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# Data analysis05 clustering

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### Data analysis05 clustering

1. 1. http://publicationslist.org/junio Data Analysis Clustering Prof. Dr. Jose Fernando Rodrigues Junior ICMC-USP
2. 2. http://publicationslist.org/junio What is it about? Clustering refers to the process of finding groups of points that are in some way “lumped together” A modality of unsupervised learning, as we do not know ahead of time where and what are the clusters – no training! Explanatorily tries to characterize the structure of a dataset
3. 3. http://publicationslist.org/junio But, what is a cluster? groups of points that are similar groups of points that are close to each other groups well-separated one from each other contiguous regions of high data point density separated by regions of lower point density
4. 4. http://publicationslist.org/junio But, what is a cluster? Any clusters here? There should not be, as they are uniformly (no two points overlap, yet) generated points. Eventhough, most algorithms would point out some clusters. It is not that there are clusters there, it is only that we do not have enough points yet.
5. 5. http://publicationslist.org/junio But, what is a cluster? Any clusters here? There should not be, as they are uniformly (no two points overlap, yet) generated points. Eventhough, most algorithms would point out some clusters. It is not that there are clusters there, it is only that we do not have enough points yet. The point here is – although one would find clusters, they definitely do not explain the phenomenon accurately.
6. 6. http://publicationslist.org/junio But, what is a cluster? Yes! Three clusters, I can see them. Distance-based algorithms can do well here. Easy huh?! No wonder, here we have convex, disjoint, and well-separated groups of points. Try the next ones!
7. 7. http://publicationslist.org/junio But, what is a cluster? Non-convex clusters – simple distance-based algorithms would have trouble here. A cluster is convex if the line connecting any two points lies entirely within the cluster itself.
8. 8. http://publicationslist.org/junio But, what is a cluster? Non-convex clusters – simple distance-based algorithms would have trouble here. A cluster is convex if the line connecting any two points lies entirely within the cluster itself. There are also the star-convex clusters: in such case, the connecting line from the spatial center of the cluster to any other point lies entirely within the cluster.
9. 9. http://publicationslist.org/junio But, what is a cluster? Intersecting clusters – quite a challenge!
10. 10. http://publicationslist.org/junio But, what is a cluster? No general clustering algorithm can solve this.The clustering is given by the global properties observed in the points – distance or neighbor based algorithms would yield a single cluster.
11. 11. http://publicationslist.org/junio But, what is a cluster? No general clustering algorithm can solve this.The clustering is given by the global properties observed in the points – distance or neighbor based algorithms would yield a single cluster. In this case, for any algorithm that considers a single point (or a single pair of points) at a time, this leads to a problem: to determine cluster membership, we need the property of the whole cluster; but to determine the properties (vertical, horizontal, and pairwise orthogonal) of the cluster, we must ﬁrst assign points to clusters.
12. 12. http://publicationslist.org/junio But, what is a cluster? To handle such situations, we would need to perform some kind of global structure analysis—a task our minds are incredibly good at (which is why we tend to think of clusters this way) but that we have a hard time teaching computers to do For problems in two dimensions, digital image processing has developed methods to recognize and extract certain features (such as edge detection) But general clustering methods deal only with local properties and therefore can’t handle problems such as these
13. 13. http://publicationslist.org/junio But, what is a cluster? If we return to our candidate definitions of cluster, we can verify that none of them survives the possibilities just presented – try it!  groups of points that are similar  groups of points that are close to each other  groups well-separated one from each other  contiguous regions of high data point density separated by regions of lower point density
14. 14. http://publicationslist.org/junio But, what is a cluster? If we return to our candidate definitions of cluster, we can verify that none of them survives the possibilities just presented – try it!  groups of points that are similar  groups of points that are close to each other  groups of points well-separated one from each other  contiguous regions of high data point density separated by regions of lower point density So this is it. • No mathematical, nor universal definition of a cluster • Rather, we have our intuition and it could be quite useful provided we have a good comprehension of the data properties – structural, statistical, and domain- related • Having, as much as possible, well-defined goals is also a demand • Just as for any other data analysis approach, do not try to use it as a magic black box – doing so will fail with high probability!
15. 15. http://publicationslist.org/junio Distances Clustering does not actually require data points to be embedded into a geometric space: all that is required is a distance or (equivalently) a similarity measure for any pair of points  This makes it possible to perform clustering on a set of strings, for example  However, if the data points have properties of a vector space then we can develop more efﬁcient algorithms that exploit these properties
16. 16. http://publicationslist.org/junio Distances – what is it? A distance is any function d(x, y) that takes two points and returns a scalar value that is a measure for how different these points are: the more different, the larger the distance A distance function – or, a similarity function:  s(x, y) = 1-d(x,y), for 0 ≤ d(x,y) ≤ 1  s(x,y) = 1/d(x,y)  s(x,y) = e-d For some problems, a particular distance measure will present itself naturally - if the data points are points in space, then we will most likely employ the Euclidean distance or a measure similar to it, but for other problems, we have more freedom to deﬁne our own metric
17. 17. http://publicationslist.org/junio Distances – metric distances  There are certain properties that a distance (or similarity) function should have. Mathematicians have developed a set of properties that a function must possess to be considered a metric (or distance) in a mathematical sense  d(x, y) = 0  d(x, y) = 0 if and only if x = y  d(x, y) = d(y, x)  d(x, y) + d(y, z) ≥ d(x, z)  These conditions are not necessarily fulﬁlled in practice. A funny example for an asymmetric distance occurs if you ask everyone in a group of people how much they like every other member of the group and then use the responses to construct a distance measure: it is not at all guaranteed that the feelings of person A for person B are requited by B
18. 18. http://publicationslist.org/junio Distances – metric distances  There are certain properties that a distance (or similarity) function should have. Mathematicians have developed a set of properties that a function must possess to be considered a metric (or distance) in a mathematical sense.  d(x, y) = 0  d(x, y) = 0 if and only if x = y  d(x, y) = d(y, x)  d(x, y) + d(y, z) ≥ d(x, z)  These conditions are not necessarily fulﬁlled in practice. A funny example for an asymmetric distance occurs if you ask everyone in a group of people how much they like every other member of the group and then use the responses to construct a distance measure: it is not at all guaranteed that the feelings of person A for person B are requited by B For technical reasons, the symmetry property is usually highly desirable. You can always construct a symmetric distance function from an asymmetric one: dS(x, y) = d(x, y) + d(y, x) 2
19. 19. http://publicationslist.org/junio Distances – common distances  Commonly used distance and similarity measures for numeric data
20. 20. http://publicationslist.org/junio Distances – common distances  Distances Manhattan, Euclidean, Maximum, and Minkowski have all similar properties, the application of each may depend on empirical testing, or on subtle details of the data-domain Minkowski (L metric) Maximum (L infinity) Minkowski (L metric)
21. 21. http://publicationslist.org/junio Distances – correlation-based  Correlation-based measures: used if the data is numeric but not mixable (so that it does not make sense to add a random fraction of one data set to a random fraction of a different data set), as for example, in time series The dot product of two points is the cosine of the angle that the two vectors make with each other - if they are perfectly aligned then the angle is 0 and the cosine (and the correlation) is 1; If they are at right angles to each other, the cosine is 0  The only difference between the dot product and the correlation coefﬁcient is that for the second, we ﬁrst center both data points by subtracting their respective means  By construction, the value of a dot product always falls in the interval [0, 1], and the correlation coefﬁcient always falls in the interval [−1, 1]
22. 22. http://publicationslist.org/junio Distances – binary and sparse If the data is categorical, then we can count the number of features that do not agree in both data points (i.e., the number of mismatched features); this is the Hamming distance As an example, imagine a patient’s health record: each possible medical condition constitutes a feature, and we want to know whether the patient has ever suffered from it In situations where the features are categorical, binary, and sparse (just a few are On), we may be interested in matches between features that are On than those that are Off; this leads us to the Jaccard coefﬁcient s: the number of matches between features that are On for both points, divided by the number of features that are On in at least one of the data points The Jaccard coefﬁcient is a similarity measure; the corresponding distance function is the Jaccard distance dj = 1-sj
23. 23. http://publicationslist.org/junio Distances – binary and sparse If the data is categorical, then we can count the number of features that do not agree in both data points (i.e., the number of mismatched features); this is the Hamming distance As an example, imagine a patient’s health record: each possible medical condition constitutes a feature, and we want to know whether the patient has ever suffered from it In situations where the features are categorical, binary, and sparse (just a few are On), we may be interested in matches between features that are On than those that are Off; this leads us to the Jaccard coefﬁcient s: the number of matches between features that are On for both points, divided by the number of features that are On in at least one of the data points The Jaccard coefﬁcient is a similarity measure; the corresponding distance function is the Jaccard distance dj = 1-sj The Jaccard distance: As an example, imagine graph data.The similarity of two vertices is given by how many neighbors they have in common (On) – what is usually sparse, as just a few vertices are neighbors of a given vertex
24. 24. http://publicationslist.org/junio Distances – strings If we are dealing with many strings that are rather similar to each other (distorted through typos, for instance), then we can use a more detailed measure of the difference between them—namely the edit or Levenshtein distance. The Levenshtein distance is the minimum number of single-character operations (insertions, deletions, and substitutions) required to transform one string into the other Another approach is to ﬁnd the length of the longest common subsequence; this metric is often used for gene sequence analysis in computational biology
25. 25. http://publicationslist.org/junio Distances – strings If we are dealing with many strings that are rather similar to each other (distorted through typos, for instance), then we can use a more detailed measure of the difference between them—namely the edit or Levenshtein distance. The Levenshtein distance is the minimum number of single-character operations (insertions, deletions, and substitutions) required to transform one string into the other Another approach is to ﬁnd the length of the longest common subsequence; this metric is often used for gene sequence analysis in computational biology The best distance measure to use does not follow automatically from data type; rather, it depends on the semantics of the data—or, more precisely, on the semantics that you care about for your current analysis! In some cases, a simple metric that only calculates the difference in string length may be perfectly sufﬁcient. In another case, you might want to use the Hamming distance. If you really care about the details of otherwise similar strings, the Levenshtein distance is most appropriate.You might even want to calculate how often each letter appears in a string and then base your comparison on that. It all depends on what the data means and on what aspect of it you are interested at the moment (which may also change as the analysis progresses). Similar considerations apply everywhere—there are no “cookbook” rules.
26. 26. http://publicationslist.org/junio Clustering methods Different algorithms are suitable for different kinds of problems— depending, for example, on the shape and structure of the clusters Some require vector-like data, whereas others require only a distance function Different algorithms tend to be misled by different kinds of pitfalls, and they all have different performance (i.e., computational complexity) characteristics There are tree main categories of clustering algorithms: center seekers, tree builders, and neighborhood growers – I said three main, not only three (check Survey Of Clustering Data Mining Techniques of author Pavel Berkhin)
27. 27. http://publicationslist.org/junio Clustering methods – k-means One of the most popular clustering methods is the k-means algorithm; the k-means algorithm requires the number of expected clusters k as input, and works in an iterative scheme to search for the correct center of each cluster The main idea is to calculate the position of each cluster’s center (or centroid) from the positions of the points belonging to the cluster and then to assign points to their nearest centroid – this process is repeated until sufﬁcient convergence is achieved The algorithm is as follows: choose initial positions for the cluster centroids repeat: for each point: calculate its distance from each cluster centroid assign the point to the nearest cluster recalculate the positions of the cluster centroids
28. 28. http://publicationslist.org/junio Clustering methods – k-means  The k-means algorithm is nondeterministic: a different choice of starting values may result in a different assignment of points to clusters; for this reason, it is customary to run the k-means algorithm several times and then compare the results  If you have previous knowledge of likely positions for the cluster centers, you can use it to precondition the algorithm; otherwise, choose random data points as initial values.  What makes this algorithm efﬁcient is that you don’t have to search the existing data points to ﬁnd one that would make a good centroid—instead you are free to construct a new centroid position; this is usually done by calculating the cluster’s center of mass:
29. 29. http://publicationslist.org/junio Clustering methods – k-means  The k-means algorithm is nondeterministic: a different choice of starting values may result in a different assignment of points to clusters; for this reason, it is customary to run the k-means algorithm several times and then compare the results  If you have previous knowledge of likely positions for the cluster centers, you can use it to precondition the algorithm; otherwise, choose random data points as initial values.  What makes this algorithm efﬁcient is that you don’t have to search the existing data points to ﬁnd one that would make a good centroid—instead you are free to construct a new centroid position; this is usually done by calculating the cluster’s center of mass: If we are using categorical data, then the k-mean algorithm cannot be used (one cannot calculate the mass center), in this case we must use the k-medoids algorithm The only difference is that instead of calculating a new centroid, it is necessary to search all the points in the cluster to find the data point that has the smallest average distance to all other points in its cluster For this reason, the k-medoids algorithm is O(n2), meanwhile the k- means algorithm is O(k*n), where k is the number of clusters For performance,it is possible to run k-medoids in a sample of the dataset to have an idea of the cluster centers, and then run it on the entire dataset
30. 30. http://publicationslist.org/junio Clustering methods – k-means  Despite its cheap-and-cheerful appearance, the k-means algorithm works surprisingly well. It is pretty fast and relatively robust. Convergence is usually quick. Because the algorithm is simple and highly intuitive, it is easy to augment or extend it—for example, to incorporate points with different weights. You might also want to experiment with different ways to calculate the centroid, possibly using the median position rather than the mean, and so on.  In summary:  The k-means algorithms and its variants work best for globular (at least star-convex) clusters; the results will be meaningless for clusters with complicated shapes and for nested clusters  The expected number of clusters is required as an input; if this number is not known, it will be necessary to repeat the algorithm with different values and compare the results  The algorithm is iterative and nondeterministic; the speciﬁc outcome may depend on the choice of starting values  The k-means algorithm requires vector data; use the k-medoids algorithm for categorical data  The algorithm can be misled if there are clusters of highly different size or different density  The k-means algorithm is linear in the number of data points; the k-medoids algorithm is quadratic in the number of points
31. 31. http://publicationslist.org/junio Clustering methods – DBSCAN Neighborhood growers work by connecting points that are “sufﬁciently close” to each other to form a cluster and then keep doing so until all points have been classiﬁed Based on the idea (definition) of a cluster as a region of high density, and it makes no assumptions about the overall shape of the cluster More robust than k-means variations in respect to the structure of the clusters
32. 32. http://publicationslist.org/junio Clustering methods – DBSCAN The DBSCAN algorithm is an example of Neighborhood grower It is based on two metrics:  The minimum density accepted for the points that define the cluster  The size of the region over which we expect the minimum density to be verified  In practice, the algorithm asks for:  The neighborhood radius r  The minimum number of points n that we expect to find within the neighborhood of each point
33. 33. http://publicationslist.org/junio Clustering methods – DBSCAN DBSCAN distinguishes between three types of points: noise, core, and edge points:  A noise point is a point which has fewer than n points in its neighborhood of radius r, such a point does not belong to any cluster – background data A core point has more than n neighbors An edge point is a point that has fewer neighbors than required for a core point but that is itself the neighbor of a core point - the algorithm discards noise points and concentrates on core points Whenever the algorithm ﬁnds a core point, it assigns a cluster label to that point and then continues to add all its neighbors, and their neighbors recursively to the cluster, until all points have been classiﬁed
34. 34. http://publicationslist.org/junio Clustering methods – DBSCAN DBSCAN distinguishes between three types of points: noise, core, and edge points:  A noise point is a point which has fewer than n points in its neighborhood of radius r, such a point does not belong to any cluster A core point has more than n neighbors An edge point is a point that has fewer neighbors than required for a core point but that is itself the neighbor of a core point - the algorithm discards noise points and concentrates on core points Whenever the algorithm ﬁnds a core point, it assigns a cluster label to that point and then continues to add all its neighbors, and their neighbors recursively to the cluster, until all points have been classiﬁed Finally, the basic algorithm lends itself to elegant recursive implementations, but keep in mind that the recursion will not unwind until the current cluster is complete.This means that, in the worst case (of a single connected cluster), you will end up putting the entire data set onto the stack!
35. 35. http://publicationslist.org/junio Clustering methods – DBSCAN DBSCAN is sensitive to the choice of parameters For example, if a data set contains several clusters with widely varying densities, then a single set of parameters may not be sufﬁcient to classify all of the clusters A possible workaround it to use k-means first to identify cluster candidates, and then to extract statistics that will help parametrize DBSCAN The computational complexity of DBSCAN is O(n2), what can be ameliorated by indexing structures able to quickly find the neighbors of each point
36. 36. http://publicationslist.org/junio Clustering methods – tree builders Another way to ﬁnd clusters is by successively combining clusters that are “close” to each other into a larger cluster until only a single cluster remains; this approach is known as agglomerative hierarchical clustering, and it leads to a treelike hierarchy of clusters The distance between clusters is given is respect to representative points within each cluster, the possibilities are:  Minimum or single link: the two points, one from each cluster that are closest to each other; handles thinly connected clusters with complicated shapes, but it is sensible to noise  Maximum or complete link: considers the points the farthest away from each other, favors compact globular clusters  Average:considers the average between all pairs of points  Centroid: considers the centroids of each cluster  Ward’s method: combiners clusters whose coherence is higher; coherence can be the average distance of all pairs, for example
37. 37. http://publicationslist.org/junio Clustering methods – tree builders The result of hierarchical clustering is not actually a set of clusters; instead, we obtain a treelike structure that contains the individual data points at the leaf nodes - this structure can be represented graphically in a dendrogram Tree builder algorithms are expensive, on the order of O(n3)  One outstanding feature of hierarchical clustering is that it does more than produce a ﬂat list of clusters; it also shows their relationships in an explicit way  Tree builder can benefit from algorithms that are center seeker or neighborhood growers
38. 38. http://publicationslist.org/junio Pre-processing The core algorithm for grouping data points into clusters is usually only part (though the most important one) of the whole strategy Some data sets may require some cleanup or normalization before they are suitable for clustering: that’s the ﬁrst topic in this section For example, look at the two plots below and answer: which one has well-defined clusters?
39. 39. http://publicationslist.org/junio Pre-processing For example, look at the two plots below and answer: which one has well-defined clusters?  Well, as a matter of fact, both plots show the same dataset, but with different aspect ratios  The same applies to datasets that spam to very different ranges – in such cases, it is necessary to normalize the data  Problems like these are not observed in correlation-based distance
40. 40. http://publicationslist.org/junio Pre-processing  The simplest normalization can be achieved by: x’ = (x – xmin)/(xmax – xmin)  Or, otherwise, if the data is reasonably Gaussian, it is possible to use the Z- score normalization: x’ = (x – xmean)/xStdDev But first, use an Interquartile Range analysis to get rid of outliers  Actually, normalization is very sensitive to outliers and distributions that are too skewed – for these cases, there are many other normalization techniques, check for instance: http://stn.spotfire.com/spotfire_client_help/norm/norm_normalizing_columns. htm
41. 41. http://publicationslist.org/junio Pre-processing  The simplest normalization can be achieved by: x’ = (x – xmin)/(xmax-xmin)  Or, otherwise, if the data is reasonably Gaussian, it is possible to use the Z- score normalization: x’ = (x - xmean)/xStdDev But first, use an Interquartile Range analysis to get rid of outliers  Actually, normalization is very sensitive to outliers and distributions that are too skewed – for these cases, there are many other normalization techniques, check for instance: http://stn.spotfire.com/spotfire_client_help/norm/norm_normalizing_columns. htm http://stn.spotfire.com/spotfire_client_help/norm/norm_normalizing_columns.htm Normalization by Mean Normalization byTrimmed Mean Normalization by Percentile Scale between 0 and 1 Subtract the Mean Subtract the Median Normalization by Signed Ratio Normalization by Log Ratio Normalization by Log Ratio in Standard Deviation Units Z-score Calculation Normalization by Standard Deviation Also, the Mahalanobis distance is less susceptible to normalization issues
42. 42. http://publicationslist.org/junio Post-processing (cluster evaluation)  It is also necessary to inspect the results of every clustering algorithm in order to validate and characterize the clusters that have been found  Given a set of clusters whose centroids are known, we can think of two metrics:  Mass: the number of points in the cluster  Radius: the standard deviation of the distances of all points in relation to the center of a given cluster; for two dimensions, we would have: r2 = ∑i (xc – xi)2 + (yc – yi)2 (xc,yc ) the center of a cluster  We can also have the density of a cluster given by: density = mass/radius
43. 43. http://publicationslist.org/junio Post-processing (cluster evaluation)  Besides density, there are:  Cohesion: the average distance between all points in a cluster, the smaller the more compact  Separation: the average distance between all point in one cluster, and all the points in another cluster – if we know the centroids, we could use them to simplify calculi  For a set of clusters, we can calculate the average cohesion and separation for all clusters, and have an idea of the overall quality  If a data set can be clearly grouped into clusters, then we expect the distance between the clusters to be large compared to the radii of the clusters; therefore, we can think of an interesting metric based on cohesion and separation: cluster_quality = separation/cohesion
44. 44. http://publicationslist.org/junio Post-processing (cluster evaluation)  One the most used metrics for clustering is the Silhouette coefficient, which for a sigle point i is given by: Si = bi – ai . max(ai,bi) where ai is the average distance from point i to all other points in its cluster (this is point i’s cohesion), bi is the smallest average distance from point i to all the points in each of the other clusters (this is point i’s separation from the closest other cluster)  The numerator is a measure for the “empty space” between clusters, the denominator is the biggest between radius and distance between clusters  Next, average the silhouette for all points in each cluster – this is the cluster’s silhouette; average it for all clusters, this is the clustering’s silhouette  The silhouette coefﬁcient ranges from −1 to 1; negative values indicate that the cluster radius is greater than the distance between clusters, so that clusters overlap; this suggests poor clustering. Large values of S suggest good clustering
45. 45. http://publicationslist.org/junio Post-processing (cluster evaluation)  One the most used metrics for clustering is the Silhouette coefficient, which for a sigle point i is given by: Si = bi – ai . max(ai,bi) where ai is the average distance from point i to all other points in its cluster (this is point i’s cohesion), bi is the smallest average distance from point i to all the points in each of the other clusters (this is point i’s separation from the closest other cluster)  The numerator is a measure for the “empty space” between clusters, the denominator is the biggest between radius and distance between clusters  Next, average the silhouette for all points in each cluster – this is the cluster’s silhouette; average it for all clusters, this is the clustering’s silhouette  The silhouette coefﬁcient ranges from −1 to 1; negative values indicate that the cluster radius is greater than the distance between clusters, so that clusters overlap; this suggests poor clustering. Large values of S suggest good clustering The silhouette can be used to toss background points from the clustering process, that is, points that notoriously exceed the average cohesion within a given cluster. This process can be used iteratively – once some points are tossed off, the clustering can be repeated and hopefully produce better results; and again.
46. 46. http://publicationslist.org/junio Post-processing (cluster evaluation)  The clustering silhouette is very important, it not only tells us the quality of a clustering, it can also tell us what is the correct clustering; for example, consider the following dataset:
47. 47. http://publicationslist.org/junio Post-processing (cluster evaluation)  The clustering silhouette is very important, it not only tells us the quality of a clustering, it can also tell us what is the correct clustering; for example, consider the following dataset: Clearly we have clusters, but how many?Visually, we can track from 6 to 8 clusters, depending on the observation. What to do?
48. 48. http://publicationslist.org/junio Post-processing (cluster evaluation)  One way to solve this problem is to use the k-mean algorithm and calculate the Silhoutte different numbers of clusters  In our example, we would get the following curve: 6 7
49. 49. http://publicationslist.org/junio Post-processing (cluster evaluation)  One way to solve this problem is to use the k-mean algorithm and calculate the Silhoutte different numbers of clusters  In our example, we would get the following curve: 6 7 The plot indicates that 6 or 7 clusters are acceptable answers, the next stage is to consider the data characteristics in order to define what the best answer is.
50. 50. http://publicationslist.org/junio Warning  Just like any other analytical technique, clustering can lead you to unproductive circumstances (waste of time) if not used with caution; some points must be of concern:  Most algorithms depend on heuristic parameters that may demand hours for one to find the most appropriate values  Also, the algorithm lend themselves to modifications that, although may sound intuitively right, are taking you nowhere  It is reasonably possible that, although you are looking for, the data has no clusters at all; it is not such an improbable circumstance because clustering algorithms usually are treated as black boxes – be circumspect, attention with the evidences!  Despite the fact that there are evaluation methods and visualization tools, still the clustering result may be flawed; remember, there are no formal theory behind cluster concepts  Finally, this review is mostly addressed for practitioners, and not for academic personnel; for those, there are many other aspects that must be considered – for more details, please check the paper “Survey Of Clustering Data MiningTechniques” of author Pavel Berkhin, among other sources
51. 51. http://publicationslist.org/junio References  Philipp K. Janert, Data Analysis with Open Source Tools, O’Reilly, 2010.  Wikipedia, http://en.wikipedia.org  Wolfram MathWorld, http://mathworld.wolfram.com/