Fused Lasso Additive Model

Combining Metabolomics and Experimental Evolution Reveals Key Mechanisms Underlying Longevity Differences in Laboratory Evolved Drosophila melanogaster Populations

Experimental evolution with Drosophila melanogaster has been used extensively for decades to study aging and longevity. In recent years, the addition of DNA and RNA sequencing to this framework has allowed researchers to leverage the statistical power inherent to experimental evolution to study the genetic basis of longevity itself. Here, we incorporated metabolomic data into to this framework to generate even deeper insights into the physiological and genetic mechanisms underlying longevity differences in three groups of experimentally evolved D. melanogaster populations with different aging and longevity patterns. Our metabolomic analysis found that aging alters mitochondrial metabolism through increased consumption of NAD+ and increased usage of the TCA cycle. Combining our genomic and metabolomic data produced a list of biologically relevant candidate genes. Among these candidates, we found significant enrichment for genes and pathways associated with neurological development and function, and carbohydrate metabolism. While we do not explicitly find enrichment for aging canonical genes, neurological dysregulation and carbohydrate metabolism are both known to be associated with accelerated aging and reduced longevity. Taken together, our results provide plausible genetic mechanisms for what might be driving longevity differences in this experimental system. More broadly, our findings demonstrate the value of combining multiple types of omic data with experimental evolution when attempting to dissect mechanisms underlying complex and highly polygenic traits such as aging.

Generalized Sparse Additive Models

We present a unified framework for estimation and analysis of generalized additive models in high dimensions. The framework defines a large class of penalized regression estimators, encompassing many existing methods. An efficient computational algorithm for this class is presented that easily scales to thousands of observations and features. We prove minimax optimal convergence bounds for this class under a weak compatibility condition. In addition, we characterize the rate of convergence when this compatibility condition is not met. Finally, we also show that the optimal penalty parameters for structure and sparsity penalties in our framework are linked, allowing cross-validation to be conducted over only a single tuning parameter. We complement our theoretical results with empirical studies comparing some existing methods within this framework.

Reluctant Generalised Additive Modelling

Sparse generalised additive models (GAMs) are an extension of sparse generalised linear models that allow a model's prediction to vary non-linearly with an input variable. This enables the data analyst build more accurate models, especially when the linearity assumption is known to be a poor approximation of reality. Motivated by reluctant interaction modelling, we propose a multi-stage algorithm, called reluctant generalised additive modelling (RGAM), that can fit sparse GAMs at scale. It is guided by the principle that, if all else is equal, one should prefer a linear feature over a non-linear feature. Unlike existing methods for sparse GAMs, RGAM can be extended easily to binary, count and survival data. We demonstrate the method's effectiveness on real and simulated examples.

Adaptive nonparametric regression with the K-nearest neighbour fused lasso

The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the [Formula: see text]-nearest-neighbours fused lasso, involves computing the [Formula: see text]-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the [Formula: see text]-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an [Formula: see text]-graph rather than a [Formula: see text]-nearest-neighbours graph and contrast it with the [Formula: see text]-nearest-neighbours fused lasso.

Nonparametric regression with adaptive truncation via a convex hierarchical penalty

We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

Drosophila transcriptomics with and without ageing

The genomic basis of ageing still remains unknown despite being a topic of study for many years. Here, we present data from 20 experimentally evolved laboratory populations of Drosophila melanogaster that have undergone two different life-history selection regimes. One set of ten populations demonstrates early ageing whereas the other set of ten populations shows postponed ageing. Additionally, both types of populations consist of five long standing populations and five recently derived populations. Our primary goal was to determine which genes exhibit changes in expression levels by comparing the female transcriptome of the two population sets at two different time points. Using three different sets of increasingly restrictive criteria, we found that 2.1-15.7% (82-629 genes) of the expressed genes are associated with differential ageing between population sets. Conversely, a comparison of recently derived populations to long-standing populations reveals little to no transcriptome differentiation, suggesting that the recent selection regime has had a larger impact on the transcriptome than its more distant evolutionary history. In addition, we found very little evidence for significant enrichment for functional attributes regardless of the set of criteria used. Relative to previous ageing studies, we find little overlap with other lists of aging related genes. The disparity between our results and previously published results is likely due to the high replication used in this study coupled with our use of highly differentiated populations. Our results reinforce the notion that the use of genomic, transcriptomic, and phenotypic data to uncover the genetic basis of a complex trait like ageing can benefit from experimental designs that use highly replicated, experimentally-evolved populations.

Genome-Wide Mapping of Gene-Phenotype Relationships in Experimentally Evolved Populations

Model organisms subjected to sustained experimental evolution often show levels of phenotypic differentiation that dramatically exceed the phenotypic differences observed in natural populations. Genome-wide sequencing of pooled populations then offers the opportunity to make inferences about the genes that are the cause of these phenotypic differences. We tested, through computer simulations, the efficacy of a statistical learning technique called the "fused lasso additive model" (FLAM). We focused on the ability of FLAM to distinguish between genes which are differentiated and directly affect a phenotype from differentiated genes which have no effect on the phenotype. FLAM can separate these two classes of genes even with relatively small samples (10 populations, in total). The efficacy of FLAM is improved with increased number of populations, reduced environmental phenotypic variation, and increased within-treatment among-replicate variation. FLAM was applied to SNP variation measured in both twenty-population and thirty-population studies of Drosophila subjected to selection for age-at-reproduction, to illustrate the application of the method.

Data-adaptive additive modeling

In this paper, we consider fitting a flexible and interpretable additive regression model in a data-rich setting. We wish to avoid pre-specifying the functional form of the conditional association between each covariate and the response, while still retaining interpretability of the fitted functions. A number of recent proposals in the literature for nonparametric additive modeling are data adaptive, in the sense that they can adjust the level of flexibility in the functional fits to the data at hand. For instance, the sparse additive model makes it possible to adaptively determine which features should be included in the fitted model, the sparse partially linear additive model allows each feature in the fitted model to take either a linear or a nonlinear functional form, and the recent fused lasso additive model and additive trend filtering proposals allow the knots in each nonlinear function fit to be selected from the data. In this paper, we combine the strengths of each of these recent proposals into a single proposal that uses the data to determine which features to include in the model, whether to model each feature linearly or nonlinearly, and what form to use for the nonlinear functions. We establish connections between our approach and recent proposals from the literature, and we demonstrate its strengths in a simulation study.

Flexible and Interpretable Models for Survival Data

As datasets continue to increase in size, there is growing interest in methods for prediction that are both Received January 2018 flexible and interpretable. A flurry of recent work on this topic has focused on additive modeling in the Revised February 2019 regression setting, and in particular, on the use of data-adaptive nonlinear functions that can be used to flexibly model each covariate's effect, conditional on the other features in the model. In this article, we extend this recent line of work to the survival setting. We develop an additive Cox proportional hazards model, in which each additive function is obtained by trend filtering, so that the fitted functions are piece-wise polynomial with adaptively chosen knots. An efficient proximal gradient descent algorithm is used to fit the model. We demonstrate its performance in simulations and in application to a primary biliary cirrhosis data set, as well as a dataset consisting of time to publication for clinical trials in the biomedical literature. Supplementary materials for this article are available online.

Saturating Splines and Feature Selection

We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data via a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the conditional gradient method. Unlike many existing approaches, our algorithm solves the original infinite-dimensional (for splines of degree at least two) optimization problem without pre-specified knot locations. We then adapt our algorithm to fit generalized additive models with saturating splines as coordinate functions and show that the saturation requirement allows our model to simultaneously perform feature selection and nonlinear function fitting. Finally, we briefly sketch how the method can be extended to higher order splines and to different requirements on the extension outside the data range.

Convex Regression with Interpretable Sharp Partitions

We consider the problem of predicting an outcome variable on the basis of a small number of covariates, using an interpretable yet non-additive model. We propose convex regression with interpretable sharp partitions (CRISP) for this task. CRISP partitions the covariate space into blocks in a data-adaptive way, and fits a mean model within each block. Unlike other partitioning methods, CRISP is fit using a non-greedy approach by solving a convex optimization problem, resulting in low-variance fits. We explore the properties of CRISP, and evaluate its performance in a simulation study and on a housing price data set.