Represent the numerical data of time series by a mathematical curve. Go through the paper

Dhritikesh Chakrabarty (2016): “Recent Developments on Representation of Numerical Data by a Polynomial Curve”, International Journal of Electronics and Applied Research (ISSN : 2395 – 0064), 3(2), 125 – 158.

Time series data can be represented by mathematical curve. Mathematical curve can be fitted to time series data by (1) least squares method and (2) by variate difference method. These are available in standard book of time series analysis.

Recently, method of fitting of mathematical curve to numerical data has been developed based on elimination-minimization principle. This method can also be applied in fitting of mathematical curve to time series data. Following papers can be consulted for the purpose:

1. Atwar Rahman & Dhritikesh Chakrabarty (2009) : “Linear Curve : A Simpler Method of Obtaining Least squares Estimates of Parameters ”, Int. J. Agricult. Stat. Sci., (ISSN : 0973 - 1903), 5(2), 415 – 424.

2. Atwar Rahman & Dhritikesh Chakrabarty (2011) : “General Linear Curve : A Simpler Method of Obtaining Least squares Estimates of Parameters ”, Int. J. Agricult. Stat. Sci., (ISSN : 0973 - 1903), 7(2), 429 – 440.

3. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “Elimination of Parameters and Principle of Least Squares: Fitting of Linear Curve to Average Minimum Temperature Data in the Context of Assam ”, International Journal of Engineering Sciences & Research Technology, 4(2), (ISSN : 2277 - 9655), 255 – 259.

4. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “ Elimination of Parameters and Principle of Least Squares: Fitting of Linear Curve to Average Maximum Temperature Data in the Context of Assam ”, AryaBhatta J. Math. & Info. (ISSN (Print): 0975 – 7139, ISSN (Online): 2394 – 9309), 7(1), 23 – 28, Also available in www.abjni.com .

5. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “ Basian-Markovian Principle in Fitting of Linear Curve ”, The International Journal Of Engineering And Science, {ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805} www.theijes.com), 4(6), 31 – 43.

6. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “Basian-Markovian Principle in Fitting of Quadratic Curve ”, International Research Journal of Natural and Applied Sciences (ISSN: 2349 - 4077), www.aarf.asia , 2(6), 186 – 210.

7. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “Method of Least Squares in Reverse Order: Fitting of Linear Curve to Average Maximum Temperature Data at Guwahati and Tezpur”, International Journal in Physical & Applied Sciences (ISSN: 2394 - 5710), www.ijmr.net.in, 2(9), 24 – 38.

8. Atwar Rahman & Dhritikesh Chakrabarty (2015) : “ Method of Least Squares in Reverse Order: Fitting of Linear Curve to Average Minimum Temperature Data at Guwahati and Tezpur”, AryaBhatta J. Math. & Info. {ISSN (Print): 0975 – 7139, ISSN (Online): 2394 – 9309}, 7(2), 305 – 312, Also available in www.abjni.com .

Dear Prof. Chakrabarty ,

Thank you for active help regarding this issue. I'm working on a real life volatility proxy, which is generated tick by tick (high frequency), hence very noisy in nature. Hence, polynomial spline could help me to build a model. I came to this conclusion after reading almost 67 papers in this regard.

Thank you once again.

Regards

Bikramaditya Ghosh, PhD

Welcome.

If your data is highly volatile and contains some underlying frequency assumptions, I'll advice you try to identify and separate the high and low frequency component using e.g., wavelet decomposition of the time series. Then you can model the low frequency component using splines, and identify a statistical distribution of the high component, which you may add on as noise. Alternatively, if you have a priori information about the nature of the data (e.g., monotonicity), then you may solve the problem using splines, where the coefficients are determined by constrained optimization. You may want to read my paper