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C M S 2005 Workshop K S Wavelet transform oriented methodologies with applications to time series analysis Wavelet Analysis (WA) Filtration Approximation Periodicity Identification Forecasting Bartosz Kozłowski, kozlow@iiasa.ac.at International Institute for Applied Systems Analysis Institute of Control and Computation Engineering, WUT

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C M S 2005 Workshop K S Wavelets Background Foundations Time and Frequency Inversible

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C M S 2005 Workshop K S Analysis with WT Original wavelet coefficients New signal Original signal New wavelet coefficients WT Inverse WT Analysis Original wavelet coefficients New signal Original signal Analysis WT

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C M S 2005 Workshop K S WA Background Characteristics Fast Spatial Localization Frequency Localization Energy Applications Acoustics Economics Geology Health Care Image Processing Management Data Mining...

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C M S 2005 Workshop K S WaveShrink –1 Network Traffic

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C M S 2005 Workshop K S WaveShrink –1 Network Traffic

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C M S 2005 Workshop K S WaveShrink –2 Network Traffic

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C M S 2005 Workshop K S WaveShrink –2 Network Traffic

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C M S 2005 Workshop K S WaveShrink –3 Network Traffic

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C M S 2005 Workshop K S WNS Approach Network Traffic

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C M S 2005 Workshop K S Trend Approximation Crop Yields

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C M S 2005 Workshop K S Trend Approximation Crop Yields

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C M S 2005 Workshop K S Periodicity Identification

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C M S 2005 Workshop K S Periodicity Identification Measures of Regularity

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C M S 2005 Workshop K S Periodicity Identification Sales

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C M S 2005 Workshop K S Periodicity Identification Weather

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C M S 2005 Workshop K S Forecasting Share Prices

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C M S 2005 Workshop K S Forecasting Sales

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C M S 2005 Workshop K S Forecasting Sales

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C M S 2005 Workshop K S Evaluations DirectSeasonal Std. Dev.5,81544899637675615,83 Max. Err.0,1833063370,172322659 Min. Err.0,0045566360,000310097 Avg. Err.0,0560005130,036521327

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C M S 2005 Workshop K S Another Forecasts Accuracy Measure How many times (%) the method correctly forecasted the raise / fall of the time series Direct Wavelet Approach for Shares ~55% Seasonal Wavelet Approach for Sales ~75%

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C M S 2005 Workshop K S Summary Allow to use standard approaches and combine them Various application domains Open possibilities for new approaches Provide multiresolutional analysis Do not increase computational order of complexity Improve results

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C M S 2005 Workshop K S

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Haar Wavelet Function

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C M S 2005 Workshop K S Other Wavelet Functions

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S Seasonal Time Series Split process into subprocesses If for each, each, and each condition is satisfied, then with accuracy of process is seasonal.

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C M S 2005 Workshop K S Why Wavelets?

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C M S 2005 Workshop K S Why Wavelets?

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C M S 2005 Workshop K S

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Wavelet Function

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C M S 2005 Workshop K S Haar Wavelet Function

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C M S 2005 Workshop K S Other Wavelet Functions

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S DWT – example

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C M S 2005 Workshop K S Filtering A part of preprocessing Altering original data to remove potential outliers or noise, which would negatively influence further-applied algorithms Kalman, Chebyshev, Hodrick-Prescott, Fourier,...

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C M S 2005 Workshop K S Filtering – WaveShrink Signal transformation into the wavelet domain Modification of each wavelet using specified shrinkage function Inverse transformation of modified wavelets into the time domain

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C M S 2005 Workshop K S Filtering – WaveShrink Signal transformation into the wavelet domain Modification of each wavelet using specified shrinkage function Inverse transformation of modified wavelets into the time domain

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C M S 2005 Workshop K S Filtering – WaveShrink Signal transformation into the wavelet domain Modification of each wavelet using specified shrinkage function Inverse transformation of modified wavelets into the time domain

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C M S 2005 Workshop K S Filtering – WaveShrink Signal transformation into the wavelet domain Modification of each wavelet using specified shrinkage function Inverse transformation of modified wavelets into the time domain

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C M S 2005 Workshop K S Filtering – WaveShrink Signal transformation into the wavelet domain Modification of each wavelet using specified shrinkage function Inverse transformation of modified wavelets into the time domain

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C M S 2005 Workshop K S Soft shrinkage functionNon-negative garrote shrinkage function WaveShrink – example Hard shrinkage function

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C M S 2005 Workshop K S WaveShrink –1

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C M S 2005 Workshop K S WaveShrink –1

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C M S 2005 Workshop K S WaveShrink –2

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C M S 2005 Workshop K S WaveShrink –2

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C M S 2005 Workshop K S WaveShrink –3

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C M S 2005 Workshop K S Wavelet-based denoising Identify distortions in signal Perform DWT of signal For each noise Check how deep does the noise propagate Shrink the noise by applying a shrinkage function

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C M S 2005 Workshop K S WBD - example j=0j=1j=2j=3j=4j=5... 920000001... 930010001... 940010001... 951110001... 960110001... 970000101... 980000101... 990000101... 1000000101... 1011110101... 1020110101... 1030010101... 1040010101... 1050001101... 1061001101... 1070001101... 1740111110... 1751111110... 1760011110...

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C M S 2005 Workshop K S WBD - example

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C M S 2005 Workshop K S TS Forecasting Problem Given a time series X find its assumed state E in next time moment Approximation of X followed by extrapolation based on established approximation function

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C M S 2005 Workshop K S TS Forecasting using WA

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C M S 2005 Workshop K S CS 1 – Market Shares

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C M S 2005 Workshop K S CS 1 – WA Forecasts

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C M S 2005 Workshop K S CS 1 – Basic Evaluation

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C M S 2005 Workshop K S Seasonal Time Series Split process into subprocesses If for each, each, and each condition is satisfied, then with accuracy of process is seasonal.

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C M S 2005 Workshop K S Why Wavelets?

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C M S 2005 Workshop K S Why Wavelets?

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C M S 2005 Workshop K S CS 2 – Sales

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C M S 2005 Workshop K S CS 2 – Seasons

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C M S 2005 Workshop K S CS 2 – WA Forecasts

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C M S 2005 Workshop K S CS 2 – Basic Evaluation

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C M S 2005 Workshop K S Another Measure of Accuracy How many times (%) the method correctly forecasted the raise / fall of the time series Shares ~55% Seasonal Time Series SWF 75%

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C M S 2005 Workshop K S Seasonality Identification Identify the L

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C M S 2005 Workshop K S Measures of Regularity

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C M S 2005 Workshop K S Measures of Regularity Interpretation tt vv t0t0 t1t1 t0t0 t1t1

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C M S 2005 Workshop K S Example 1 Original Time Series

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C M S 2005 Workshop K S Example 1 Wavelets

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C M S 2005 Workshop K S Example 1 Measures of Regularity

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C M S 2005 Workshop K S Example 2 Original Time Series

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C M S 2005 Workshop K S Example 2 Wavelets

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C M S 2005 Workshop K S Example 2 Measures of Regularity

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