By Thomas A. Runkler
This booklet is a complete creation to the equipment and algorithms of contemporary info analytics. It presents a legitimate mathematical foundation, discusses benefits and downsides of other techniques, and allows the reader to layout and enforce facts analytics ideas for real-world purposes. This publication has been used for greater than ten years within the information Mining path on the Technical collage of Munich. a lot of the content material relies at the result of business learn and improvement tasks at Siemens.
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Extra info for Data Analytics: Models and Algorithms for Intelligent Data Analysis
18) iDqC1 PCA chooses the eigenvectors with the largest eigenvalues, so PCA yields the projection with not only the largest variance but also the smallest quadratic transformation error. In our derivation of PCA we maximized the variance and found that the same method minimizes the quadratic transformation error. PCA can also be derived in the reverse order: if we minimize the quadratic transformation error, then we will find that the same method maximizes the variance. We illustrate PCA with a simple example.
It has to be small enough to achieve a sufficient filter effect but large enough to maintain the essential characteristics of the original data. The moving mean and the exponential filter are special cases of the more general family of discrete linear filters. 12) i iD0 with the filter coefficients a0 ; : : : ; aq 1 ; b0 ; : : : ; bq 1 2 R . 13) i For simplicity we consider only the asymmetric case here. The reader may easily modify the indices to obtain the equations for the symmetric discrete linear filter.
49) D kyi yj k D :0 @yk @yk otherwise so we obtain @E3 2 D n n P P @yk iD1 jDiC1 @2 E3 2 D n n 2 P P @yk iD1 jDiC1 n X dijx jD1 j¤k n X dijx jD1 j¤k 1 dkjx 1 y dkj 1 dkjx 1 y dkj ! dkj /3 ! 51) Consider again the data set from Fig. 5. 41). We initialize Y D f1; 2; 3; 4g, which corresponds to the bottom row in Fig. 5 0 0 1 2 3 4 y1 1 y2 2 y3 3 y4 4 0 2 4 6 8 10 Fig. 53) This is the first (leftmost) value of the Sammon error function shown in Fig. 9 (right). For this initialization, the error gradients are 2 @E3 D p p @y1 3 1C2 2C 5 2 @E3 D p p @y2 3 1C2 2C 5 2 @E3 D p p @y3 3 1C2 2C 5 2 @E3 D p p @y4 3 1C2 2C 5 p p !