By Iske A , Levesley J (Eds)

ISBN-10: 1402055722

ISBN-13: 9781402055720

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**Extra info for Algorithms For Approximation Proc, Chester 2005**

**Example text**

One of the important factors in partitional clustering is the criterion function [40], and the sum of squared error function is one of the most widely used, which aims to minimize the cost function. The K-means algorithm is the best-known squared error-based clustering algorithm, which is very simple and can be easily implemented in solving many practical problems [54]. It can work very well for compact and hyperspherical clusters. The time complexity of K-means is O(N Kd), which makes it scale well for large data sets.

K randomly; 2. e. J = arg minj { x − mj }; 3. Update prototype vectors mi (t + 1) = mi (t) + hci (t)[x − mi (t)], where hci (t) is the neighborhood function that is often defined as hci (t) = rc −ri 2 ), where α(t) is the monotonically decreasing learning α(t) exp( − 2σ 2 (t) rate, r represents the position of corresponding neuron, and σ(t) is the monotonically decreasing kernel width function, or hci (t) = α(t) if node c belongs to neighborhood of winning node J 0 otherwise 4. Repeat steps 2 and 3 until no change of neuron position that is more than a small positive number is observed.

Note that by this construction the approximant is continuous, but is not a polynomial. 3 Two Examples In Figure 7 we demonstrate the operation of our algorithm in case of three domain singularities. This example indicates that the approximant generated by the dimension-elevation algorithm is superior to the bivariate polynomial approximation, in particular along the boundaries of the domain singularities. Figure 8 displays an example, showing that the approximant generated by the dimension-elevation algorithm is better than the approximant generated by the geometry-driven binary partition algorithm, and that it has a better visual quality (by avoiding the introduction of the artificial discontinuities along the partition lines).

### Algorithms For Approximation Proc, Chester 2005 by Iske A , Levesley J (Eds)

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