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K-means and Kohonen Maps Unsupervised Clustering Techniques Steve Hookway 4/8/04

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What is a DNA Microarray? An experiment on the order of 10k elements A way to explore the function of a gene A snapshot of the expression level of an entire phenotype under given test conditions

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Some Microarray Terminology Probe: ssDNA printed on the solid substrate (nylon or glass) These are the genes we are going to be testing Target: cDNA which has been labeled and is to be washed over the probe

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Microarray Fabrication Deposition of DNA fragments Deposition of PCR-amplified cDNA clones Printing of already synthesized oligonucleotieds In Situ synthesis Photolithography Ink Jet Printing Electrochemical Synthesis From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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cDNA Microarrays and Oligonucleotide Probes From Data Analysis Tools for DNA Microarrays by Sorin Draghici cDNA ArraysOligonucleotide Arrays Long Sequences Spot Unknown Sequences More variability Short Sequences Spot Known Sequences More reliable data

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In Situ Synthesis Photochemically synthesized on the chip Reduces noise caused by PCR, cloning, and Spotting As previously mentioned, three kinds of In Situ Synthesis Photolithography Ink Jet Printing Electrochemical Synthesis From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Photolithography Similar to process used to build VLSI circuits Photolithographic masks are used to add each base If base is present, there will be a hole in the corresponding mask Can create high density arrays, but sequence length is limited From Data Analysis Tools for DNA Microarrays by Sorin Draghici Photodeprotection mask C

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Ink Jet Printing Four cartridges are loaded with the four nucleotides: A, G, C,T As the printer head moves across the array, the nucleotides are deposited where they are needed From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Electrochemical Synthesis Electrodes are embedded in the substrate to manage individual reaction sites Electrodes are activated in necessary positions in a predetermined sequence that allows the sequences to be constructed base by base Solutions containing specific bases are washed over the substrate while the electrodes are activated From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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http://www.bio.davidson.edu/courses/genomics/chip/chip.html

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Application of Microarrays We only know the function of about 20% of the 30,000 genes in the Human Genome Gene exploration Faster and better Can be used for DNA computing http://www.gene-chips.com/sample1.html From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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A Data Mining Problem On a given Microarray we test on the order of 10k elements at a time Data is obtained faster than it can be processed We need some ways to work through this large data set and make sense of the data

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Grouping and Reduction Grouping: discovers patterns in the data from a microarray Reduction: reduces the complexity of data by removing redundant probes (genes) that will be used in subsequent assays

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Unsupervised Grouping: Clustering Pattern discovery via grouping similarly expressed genes together Three techniques most often used k-Means Clustering Hierarchical Clustering Kohonen Self Organizing Feature Maps

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Clustering Limitations Any data can be clustered, therefore we must be careful what conclusions we draw from our results Clustering is non-deterministic and can and will produce different results on different runs

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K-means Clustering Given a set of n data points in d- dimensional space and an integer k We want to find the set of k points in d- dimensional space that minimizes the mean squared distance from each data point to its nearest center No exact polynomial-time algorithms are known for this problem A Local Search Approximation Algorithm for k-Means Clustering by Kanungo et. al

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K-means Algorithm (Lloyds Algorithm) Has been shown to converge to a locally optimal solution But can converge to a solution arbitrarily bad compared to the optimal solution K-means-type algorithms: A generalized convergence theorem and characterization of local optimality by Selim and Ismail A Local Search Approximation Algorithm for k-Means Clustering by Kanungo et al. K=3 Data Points Optimal Centers Heuristic Centers

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Euclidean Distance Now to find the distance between two points, say the origin and the point (3,4): Simple and Fast! Remember this when we consider the complexity!

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Finding a Centroid We use the following equation to find the n dimensional centroid point amid k n dimensional points: Lets find the midpoint between 3 2D points, say: (2,4) (5,2) (8,9)

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K-means Algorithm 1. Choose k initial center points randomly 2. Cluster data using Euclidean distance (or other distance metric) 3. Calculate new center points for each cluster using only points within the cluster 4. Re-Cluster all data using the new center points 1. This step could cause data points to be placed in a different cluster 5. Repeat steps 3 & 4 until the center points have moved such that in step 4 no data points are moved from one cluster to another or some other convergence criteria is met From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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An example with k=2 1. We Pick k=2 centers at random 2. We cluster our data around these center points Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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K-means example with k=2 3. We recalculate centers based on our current clusters Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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K-means example with k=2 4. We re-cluster our data around our new center points Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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K-means example with k=2 5. We repeat the last two steps until no more data points are moved into a different cluster Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Choosing k Use another clustering method Run algorithm on data with several different values of k Use advance knowledge about the characteristics of your test Cancerous vs Non-Cancerous

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Cluster Quality Since any data can be clustered, how do we know our clusters are meaningful? The size (diameter) of the cluster vs. The inter- cluster distance Distance between the members of a cluster and the clusters center Diameter of the smallest sphere From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Cluster Quality Continued size=5 distance=20 distance=5 Quality of cluster assessed by ratio of distance to nearest cluster and cluster diameter Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Cluster Quality Continued Quality can be assessed simply by looking at the diameter of a cluster A cluster can be formed even when there is no similarity between clustered patterns. This occurs because the algorithm forces k clusters to be created. From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Characteristics of k-means Clustering The random selection of initial center points creates the following properties Non-Determinism May produce clusters without patterns One solution is to choose the centers randomly from existing patterns From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Algorithm Complexity Linear in the number of data points, N Can be shown to have time of cN c does not depend on N, but rather the number of clusters, k Low computational complexity High speed From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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The Need for a New Algorithm -Each data point is assigned to the correct cluster -Data points that seem to be far away from each other in heuristic are in reality very closely related to each other Figure Reproduced From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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The Need for a New Algorithm Eisen et al., 1998

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Kohonen Self Organizing Feature Maps (SOFM) Creates a map in which similar patterns are plotted next to each other Data visualization technique that reduces n dimensions and displays similarities More complex than k-means or hierarchical clustering, but more meaningful Neural Network Technique Inspired by the brain From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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SOFM Description Each unit of the SOFM has a weighted connection to all inputs As the algorithm progresses, neighboring units are grouped by similarity Input Layer Output Layer From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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SOFM Algorithm Initialize Map For t from 0 to 1 //t is the learning factor Randomly select a sample Get best matching unit Scale neighbors Increase t a small amount //decrease learning factor End for From: http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color Three dimensional data: red, blue, green Will be converted into 2D image map with clustering of Dark Blue and Greys together and Yellow close to Both the Red and the Green From http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color Each color in the map is associated with a weight From http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color 1. Initialize the weights Random Values Colors in the Corners Equidistant From http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color Continued 2. Get best matching unit After randomly selecting a sample, go through all weight vectors and calculate the best match (in this case using Euclidian distance) Think of colors as 3D points each component (red, green, blue) on an axis From http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color Continued 2. Getting the best matching unit continued… For example, lets say we chose green as the sample. Then it can be shown that light green is closer to green than red: Green: (0,6,0) Light Green: (3,6,3) Red(6,0,0) This step is repeated for entire map, and the weight with the shortest distance is chosen as the best match From http://davis.wpi.edu/~matt/courses/soms/

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An Example Using Color Continued 3. Scale neighbors 1. Determine which weights are considred nieghbors 2. How much each weight can become more like the sample vector From http://davis.wpi.edu/~matt/courses/soms/ 1. Determine which weights are considered neighbors In the example, a gaussian function is used where every point above 0 is considered a neighbor

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An Example Using Color Continued From http://davis.wpi.edu/~matt/courses/soms/ 2.How much each weight can become more like the sample When the weight with the smallest distance is chosen and the neighbors are determined, it and its neighbors learn by changing to become more like the sample…The farther away a neighbor is, the less it learns

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An Example Using Color Continued NewColorValue = CurrentColor*(1-t)+sampleVector*t For the first iteration t=1 since t can range from 0 to 1, for following iterations the value of t used in this formula decreases because there are fewer values in the range (as t increases in the for loop) From http://davis.wpi.edu/~matt/courses/soms/

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Conclusion of Example Samples continue to be chosen at random until t becomes 1 (learning stops) At the conclusion of the algorithm, we have a nicely clustered data set. Also note that we have achieved our goal: Similar colors are grouped closely together From http://davis.wpi.edu/~matt/courses/soms/

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SOFM Applied to Genetics Consider clustering 10,000 genes Each gene was measured in 4 experiments Input vectors are 4 dimensional Initial pattern of 10,000 each described by a 4D vector Each of the 10,000 genes is chosen one at a time to train the SOM From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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SOFM Applied to Genetics The pattern found to be closest to the current gene (determined by weight vectors) is selected as the winner The weight is then modified to become more similar to the current gene based on the learning rate (t in the previous example) The winner then pulls its neighbors closer to the current gene by causing a lesser change in weight From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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SOFM Applied to Genetics This process continues for all 10,000 genes Process is repeated until over time the learning rate is reduced to zero From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Our Favorite Example With Yeast Reduce data set to 828 genes Clustered data into 30 clusters using a SOFM Interpresting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al. Each pattern is represented by its average (centroid) pattern Clustered data has same behavior Neighbors exhibit similar behavior

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A SOFM Example With Yeast Interpresting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al.

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Benefits of SOFM SOFM contains the set of features extracted from the input patterns (reduces dimensions) SOFM yields a set of clusters A gene will always be most similar to a gene in its immediate neighborhood than a gene further away From Data Analysis Tools for DNA Microarrays by Sorin Draghici

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Conclusion K-means is a simple yet effective algorithm for clustering data Self-organizing feature maps are slightly more computationally expensive, but they solve the problem of spatial relationship Interpresting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al.

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References Basic microarray analysis: grouping and feature reduction by Soumya Raychaudhuri, Patrick D. Sutphin, Jeffery T. Chang and Russ B. Altman; Trends in Biotechnology Vol. 19 No. 5 May 2001 Self Organizing Maps, Tom Germano, http://davis.wpi.edu/~matt/courses/soms http://davis.wpi.edu/~matt/courses/soms Data Analysis Tools for DNA Microarrays by Sorin Draghici; Chapman & Hall/CRC 2003 Self-Organizing-Feature-Maps versus Statistical Clustering Methods: A Benchmark by A. Ultsh, C. Vetter; FG Neuroinformatik & Kunstliche Intelligenz Research Report 0994

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References Interpreting patterns of gene expression with self- organizing maps: Methods and application to hematopoietic differentiation by Tamayo et al. A Local Search Approximation Algorithm for k-Means Clustering by Kanungo et al. K-means-type algorithms: A generalized convergence theorem and characterization of local optimality by Selim and Ismail

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