london presentation - a least square ratio (lsr) approach to fuzzy linear regression anaysis
TRANSCRIPT
The Weighted Least Squares Ratio (WLSR) Methodto M-Estimators
SAI Computing Conference, July 2016, London, United Kingdom
e-mail: [email protected]
MURAT YAZICI, M.Sc.Sr. Data Scientist & Researcher
Murat YAZICI, M.Sc.
Contents
Introduction: The Regression Model and Its Usage Areas
What is the problems while setting up a regression model?
The Least Square Ratio (LSR) Method and Its Benefits
The M-Estimators and The Proposed WLSR Method
A Simulation Study of The LSR vs. The OLS Approach to M-Estimators
Conclusion and Future Work
Who is JForce IT Company?
References
Thanks…
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Introduction: The Regression Model
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Introduction: Its Usage Areas
Some of usage areas of The Regression Analysis;
Economy Finance Business Law Meteorology Medicine Biology
Chemistry Engineering Education Sports History Sociology Psychology
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
What is the problems while setting up a regression model?
While setting up a linear regression model, Ordinary Least Square (OLS) Method is generally used.
The OLS Method: One of the biggest problems of the OLSmethod is that it could not successfullyestimate coefficients in case of outliersand/or extrem values.
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
The Least Square Ratio (LSR) Method and Its Benefits
The OLS method aims to estimate observed values with zero error: we can indicate this goal by ,or . Hence, the ordinary least squares approach satisfies this aim by finding the regressionparameters minimizing the sum of . From the error definition , it is clear that, the sizeof error does not depend on the size of . For example, consider estimating 100 as 200 and 1,000 as1,100: we get the same error −100. However, another point of view says that for the first estimationthere is 100% error, but only 10% for second.*
The Least Square Ratio (LSR) starts with the same goal as in OLS. However, it proceeds by dividingthrough by and so is obtained under an assumption of . Hence, it is obvious that,equations and are raised by basic mathematical operations. This finalequation is taken into account as the origin of the LSR which minimizes the sum of .*
* O. Akbilgic, E.D. Akinci, A Novel Regression Approach: Least Squares Ratio, Communications in Statistics - Theory and Methods 38:9 (2009) 1539-1545.
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
The Least Square Ratio (LSR) Method and Its Benefits
The LSR Method: One of the biggest problems of the OLSmethod is that it could not successfullyestimate coefficients in case of outliersand/or extrem values.
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
The M-Estimators and The Proposed WLSR Method
M-estimation was first proposed by Huber (1964, 1973, 2004). M-estimation for regression is a relativelystraightforward extension of M-estimation for location. It represents one of the first attemps at acompromise between the efficiency of the least squares estimators and the resistance of the LAVestimators, both of which can be seen as special cases of M-estimation. In simplest terms, the M-estimator minimizes some function of the residuals. As in the case of M-estimation location, therobustness of the estimator is determined by the choice of weight function [3].
SAI Computing Conference, July 2016, London, United Kingdom
OLS LSR
Murat YAZICI, M.Sc.
The M-Estimators and The Proposed WLSR Method
SAI Computing Conference, July 2016, London, United Kingdom
Huber’sweighting function:
Andrew’s weight functions:
Tukey’sweighting function:
Ramsay’sweighting function:
Murat YAZICI, M.Sc.
The M-Estimators and The Proposed WLSR Method
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
A Simulation Study of The LSR vs. The OLS Approach to M-Estimators
SAI Computing Conference, July 2016, London, United Kingdom
The simulation study appraises linear multiple regression analysis by using two independent variables as follows. WLS and WLSR methods are checked against according to the MAE of and the MAE of .
In the simulation process, the independent variables x1 and x2 are randomly generated from a normaldistribution with are equal to 1, so Thus, the regression model becomes as follows:
Finally, errors are randomly generated as Gaussian white noise with variance . Therefore, thedependent variable has a normal distribution with mean 201 and variance 200 + .
Murat YAZICI, M.Sc.
A Simulation Study of The LSR vs. The OLS Approach to M-Estimators
SAI Computing Conference, July 2016, London, United Kingdom
The simulations were performed by R by using various sample sizes and error variances. Duringcalculation of m-estimators, OLS and LSR methods were used to fit initial regression model; initialresiduals were found, and they were scaled by MAD; a chosen weight function was applied to obtainpreliminary weights. The preliminary weights were used in iteratively reweighted least squares anditeratively reweighted least squares ratio methods to obtain regression parameters; secondary residualswere found during the first iteration. In the second and other iterations, the residuals were scaled byHuber proposal 2 until the best model was found. The following criteria were used to obtain the finalestimates;
where refers to the number of iterations; indicates avery small positive number. In this study, took thevalue of 0.0001.
Murat YAZICI, M.Sc.
A Simulation Study of The LSR vs. The OLS Approach to M-Estimators
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Conclusion and Future Work
In this study, it is shown which method (WLS and WLSR) gives better results in M-estimationaccording to statistics values of the mean absolute errors (MAE) of the estimated regressionparameters and dependent value through a simulation study using various sample sizes and errorvariances. It was studied on Huber, Tukey, Andrew and Ramsay’s weighting functions in this paper.Based on the simulation results, WLSR Method outperforms than WLS Method in case of thepresence of outliers and increased error variance apart from Andrew’s weighting function. For futurework, other weighting functions in the literature can be examined for which method gives betterresults.
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
About JForce IT Company
JFORCE is founded in 2003 with a Focus in Insurance, Banking and IBM solutions.
Acting as a software house and system integrator JForce is delivering innovativesolutions with state of the art technologies.
With a team of 40 and over 85 Certifications JForce is one of the biggest solutionproviders of IBM in Turkey.
Key Technology focus is in Systems&Middleware : Application Servers, Integration,Business Process Management, Business Rules Management, Complex Events ,Statistical Modelling
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
JFORCE Key Solution Areas for Insurance
JForce is serving Insurance Market with
• Core Insurance Solutions (NTT Data Partnership)• Claims Process Automation & Analytic Dashboards• Fraud & Leakage Management• Underwriting Automation / Contract Management• Dynamic Pricing• Telematics and Related Mobile Applications• Predictive Customer Intellegence, Realtime Marketing and Event Management• Service Integration• Provision Automation and• Online Pharmacy Automation Solutions.
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Our Global Partnerships
International Conference for Engineering and Technology, May 2016, Toronto
Murat YAZICI, M.Sc.
Awards & Recognitions
2003 IBM Best Project of The Year2004 IBM Best System i Partner 2004 INDEX Best Performing Partner2005 IBM Best System i Partner2006 IBM Best System i Partner2006 ORACLE Partner Network2007 IBM Best System i Partner2008 IBM TÜRK 70. YIL ÖZEL ÖDÜLÜ
2009 IBM Best Performance Websphere2009 Best Performance – Power i2009 VISION SOLUTIONS Quota Achievement2012 IBM Best Project of The Year2013 Best Performing DB2 Partner2014 IBM Most Competitive Project of The Year2015 IBM Technical Accelence Award in 3 Categories
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Some of Our References
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
Contact Us
www.jforce.com.tr
Göztepe Mh. Göksuevleri Sit. Sardunya Sk. B212B 34810 Anadoluhisari / Istanbul, Turkey
Phone: 0090 216 668 0290 Fax: +90 216 668 02 95E-mail: [email protected]
SAI Computing Conference, July 2016, London, United Kingdom
Murat YAZICI, M.Sc.
References
1. R. Muthukrishnan, M. Radha, “M-estimators in regression models,” Journal of Mathematics Research, 2010, vol. 2, no. 4, pp. 23-27.
2. O. Akbilgic, E. D. Akinci, “A novel regression approach: Least squares ratio. Communications in Statistics - Theory and Methods, 2009, vol. 38, no. 9, pp.1539-1545.
3. R. Andersen, Modern Methods for Robust Regression. Sage Publications, 2007.
4. A. Ali, M. F. Quadir, “A modified M-estimator for the detection of outliers,” Pakistan Journal of Statistics and Operation Research, 2005, vol.1, no. 1, pp. 49-64.
5. D. C. Hoaglin, F. Mosteller, and J. W. Tukey, Understanding robust and exploratory data analysis. John Wiley and Sons, New York, 1983.
SAI Computing Conference, July 2016, London, United Kingdom
Thank you!..MURAT YAZICI, M.Sc.
Sr. Data Scientist & Researcher
LinkedIn: https://tr.linkedin.com/in/muraty1E-mail: [email protected]
Phone: 0090 539 601 6854