Representative Approach for Big Data Dimension Reduction with Binary Responses

Sufficient dimension reduction (SDR) reduces the data dimensionality without specifying a regression model. Since it was first introduced by Li, 1991, SDR has been popular and many SDR methods have been proposed and studied. Among those methods, we focus on Sliced Inverse Regression (SIR) and Sliced Average Variance Estimation (SAVE), which are inverse-moment based methods. Those methods work well with continuous responses, but not with binary cases due to the limited number of levels of the response. In order to solve the issue, Shin et al., 2014 have proposed a solution for SDR methods on binary data called Probability Enhanced SDR (PRE-SDR). The PRE-SDR works well under a binary dataset. But it becomes time-consuming when a dataset is large, e.g., N > 10000, because of its computational intensity. In this thesis, motivated by the existing solution and its limitation on large data, we investigate and improve the SIR and SAVE from different perspectives. Firstly, we incorporate an online algorithm, which helps to reduce the usage of computer memory when a dataset is large. The general idea of this method is to scan the data chunk by chunk, calculate intermediate statistics, and combine intermediate results to get the final result. We develop online algorithms for SIR and SAVE and show that the online method’s result is the same as it calculated from using the full data at once. Besides, we enhance those algorithms with a parallel computation framework so that it could process multiple chunks at the same time. Simulation results suggest that the online algorithm reduces the computational time at least by 3-5 times compared with the original methods. Secondly, we propose a novel SDR approach, named as Mean Representative approach (MRDR), for binary responses. The main idea is to partition the data into blocks, calculate representatives for each block, and use the representatives as our new dataset for the following SDR analysis. By converting a block of data points into a representative data point, the corresponding binary responses become continuous, and the size of the data is reduced significantly because the number of the block is much smaller than the original observations. Therefore, the proposed representative approach provides an ideal solution for large data dimension reduction and can be incorporated with the classical SDR approaches naturally. The details of MRDR are introduced and discussed in Chapters 1 and 3. We study the asymptotic properties of MRDR in Chapter 4 and show that the proposed approach can recover the central subspace better than SIR and SAVE. Besides, we also discuss the optimal choice of the number of blocks in Section 4.3. The simulation studies in Chapter 5 verify the advantage of the proposed method over the original SIR and SAVE in estimating the central subspace and demonstrates the time efficiency compared to PRE-SIR. In the end, we apply the proposed method on the Electrical Grid Stability (EGS) data and simulated data based on the EGS data. The result shows the advantage of the proposed method over the several existing methods on sufficient dimension reduction with large data

Reconstruction of the Gene Regulatory Network Involved in the Sonic Hedgehog Pathway with a Potential Role in Early Development of the Mouse Brain

<div><p>The Sonic hedgehog (Shh) signaling pathway is crucial for pattern formation in early central nervous system development. By systematically analyzing high-throughput in situ hybridization data of E11.5 mouse brain, we found that Shh and its receptor Ptch1 define two adjacent mutually exclusive gene expression domains: Shh⁺Ptch1⁻ and Shh⁻Ptch1⁺. These two domains are associated respectively with Foxa2 and Gata3, two transcription factors that play key roles in specifying them. Gata3 ChIP-seq experiments and RNA-seq assays on Gata3-knockdown cells revealed that Gata3 up-regulates the genes that are enriched in the Shh⁻Ptch1⁺ domain. Important Gata3 targets include <i>Slit2</i> and <i>Slit3</i>, which are involved in the process of axon guidance, as well as <i>Slc18a1</i>, <i>Th</i> and <i>Qdpr</i>, which are associated with neurotransmitter synthesis and release. By contrast, Foxa2 both up-regulates the genes expressed in the Shh⁺Ptch1⁻ domain and down-regulates the genes characteristic of the Shh⁻Ptch1⁺ domain. From these and other data, we were able to reconstruct a gene regulatory network governing both domains. Our work provides the first genome-wide characterization of the gene regulatory network involved in the Shh pathway that underlies pattern formation in the early mouse brain.</p></div

Genome-wide characterization of Gata3 binding sites in ChIP-seq.

<p>(A) Top binding motif identified by MEME-ChIP. (B) Conservation plot of Gata3 binding sites in vertebrate species. (C) The genome-wide distribution of Gata3-binding sites. (D) The enriched biological processes in Gata3-binding targets revealed by GO analysis.</p

Illustration of Gata3 ChIP-seq binding sites on the selected genes.

<p>Red boxes indicate the binding peak locations called by MACS program. (A) <i>Sufu</i> and <i>Gsk3b</i> in the Shh signaling pathway. (B) <i>Slit2/3</i> involved in axon guidance. (C) <i>Nfasc, Mapt</i> and <i>Limk1</i> regulating brain development. (D) <i>Qdpr, Th</i> and <i>Slc18a1</i> involved in neurotransmitter synthesis and release.</p

The relationship of Gata3 and Foxa2 targets with Shh⁺Ptch1⁻-/Shh⁻Ptch1⁺-pattern genes.

<p>(A) The overlaps between Foxa2 targets in ChIP-seq with ABA ISH annotations and Shh⁺Ptch1⁻-/Shh⁻Ptch1⁺-pattern genes. (B) The overlaps between 84 Gata3 targets in ChIP-seq with ABA ISH annotations and Shh⁺Ptch1⁻-/Shh⁻Ptch1⁺-pattern genes. (C–D) The overlaps between genes with ABA ISH annotations down- or up-regulated by Gata3-knockdown and Shh⁺Ptch1⁻-/Shh⁻Ptch1⁺-pattern genes. (* indicates the statistical significance of the overlap, <i>P</i><0.05, in Fisher's exact test).</p

Regenerative capability of zebrafish mandible.

<p>The regenerated bone and cartilage are shown by Alcian blue staining. The first occurrence of the mandible regeneration was the chondrogenic differentiation indicated by cartilage blue-staining (10 dpa). At 21days after amputation, new bone formation began in an anterior direction and maintained the shape of the mandibular arch. In the following days, the cartilage blue-staining was reduced while red-skeletal stains increased. By two months the regenerated two ridges (mandibles) converged medially. Different from the uncut mandibular bones that were separated by a cartilaginous median symphysis, the regenerated mandibles appeared to fuse at the midline. Ms, mandibular symphysis; Ds, dorsal; V, ventral. Scale bars, 500 µm.</p

A schematic model of two origins of blastemal progenitors in lower jaw regeneration.

<p>Two origins of blastemal progenitors (<i>foxi1</i>-positive neural crest cells and <i>isl1</i>-positive mesodermal progenitors) arise in the lower jaw blastema. They undergo mesenchymalization, chondrogenic differentiation and tissue respecifications toward skeletal, connective tissues and muscle regeneration through specific signaling pathways. The possible signaling communications are indicated. For instance, positional signaling mediated by local cues orchestrates blastema reformation; the activation <i>hoxa2b</i> and <i>hoxa11b</i> is induced/enhanced by the second arch signaling and blood signaling respectively.</p

Anatomical structure and regenerative capability of zebrafish lower jaw.

<p>C–F, Alcian blue staining of bone and cartilage of lower jaw. The dotted line demarcates the amputation plane. Md, mandible; Ms, mandibular symphysis; Mc, Meckel's cartilage, Mm, mandibular muscle. Scale bars, 1000 µm.</p

Two Origins of Blastemal Progenitors Define Blastemal Regeneration of Zebrafish Lower Jaw

<div><p>Zebrafish possess a remarkable ability to regenerate complicated structures by formation of a mass of undifferentiated mesenchymal cells called blastema. To understand how the blastema retains the original structural form, we investigate cellular transitions and transcriptional characteristics of cell identity genes during all stages of regeneration of an amputated lower jaw. We find that mesenchymal blastema originates from multiple sources including nucleated blood cells, fibroblasts, damaged muscle cells and pigment cells. These cells are transformed into two populations of blastemal progenitors: <i>foxi1</i>-expression and <i>isl1</i>-expression, before giving rise to cartilage, bone, and muscle. Time point- based transcriptomal analysis of 45 annotated Hox genes reveal that five 3′-end Hox genes and an equal number of 5′-end Hox genes are activated largely at the stage of blastema reformation. RNA <i>in situ</i> hybridization shows that <i>foxi1</i> and <i>pax3a</i> are respectively expressed in the presumptive mandible skeletal region and regenerating muscle at 5 dpa. In contrast, <i>hoxa2b</i> and <i>hoxa11b</i> are widely expressed with different domain in chondrogenic blastema and blastema mesenchyme. Knockdown <i>foxi1</i> changes the expression patterns of <i>sox9a</i> and <i>hoxa2b</i> in chondrogenic blastema. From these results we propose that two origins of blastemal progenitors define blastema skeleton and muscle respecifications through distinct signaling pathways. Meanwhile, the positional identity of blastema reformation is implicated in mesenchymal segmentation and characteristic expression pattern of Hox genes.</p></div

Histological observation of blastema formation and reformation.

<p>Figure A shows that regeneration of the mandible retains the original mandibular arch. The cut arch (arrow) at 2 dpa was restored to the original shape (uncut) at 60 dpa. HE staining. Scale bars, 250 µm. Figure B shows blastemal structure and extracellular matrix composition. At 2 hpa, wound epidermis reconstitution started (blue arrow), and blastema was not yet formed between wound epidermis and the injured muscle. At 5 dpa, the blastemal ECM began to reorganize toward hypodermis (vertical red arrow) and the chondrogenic center (horizon red arrow). In comparison with HE stains, Alcian blue-Ponceau S staining shows hyaluronic acid and hyaline cartilage as blue, collagen and mineralized bone matrix as red; In a modification of Masson' trichromal staining, green or blue staining is collagen; brown staining is elastic fibers; red is cytoplasm, muscle, nerve sheathe, fibronectin and erythrocytes). Scale bars, 50 µm. Figure C shows two types of chondrogenic ossification. After blastema formation (8 dpa), chondrogenic blastema was composed of three chondrogenic ossification centers: two Meckel-lateral centers (Mlc) and one median symphysis center (Msc). In the Meckel-lateral centers, perichondral ossification was more like atypical or incomplete endochondral ossification. The surrounding matrix became calcified while the central portion of ossification center developed to bone cysts (BC), a similar structure like bone marrow cavity filled with little connective tissues (120 dpa). The median symphysis center, however, adopted perichondral ossification. Scale bars, 100 µm.</p