TIME SERIES ANALYSIS

Effective Method for analyzing Time Series Data

1. Bursty Process vs. Random Process

There is a generalized queuing process, using a parameter in Barabasi's Burst paper.
("The origin of bursts and heavy tails in human dynamics, 2005" : https://www.nature.com/articles/nature03459).

• : Control Parameter of Priority array's components' exponent. (like as fitness of node or task.)

• : Random Process.

• : Bursty Process. (Or Deterministic Process)

Event & Inter Event Time (IET) & Inter Event Time Distribution (IETD)

1. Event : Clear definition of event is needed. (eg. In restaurant, food coming out can be defined as an event.)

2. Inter Event Time (IET) : The time interval between events defined above.

3. Inter Event Time Distribution (IETD) : The time interval between events defined above.

For more details, please refer to the following paper : ("The origin of bursts and heavy tails in human dynamics, 2005" : https://www.nature.com/articles/nature03459)

How to analyze time series data from this theory.

1. Prepare time-series data with clear definition of events.

2. Calculate IET & IETD from prepared data.

3. If IETD follows power-law distribution, measure exponent of IETD.
   If it follows exponential distribution, this time-seris data might be a randomly generated data from iid.

4. If you want to classify time-series data by generating method of data, it might be okay to compare exponents.

2. Record Statistics

There is a Good statistics for analyzing time series data. (If time series data is generated by probability density function) : Record Statistics

If you want to see universal result of record statistics for random walk time series, see
: "Universal record statistics for random walks and Lévy flights with a nonzero staying probability, Satya N Majumdar, Philippe Mounaix and Grégory Schehr, 2021"

(https://iopscience.iop.org/article/10.1088/1751-8121/ac0a2f/meta?casa_token=A6gUJFqk4dcAAAAA:l7Ou2iTFRK2j7ADeCi9SXsRETYH41F9bomll9YAkt3x7Dlt-hY1Pvm56UjCWRf0suhsEOM1imMU)

Definition of Record Event

Time Series Data Set :
For each , when at timing , a Record Event occurs.

Some Important Variables for Record Statistics

• : Binary Indicator for Record Event Occurs.
  
  

• : Record Number (The number of Record Events occurred)
  
  

• : Average Record Number (The mean value of Record Number for many time series samples)
  
  

• : Record Rate (Average Binary Indicator for Record Event Occurs).
  
  

• : Variance of Record Number

• : Fano Factor

• : Connected Two-time Correlation Function

Record Analysis (Real Case : Covid19, the number of Seoul's confirmed people)

Dataset from http://data.seoul.go.kr/dataList/OA-20279/S/1/datasetView.do
x-axis = [2020-01-24 ~ 2021-07-15]
y-axis = "Record Number"

Currently, Record Number of Seoul is going to increase until it reach to analytic result...(ㅠㅠㅠ)