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, kozlow@iiasa.ac.at 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.5,81544899637675615,83 Max. Err.0,1833063370,172322659 Min. Err.0,0045566360,000310097 Avg. Err.0,0560005130,036521327

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=5... 920000001... 930010001... 940010001... 951110001... 960110001... 970000101... 980000101... 990000101... 1000000101... 1011110101... 1020110101... 1030010101... 1040010101... 1050001101... 1061001101... 1070001101... 1740111110... 1751111110... 1760011110...

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