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data_mining:pca [2014/08/24 01:34] – angelegt phreazer | data_mining:pca [2014/08/24 02:10] – [Parameterwahl (k)] phreazer | ||
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Für k Dimensionen: | Für k Dimensionen: | ||
+ | |||
+ | ===== Algorithmus ===== | ||
+ | |||
+ | Berechnung der Kovarianzmatrix: | ||
+ | $\Sigma = \frac{1}{m} \sum_{i=1}^n x^{(i)} (x^{(i)})^T = 1/m X^T X$ | ||
+ | |||
+ | Berechnung der Eigenvektoren der Matrix $\sigma$: | ||
+ | |||
+ | svd-Funktion: | ||
+ | |||
+ | $U_{\text{reduce}}$ : k-Spalten der U-Matrix ($n \times n$) | ||
+ | |||
+ | $z = U_{\text{reduce}}^T x$ | ||
+ | |||
+ | ===== Parameterwahl (k) ===== | ||
+ | |||
+ | 99% der Varianz bleibt erhalten. | ||
+ | |||
+ | $$ | ||
+ | \frac{\frac{1}{m} \sum_{i=1}^m || x^{(i)} - x_{\text{approx}}^{(i)} ||^2}{\frac{1}{m} \sum_{i=1}^m || x^{(i)}||^2} \leq 0.01 | ||
+ | $$ | ||
+ | |||
+ | [U,S,V] mit S als diagonale Matrix. | ||
+ | |||
+ | Für ein k, kann $1-\frac{\sum_{i=1}^k S_{ii}}{\sum_{i=1}^n S_{ii}} \leq 0.01$. | ||
+ | |||
+ | ===== Decompression ===== | ||
+ | $x_\text{approx} = U_\text{reduce} z$ |