MATHEMATICAL SIGNIFICANCE IN ANGULAR STATISTICS
This virtual conference, organized by Prof. CR Rao's School of Thought on Statistical Sciences at Pondicherry University, will focus on the mathematical significance in angular statistics. The event, taking place on 21st, 22nd, 28th, and 29th November 2020, aims to delve deep into the applications and implications of angular statistics in various fields. Participants can expect insightful discussions, presentations, and networking opportunities related to this specialized area of statistical science.
Uploaded on Feb 18, 2025 | 0 Views
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
MATHEMATICAL SIGNIFICANCE IN ANGULAR STATISTICS International Virtual Conference on Prof. CR Rao s School of Thought on Statistical Sciences Pondicherry University 21, 22, 28, 29 Nov 2020 Prof. S.V.S. GIRIJA Dept. of Mathematics Hindu College, GUNTUR Andhra Pradesh Email: sagivsg@gmail.com
Theme Applications of Mathematical Tools in Construction of New Circular Models Developing New Construction Procedures
DIRECTIONAL DATA ANALYSIS To model angular data Bird migration problem ,...., , 2 1 n are angles of extinction
APPLICATION AREAS Birds navigational problems (Kendall.D.G. (1974) , Schimdt Koeing, (1963) ) Interpreting paleomagnetic currents (Jammalamadaka and Sen Gupta, (1972)) Assessing variation in the onset of leukemia (Hrushesky,W.J.M. and Morgan (1990) ) Analyzing wind directions ( Breckling,J. (1989) ) Angle of knee flexion to assess the recovery of orthopedic problems of patients ( Rao,J.S., et al, (1986) ) Occurrence of flight accidents to assess whether they are uniformly distributed over different seasons in a year and so on. (Jammalamadaka ,S.R., et al)
Plant Phenology (Patricia et al. 2010) Study of Muscular Activity (Momeni and Faghri 2014) Spatial and Temporal Performance Analysis (Robert 1991) Study of Neurology (Drew and Doucet 1991) Political Science (Jeff and Dominik 2010) Psychophysical Research (Pearl et al. 2009) Geography (Li et al. 2010) Archaeology (Bernard 2007) Remote Sensing (Cuartero et al. 2010) Forecasting BSE Sensex (Samanthi 2019)
LINEAR DATA Vs ANGULAR DATA Existing linear statistical methods/inference will not yield reasonable results when applied on angular data.
Need for Distinct Statistical Treatment N 600 or 300 Q 150 1800 L M E 3450 P Angle The arithmetic mean points the wrong way Arc length
Need to construct new circular models which are invariant of Zero direction and sense of rotation
Circular Model is circular density function iff has the following basic properties ( ) , 0 g k g g + = ) g : 0, 2 g 2 d = ( ) 1 g 0 for any integer k (i.e. g is periodic(Mardia,1972)) ( ) ( 2 )
EXISTING METHODS OF CONSTRUCTION 1. Geometrical Considerations. 2. Wrapping a Linear Distribution around the unit circle. 3. Characterization Properties such as Maximum Entropy or Maximum Likelihood, etc., 4. Transforming a bivariate random variate to just its directional component, offsetdistributions . 5. Stereographic Projection to a Linear model. namely
Methodology of Wrapping Moduloreduction 2 ( ) XW 2 mod X It is clearly a many valued function given by ( ) ( ) = + 2 / integer an is k XW X k ( The wrapped pdf ) g = ( ) , 2 , 0 [ ) g = + ( 2 ) f k k
TheCharacteristic Function of the Circular Model The characteristic function ( ) ) + ( 2 ) it it = = ( ) ( ) t E e E e 2 2 it it it = = ( ( ) e E e e t 2 it = = either ( ) 0 or 1 integer. an is t t e ( ) 2 ( ) ( ) i ip ip = = = p E e e dF e p p 0 = whe re integer an is p = , 1 where ( is ) the cdf. F 0 p p
Trigonometric moments The value of the characteri stic function integer an at is p also called p the th trigono p metric moment of . The real part and the imaginary part of denoted are by and respective ly. p p p We can E = also view p these p trigonome E tric moments in terms of ( ) ( ) = p cos , sin for , being p integer. an p The first trig onometric moment e i = + i = 1 1 1 1 1 plays a prominent role in determining the mean direction and resultant length.
TRIGONOMETRIC MOMENTS FROM THE CHARACTERISTIC FUNCTION OF A LINEAR MODEL The trigonometric moments can be obtained using the following Proposition [c.f. p.31, Rao Jammalamadaka, and SenGupta (2001)]. p Proposition 1 The trigonometric moment of order for a wrapped circular distribution corresponds to the value of the characteristic function of the unwrapped r.v. X = (p ) p for p X
DENSITY FUNCTION IN TERMS OF TRIGONOMETRIC MOMENTS (Fourier Representation) The random variable density a has which g defined is by 1 1 ( ) = ip = = + + 1 2 ( cos sin ) g e p p p p p 2 2 1 p 2 , 0 ) , being p integer an
CHARACTERISTICS OF ANGULAR MODELS The central trigonome moments tric The mean direction defined are = by the equation = ( ) ip X + * p * p Ee i 0 W = = 1 o 1 tan p 1 o The central trigonome moments tric 1 ( ) = o p cos , p The resultant length p p o ( ) = = + 2 1 2 = o p sin p 1 1 p p o The V circular v = ariance Skewness ( ) 2 2 2 2 1 = = 3 2 o 2 1 1 2 1 V O 1 2 o 2 3 2 1 ( ) The standard deviation The circular kurtosis = 2 ln( 1 ) V 4 1 ( V ) V = o o o 2 o 2 2 o
New Wrapped Circular Models 1. Wrapped Logistic model 2. Wrapped Lognormal model 3. Wrapped Extreme value model 4. Wrapped Weibull model 5. Wrapped Halflogistic model
New Wrapped Circular Models 6. Wrapped Binormal model 7. Wrapped Gamma model 8. Wrapped Exponentiated Inverted Weibull model 9. Wrapped New Weibull-Pareto model
Wrapped Lognormal Model (Dattatreya Rao et al. 2007) density The function ( ) 2 1 {ln } x x , exp , 0 ( ) 0, = ( ) f x 2 2 2 x x , 0 The cdf ( ) 1 ln x = + ( ) 1 , 0 F x erf 2 2
By applying the methodology of wrapping the pdf and the cdf of Wrapped Lognormal distribution are 2 + k 1 {ln( 2 )} = k ( ) exp , g 2 + k ( 2 ) 2 2 = 0 2 , 0 ) where , , , 0 ( ) ( ) + k k 1 log 2 log 2 = k ( ) G erf erf 2 2 2 = 0 2 , 0 ) , , , 0 where The Wrapped Lognormal density admits the Fourier representation 2 , 0 ) 1 =1 p , , , , 0 = + + where ( ) 1 2 cos sin g p p p p 2 p
wrapped lognormal pd f (circular representation) 0.7 = 0.5 = 0.75 = 1 = 1.5 0.6 0.5 0.4 y=g*sin( ) 0.3 0.2 0.1 0 -0.1 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x=g*cos( )
CHARACTERISTIC FUNCTION OF LOGNORMAL (Ramabhadra Sarma et al. 2011) ( ) 2 itx {ln } 2 e x ( ) t dx = = itx exp E e X 2 2 x 0 1 ln x 2 2 y = = ite y for being real. ( put ) e e dy t y 2 Using n-point Hermite Gauss quadrature. According to the proposition 1 the values of the characteristic function of the Wrapped Lognormal distribution can be obtained by choosing an integer for t .
Characteristic Function of Lognormal distribution with = 0.25 Characteristic Function of Lognormal distribution with = 1 1 1.2 real( ) imag( ) real( ) imag( ) 0.8 1 0.6 0.8 0.4 0.6 0.2 0 0.4 -0.2 0.2 -0.4 0 -0.6 -0.8 -0.2 0 2 4 6 8 10 t 12 14 16 18 20 0 2 4 6 8 10 t 12 14 16 18 20
By applying the methodology of wrapping the pdf and the cdf of Wrapped Weibull distribution are ,
The Characteristic Function of the Weibull distribution t ( ) ( ) t c 1 itx itx c x = = dx 0 E e e cx e c X 0 / 1 / 1 c c ity y ity y = = e e dy e e dy 0 0 ( ) / 1 y 1 c c c = = = put and x y x cx dx dy We use n point Gauss Laguerre quadrature
Stereographic Projection is one way to make a flat map of the earth Stereographic projection to plot planar and linear data in structural geology http://www.youtube.com/watch?v=- fA27IIuz7Q#
Link between Linear and Circular :circle line (onetoonecorrespondence) T 1 :line circle T ( ) = = tan x T 2 d d ( ) ( ) ( ) ( ) ( ) ( ) = = . g F T f T T d d 1sec 2 2 = tan f 2 2
THE CHARACTERISTIC FUNCTION OF A STEREOGRAPHIC CIRCULAR MODEL
New Stereographic models 1. Stereographic Logistic Distribution 2. Stereographic Lognormal Distribution 3. Stereographic Extreme-value Distribution 4. Stereographic Double Weibull Distribution 5. Stereographic Reflected Gamma model 6. Stereographic Arc tan Exponential type models 7. Stereographic Double Exponential model 8. Stereographic Circular Normal Moment model
Stereographic Logistic Model (Phani 2013)
Why semicircular / angular models? Need Occurrence in natural phenomenon
Semicircular models 1. Stereographic Semicircular Weibull model 2. Stereographic Semicircular Half Logistic model 3. Stereographic Semicircular Exponential Inverted Weibull model 4. Arc offset Beta 5. Stereographic Semicircular Exponential model 6. Stereographic Semicircular Gamma model 7. Stereographic Semicircular New Weibull- Pareto model 8. Offset Semicircular Cauchy model 46
Stereographic Semicircular Weibull Model (Phani et al. 2013)
A random variable XSC on the unit semicircle is said to have the Stereographic Semicircular Weibull Distribution with shape . parameter c > 0, location parameter and concentration parameter > 0 denoted by SSCWEB(c, , ), if the probability density and the cumulative distribution functions are given by c 1 c c c ( ) = 2 sec tan exp tan , g 2 2 2 2 = = where 0 , 0, 0 and c v v c ( ) 1 exp = tan G 2
Trigonometric moments of the Stereographic Semicircular Weibull model First two trigonometric moments are evaluated by applying numerical integration in MATLAB.
