By Zhu Y.
The 1st a part of this dissertation describes the framework for prime functionality time sequence facts mining in line with vital primitives. info relief rework equivalent to the Discrete Fourier remodel, the Discrete Wavelet remodel, Singular price Decomposition and Random Projection, can lessen the dimensions of the knowledge with out monstrous lack of info, for this reason offers a synopsis of the knowledge. Indexing tools set up facts in order that the time sequence information should be retrieved e+ciently. Transformation on time sequence, corresponding to transferring, scaling, time moving, time scaling and dynamic time warping, allows the invention of versatile styles from time sequence. the second one a part of this dissertation integrates the above primitives into valuable functions starting from track to physics to finance to drugs.
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Extra info for High Performance Data Mining in Time Series: Techniques and Case Studies
Sample text
This requires the Fourier Transform. The formal definition of Fourier transform is stated in the following theorem. 28) −∞ f (t) and F (ω) are a Fourier transform pair, we denote this as F (ω) = F[f (x)] f (t) = F −1 [F (ω)] We will not discuss Fourier Transform here. Instead, we will discuss Discrete Fourier Transform for time series in the next section. It should be noted that many properties Fourier Transform are similar to the corresponding properties of Discrete Fourier Transform. 4 Discrete Fourier Transform Both the Fourier series and Fourier transform deal with functions.
In our discussion, we assume a function to be a real function unless otherwise noted. A function f is said to be square integrable on an interval [a, b] if the following holds: b f 2 (x)dx < ∞. 1) a This is denoted by f ∈ L2 ([a, b]). 1 (Orthogonal Function Family) An infinite square integrable function family {φi }∞ i=0 is orthogonal on interval [a, b] if b φi (x)φj (x)dx = 0, i = j, i, j = 0, 1, ... 2) a and b |φi (x)|2 dx = 0, i = 0, 1, ... 3) a The integral above is called the inner product of two functions.
12, DF T (x(−i)) = X(−F ). 51) X(F ) = DF T (x(i)) = DF T (x(−i)) = X(−F ). 52) Therefore So by definition, X(F ) is even. (b) Similar to (a). 14 (Symmetry 3) (a)If x(i) is real, then X(F ) = X ∗ (−F ). (b) If X(F ) is real, then x(i) = x∗ (−i) Proof (a) N −1 1 X(−F ) = √ N x(i)WNF i i=0 ∗ Because x(i) is real, x (i) = x(i). 1 X (−F ) = √ N N −1 x∗ (i)WN−F i ∗ 1 =√ N i=0 N −1 x(i)WN−F i i=0 = X(F ) 2 3 A sequence x(i) is even if and only if x(i) = x(−i). A sequence x(i) is odd if and only if x(i) = −x(−i).
