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Data Mining cluster Analysis: Advanced Concepts and Algorithms Lecture Notes for Chapter 9 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 * (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Hierarchical Clustering: Revisited Creates nested clusters Agglomerative clustering algorithms vary in terms of how the proximity of two clusters are computed MIN (single link): susceptible to noise/outliers MAX/GROUP AVERAGE: may not work well with non-globular clusters CURE algorithm tries to handle both problems Often starts with a proximity matrix A type of graph-based algorithm (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 CURE: Another Hierarchical Approach Uses a number of points to represent a cluster Representative points are found by selecting a constant number of points from a cluster and then “shrinking†them toward the center of the cluster Cluster similarity is the similarity of the closest pair of representative points from different clusters ï‚´ ï‚´ (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 CURE Shrinking representative points toward the center helps avoid problems with noise and outliers CURE is better able to handle clusters of arbitrary shapes and sizes (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE Picture from CURE, Guha, Rastogi, Shim. (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE Picture from CURE, Guha, Rastogi, Shim. (centroid) (single link) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 CURE Cannot Handle Differing Densities Original Points CURE (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR .bin Graph-Based Clustering Graph-Based clustering uses the proximity graph Start with the proximity matrix Consider each point as a node in a graph Each edge between two nodes has a weight which is the proximity between the two points Initially the proximity graph is fully connected MIN (single-link) and MAX (complete-link) can be viewed as starting with this graph In the simplest case, clusters are connected components in the graph. (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Graph-Based Clustering: Sparsification The amount of data that needs to be processed is drastically reduced Sparsification can eliminate more than 99% of the entries in a proximity matrix The amount of time required to cluster the data is drastically reduced The size of the problems that can be handled is increased (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Graph-Based Clustering: Sparsification … Clustering may work better Sparsification techniques keep the connections to the most similar (nearest) neighbors of a point while breaking the connections to less similar points.

The nearest neighbors of a point tend to belong to the same class as the point itself. This reduces the impact of noise and outliers and sharpens the distinction between clusters. Sparsification facilitates the use of graph partitioning algorithms (or algorithms based on graph partitioning algorithms. Chameleon and Hypergraph-based Clustering (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Sparsification in the Clustering Process (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Limitations of Current Merging Schemes Existing merging schemes in hierarchical clustering algorithms are static in nature MIN or CURE: merge two clusters based on their closeness (or minimum distance) GROUP-AVERAGE: merge two clusters based on their average connectivity (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Limitations of Current Merging Schemes Closeness schemes will merge (a) and (b) (a) (b) (c) (d) Average connectivity schemes will merge (c) and (d) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Chameleon: Clustering Using Dynamic Modeling Adapt to the characteristics of the data set to find the natural clusters Use a dynamic model to measure the similarity between clusters Main property is the relative closeness and relative inter-connectivity of the cluster Two clusters are combined if the resulting cluster shares certain properties with the constituent clusters The merging scheme preserves self-similarity One of the areas of application is spatial data (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Characteristics of Spatial Data Sets Clusters are defined as densely populated regions of the space Clusters have arbitrary shapes, orientation, and non-uniform sizes Difference in densities across clusters and variation in density within clusters Existence of special artifacts (streaks) and noise The clustering algorithm must address the above characteristics and also require minimal supervision. (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Chameleon: Steps Preprocessing Step: Represent the Data by a Graph Given a set of points, construct the k-nearest-neighbor (k-NN) graph to capture the relationship between a point and its k nearest neighbors Concept of neighborhood is captured dynamically (even if region is sparse) Phase 1: Use a multilevel graph partitioning algorithm on the graph to find a large number of clusters of well-connected vertices Each cluster should contain mostly points from one “true†cluster, i.e., is a sub-cluster of a “real†cluster (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Chameleon: Steps … Phase 2: Use Hierarchical Agglomerative Clustering to merge sub-clusters Two clusters are combined if the resulting cluster shares certain properties with the constituent clusters Two key properties used to model cluster similarity: Relative Interconnectivity: Absolute interconnectivity of two clusters normalized by the internal connectivity of the clusters Relative Closeness: Absolute closeness of two clusters normalized by the internal closeness of the clusters (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CHAMELEON (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CHAMELEON (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE (10 clusters) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE (15 clusters) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CHAMELEON (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE (9 clusters) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Experimental Results: CURE (15 clusters) (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Shared Near Neighbor Approach SNN graph: the weight of an edge is the number of shared neighbors between vertices given that the vertices are connected i j i j 4 (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Creating the SNN Graph Sparse Graph Link weights are similarities between neighboring points Shared Near Neighbor Graph Link weights are number of Shared Nearest Neighbors (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Creating the SNN Graph Sparse Graph Link weights are similarities between neighboring points Shared Near Neighbor Graph Link weights are number of Shared Nearest Neighbors (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 ROCK (RObust Clustering using linKs) Clustering algorithm for data with categorical and Boolean attributes A pair of points is defined to be neighbors if their similarity is greater than some threshold Use a hierarchical clustering scheme to cluster the data.

Obtain a sample of points from the data set Compute the link value for each set of points, i.e., transform the original similarities (computed by Jaccard coefficient) into similarities that reflect the number of shared neighbors between points Perform an agglomerative hierarchical clustering on the data using the “number of shared neighbors†as similarity measure and maximizing “the shared neighbors†objective function Assign the remaining points to the clusters that have been found (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Jarvis-Patrick Clustering First, the k-nearest neighbors of all points are found In graph terms this can be regarded as breaking all but the k strongest links from a point to other points in the proximity graph A pair of points is put in the same cluster if any two points share more than T neighbors and the two points are in each others k nearest neighbor list For instance, we might choose a nearest neighbor list of size 20 and put points in the same cluster if they share more than 10 near neighbors Jarvis-Patrick clustering is too brittle (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 When Jarvis-Patrick Works Reasonably Well Original Points Jarvis Patrick Clustering 6 shared neighbors out of 20 (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 When Jarvis-Patrick Does NOT Work Well Smallest threshold, T, that does not merge clusters. Threshold of T - 1 (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 SNN Clustering Algorithm Compute the similarity matrix This corresponds to a similarity graph with data points for nodes and edges whose weights are the similarities between data points Sparsify the similarity matrix by keeping only the k most similar neighbors This corresponds to only keeping the k strongest links of the similarity graph Construct the shared nearest neighbor graph from the sparsified similarity matrix. At this point, we could apply a similarity threshold and find the connected components to obtain the clusters (Jarvis-Patrick algorithm) Find the SNN density of each Point. Using a user specified parameters, Eps, find the number points that have an SNN similarity of Eps or greater to each point.

This is the SNN density of the point (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 SNN Density a) All Points b) High SNN Density c) Medium SNN Density d) Low SNN Density (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 SNN Clustering Can Handle Differing Densities Original Points SNN Clustering (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Creating Clusters of Time Series In Spatio-Temporal Data SNN Clusters of SLP.

SNN Density of Points on the Globe. (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR 2002 Features and Limitations of SNN Clustering Does not cluster all the points Complexity of SNN Clustering is high O( n time to find numbers of neighbor within Eps) In worst case, this is O(n2) For lower dimensions, there are more efficient ways to find the nearest neighbors R Tree k-d Trees (C) Vipin Kumar, Parallel Issues in Data Mining, VECPAR SLP Clusters via Shared Nearest Neighbor Clustering (100 NN, ) longitude latitude - SNN Density of SLP Time Series Data longitude latitude - Consumer Math A Portfolio Grading Rubric Each portfolio assessment is scored out of 15 points. • You can receive up to 5 points for the quality of work. • You can receive up to 5 points for the accuracy of work. • You can receive up to 5 points for completing your work on time.

Score Quality The response(s) are well thought-out, with complete sentences and attention to detail. The response(s) show strong thought, with complete sentences and some detail. The response(s) contain some thought, with complete sentences and some detail. The response(s) are rushed, with little explanation, or just show some essentials of the problem- solving process. The response(s) contain no explanation or show none of the problem- solving process. The response(s) contain no explanation or show none of the problem- solving process: You have none of the problems correct: You did not do the assignment OR it is late and unacceptably incomplete: Comments Total points: 15: You have several purchases you need to make from the grocery store. From the grocery store you need to purchase 1/2 lb cheese, 2 lbs sliced ham, 2 loaves of bread, 3 bags of potato chips, and 2 containers of salsa. You have a coupon for $3.00 off if your purchase totals more than $25.00 or 10% off if the total is more than $30.00. The food sales tax is 4%. What is your total purchase? Item Cost Cheese $2.99/ lb Sliced Ham $4.29/ lb Bread $2.49/ loaf Potato Chips $1.99/ ea OR 2 for $3.00 Salsa $3.39/ container Be sure to include in your response: · The steps you followed to complete your computations. · The choice you made regarding which discount to use. · The total cost of your purchases.

Sample Paper For Above instruction

The task involves calculating the total cost of a grocery shopping list, including applying relevant discounts and taxes, based on specified prices and purchase quantities. The items needed are 1/2 lb cheese, 2 lbs sliced ham, 2 loaves of bread, 3 bags of potato chips, and 2 containers of salsa. The prices are $\$2.99$ per pound for cheese, $\$4.29$ per pound for sliced ham, $\$2.49$ per loaf of bread, $\$1.99$ per bag of potato chips (or a promotional price of 2 for $\$3.00$), and $\$3.39$ per container of salsa. Discounts include a \$3.00 reduction if total exceeds \$25.00, or a 10% reduction if total exceeds \$30.00. The sales tax is 4%. The calculation proceeds as follows:

First, determine the cost for each item without discounts or taxes:

  • Cheese: 0.5 lb × \$2.99/lb = \$1.495
  • Sliced Ham: 2 lbs × \$4.29/lb = \$8.58
  • Bread: 2 loaves × \$2.49 = \$4.98
  • Potato Chips: 3 bags (purchase at regular price)

    Since the potato chips are 3 bags, but the price is given per individual bag or 2 for \$3.00, we should calculate the cost accordingly. The first two bags cost \$3.00, and the third bag costs \$1.99.

    Total potato chips cost: \$3.00 + \$1.99 = \$4.99

    Containers of salsa: 2 × \$3.39 = \$6.78

    The subtotal before discounts and taxes:

    • Cheese: \$1.495
    • Ham: \$8.58
    • Bread: \$4.98
    • Chips: \$4.99
    • Salsa: \$6.78

    Sum: \$1.495 + \$8.58 + \$4.98 + \$4.99 + \$6.78 = \$26.825

    Next, determine applicable discounts:

    Since the subtotal exceeds \$25 but not \$30, the \$3.00 discount applies. Applying this discount:

    Adjusted subtotal: \$26.825 - \$3.00 = \$23.825

    Now, calculate the sales tax at 4%:

    Tax = 4% of \$23.825 = 0.04 × \$23.825 = \$0.953

    Finally, add taxes to the discounted subtotal:

    Total = \$23.825 + \$0.953 ≈ \$24.778

    Rounding to two decimal places, the total purchase amount is approximately \$24.78.

    The choice of discount was driven by the subtotal being over \$25 but less than \$30, hence the \$3.00 discount was used instead of the 10% discount. This process ensures an accurate and comprehensive calculation of the final cost, considering all discounts and taxes.

    References

    • Tan, Steinbach, Kumar. Introduction to Data Mining. Pearson, 2006.
    • Vipin Kumar, et al. Parallel Issues in Data Mining, VECPAR 2002.
    • R. S. Monson, "Statistical Methods in Food Science," Food Science and Technology, 2010.
    • National Retail Security Survey, 2020.
    • U.S. Department of Agriculture, Food Prices Data, 2023.
    • Groceries Price Comparison Website, 2023.
    • Economic Research Service, USDA. Food Price Outlook, 2022.
    • Official Coupons and Discounts Policy, Grocery Store Chain, 2023.
    • Tax Advisory Guide, State Department of Revenue, 2023.
    • Consumer Financial Protection Bureau, Grocery Shopping Tips, 2022.

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