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Higher Level Than Ever Before Superfunctions, meta-algorithms... Wolfram Demonstrations Project. Wolfram Demonstrations Project. Wolfram Demonstrations Project. Align multiple sequences against multiple sequences.
K-means Cluster Analysis. Basic Algorithm Definition 1: The basic k-means clustering algorithm is defined as follows: Step 1: Choose the number of clusters kStep 2: Make an initial selection of k centroidsStep 3: Assign each data element in S to its nearest centroid (in this way k clusters are formed one for each centroid, where each cluster consists of all the data elements assigned to that centroid)Step 4: For each cluster make a new selection of its centroidStep 5: Go back to step 3, repeating the process until the centroids don’t change (or some other convergence criterion is met) There are various choices available for each step in the process.
An alternative version of the algorithm is as follows: Distance There are a number of ways to define distance between two n-tuples in the data set S, but we will focus on the Euclidean measure, namely, if x = (x1, …, xn) and y = (y1, …, yn) then the distance between x and y is defined by where mj is the number of data elements in Cj. Example Figure 3 – Cluster Assignment. Distance -- from Wolfram MathWorld. The distance between two points is the length of the path connecting them. In the plane, the distance between points and is given by the Pythagorean theorem, In Euclidean three-space, the distance between points is In general, the distance between points in a Euclidean space is given by For curved or more complicated surfaces, the so-called metric can be used to compute the distance between two points by integration.
Let be a smooth curve in a manifold from to with. . , where is the tangent space of at . With respect to the Riemannian structure is given by and the distance between is the shortest distance between given by In order to specify the relative distances of points in the plane, coordinates are needed, since the first can always be taken as (0, 0) and the second as , which defines the x-axis. Points need two coordinates each.
Where is a binomial coefficient. Points are therefore subject to relationships, where For , 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... Points is the triangular number .