library EC to compute the empirical calibration weights. The problem is formulated as convex optimization and solved efficiently in the dual form. Compared to existing software, EC is both more efficient and robust. EC also accommodates different optimization objectives, supports weight clipping, and allows inexact calibration, which improves usability. We demonstrate its usage across various experiments with both simulated and real-world data.
We generalize Schwenks result that almost all trees contain any given limb to trees with positive integer vertex weights. The concept of characteristic polynomial is extended to such weighted trees and we prove almost all weighted trees have a cospec
tral mate. We also prove almost all trees contain $k$ cospectral vertices for any integer $kge2$.
Item response theory (IRT) models have been widely used in educational measurement testing. When there are repeated observations available for individuals through time, a dynamic structure for the latent trait of ability needs to be incorporated into
the model, to accommodate changes in ability. Other complications that often arise in such settings include a violation of the common assumption that test results are conditionally independent, given ability and item difficulty, and that test item difficulties may be partially specified, but subject to uncertainty. Focusing on time series dichotomous response data, a new class of state space models, called Dynamic Item Response (DIR) models, is proposed. The models can be applied either retrospectively to the full data or on-line, in cases where real-time prediction is needed. The models are studied through simulated examples and applied to a large collection of reading test data obtained from MetaMetrics, Inc.
