Modeling time series and analyzing its fractal dimension using fractal interpolation function

소개글

"Modeling time series and analyzing its fractal dimension using fractal interpolation function"에 대한 내용입니다.

목차

I Introduction 3

II Background Information 5
2 1 Notations and terms explained 5
2 2 Time series 5
2 3 Fractal dimension 6
2 4 Box-counting dimension 8
2 5 Holder condition 10

III Fractal Interpolation Function 12
3 1 Fractal interpolation 14
3 2 Constructing function 18
3 3 Constructing function 19
3 4 Interpolating real time series data 25
3 5 Deriving the contraction factor 27

IV Analyzing fractal dimension using fractal interpolation function 33
4 1 Defining the functions and in terms of ,, and ' 33
4 2 Analyzing the box-counting dimension of fractal interpolation function 35

V Conclusion 53

VI Works Cited 55

본문내용

A fractal is an object that exhibits self-similarity (Falconer 22). Self-similarity is a property that a similar shape can be identified at different scales looking at the object (Falconer 22). The more self-similar an object is, the more complex it looks (Chaos, Fractals and Dynamics). A lot of time series data, such as which are data ordered by time, display this feature and this feature commonly correlates with real-life phenomena (Kantelhardt 4). For example, in electroencephalogram (EEG) signals, if the patient is has an epileptic seizure, his EEG signal looks more complex than a non-epileptic patient’s EEG signal(Neurocomputing). To incorporate this feature of self-similarity when modeling EEG signals, functions called fractal interpolation functions can be used. These fractal interpolation functions can convert the raw data into an analyzable function that displays self-similarity (Manousopoulos 1). Not only they can model these signals, they can also measure fractal dimension, which is a measure to quantify the level of self-similarity (Barnsley 223).

By doing this, fractal interpolation functions can take data points that are not even part of the raw data points into consideration. There have been many algorithms devised in academics to measure a fractal’s fractal dimension using the raw data only, but there has not been much development in measuring the fractal dimension using fractal interpolation functions (Navascués 2).

참고 자료

Akhtar, Md. Nasim, et al. “BOX DIMENSIONS OF α-FRACTAL
FUNCTIONS.” Fractals, vol. 24, no. 03, 2016, p. 1650037.,
doi:10.1142/s0218348x16500377.
Barnsley, Michael F. “Fractal Functions and Interpolation.” Constructive
Approximation, vol. 2, no. 1, 1986, pp. 303–329., doi:10.1007/bf01893434.
“Epileptic Seizure Detection by Analyzing EEG Signals Using Different
Transformation Techniques.” Neurocomputing, Elsevier, 2 June 2014,
www.sciencedirect.com/science/article/pii/S0925231214006936.
Falconer, Kenneth. Fractal Geometry: Mathematical Foundations and
Applications, 3rd Edition. John Wiley & Sons, 2014.
Gayatri, Ramesh. FRACTAL INTERPOLATION . Dec. 2008,
www.researchgate.net/publication/47714303_FRACTAL_INTERPOLATION.
“January 21 - Self-Similarity, Fractal Dimension.” Commentary: MAT335,
www.sfu.ca/~rpyke/335/W00/21jan.html.
Kantelhardt, Jan W. Fractal and Multifractal Time series, Institute of Physics, Martin-Luther-University, 4 April 2008.
Mandelbrot Benoît. The Fractal Geometry of Nature. W.H. Freeman and Company, 2006.
Manousopoulos, Polychronis. “Curve fitting by fractal interpolation.” Department of Informatics and Telecommunications, University of Athens.
Navascués, María Antonia. “Box Dimensions of !-Fractal Functions.” World Scientific, 22 August 2016.
Navascués, María Antonia. “Fractal Polynomial Interpolation.” Zeitschrift Für
Analysis Und Ihre Anwendungen, 2005, pp. 401–418., doi:10.4171/zaa/1248.
Ullah, Muhammad Imdad. “Time Series Analysis and Forecasting.” Basic
Statistics and Data Analysis, 4 Jan. 2014, itfeature.com/time-seriesanalysis-and-forecasting/time-series-analysis-forecasting.