Functional Data Analysis in Orthogonal Designs with Applications to Gait Patterns

PhDThis thesis presents a contribution to the active research area of functional data analysis (FDA) and is concerned with the analysis of data from complex experimental designs in which the responses are curves. High resolution, closely correlated data sets are encountered in many research fields, but current statistical methodologies often analyse simplistic summary measures and therefore limit the completeness and accuracy of conclusions drawn. Specifically the nature of the curves and experimental design are not taken into account. Mathematically, such curves can be modelled either as sample paths of a stochastic process or as random elements in a Hilbert space. Despite this more complex type of response, the structure of experiments which yield functional data is often the same as in classical experimentation. Thus, classical experimental design principles and results can be adapted to the FDA setting. More specifically, we are interested in the functional analysis of variance (ANOVA) of experiments which use orthogonal designs. Most of the existing functional ANOVA approaches consider only completely randomised designs. However, we are interested in more complex experimental arrangements such as, for example, split-plot and row-column designs. Similar to univariate responses, such complex designs imply that the response curves for different observational units are correlated. We use the design to derive a functional mixed-effects model and adapt the classical projection approach in order to derive the functional ANOVA. As a main result, we derive new functional F tests for hypotheses about treatment effects in the appropriate strata of the design. The approximate null distribution of these tests is derived by applying the Karhunen- Lo`eve expansion to the covariance functions in the relevant strata. These results extend existing work on functional F tests for completely randomised designs. The methodology developed in the thesis has wide applicability. In particular, we consider novel applications of functional F tests to gait analysis. Results are presented for two empirical studies. In the first study, gait data of patients with cerebral palsy were collected during barefoot walking and walking with ankle-foot orthoses. The effects of ankle-foot orthoses are assessed by functional F tests and compared with pointwise F tests and the traditional univariate repeated-measurements ANOVA. The second study is a designed experiment in which a split-plot design was used to collect gait data from healthy subjects. This is commonly done in gait research in order to better understand, for example, the effects of orthoses while avoiding confounded analysis from the high variability observed in abnormal gait. Moreover, from a technical point of view the study may be regarded as a real-world alternative to simulation studies. By using healthy individuals it is possible to collect data which are in better agreement with the underlying model assumptions. The penultimate chapter of the thesis presents a qualitative study with clinical experts to investigate the utility of gait analysis for the management of cerebral palsy. We explore potential pathways by which the statistical analyses in the thesis might influence patient outcomes. The thesis has six chapters. After describing motivation and introduction in Chapter 1, mathematical representations of functional data are presented in Chapter 2. Chapter 3 considers orthogonal designs in the context of functional data analysis. New functional F tests for complex designs are derived in Chapter 4 and applied in two gait studies. Chapter 5 is devoted to a qualitative study. The thesis concludes with a discussion which details the extent to which the research question has been addressed, the limitations of the work and the degree to which it has been answered

Certified Robustness for Large Language Models with Self-Denoising

Although large language models (LLMs) have achieved great success in vast real-world applications, their vulnerabilities towards noisy inputs have significantly limited their uses, especially in high-stake environments. In these contexts, it is crucial to ensure that every prediction made by large language models is stable, i.e., LLM predictions should be consistent given minor differences in the input. This largely falls into the study of certified robust LLMs, i.e., all predictions of LLM are certified to be correct in a local region around the input. Randomized smoothing has demonstrated great potential in certifying the robustness and prediction stability of LLMs. However, randomized smoothing requires adding noise to the input before model prediction, and its certification performance depends largely on the model's performance on corrupted data. As a result, its direct application to LLMs remains challenging and often results in a small certification radius. To address this issue, we take advantage of the multitasking nature of LLMs and propose to denoise the corrupted inputs with LLMs in a self-denoising manner. Different from previous works like denoised smoothing, which requires training a separate model to robustify LLM, our method enjoys far better efficiency and flexibility. Our experiment results show that our method outperforms the existing certification methods under both certified robustness and empirical robustness. The codes are available at https://github.com/UCSB-NLP-Chang/SelfDenoise

Decomposing Uncertainty for Large Language Models through Input Clarification Ensembling

Uncertainty decomposition refers to the task of decomposing the total uncertainty of a model into data (aleatoric) uncertainty, resulting from the inherent complexity or ambiguity of the data, and model (epistemic) uncertainty, resulting from the lack of knowledge in the model. Performing uncertainty decomposition for large language models (LLMs) is an important step toward improving the reliability, trustworthiness, and interpretability of LLMs, but this research task is very challenging and remains unresolved. The existing canonical method, Bayesian Neural Network (BNN), cannot be applied to LLMs, because BNN requires training and ensembling multiple variants of models, which is infeasible or prohibitively expensive for LLMs. In this paper, we introduce an uncertainty decomposition framework for LLMs, called input clarifications ensemble, which bypasses the need to train new models. Rather than ensembling models with different parameters, our approach generates a set of clarifications for the input, feeds them into the fixed LLMs, and ensembles the corresponding predictions. We show that our framework shares a symmetric decomposition structure with BNN. Empirical evaluations demonstrate that the proposed framework provides accurate and reliable uncertainty quantification on various tasks. Code will be made publicly available at https://github.com/UCSB-NLP-Chang/llm_uncertainty .Comment: 15 pages, 3 figure

Optimal experimental design for efficient toxicity testing in microphysiological systems: A bone marrow application

Introduction: Microphysiological systems (MPS; organ-on-a-chip) aim to recapitulate the 3D organ microenvironment and improve clinical predictivity relative to previous approaches. Though MPS studies provide great promise to explore treatment options in a multifactorial manner, they are often very complex. It is therefore important to assess and manage technical confounding factors, to maximise power, efficiency and scalability.Methods: As an illustration of how MPS studies can benefit from a systematic evaluation of confounders, we developed an experimental design approach for a bone marrow (BM) MPS and tested it for a specified context of use, the assessment of lineage-specific toxicity.Results: We demonstrated the accuracy of our multicolour flow cytometry set-up to determine cell type and maturity, and the viability of a “repeated measures” design where we sample from chips repeatedly for increased scalability and robustness. Importantly, we demonstrated an optimal way to arrange technical confounders. Accounting for these confounders in a mixed-model analysis pipeline increased power, which meant that the expected lineage-specific toxicities following treatment with olaparib or carboplatin were detected earlier and at lower doses. Furthermore, we performed a sample size analysis to estimate the appropriate number of replicates required for different effect sizes. This experimental design-based approach will generalise to other MPS set-ups.Discussion: This design of experiments approach has established a groundwork for a reliable and reproducible in vitro analysis of BM toxicity in a MPS, and the lineage-specific toxicity data demonstrate the utility of this model for BM toxicity assessment. Toxicity data demonstrate the utility of this model for BM toxicity assessment

Novel Cytokinin Derivatives Do Not Show Negative Effects on Root Growth and Proliferation in Submicromolar Range

BACKGROUND: When applied to a nutrition solution or agar media, the non-substituted aromatic cytokinins caused thickening and shortening of the primary root, had an inhibitory effect on lateral root branching, and even showed some negative effects on development of the aerial part at as low as a 10 nanomolar concentration. Novel analogues of aromatic cytokinins ranking among topolins substituted on N9-atom of adenine by tetrahydropyranyl or 4-chlorobutyl group have been prepared and tested in standardized cytokinin bioassays [1]. Those showing comparable activities with N(6)-benzylaminopurine were further tested in planta. METHODOLOGY/PRINCIPAL FINDINGS: The main aim of the study was to explain molecular mechanism of function of novel cytokinin derivatives on plant development. Precise quantification of cytokinin content and profiling of genes involved in cytokinin metabolism and perception in treated plants revealed several aspects of different action of m-methoxytopolin base and its substituted derivative on plant development. In contrast to standard cytokinins, N9- tetrahydropyranyl derivative of m-topolin and its methoxy-counterpart showed the negative effects on root development only at three orders of magnitude higher concentrations. Moreover, the methoxy-derivative demonstrates a positive effect on lateral root branching and leaf emerging in a nanomolar range of concentrations, in comparison with untreated plants. CONCLUSIONS/SIGNIFICANCE: Tetrahydropyranyl substitution at N9-position of cytokinin purine ring significantly enhances acropetal transport of a given cytokinins. Together with the methoxy-substitution, impedes accumulation of non-active cytokinin glucoside forms in roots, allows gradual release of the active base, and has a significant effect on the distribution and amount of endogenous isoprenoid cytokinins in different plant tissues. The utilization of novel aromatic cytokinin derivatives can distinctively improve expected hormonal effects in plant propagation techniques in the future

Dynamics of a Predator-Prey Model with Fear Effect and Time Delay

In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. The dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. The explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. Thus, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions