Replacement of Missing Data and Outliers Using Wavelet Transform Methods
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1 Replacement of Missing Data and Outliers Using Wavelet Transform Methods Liqian Zhang, Research Associate Department of Chemical and Materials Engineering University of Alberta
2 Outline 2 1. Motivation 2. Discrete wavelet transform 3. Proposed algorithms for treatment of missing data 4. Examples to illustrate reconstruction of missing data 5. Application to detect and reconstruct outliers 6. Conclusions
3 Motivation 3 Causes of missing data Failure of measurement devices and/or errors in data management Sensor breakdown Measurement outside the range of the sensor Data acquisition system malfunction Energy blackouts Interruption of transmission lines
4 Motivation (Causes of missing data) 4 Example of measurement outside the range of the sensor or sensor breakdown Measurements outside the upper bound of the sensor Sensor breakdown
5 Motivation (Causes of missing data) 5 Data may be missing because of the strategy of sampling: multi-rate or irregular sampling An example: multi-rate/irregular sampling.5 faster sampled variable x[n] slower sampled variable Missing data at irregular samples y[n]
6 Motivation (Causes of missing data) 6 Outliers can be considered as missing data: v14 from Syncrude Canada Data compression (with default settings) is still a common practice in industry
7 Motivation 7 Data driven methods are now extensively used in process industries for Process identification Process monitoring Such methods require well-conditioned data with the following features: Uncompressed data or raw data Properly time synchronized Outlier detection and replacement Reconstruction of missing data
8 Discrete wavelet transform 8 Why use wavelet transform? Existing methods of treating missing data Direct interpolation in the time-domain (with no regard for information in the frequency domain) Spectrum estimation (with no regard for information in the time domain) Wavelet transform shows how the energy of signal varies with time and frequency Applications: For de-noising and compressing signals In biology for cell membrane recognition In finance for detecting quick variation of data For machine condition monitoring and fault diagnosis,.
9 Discrete wavelet transform 9 Decomposition d1 [ π / 2, π ) Level-2 decomposition Level-3 decomposition a 1 ( n) d2 [ π / 4, π / 2) d3 [ π / 4, π / 2) a 2 ( n) Level-1 decomposition a3 [, π / 8)
10 Discrete wavelet transform 1 x is decomposed as x = d d2 + d3 a3 d1 [ π / 2, π ), d2 [ π / 4, π / 2), d3 [ π / 8, π / 4), a3 [, π / 8) DWT offers a good time resolution at high frequencies, and good frequency resolution at low frequencies.
11 Discrete wavelet transform 11 Reconstruction
12 Discrete wavelet transform 12 An example 1 original data level-1 approximation level-1 detail
13 Proposed algorithms for treatment of missing data 13 Data set considered Missing data description Regularly sampled ( n) = y( t) x t= nt y(t) is a continuous-time signal Samples at some sampling instants are missing Two types of missing data Type 1 randomly missing data: data are missing at random time instants Type 2 - gapped data: missing data constitutes some gaps
14 Proposed algorithms for treatment of missing data Type 1 randomly missing data Blue: available values Red: missing values
15 Proposed algorithms for treatment of missing data 15 Type 2: gapped data Scaled industrial data from Matrikon
16 Proposed algorithms for treatment of missing data 16 Algorithm 1: Wavelet transform + EM algorithm x = d 1 + a 1 Larger computational burden with larger data sets Sensitive to the initial choice Reliable for gapped data and more missing data Algorithm 2: Wavelet transform + least squares x a 1 Computational burden is much less No initial choice is needed Reliable for randomly missing data, but not for gapped data
17 Examples to illustrate reconstruction of missing data 17 Example 1: Scaled data from AT Plastics: 4% data were removed randomly and can be considered as missing data x[n] NSERC-Matrikon-Suncor-iCORE IRC n Seminar; 3 December 27
18 Examples to illustrate reconstruction of missing data Simulation results for the first 1 samples original data original reconstructed data data by EM reconstructed data data by by EMLS reconstructed data by LS x(n) x(n) n n
19 Examples to illustrate reconstruction of missing data Example 2: Scaled industrial data from Suncor Original data Reconstructed Original data data by EM Reconstructed data data by by least EMsquares Reconstructed data by least squares
20 Examples to illustrate reconstruction of missing data 2 Example 3: Scaled industrial data from Matrikon Original data Original data Reconstructed data by EM Reconstructed data by EM Reconstructed data by least squares Reconstructed data by least squares
21 Examples to illustrate reconstruction of missing data 21 Example 4: Experimental data from a dryer Original data Original data Reconstructed data by EM Reconstructed data by EM Reconstructed data by LS Reconstructed data by LS
22 Examples to illustrate reconstruction of missing data 22 Example 5: Simulated data: sum of two sinusoidal signals with white noise remove the samples of 31-5 and 71-8 as gapped data
23 Examples to illustrate reconstruction of missing data
24 Examples to illustrate reconstruction of missing data original data reconstructed data
25 Examples to illustrate reconstruction of missing data Example 6: Experimental data from a pilot plant remove the samples of 71-9 and as gapped data
26 Examples to illustrate reconstruction of missing data
27 Examples to illustrate reconstruction of missing data Blue: Original data Red: Reconstructed data
28 Examples to illustrate reconstruction of missing data 28 Example 7: Scaled industrial data from Matrikon notice large sections of gapped data
29 Examples to illustrate reconstruction of missing data Blue: Original signal with gapped data Red: Reconstructed signal
30 Application to detect and reconstruct outliers 3 Choose window size T and suppose that there are no outliers in the first R data Based on the first R data, Start reconstruct the next T data x ) u Yes x( i) tol. x ( i) = x( i) xˆ u ( i) < No ) Outlier: x( i) = x ( i) u R = R + T No R N Yes End
31 Application to detect and reconstruct outliers 31 Example 1: generate outliers at 31-33,76, original data without outliers data with generated outliers
32 Application to detect and reconstruct outliers 32 Suppose there is no outlier in the first 3 points, estimate the data over the interval original data with outliers estimated data
33 Application to detect and reconstruct outliers 33 Detect outliers over 31-4: data at are outliers 1 9 2σ line estimation error
34 Application to detect and reconstruct outliers 34 Replace outliers with estimated values, based on the first 4 samples, estimate samples over interval data with outliers estimated data σ line estimation error
35 Application to detect and reconstruct outliers data with outliers 2σ line estimated data estimation error
36 Application to detect and reconstruct outliers σ line data with outliers estimation error estimated data
37 Application to detect and reconstruct outliers σ line estimation error data with outliers estimated data
38 Application to detect and reconstruct outliers σ line data with outliers estimated error estimated data
39 Application to detect and reconstruct outliers σ line data with outliers estimation error estimated data
40 Application to detect and reconstruct outliers 4 Example 2: Industrial data V14 tag from Syncrude original data reconstructed data
41 Concluding Remarks Two methods of reconstructing missing data based on discrete wavelet transform have been introduced. 2. The two methods have been compared by application to industrial data sets: 1) The LS-based algorithm is simple and reliable if the data are missing randomly. 2) The EM-algorithm is more reliable for gapped data. 3. The proposed method has also been successfully applied for detecting and replacing outliers.
42 Acknowledgements 42 Dr. Sirish L. Shah and Dr. Tongwen Chen CPC Group Members NSERC-Matrikon-Suncor-iCORE for financial support
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