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Understanding Relaxed Periodicity in Object Association and Its Application in Seismology

This paper explores the concept of relaxed periodicity in correlation between various objects and its application in seismology. It discusses algorithms designed to identify tendencies with inertia in earthquake magnitudes and radon concentration in groundwater. By analyzing historical data from Sichuan Province, the researchers present methods to predict earthquake patterns while addressing noise and temporal stretching. The study highlights the importance of establishing reliable correlations that can improve pre-disaster forecasting and mitigation strategies.

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Understanding Relaxed Periodicity in Object Association and Its Application in Seismology

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  1. Discovering Tendency Associationbetween Objects with Relaxed Periodicity and its Application in Seismology • Changjie. TANG1Rynson W. H. LAU2H. YIN1Qing LI2 • L. YANG 1Z. YU1L. XIANG1T. ZHANG 1 • 1 Computer Department, Sichuan University, Chengdu, China • chjtang@scu.edu.cn • 2 Department of Computer Science, City University of Hong Kong, Hong Kong • {rynson,csqli}@cityu.edu.hk

  2. Outline • Introduction, Background • Main Idea ( • Concepts ( Relaxed periodicity ) • Algorithms • Theoretical results • Implementation and App. in Seismic Research • Summary

  3. Background • Periodic phenomena , ( repetitive phenomena) • Correlation between objects with periodicity, • such as: • Climate and its premonitions • Earthquake (Seismology )and its premonitions • Example : Earthquake magnitude and the Radon • concentration in the groundwater).

  4. Background Correlation between objects with periodicity, • earthquake • magnitude • Radon • groundwater • The repetitious is not in the mathematical sense • because noises and inflation /deflation, or stretching unevenly on the time axis. Relaxed Periodicity

  5. Background : Project RADIUS • Project RADIUS ( UN, To reduce the losses of natural disaster ,1990-1999) • Sichuan Province : Earth quake --intensive active. • Use new techniques in research of Earth quake : • such as: Internet techniques , KDD/DM,... • Hundreds researchers are working in Sichuan

  6. Background : (SCEQIS) • Sichuan Earth Quake Infor. System (SCEQIS) This work S1 Web DB KDD s2 DW Decision s3 station

  7. 1748-1998 =History, 1999-2048 =prediction( Earthquake magnitude for Area Anning River)

  8. Outline • Introduction, Background • Main idea of this work • Algorithms • Concepts ( Relaxed periodicity ) • Theoretical results • Implementation and App in Seismic Research of Annin River • Summary

  9. Main Idea : Tendency With Inertia Down Peak Up Valley

  10. Main Idea : Peak-Valley Array (Series) with Inertia • By the concept of Inertia, we can overcome • Small noise, • inflation /deflation, • Stretching unevenly on the time axis. • Peak-Valley Array (Series) with Inertia

  11. Evaluate n-Peaks length ( 110 days) • For magnitude of Anning Area, 110 days = Stable 3-Peaks length in peak-Valley series: Relaxed Periodicity

  12. Evaluate n-Peaks length ( 110 days) • Idea :Use the N-Peaks as time measure unit for life cycle of Earthquake. ( Relative time ). Ignore the uneven stretching on the time axis

  13. ！ Correlation between objects with periodicity, • earthquake • magnitude • Earth • deform • (observable) • Our Task: Confidence= ? Support=?

  14. Outline • Introduction, Background • Main Idea • Concepts ( Relaxed periodicity ) • Algorithms • Theoretical results • Implementation and App in Seismic Research of Annin River • Summary

  15. Concepts : (Tendency with inertia ) • Tendency • TendencyType={Up, Peak, Down, Valley, Unknown}. • ObjTendency: [t, t’] →TendencyType • t1, t2, t3 adjacent time points • ObjTendency(t2, obj) the tendency of obj at time t2. • defined by checking the obj value, Tab. 1

  16. Tendency at t2. defined by checking the value at t1 & t2

  17. Tendency With Inertia Peak Down Up Valley

  18. Definition 4 (Candidate periodicity with n peaks). • Definition 5 ( Relaxed confidence of candidate periodicity) CP(n) • Definition 6 ( Relaxed periodicity with • Confidence> C0 ). • C0 is specified min value

  19. Evaluate n-Peaks length ( 4 peks = 110 days ) • For magnitude of Anning Area, 110 days = Stable 3-Peaks length in peak-Valley series • 4-Peaks =110 daysRelaxed Periodicity with C > 72%

  20. Outline • Introduction, Background • Main Idea • Concepts ( Relaxed periodicity ) • Algorithms • Theoretical results • Implementation and App in Seismic Research of Annin River • Summary

  21. Algorithm 1. Get Tendency Series with inertia • Input: Database, inertia of the object. • Output: Tendency Series. • Steps: • (1) Retrieve the time series from database, stored in {a1, a2, ...,am}. • (2) Decide the tendency of a1 by comparing a1 and a2.The value falls in {Peak, Valley}. • (3) For ( i=2; i<m; i++) • {Left=ai; Middle=ai+1; Right=ai+2; • Determine the tendency of middle according to the Table-1.} • (4) Decide the tendency of am by comparing am-1 and am. The values falls in {Peak, Valley}.

  22. Algorithm 2 Get Peak-Valley Series • Input: Tendency Series. • Output: Peak _Valley series. • Steps: • (1) Result = Input; • (2) Scan Result for adjoining peak (valley). If there are k adjacent peaks (valleys), keep the middle peak (valley), i.e. the [k/2]-th peak(valley), and delete rest (k-1) peaks(valleys). • (3) Delete all Up and Down from Result. ▍

  23. Algorithm 3. Mine the relaxed periodicity • .Input: (1)Peak_Valley_Series; • (2) MaxSpan, //the maximum span for candidate periodicity - a positive integer • (3) MinSpan , //the minimum span for candidate periodicity - a positive integer • (4) MaxTimeLength, // It is a multiple of logical Chronon • (5) Min_Confidence, Gap, // two predefined numbers • Output: Relaxed Periodicity P and its confidence C.

  24. Algorithm 3. ( Continue) • Procedure GetRelaxedPeriods( ) • for (i= 2;MinSpan<=i<MaxSpan; i ++) • for (TimeLength=1; TimeLength <= MaxTimeLenth; TimeLength++) { • scan Peak_Valley_Series calculate total_Peaks; • Total_i_span = total_Peaks – i + 1; // number of i-span; see Definition 3 and • // Lemma 1. • Valid_i_Span = number of i-Span that satisfying |RP(i)- TimeLength| <= Gap; • confid = Valid_i_Span /Total_i_Span; • } • if (confid >= Min_Confidence) { • C = confid; P = TimeLength; // output • Output “P as relaxed periodicity of i peaks, confidence = C”; • }. ▍

  25. Result of GetRelaxedPeriods( ) in Earthquake Magnitude at Annin

  26. Algorithm 4 ( Association of two Obj with relaxed Periodicity) • Input: Two peak-valley series S1 and S2, Correlation • Cond1, Cond2, Logical Chronon, Gap, which is a non-negative number. • Output: Relaxed match confidence. • Steps: ( SQL-like) • (1) Select TimeStamp, attribute concerned into TempR1 • (2) From peak-valley Series S1 • (3) Where Cond1 is true. • (4) Select TimeStamp, Attribute concerned into TempR2 • (5) From peak-valley Series S2

  27. Algorithm 4 ( Continue) • (6) Where Cond2 is true. • (7) Select TimeStamp, Attribute concerned • (8) From R1, R2 • (9) Where Correlation match error is not greater than gap.

  28. Outline ！ • Introduction, Background • Main Idea • Concepts ( Relaxed periodicity ) • Algorithms • Theoretical results • Implementation and App in Seismic Research of Annin River • Summary

  29. Theoretical results • Lemma 1. Let Peaks={P1,P2,...,Pr} be the peak series of object obj and n<=r. The number of candidate periodicity with n peaks is r-n+1. • As loop control number in algorithm • Theorem 1. the complexity of Algorithm 4 is O(m*n+m+n), where m and n are the sizes of two time series

  30. Outline • Introduction, Background • Main Idea • Concepts ( Relaxed periodicity ) • Algorithms • Theoretical results • Implementation and App in Seismic Research of Annin River • Summary

  31. Environment and Data • Annin River Area , geometric fault structure , in Sichuan province. • Covers 150 KM2 , E-quake intensely active status. • DB1: “Earthquake catalog from 1970 to 1995 at Annin”, • DB2: “Earth deform database from 1986 to 1996 at Annin”. • After data cleaning, the total number of records in DW is about 12,000.

  32. Mining relaxed periodicity for earthquake magnitude • By Algorithm 1, mine out the Tendency series for each time point related to the seismic catalog. • By Algorithm 2, mine out the peak-valley Series from Tendency series. • By Algorithm 3, mine out the relaxed periodicity of earthquake and their complete support ratio from peak-valley series. The results are as shown in Fig.2.

  33. Figure 2. The relaxed periodicity of earthquake and relaxed confidence.

  34. Mining the relaxed periodicity for Earth deform

  35. Table 3. Peak_valley Series for Earthquake magnitude (Min_Sup=25%).

  36. Peak-valley Series for Earthquake magnitude (Min_Support = 40%)

  37. The association between the of Earthquake and earth deform.

  38. Performance • CPU: Pentium II 200 with 32MB RAM. • Test data: “earthquake catalog from 1970 to 1995”, 6000 records., “ earth deform database from 1986 to 1996”, 6000 records. • Algorithms: 1 2 3 4 • Execution time: 60sec. 70sec 10sec. 10s. • Not very fast, but acceptable.

  39. Summary • Mining association between objects with relaxed periodicity is useful. • Propose concepts : such as relaxed periodicity • Four algorithms, Acceptable performance. • Theorem 1 shows that Algorithm 4 is still naive, • Efficiency remains to be improved. • Much further research needs to be done.

  40. 请各位专家同行指正！ Thank you !!!

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