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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.

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Presentation on theme: "C M S 2005 Workshop K S Wavelet transform oriented methodologies with applications to time series analysis Wavelet Analysis (WA) Filtration Approximation."— Presentation transcript:

1 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

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

3 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

4 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...

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

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

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

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

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

10 C M S 2005 Workshop K S WNS Approach Network Traffic

11 C M S 2005 Workshop K S Trend Approximation Crop Yields

12 C M S 2005 Workshop K S Trend Approximation Crop Yields

13 C M S 2005 Workshop K S Periodicity Identification

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

15 C M S 2005 Workshop K S Periodicity Identification Sales

16 C M S 2005 Workshop K S Periodicity Identification Weather

17 C M S 2005 Workshop K S Forecasting Share Prices

18 C M S 2005 Workshop K S Forecasting Sales

19 C M S 2005 Workshop K S Forecasting Sales

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

21 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%

22 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

23 C M S 2005 Workshop K S

24 C M S 2005 Workshop K S Wavelet Function

25 C M S 2005 Workshop K S Wavelet Function

26 C M S 2005 Workshop K S Wavelet Function

27 C M S 2005 Workshop K S Wavelet Function

28 C M S 2005 Workshop K S Haar Wavelet Function

29 C M S 2005 Workshop K S Other Wavelet Functions

30 C M S 2005 Workshop K S DWT – example

31 C M S 2005 Workshop K S DWT – example

32 C M S 2005 Workshop K S DWT – example

33 C M S 2005 Workshop K S DWT – example

34 C M S 2005 Workshop K S DWT – example

35 C M S 2005 Workshop K S DWT – example

36 C M S 2005 Workshop K S DWT – example

37 C M S 2005 Workshop K S DWT – example

38 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.

39 C M S 2005 Workshop K S Why Wavelets?

40 C M S 2005 Workshop K S Why Wavelets?

41 C M S 2005 Workshop K S

42 C M S 2005 Workshop K S Wavelet Function

43 C M S 2005 Workshop K S Wavelet Function

44 C M S 2005 Workshop K S Wavelet Function

45 C M S 2005 Workshop K S Wavelet Function

46 C M S 2005 Workshop K S Haar Wavelet Function

47 C M S 2005 Workshop K S Other Wavelet Functions

48 C M S 2005 Workshop K S DWT – example

49 C M S 2005 Workshop K S DWT – example

50 C M S 2005 Workshop K S DWT – example

51 C M S 2005 Workshop K S DWT – example

52 C M S 2005 Workshop K S DWT – example

53 C M S 2005 Workshop K S DWT – example

54 C M S 2005 Workshop K S DWT – example

55 C M S 2005 Workshop K S DWT – example

56 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,...

57 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

58 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

59 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

60 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

61 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

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

63 C M S 2005 Workshop K S WaveShrink –1

64 C M S 2005 Workshop K S WaveShrink –1

65 C M S 2005 Workshop K S WaveShrink –2

66 C M S 2005 Workshop K S WaveShrink –2

67 C M S 2005 Workshop K S WaveShrink –3

68 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

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

70 C M S 2005 Workshop K S WBD - example

71 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

72 C M S 2005 Workshop K S TS Forecasting using WA

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

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

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

76 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.

77 C M S 2005 Workshop K S Why Wavelets?

78 C M S 2005 Workshop K S Why Wavelets?

79 C M S 2005 Workshop K S CS 2 – Sales

80 C M S 2005 Workshop K S CS 2 – Seasons

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

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

83 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%

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

85 C M S 2005 Workshop K S Measures of Regularity

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

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

88 C M S 2005 Workshop K S Example 1 Wavelets

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

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

91 C M S 2005 Workshop K S Example 2 Wavelets

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


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