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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.