Time series exhibiting linear and periodic trends is analyzed Detailed method of extracting nested periods and resulting periodic components, finding the best fit ARMA model for non-linear and non-periodic component, and setting up control boundary are presented. The concept of time scale is introduced to better understand periodic dynamic.

Abstract
 1. Overview
 2. Description of the problem
 3. Theoretical Characterization
  3.1 Normal distribution of data from stationary ARMAmodel
  3.2 Extending to PL-ARMA Model
  3.3 Three Stage Estimation Process for PL-ARMA
 4. Detection of periodic components
  4.1 Time-scale
  4.2 Extraction of Periods
   4.2.1 FFT and log Transformation
   4.2.2 Pruning Lower Part of Amplitude
   4.2.3 Smoothing Using Polynomial Weight Filter(PWF)
 5. MA-Approximation
  5.1 Theoretical Framework
 6. Setting up Control Boundary
  6.1 Control boundary for 
   6.1.1 How to calculate m in
  6.2 Control boundary for 
 7. Conclusion
 References

• Jung-Yul Suh(Department of Industrial & Systems Engineering, Kumoh National Institute of Technology)
• Sae-jae Lee(Department of Industrial & Systems Engineering, Kumoh National Institute of Technology)
• Hyun-Seung Oh(Department of Industrial and Systems Management Engineering, Hannam University)
• Ja-Hwal Koo(Department of Industrial & Systems Engineering, Kumoh National Institute of Technology)
• Lim Taek(Department of Industrial & Systems Engineering, Kumoh National Institute of Technology)
• Jin-Hyung Cho(Department of Industrial & Systems Engineering, Kumoh National Institute of Technology)