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, International Institute for Applied Systems Analysis Institute of Control and Computation Engineering, WUT

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

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

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...

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

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

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

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

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

C M S 2005 Workshop K S WNS Approach Network Traffic

C M S 2005 Workshop K S Trend Approximation Crop Yields

C M S 2005 Workshop K S Trend Approximation Crop Yields

C M S 2005 Workshop K S Periodicity Identification

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

C M S 2005 Workshop K S Periodicity Identification Sales

C M S 2005 Workshop K S Periodicity Identification Weather

C M S 2005 Workshop K S Forecasting Share Prices

C M S 2005 Workshop K S Forecasting Sales

C M S 2005 Workshop K S Forecasting Sales

C M S 2005 Workshop K S Evaluations DirectSeasonal Std. Dev.5, ,83 Max. Err.0, , Min. Err.0, , Avg. Err.0, ,

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%

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

C M S 2005 Workshop K S

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Haar Wavelet Function

C M S 2005 Workshop K S Other Wavelet Functions

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

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.

C M S 2005 Workshop K S Why Wavelets?

C M S 2005 Workshop K S Why Wavelets?

C M S 2005 Workshop K S

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Wavelet Function

C M S 2005 Workshop K S Haar Wavelet Function

C M S 2005 Workshop K S Other Wavelet Functions

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

C M S 2005 Workshop K S DWT – example

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,...

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

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

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

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

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

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

C M S 2005 Workshop K S WaveShrink –1

C M S 2005 Workshop K S WaveShrink –1

C M S 2005 Workshop K S WaveShrink –2

C M S 2005 Workshop K S WaveShrink –2

C M S 2005 Workshop K S WaveShrink –3

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

C M S 2005 Workshop K S WBD - example j=0j=1j=2j=3j=4j=

C M S 2005 Workshop K S WBD - example

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

C M S 2005 Workshop K S TS Forecasting using WA

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

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

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

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.

C M S 2005 Workshop K S Why Wavelets?

C M S 2005 Workshop K S Why Wavelets?

C M S 2005 Workshop K S CS 2 – Sales

C M S 2005 Workshop K S CS 2 – Seasons

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

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

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%

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

C M S 2005 Workshop K S Measures of Regularity

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

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

C M S 2005 Workshop K S Example 1 Wavelets

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

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

C M S 2005 Workshop K S Example 2 Wavelets

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