Time Series (1) Information Systems M

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1 Time Series () Information Systems M Prof. Paolo Ciaccia Time series are everywhere Time series, that is, sequences of observations (samples) made through time, are present in everyday s life: Temperature, rainfalls, seismic traces Weblogs Stock prices EEG, ECG, blood pressure Enrolled students at the Engineering Fac This as well as many of the following figures/examples are taken from the tutorial given by Eamonn Keogh at SBBD (XVII Brazilian Symposium on Databases) Sistemi Informativi M

2 Why is similarity search so important? Similarity search can help you in: Looking for the occurrence of known patterns Discovering unknown patterns Putting things together (clustering) Classifiying new data Predicting/extrapolating future behaviors Sistemi Informativi M 3 Why is similarity search difficult? Consider a large time series DB, e.g.: hour of ECG data: GByte Typical Weblog: 5 GBytes per week Space Shuttle DB: 58 GBytes MACHO Astronomical DB: Tbytes, updated with 3 GBytes a day ( million stars recorded nightly for 4 years) First problem: large database size Second problem: subjectivity of similarity evaluation Sistemi Informativi M 4

3 How to measure similarity Given two time series of equal length D, the usual way to measure their (dis-)similarity is based on Euclidean distance However, with Euclidean distance we have to face two basic problems. High-dimensionality: (very) large D values. Sensitivity to alignment of values s q For problem.we need to define effective lower-bounding techniques that work in a (much) lower dimensional space For problem. we will introduce a new similarity criterion D- ( s,q) = ( s ) t qt t= L Sistemi Informativi M 5 Pre-processing Diretly comparing two time series (e.g., using Euclidean distance) might lead to counter-intuitive results This is because Euclidean distance, as well as other measures, are very sensitive to some data features that might conflict with the subjective notion of similarity Thus, a good idea is to pre-process time series in order to remove such features Sistemi Informativi M 6 3

4 Offset translation Subtract from each sample the mean value of the series d(q,s) q = q - mean(q) s = s - mean(s) d(q,s) Sistemi Informativi M 7 Amplitude scaling Normalize the amplitude (divide by the standard deviation of the series) q = (q - mean(q)) / std(q) s = (s - mean(s)) / std(s) Sistemi Informativi M 8 4

5 Noise removal Average the values of each sample with those of its neighbors (smoothing) q = smooth(q) s = smooth(s) Sistemi Informativi M 9 Dimensionality reduction: DFT () The first method for reducing the dimensionality of time series, proposed in [AFS93], was based on the Discrete Fourier Transform (DFT) Remind: given a time series s, the Fourier coefficients are complex numbers (amplitude,phase), defined as: S f D = D t= s exp ( jπft/d) f =,...,D Parseval theorem: DFT preserves the energy of the signal: t Img D D ( ) = st = E( S) = E s t= f = S f S f S f Re where S f = Re(S f ) + j Img(S f ) = (Re(S f ) + Img(S f ) ) is the absolute value of S f (its amplitude), which equals the Euclidean distance of S f from the origin Sistemi Informativi M 5

6 Dimensionality reduction: DFT () The key observation is that the DFT is a linear transformation Thus, we have: D D (s,q) = Q) t= f = ( st qt ) = E( s q) = E( S Q) = Sf Qf L (S, L = where S f -Q f is the Euclidean distance between S f and Q f in the complex plane The above just says that DFT also preserves the Euclidean distance Img What can we gain from such transformation? Q f S f - Q f S f Re Sistemi Informativi M Dimensionality reduction: DFT (3) The key observation is that, by keeping only a small set of Fourier coefficients, we can obtain a good approximation of the original signal This is because most of the energy of many real-world signals concentrates in the low frequencies ([AFS93]): More precisely, the energy spectrum ( S f vs. f) behaves as O(f -b ), b > : b = (random walk or brown noise): used to model the behavior of stock movements and currency exchange rates b > (black noise): suitable to model slowly varying natural phenomena (e.g., water levels of rivers) b = (pink noise): according to Birkhoff s theory, musical scores follow this energy pattern Thus, by only keeping the first few coefficients (D << D), an effective dimensionality reduction can be obtained Note: this is the basic idea used by well-known compression standards, such as JPEG (which is based on Discrete Cosine Transform) For what we have seen, this projection technique satisfies the L-B lemma Sistemi Informativi M 6

7 An example: EEG data Sampling rate: 8 Hz Time series (4 secs, 5 points) Energy spectrum Sistemi Informativi M 3 Another example 8 points s s s = approximation of s with 4 Fourier coefficients data values Fourier coefficients First 4 Fourier coefficients Sistemi Informativi M

8 Comments on DFT Can be computed in O(DlogD)time using FFT (provided D is a power of ) Difficult to use if one wants to deal with sequences of different length Not really amenable to deal with signals with spots (time-varying energy) An alternative to DFT is to use wavelets, which takes a different perspective: A signal can be represented as a sum of contributions, each at a different resolution level Discrete Wavelet Transform (DWT) can be computed in O(D) time Also used in the JPEG standard for compressing images Experimental results however show that the superiority of DWT w.r.t. DFT is dependent on the specific dataset Good for wavelets bad for Fourier 4 6 Good for Fourier bad for wavelets 4 6 Sistemi Informativi M 5 Haar Wavelets: the intuition DWT Haar Haar Haar Haar 3 Haar 4 Haar 5 Haar 6 X X' As with DFT, the time series is decomposed into a linear combinations of base elements The Haar DWT, applied to a series of length D = n, pairs samples, stores their difference and passes their averageto the next stage This process is repeated recursively, which leads to n difference values and one final average (which is for normalized series) s = (3,,4,6,5,6,,) Differences Averages (,-,-,) (.5,5,5.5,) (-3.5,3.5) (3.75,3.75) () (3. 75) Haar 7 Sistemi Informativi M 6 8

9 Dimensionality reduction: PAA PAA(Piecewise Aggregate Approximation) [KCP+,YF] is a very simple, intuitive and fast, O(D), method to approximate time series Its performance is comparable to that of DFT and DWT We take a window of size W and segment our time series into D = D/W pieces (sub-sequences), each of size W i W For each piece, we compute the average of values, i.e. st Our approximation is therefore s = (s,,s D ) ' t= ( i ) W si = We have W L (s,q ) L (s,q) W (arguments generalize those used for the global average example) The same can be generalized to work with arbitrary Lp-norms [YF] W s' s Sistemi Informativi M 7 Some results (thanks again to Eamonn Keogh!) The graphs shows the fraction of data (indexed by an R-tree) that must be retrieved from disk to answer a -NN query mixed dataset On most datasets,.5 the three methods.4 yield very similar results The actual response time may vary, even for a given method, depending on the implementation DFT DWT PAA Sistemi Informativi M 8 9

10 Example: computing the Euclidean distance Consider a sequential -NN algorithm based on Euclidean distance, and let r be the lowest distance found so far Two basic optimization techiniques are. Avoid taking the square root, i.e., use squared Euclidean distance This does not influence the result 5. Early terminate to evaluate the distance on a seres s if the so-far accumulated distance for s is r Clearly, these optimization are not peculiar to time series Seconds 4 3 Euclid Opt Opt, 5,, Number of Objects Sistemi Informativi M 9