Cluster EMD and its Statistical Application

Cluster EMD and its Statistical Application Donghoh Kim and Heeseok Oh Sejong University and Seoul National University November 10, 2007 1/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 2/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 3/27

Multi-scale Concept Reducing Complexity and Inducing Information Balloon Data Observation 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 4/27

Multi-scale Concept Reducing Complexity and Inducing Information Coarse Scale 0.5 1.0 1.5 2.0 Balloon Data Observation 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 4/27

Multi-scale Concept Reducing Complexity and Inducing Information Coarse Scale 0.5 1.0 1.5 2.0 Balloon Data Observation 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 0.10 0.05 0.00 0.05 0.10 4/27 0.0 0.2 0.4 0.6 0.8 1.0 Fine Scale 0.0 0.2 0.4 0.6 0.8 1.0

Multi-scale Concept Reducing Complexity and Inducing Information (Continued) Solar irradiance data have a similar feature to sunspot data. Sunspot Lean's solar irradiance proxy data 0 50 100 150 1363.5 1365.0 1366.5 1600 1700 1800 1900 2000 year 5/27

Multi-scale Concept Reducing Complexity and Inducing Information (Continued) Solar irradiance data have a similar feature to sunspot data. Is there other hidden information in solar irradiance data? Sunspot Lean's solar irradiance proxy data 0 50 100 150 1363.5 1365.0 1366.5 1600 1700 1800 1900 2000 year 5/27

Multi-scale Concept Reducing Complexity and Inducing Information (Continued) Sun spot 0 50 100 150 IMF 2 IMF 6 0.6 0.2 0.2 0.6 1.0 0.0 0.5 1.0 1600 1700 1800 1900 2000 Year 6/27

Multi-scale Concept Reducing Complexity and Inducing Information (Continued) 7/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 8/27

Decomposition Introduction 1. Observe Y t R or R 2, t = 1,..., T reflecting complex phenomena of the real world. 9/27

Decomposition Introduction 1. Observe Y t R or R 2, t = 1,..., T reflecting complex phenomena of the real world. 2. Apply operation S 1,..., S J to Y so that Y t = S 1 (Y t ) + S 2 (Y t ) + + S J (Y t ). Index j is complexity or resolution level or scale or frequency. 3. Remove noise of S j by Shrinkage or Thresholding. For example, a simple thresholding D is D(S j )(Y t ) = S j (Y t ) I ( S j (Y t ) > λ j ) for some λ j. λ j are called thresholding values. 9/27

Decomposition Application 1. Reducing complexity of data and producing smoothed version Ŷ t = J D(S j )(Y t ). j=1 2. Extracting and interpreting information embedded in a data, for example, frequency, period, energy from S j or D(S j ). 10/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 11/27

Empirical Mode Decomposition (Huang, 1998) Frequency What is Frequency? 1) Oscillating and periodic patterns are repeated 2) Local mean is zero and the signal is symmetric to its local mean 3) One cycle of oscillation : sinusoidal function starting at 0 and ending at 0 with passing through zero between two zero crossings. Or starting at local maximum and terminating at consecutive local maximum with passing through two zeros and local minimum. 12/27

Empirical Mode Decomposition Decomposition A signal observed in real world consist of low and high frequencies. Suppose we observe signal Y t which is of the form Y t = 0.5t + sin(πt) + sin(2πt) + sin(6πt). 13/27

Empirical Mode Decomposition Decomposition (Continued) (a) (b) (c) (d) 14/27

Empirical Mode Decomposition Decomposition (Continued) (a) (b) (c) (d) 1. First identify the local extrema. 14/27

Empirical Mode Decomposition Decomposition (Continued) (a) (b) (c) (d) 1. First identify the local extrema. 2. Consider the two functions interpolated by local maximum and local minimum. 14/27

Empirical Mode Decomposition Decomposition (Continued) (a) (b) (c) (d) 1. First identify the local extrema. 2. Consider the two functions interpolated by local maximum and local minimum. 3. Their average, envelop mean will yields the lower frequency component than the original signal. 4. By subtracting envelop mean, from the original signal Y t, the highly oscillated pattern h is separated and is called Intrinsic Mode Function. 14/27

Empirical Mode Decomposition Decomposition (Continued) Note that as the name sifting implies, the lower frequency component is repeatedly removed from the highest frequency. The first IMF imf 1 produced by sifting is the highest frequency by its construction. Residue signal r less oscillated than the original signal. Remaining signal r = Y imf 1 still may be compound of several frequencies. The same procedure is applied on the residue signal r to obtain the next IMF. By the construction, the number of extrema will eventually decreased as the procedure continues so that a signal is sequently decomposed into the highest frequency component imf 1 to the lowest frequency component imf J, for some finite J and residue r. Finally we have J empirical mode and residue as J Y t = imf j (t) + r(t). j=1 15/27

Empirical Mode Decomposition Decomposition (Continued) First two IMF s Signal = 1 st IMF + 1 st residue 1 st residue = 2 nd IMF + 2 nd residue 2 0 2 4 6 2 0 2 4 6 1 st imf 2 nd imf 4 2 0 2 4 4 2 0 2 4 1 st residue 2 nd residue 2 0 2 4 6 2 0 2 4 6 16/27

Empirical Mode Decomposition Decomposition (Continued) Decomposition Result 1 st IMF 4 2 0 2 4 2 nd IMF 4 2 0 2 4 3 rd IMF 4 2 0 2 4 residue 2 0 2 4 6 17/27

Empirical Mode Decomposition Lennon Example Treat 2-dim. data as 1-dim. data and run EMD imf 1 imf 2 imf 3 residue 18/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 19/27

Cluster EMD Spatially inhomogeneous and multiscale images To properly capture the spatially inhomogeneous and multiscale feature of images Partition the image by equivalence relation Identifying local maximum and minimum of 2-dim. data. Run usual EMD by 2-dim. interpolation method. 20/27

Cluster EMD Spatially inhomogeneous and multiscale images (Continued) Take the 8 grey pixels are neighbors of a pixel i 0. Define equivalence relation of the pixel i and j if the colors of both pixels are the same. Partitions induced by the relation are established by checking relations of four directions. Let each partition be cluster. Rem s algorithm (Dijkstra, 1976) efficiently finds partition. Identifying extrema by comparing the color of a cluster with the color of its neighbors. For this example, we have 7 clusters and 5 black clusters are local minima. 1 2 3 4 5 6 7 8 9 i 1 i 2 i 3 i 4 i 0 i 5 i 6 i 7 i 8 j 1 j 2 j 0 j 3 j 4 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 21/27

Cluster EMD Lena Example 22/27

Cluster EMD Lena Example (Continued) The right eye of Lena image is magnified. right eye partition local minimum local maximum 1:50 23/27

Cluster EMD Lennon Example imf 1 imf 2 imf 3 residue 24/27

Cluster EMD Simulation Study For noisy image, the first IMF will capture noise. Simulation 1. 64 64 lennon image 2. Add gaussian noise of SNR=1 3. Compare the MSE of 1-dim. EMD and cluster EMD by deleting the first IMF. MSE 600 700 800 900 1000 1d 2d 25/27

Contents 1. Multi-scale Concept 2. Decomposition 3. Cluster EMD Empirical Mode Decomposition Cluster EMD 4. Summary and Future Works 26/27

Summary and Future Works EMD is a data-adaptive method capturing local properties, easy to implement and robust to presence of non-linearity and non-stationarity. We extended 1-dim EMD to 2-dim EMD called Cluster EMD. Cluster EMD efficiently captures spatially inhomogeneous and multiscale feature of images. Cluster EMD also provides promising tool for removing the noise of 2-dim. data. Future Works Interpolation vs. smoothing for extracting IMF. Statistical inference by bootstraping. 27/27