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Claremont Math Modeling Camp Problems, July 24-26, 2009 Problem Statements Problem 1: Presented by Jen-Mei Chang, Department of Mathematics and Statistics, California State University, Long Beach. Pattern Recognition/Analysis The subject of pattern recognition in data is broadly known as a sub-category of machine learning which is a scientific discipline that is concerned with the design of algorithms that allow artificial intelligence to learn, based on the information given. The general techniques of pattern analysis can be seen in numerous interesting applications, such as hurricane modeling, biometrics, understanding of brain wave patterns, stock market trend modeling, landscape ecology, medical diagnosis, natural language processing, search engines, DNA sequence, speech, and handwriting matching and recognition, etc. While the analysis of patterns in data has typically been a subject in statistics (i.e., data mining), computer science (i.e., computer vision), and engineering, recently, however, fundamental mathematical theory in areas such differential and algebraic geometry and topology have provided new mathematical frameworks and insights for understanding large data sets residing in spaces of large ambient dimensions. Consequently, understanding the geometry of the data becomes an essential ingredient in algorithm selection and information discovery. Data: 1. PatternRecData.mat which contains two variables: the 198-by-198 matrix KLDATA.mat and a row vector sub-labels of length 160. The data matrix KLDATA contains distinct images of cats and dogs in its columns. There are 99 of each animal and they are randomly placed in the columns of KLDATA. The vector sub-labels gives you the identity (with cat = 1 and dog = 0) of the first 160 patterns. 2. ./TIFFtraining which contains TIFF images for the first 160 patterns in KLDATA. Let’s explore the following: 1. Consider how we might use this information to build a pattern recognition architecture with a brief write-up of the method. It might be helpful to examine the classification errors as a 2-by-2 confusion matrix (dogs classified as dogs, dogs classified as cats, cats classified as cats, cats classified as dogs) in the analysis. (e.g., split the data into testing and training and provide classification errors on the testing set.) Some essential elements of a write-up include the following: Description of the classification method and details about how the classifier is constructed. Confusion matrix errors for the testing data. Predicted class membership for unlabeled data. Print-out of all codes used in the exploration process. 2. Once we are satisfied with our pattern recognition routine on the known data, classify the last 38 unknown columns in KLDATA as either cats or dogs. Save the result as a row vector of zeros (= dogs) and ones (= cats). Alternatively, if we wish to classify the raw data (instead of the KL data), we would design our classifier to take in 4096-by-1 column vectors and output their class labels as either cats or dogs. 3. Suggestions for Possible Approaches: Determine the covariance matrix of the cats and the covariance matrix of the dogs and construct optimal bases for each using 1. maximum noise fraction. Project new samples onto the cat basis and dog basis and see which gives a better representation. 2. Use vector quantization, e..g, Kohonen’s self-organizing map on a 2D lattice. 3. Use symmetric eigen-cats and eigen-dogs and the Principal Component Analysis (PCA) on the raw data. 4. 2D Discrete Wavelet Analysis (DWT) or Fourier Analysis on the raw data for frequency content information. 5. Radial Basis Function (map cats to ones and dogs to zeros). 6. Fisher’s Linear Discriminant Analysis (FDA). 7. Labeled Voronoi cell classification. 8. One-sided or two-sided tangent distances. 9. Set-to-set comparison with principal angles and Grassmannian distances. Problem 2: Presented by Mario Martelli, School of Mathematical Sciences, Claremont Graduate University Convergence problems for forward neural networks. In any neural network one can easily recognize three levels of neurons: 1. Entry (or input) level; 2. Hidden level; 3. Exit (or output) level. The net receives an input signal with its entry (or input) level. The answer of the net to the problem of interpreting and recognizing the signal is provided by the exit (or output) level. The net is called forward when the connectivity matrix T (a matrix with entries in the interval [1,1]) is lower triangular, with, possibly, zero diagonal elements. The main problem facing any neural network (including the forward ones) is represented by the convergence and the possibility of errors. The answer to an input should be fast and accurate. Hence, the speed and the accuracy of the answer are important. Is it possible to construct a forward neural network to perform a fixed number of tasks in a short time and without errors? We shall discuss some possible solutions. Problem 3: Presented by Ruye Wang, Engineering Department, Harvey Mudd College Many biological processes can be modeled by a network composed of many interacting component units (e.g. genes, proteins, neurons). It is of great biological interest to learn the interactions between these components involved in a biological process by circuit inference methods. However, one main obstacle in model-based circuit inference is the high dimensionality due to the large scale of the network model of the biological system under study. In particular, for those models based on differential equations, the large number of model parameters that need to be optimally estimated pose a major challenge due to the high computational costs in the optimization search process. This paper presents a new algorithm that decomposes the task of global search for the optimal parameters for the entire network model into a set of sub-tasks each searching for a subset of parameters associated with a single unit in the network. Such an algorithm can therefore be easily implemented in parallel by a multiprocessor computer system with each processor responsible for the computation for one or more of the units in the network. The combination of the improved search algorithm and its parallel implementation makes it possible to apply the algorithm to large scale networks in realistic problems involving large number of components. The implementation details of the algorithm are discussed and the simulation results are presented. Introduction: Circuit inference is a general approach for modeling and analyzing networks composed of interacting component units in various fields in science and engineering. In particular, examples in biology include gene regulation networks, protein phosphoregulation networks, and neural networks in the nervous system. In all these cases, the component units (the genes, proteins or neurons) can be activated (turned on, excited) or deactivated (turned off, inhibited) to varying degrees at different times by other units in the network. For example, when a gene is turned on it is transcribed to produce messenger RNA (mRNA) which is subsequently translated into protein molecules. Some of these proteins are transcription factors which can bind to specific sites (promoter regions) of the DNA and affect the corresponding genes to turn them on or off. As another example, large number of neurons in the brain form networks responsible for various functions (such as learning). A neuron may be either excited or inhibited by the synaptic inputs received by its dendrites from many other neurons and its response to these inputs is sent through its axon to interact with still other neurons. The interplay of these component units in such networks can be described as a circuit or network model, which controls the relevant biological processes, such as gene expression, signal transduction, or neural signal processing. It is therefore of great biological interest to learn how the component units in the network of a specific biological process interact with each other, based on the observed data (gene expressions, protein activities, neural activations, etc.), by various circuit inference methods. Modeling and circuit inference can be considered as a reverse engineering problem. Based on the understanding of the biological process under study, a mathematical or computational model is to be developed to produce output data consistent with the observed biological data, such as the expression levels of genes or firing rates of neurons. A model should be able to simulate the behaviors of the gene or neural networks and reveal the interactions between the genes or neurons. Moreover, a good network model should be able to make predictions of the behaviors of the actual biological system, and the prediction should be verified by the observed data. The goal of such modeling efforts is to gain insights into biological process under study. Various types of models for circuit inference have been proposed, such as the Boolean network models (Liang et al [9], Akutsu et al [1]) which simulate the genes by a set of binary nodes interacting with each other following some simple logical operations; the linear and quasi-linear models (D'Haeseleer [5], ven Someren [15]) based on the assumption that the genes are all linearly related in the network; and the differential equation (DE) models (Cohen and Grossberg [4], Hopfield [8], Mjolsness and Rienitz [12], Chen [3], Mjolsness et al [11]) which simulate the dynamics of biological network by a set of differential equations and find the interaction of the units in the network qualitatively. Compared to other model types, the differential equation models are more realistic biologically, but are also most challenging computationally, not only because a large set of differential equations (one for each unit in the network) need to be solved, but also, a large number of parameters proportional to the number differential equations need to be optimally estimated to fit the observed data. Problem 4: presented by Ashish Bhan, Keck Graduate Institute DNA methylation is a major epigenetic mechanism that causes heritable alterations in genomic functions including gene expression without changes in the coding or promoter sequence of a gene. Methylation can occur in CpG dinucleotides in promoter regions – this can have the effect of changing the affinity of transcription factor binding and can result in silencing gene transcription. This phenomenon is presumed to be involved in many neurological development disorders like Angelman and Rett syndromes and hence there is a great deal of interest in gaining a deeper understanding of the mechanisms involved. Aberrant denovo methylation of tumor suprresor genes early in tumorigenesis is a hallmark of human cancers and autoimmune diseases and aging. We have access to data generated from the Perera Lab at http://www.memorialhealth.com/aci/research/perera.aspx The data consists of microarray data for tens of thousands of mammalian genes/probes that are treated with various chemicals that induce methylation or suppress it. In addition we have the sequence information for each probe. The challenge now is to look for correlations between the CpG island content in the probes and the changes in expression under conditions of methylation and try and unravel any connections between the CpG islands (that are ostensibly involved in silencing gene expression) and the effects of methylation on gene expression. Problem 5: presented by Michael Gratton, Engineering Sciences and Applied Mathematics, Northwestern University The effect of external chemicals on a cell can depend strongly on concentration. Applications in cell assays (to determine information about a cell) and drug testing (to determine safe and effective dosages) require experiments to be run subjecting cells to a range of concentration of the "analyte." Traditionally, large cell colonies are grown in separate reactors, each with a fixed concentration of the analyte. Once uptake is complete, the cells are then removed from culture and tested. Advancements in microfluidics offer a more efficient method of studying analyte concentration by building a lab-on-a-chip. In our device, analyte in two concentrations is loaded into two reservoir and driven by a syringe pump. It first passes through a mixer segment that uses special channel branching to obtain several intermediate concentrations. These channels all empty into n open interrogation" segment consisting of a chemically patterned substrate. The pattern only allows cells to adhere to certain cell spots in tiny colonies; other regions prohibit cell attachment. The spots are organized into the lanes of intermediate analyte concentration. Our design offers several advantages. Analyte mixing is carried out inside the device, reducing labor. The smaller cell colonies require only small samples be provided. These cells are in a continuous flow of nutrients, an environment well-suited for cell cultures. Moreover, by making the interrogation segment thin, optical methods may be used to perform some tests noninvasively. The small size and modular design of the device make it a flexible and mobile apparatus. We anticipate that the interrogation segment is no larger than 10cm x 10cm, and the mixer is shorter still. Both the mixer and interrogation segments are modular, so that different concentration profiles and cell spot layouts may be swapped into the device. By minimizing the fabrication involved in the interrogation segment, we can make the part of the device in contact with the cells inexpensive enough to be disposable, or for multiple interrogators to be available with each device. However, several key design parameters remain unknown. First, what velocity should the flow be forced through the device? On the one hand, the mixer functions well at low flow velocities. On the other, the interrogation segment has no walls, so the lanes of the concentration profile may mix. A faster velocity would control this cross-mixing. Second, how many cells may be put into a lane? Other cells will deplete the concentration of analyte locally and emit effluent into the environment. Both of these effects must be minimized to ensure experimental control. There are competing design goals: good control, good mixing, lots of spots. Obviously these cannot all be met. Rather, we are interested in understanding the space of feasible solutions. Problem 6: presented by Yousef Daneshbod, Department of Mathematics, University of La Verne RapidLyser Noise Reduction Headquartered in Upland, CA, the company Claremont BioSolutions LLC (CBS) offers a family of instruments to break up biological cells and spores in order to release their contents for analysis or purification. One of the instruments, the RapidLyser, has an oscillating arm that moves a cartridge containing the liquid sample in a packed bed of beads at a very high frequency. The motion is similar to a “metronome” but at much higher oscillation rates. In order to reduce the noise produced during the operation of this device, engineers at CBS are using a viscoelastic material called Sorbothane to attach 4 circular legs as “bumpers” at the corners of the rectangular metal base that forms the bottom of the instrument. Material properties of Sorbothane are available at www.sorbothane.com. In order to reduce manufacturing costs, it is desired to use the minimal amount of Sorbothane that still provides adequate noise reduction. The participants at the math-in-industry workshop could consider whether the use of 4 legs at the corners is optimal or whether other arrangements might work better. The number, shape, size, thickness and placement of the bumpers can all be varied. Parameters such as the mass of the RapidLyser, the length of the oscillating arm, the weight of the moving cartridge, the range of oscillation frequencies, etc. will be provided at the workshop. Also, although Sorbothane is the material of choice, it would be nice to have a model that is applicable to any viscoelastic polymeric material.