Feature Selection Based on SVM in Photo-Thermal Infrared (IR) Imaging Spectroscopy Classification With Limited Training Samples

Size: px
Start display at page:

Download "Feature Selection Based on SVM in Photo-Thermal Infrared (IR) Imaging Spectroscopy Classification With Limited Training Samples"

Transcription

1 Na Zhag, Keea Leatham Feature Selecto Based o SVM Photo-Thermal Ifrared (IR) Imagg Spectroscopy Classfcato Wth Lmted Trag Samples NIAN ZHANG ad KEENAN LEATHAM Departmet of Electrcal ad Computer Egeerg, Uversty of the Dstrct of Columba 400 Coectcut Aveue, NW, Washgto, D.C USA zhag@udc.edu, keea.leatham@udc.edu Abstract: - I ths paper, we propose a kerel based SVM algorthm wth varable models to adapt to the hgh-dmesoal but relatvely small samples for remote explosve detecto o photo-thermal frared magg spectroscopy (PT-IRIS) classfcato. The algorthms of the represetatve lear ad olear SVM are preseted. The respose plot, predcted vs. actual plot, ad resduals plot of the lear, quadratc, ad coarse Gaussa SVM are demostrated. A comprehesve comparso of Lear SVM, Quadratc SVM, Cubc SVM, Fe Gaussa SVM, Meda Gaussa SVM, Coarse Gaussa SVM s performed terms of root mea square error, R-squared, mea squared error, ad mea absolute error. The excellet expermetal results demostrated that the kerel based SVM models provde a promsg soluto to hgh-dmesoal data sets wth lmted trag samples. Key-Words: - Feature selecto, Support vector mache, SVM, Hgh-dmesoal, Classfcato, Photothermal frared magg spectroscopy 1 Itroducto Recet advaces moder techologes, such as photo-thermal frared (IR) magg spectroscopy techology the applcato of remote explosve detecto, 4D CT-scas techology, ad DNA mcroarrays have produced umerous massve ad mbalaced data. The eeds of classfcato ubqutously exst real-world data-tesve applcatos, ragg from cvla applcatos such as cacer dagoses ad outler detecto stock market tme seres, to homelad securty or defese related applcatos such as remote explosve detecto, llegal drug detecto, ad abormal behavor recogto. I the stuato whe the dmesoalty of data s hgh but wth few data, feature selecto usually becomes mperatve to the learg algorthms because hgh-dmesoal data teds to egatvely affect the effcecy of most learg algorthms. Feature selecto s a effcet dmesoalty reducto techque that selects a optmal subset of the orgal features that provde the best predctve power modelg the data. They are the most dstct features that ca be used to dfferetate samples to dfferet classes. There are a large umber of state-of-the-art feature selecto methods. A smultaeous spectral-spatal feature selecto ad extracto algorthm was proposed for hyperspectral mages spectral-spatal feature represetato ad classfcato. However, t lacks of kerel verso ad thus ts performace o complex datasets s ukow [1]. A regularzed regresso based feature selecto classfer was modfed to a cost-sestve classfer by geeratg ad assgg dfferet costs to each class. Features wll be selected accordg to the classfer wth optmal F-measure order to solve the class mbalace problem []. A feature selecto algorthm usg AdaBoost was preseted to deal wth Haar-lke features for vehcle detecto. The ormalzed feature vector set s used to tra the RBF-SVM classfer wth cross-valdato to select the optmal parameters [3]. A support vector mache (SVM) was appled as a classfer to detfy resdual fuctoal abormaltes athletes sufferg from cocusso usg a multchael EEG data set. The total accuracy of the classfer usg the 10 features was 77.1% [4]. A multple stace learg (MIL) was adopted to descrbe dfferet kds of actos from complexty data sources E-ISSN: Volume 13, 017

2 Na Zhag, Keea Leatham ad preset a boosted exemplar learg (BEL) method to lear the smlarty metrc ad select some represetatve exemplars from the Web for acto recogto. It takes about 98 ms to tra a multple stace SVM (m-svm) for oe exemplar. The proposed m-svm has much better result 65.37% tha the 61.53% usg SVM classfer [5]. Dfferet from the above SVM based learg algorthm, we propose a kerel based SVM algorthm wth varable models to adapt to the hgh-dmesoal but relatvely small samples for remote explosve detecto o photo-thermal frared magg spectroscopy (PT-IRIS) classfcato. The rest of the paper s orgazed as follows. I Secto, the SVM models cludg the lear SVM, quadratc SVM, ad coarse Gaussa SVM are dscussed. I Secto 3, photo-thermal frared magg spectroscopy (PT-IRIS) data set s troduced. I Secto 4, the classfcato performace of lear, quadratc, ad coarse Gausa SVM are demostrated. I addto, a comparso of Lear SVM, Quadratc SVM, Cubc SVM, Fe Gaussa SVM, Meda Gaussa SVM, Coarse Gaussa SVM s preseted terms of varous model statstcs. Fally, the paper s cocluded Secto 5. Types of SVM Algorthms Support Vector Mache (SVM) learg algorthms has bee a actve research topc wth the computer tellgece commuty. Support vector mache (SVM) aalyss s a popular mache learg tool for classfcato ad regresso, frst detfed by Vladmr Vapk ad hs colleagues 199 [6]. SVM regresso s cosdered a oparametrc techque because t reles o kerel fuctos. Support Vector Mache algorthms are utlzed may real world applcatos such as Face Detecto, Text Categorzato, ad Boformatcs. Support Vector Mache algorthm (SVM) s a supervsed mache learg algorthm, whch ca be used for ether classfcato or regresso challeges. The dfferet types of SVM learg algorthms are Lear SVM, Quadratc SVM, ad Cubc SVM. Each Support Vector Mache Algorthm has ther advatages terms of provdg solutos o a data set. For each algorthm we wll be (1) Trag a data set wth Lear SVM, Quadratc SVM, ad Cubc SVM, () Plottg the behavor of each algorthm fgurg out the RSME, R- Squared Value, MSE, MAE, Predcto Speed, Trag Tme, ad (3) Aalyzg the results of each Support Mache Algorthm to see the smlartes ad dffereces of the data. The purpose of these trals s to see f we ca fd some terestg behavors, so we ca fd dfferet methods to optmze SVM algorthms. Show below are the dfferet behavors of each SVM..1 Lear SVM Lear SVM s the ewest fast mache learg data mg algorthm for solvg multclass classfcato problems from ultra large data sets; that mplemets a orgal propretary verso of a cuttg plae algorthm for desgg a lear support vector mache. Lear SVM s a learly scalable route meag that t creates a SVM model a CPU tme, whch scales the sze of the trag data set learly. Our comparsos wth other kow SVM models clearly show ts performace s hghly accurate, mplemeted wth large data sets. It s deal for a Lear SVM to be utlzed cotemporary applcatos such as dgtal advertsemet-commerce, web page categorzato, text classfcato, boformatcs, proteomcs, bakg servces ad may other areas. It provdes solutos of multclass classfcato problems wth ay umber of classes wth hgh dmesoal data both sparse ad dese formats. There s o eed for expesve computg resources other tha a stadard platform whle mplemetg ths algorthm. The algorthm of the Lear SVM s llustrated as follows. Algorthm of the Lear SVM Iput: 1. A trag data set of the form: ( x1, y1),,( x, y). A y that s ether 1 or -1. Procedure: E-ISSN: Volume 13, 017

3 Na Zhag, Keea Leatham 1. Let the gve trag data set of pots be the form of: x, y ),,( x, y ) ( 1 1. Each x s a dmesoal vector. 3. We wat to fd the maxmum marg hyperplae that groups pot of x 1 to the group of pots where y =1. 4. The pots of x have to be satsfed by: ( w x b) = 0 Where w s the ormal vector to the hyperplae.. Quadratc SVM If data sets are ot learly separable, Quadratc SVM s utlzed to pck out a terval betwee two classes. To solve ths problem the data s mapped o to a hgher dmesoal space ad the uses a lear classfer the hgher dmesoal space. For example, a lear separator ca easly classfy the data f we use a quadratc fucto to map the data to two dmesos. The geeral dea s to map the orgal feature space to a hgher-dmesoal feature space where the trag set s separable. As the expaso creases th degrees t allows the data set to be traed a effcet maer. The algorthm of the quadratc SVM s llustrated as follows. Algorthm of the Quadratc SVM Iput: 1. A trag data set of the form: ( x1, y1),,( x, y). A kerel mappg: K( x, = ϕ ( x), ϕ( Procedure: 1. Let the gve trag data set of pots be the form of: ( x 1, y 1),,( x, y ). The polyomal kerel s defed as: K( x, = ϕ( x), ϕ( where c > 0 3. For polyomals the kerel s defed by: T d K ( x, = ( x y + c) d = 4. Usg the multomal theorem the expaso becomes: K( x, = ( + + = = 1 1 j= ( ( = 1 x c x )( x j )( x y + c) = = 1 y c y ) + c ( x y y.3 Gaussa SVM The Gaussa kerel oly depeds o the Eucldea dstace betwee x ad x, ad s based o the assumpto that smlar pots are close oe to each other the feature space ( terms of Eucldea dstace). The algorthm of the cubc SVM s llustrated as follows. Algorthm of the Gaussa SVM Iput: 1. A trag data set of the form: ( x1, y1),,( x, y). A kerel mappg: K( x, = ϕ ( x), ϕ( Procedure: 1. Let the gve trag data set of pots be the form of: ( x1, y1),,( x, y). The Gaussa kerel s defed as: x y K( x, = e γ for a gve parameter γ > 0 3 Photo-Thermal Ifrared (IR) Imagg Spectroscopy (PT-IRIS) Data Set A photo-thermal frared magg spectroscopy (PT-IRIS) techque has recetly bee developed by the Naval Research Laboratory (NRL), Washgto, DC wth uprecedeted spatal resoluto at ~1 mcro [8]. I ths data set, the mxg Ifrared IR absorpto/emsso features causes some complcated ad overlappg samples, whch leads to grad challeges to mult-class classfcato. Specfcally, frared quatum cascade lasers are used to llumate a surface potetally scattered wth samples of terest. If the wavelegth of the thermal emsso of lght s resoat wth collecto features of surface j ) ) E-ISSN: Volume 13, 017

4 Na Zhag, Keea Leatham samples, the sample heats by ~1oC. By varyg the cdet wavelegth, ay samples of terest could be maged [9]. The feature of the PT-IRIS sgal s the temperature crease ormalzed to the average power of the laser pulse at the ed of the laser pulse,.e. Tmax. Tmax as a fucto of exctato ad collecto wavelegth are bult to a feature vector [10], as show Fg. 1. Fg. 1 Data matrx wth feature vectors. Each square s a feature value, Tmax, whch s a fucto of exctato ad collecto wavelegth. Smulated samples clude 4 dfferet partcle dameters (5, 3, ad 7 mm) ad 4 aalytes (RDX, TNT, AN, Sucrose) o 4 substrates (whte pat, steel, glass, polyethylee) usg 38 exctato wavelegths ad 33 collecto wavelegths. Thus, wth 38 exctato wavelegths ad 33 collecto wavelegths, they would geerate 154 features (predctor varables). Therefore, each colum cotas 154 features ad there are oly 13 samples. We may demostrate the sgal matrx for all the 13 samples. Ths ca be see by dsplay the data set false color plot whch wll show vsble or o-vsble parts of the electromagetc spectrum. The false color plot of the data set s show Fg.. The color of the data pot s proportoal to sgal stregth,.e. red represets hgh, ad blue represets low. 4 Smulato Aalyss 4.1 Explore Data ad Results Respose Plot After a regresso model s traed, the regresso model results ca be dsplayed by the respose plot,.e. the predcted respose versus record umber. Holdout or cross-valdato s used, thus each predcto s obtaed usg a model that was traed wthout usg the correspodg observato. Therefore, these predctos are the predctos o the held-out observatos. 80% of the data s used to tra the etwork ad the remag 0% data pots are used as the testg data. The respose plot of lear SVM, quadratc SVM, ad coarse Gaussa SVM are show Fg. 3, Fg. 4, ad Fg. 5, respectvely. Fg. 3 The respose plot of lear SVM. Fg. 4 The respose plot of quadratc SVM. Fg. False color plot of data set. The data set has 13 samples ad 154 features. E-ISSN: Volume 13, 017

5 Na Zhag, Keea Leatham Fg. 5 The respose plot of coarse Gaussa SVM. 4. Plot Predcted vs. Actual Respose The Predcted vs. Actual plot s used to check model performace after trag a model. Use ths plot to uderstad how well the regresso model makes predctos for dfferet respose values. Whe the plot s ope, the predcted respose of our model s plotted agast the actual, true respose. A perfect regresso model has a predcted respose equal to the true respose, so all the pots le o a dagoal le. The vertcal dstace from the le to ay pot s the error of the predcto for that pot. A good model has small errors, ad so the predctos are scattered ear the le. Usually a good model has pots scattered roughly symmetrcally aroud the dagoal le. If we ca see ay clear patters the plot, t s lkely that we ca mprove the model. The predcted vs. actual plot of lear SVM, quadratc SVM, ad coarse Gaussa SVM are show Fg. 6, Fg. 7, ad Fg. 8, respectvely. Fg. 7 The Predcted vs. Actual plot of quadratc SVM. Fg. 8 The Predcted vs. Actual plot of coarse Gaussa SVM. 4.3 Evaluate Model Usg Resduals Plot We further evaluate the model performace by usg the resduals plot after trag a model. The resduals plot dsplays the dfferece betwee the predcted ad true resposes. Usually a good model has resduals scattered roughly symmetrcally aroud 0. If we ca see ay clear patters the resduals, t s lkely that we ca mprove the model. We eapecally look for the followg patters: Resduals are ot symmetrcally dstrbuted aroud 0. Resduals chage sgfcatly sze from left to rght the plot. Outlers occur, that s, resduals that are much larger tha the rest of the resduals. Fg. 6 The Predcted vs. Actual plot of lear SVM. E-ISSN: Volume 13, 017

6 Na Zhag, Keea Leatham Clear, olear patter appears the resduals. The resdual plots of lear SVM, quadratc SVM, ad coarse Gaussa SVM are show Fg. 9, Fg. 10, ad Fg. 11, respectvely. Fg. 9 The resduals plot of lear SVM. 4.4 Model Statstcs The model parameters are very useful ad mportat to evaluate the performace of dfferet models. They are defed as follows. RMSE (Root mea square error). The RMSE s always postve ad ts uts match the uts of the respose. Look for smaller values of the RMSE. R-Squared. Coeffcet of determato. R- squared s always smaller tha 1 ad usually larger tha 0. It compares the traed model wth the model where the respose s costat ad equals the mea of the trag respose. If the model s worse tha ths costat model, the R-Squared s egatve. Look for a R- Squared close to 1. MSE (Mea squared error). The MSE s the square of the RMSE. Look for smaller values of the MSE. MAE (Mea absolute error). The MAE s always postve ad smlar to the RMSE, but less sestve to outlers. Look for smaller values of the MAE. For each SVM algorthm, after the etwork has bee well traed, we evaluate the performace of each featured subset. The comprehesve comparso s show Table 1. TABLE 1 COMPASIRON OF DIFFERENT SVM MODELS Fg. 10 The resduals plot of quadratc SVM. RSME R- Sq MSE MAE Tra Tme (sec) Lear 6.61* * * Quadratc 1.79* * * Cubc 3.65* * * Fe 3.77* * * Gaussa Meda 6.38* * * Gaussa Coarse Gaussa 7.83* * * Fg. 11 The resduals plot of coarse Gaussa SVM. E-ISSN: Volume 13, 017

7 Na Zhag, Keea Leatham 5 Coclusos We propose a kerel based SVM algorthm wth varable models to adapt to the hgh-dmesoal but relatvely small samples for remote explosve detecto o photo-thermal frared magg spectroscopy (PT-IRIS) classfcato. For each SVM algorthm t reveals classfcato accuracy ad mmum feature umber objectves. After the etwork has bee well traed, we evaluate the performace of each featured subset. The respose plot,predcted vs. actual plot, ad resduals plot of the lear, quadratc, ad coarse Gaussa SVM are demostrated. A comprehesve comparso of lear SVM, quadratc SVM, cubc SVM, fe Gaussa SVM, meda Gaussa SVM, coarse Gaussa SVM s performed terms of root mea square error, R-squared, mea squared error, ad mea absolute error. The excellet expermetal results demostrated that the kerel based SVM models provde a promsg soluto to hgh-dmesoal data sets wth lmted trag samples. ACKNOWLEDGMENT Ths work was supported by the Natoal Scece Foudato (NSF) grats: HRD # , DUE # , ad HRD # Refereces: [1] L. Zhag, Q. Zhag, B. Du, X. Huag, Y. Y. Tag ad D. Tao, "Smultaeous Spectral- Spatal Feature Selecto ad Extracto for Hyperspectral Images," IEEE Trasactos o Cyberetcs, vol. 48, o. 1, pp. 16-8, Ja [] M. Lu, C. Xu, Y. Luo, C. Xu, Y. We ad D. Tao, "Cost-Sestve Feature Selecto by Optmzg F-Measures," IEEE Trasactos o Image Processg, vol. 7, o. 3, pp , March 018. [3] X. We, L. Shao, W. Fag ad Y. Xue, "Effcet Feature Selecto ad Classfcato for Vehcle Detecto," IEEE Trasactos o Crcuts ad Systems for Vdeo Techology, vol. 5, o. 3, pp , March 015. [4] C. Cao, R. L. Tutwler ad S. Slobouov, "Automatc Classfcato of Athletes Wth Resdual Fuctoal Defcts Followg Cocusso by Meas of EEG Sgal Usg Support Vector Mache," IEEE Trasactos o Neural Systems ad Rehabltato Egeerg, vol. 16, o. 4, pp , Aug [5] T. Zhag, J. Lu, S. Lu, C. Xu ad H. Lu, "Boosted Exemplar Learg for Acto Recogto ad Aotato," IEEE Trasactos o Crcuts ad Systems for Vdeo Techology, vol. 1, o. 7, pp , July 011. [6] Vapk, V. The Nature of Statstcal Learg Theory. Sprger, New York, [7] Fursteberg, R., Kedzora, C. A., Stepowsk, J., Stepowsk, S. V., Rake, M., Papatoaks, M. R., Nguye. V., Hubler, G. K., ad McGll, R. A., Stad-Off Detecto of Trace Explosves va Resoat Ifrared Photothermal Imagg, Appl. Phys. Lett., vol. 93, o., 008. [9] Fursteberg, R., Kedzora, C. A., Stepowsk, J., Stepowsk, S. V., Rake, M., Papatoaks, M. R., Nguye. V., Hubler, G. K., ad McGll, R. A., Stad-Off Detecto of Trace Explosves va Resoat Ifrared E-ISSN: Volume 13, 017

8 Na Zhag, Keea Leatham Photothermal Imagg, Appl. Phys. Lett., vol. 93, o., 008. [10] C. A. Kedzora, R. Fursteberg, M. Papatoaks, Ifrared Photothermal Imagg of Trace Explosves o Relevat Substrates, Proceedgs of SPIE, vol. 8709, 013. E-ISSN: Volume 13, 017

An Introduction to. Support Vector Machine

An Introduction to. Support Vector Machine A Itroducto to Support Vector Mache Support Vector Mache (SVM) A classfer derved from statstcal learg theory by Vapk, et al. 99 SVM became famous whe, usg mages as put, t gave accuracy comparable to eural-etwork

More information

Kernel-based Methods and Support Vector Machines

Kernel-based Methods and Support Vector Machines Kerel-based Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to Kerel-Based Learg

More information

Introduction to local (nonparametric) density estimation. methods

Introduction to local (nonparametric) density estimation. methods Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest

More information

Analysis of Lagrange Interpolation Formula

Analysis of Lagrange Interpolation Formula P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. www.jset.com ISS 348 7968 Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal

More information

Research on SVM Prediction Model Based on Chaos Theory

Research on SVM Prediction Model Based on Chaos Theory Advaced Scece ad Techology Letters Vol.3 (SoftTech 06, pp.59-63 http://dx.do.org/0.457/astl.06.3.3 Research o SVM Predcto Model Based o Chaos Theory Sog Lagog, Wu Hux, Zhag Zezhog 3, College of Iformato

More information

Bayes (Naïve or not) Classifiers: Generative Approach

Bayes (Naïve or not) Classifiers: Generative Approach Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg

More information

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632

More information

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract

More information

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information

A Robust Total Least Mean Square Algorithm For Nonlinear Adaptive Filter

A Robust Total Least Mean Square Algorithm For Nonlinear Adaptive Filter A Robust otal east Mea Square Algorthm For Nolear Adaptve Flter Ruxua We School of Electroc ad Iformato Egeerg X'a Jaotog Uversty X'a 70049, P.R. Cha rxwe@chare.com Chogzhao Ha, azhe u School of Electroc

More information

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn: Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:

More information

CS 1675 Introduction to Machine Learning Lecture 12 Support vector machines

CS 1675 Introduction to Machine Learning Lecture 12 Support vector machines CS 675 Itroducto to Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Mdterm eam October 9, 7 I-class eam Closed book Stud materal: Lecture otes Correspodg chapters

More information

Dimensionality reduction Feature selection

Dimensionality reduction Feature selection CS 750 Mache Learg Lecture 3 Dmesoalty reducto Feature selecto Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 750 Mache Learg Dmesoalty reducto. Motvato. Classfcato problem eample: We have a put data

More information

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur

More information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The

More information

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971)) art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the

More information

Binary classification: Support Vector Machines

Binary classification: Support Vector Machines CS 57 Itroducto to AI Lecture 6 Bar classfcato: Support Vector Maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI Supervsed learg Data: D { D, D,.., D} a set of eamples D, (,,,,,

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of

More information

Multiple Linear Regression Analysis

Multiple Linear Regression Analysis LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple

More information

Principal Components. Analysis. Basic Intuition. A Method of Self Organized Learning

Principal Components. Analysis. Basic Intuition. A Method of Self Organized Learning Prcpal Compoets Aalss A Method of Self Orgazed Learg Prcpal Compoets Aalss Stadard techque for data reducto statstcal patter matchg ad sgal processg Usupervsed learg: lear from examples wthout a teacher

More information

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal

More information

Support vector machines II

Support vector machines II CS 75 Mache Learg Lecture Support vector maches II Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Learl separable classes Learl separable classes: here s a hperplae that separates trag staces th o error

More information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

A tighter lower bound on the circuit size of the hardest Boolean functions

A tighter lower bound on the circuit size of the hardest Boolean functions Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the

More information

Supervised learning: Linear regression Logistic regression

Supervised learning: Linear regression Logistic regression CS 57 Itroducto to AI Lecture 4 Supervsed learg: Lear regresso Logstc regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI Data: D { D D.. D D Supervsed learg d a set of eamples s

More information

Chapter 13 Student Lecture Notes 13-1

Chapter 13 Student Lecture Notes 13-1 Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato

More information

PROJECTION PROBLEM FOR REGULAR POLYGONS

PROJECTION PROBLEM FOR REGULAR POLYGONS Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos

More information

L5 Polynomial / Spline Curves

L5 Polynomial / Spline Curves L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a

More information

Block-Based Compact Thermal Modeling of Semiconductor Integrated Circuits

Block-Based Compact Thermal Modeling of Semiconductor Integrated Circuits Block-Based Compact hermal Modelg of Semcoductor Itegrated Crcuts Master s hess Defese Caddate: Jg Ba Commttee Members: Dr. Mg-Cheg Cheg Dr. Daqg Hou Dr. Robert Schllg July 27, 2009 Outle Itroducto Backgroud

More information

Cubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem

Cubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs

More information

Handout #1. Title: Foundations of Econometrics. POPULATION vs. SAMPLE

Handout #1. Title: Foundations of Econometrics. POPULATION vs. SAMPLE Hadout #1 Ttle: Foudatos of Ecoometrcs Course: Eco 367 Fall/015 Istructor: Dr. I-Mg Chu POPULATION vs. SAMPLE From the Bureau of Labor web ste (http://www.bls.gov), we ca fd the uemploymet rate for each

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

Simple Linear Regression

Simple Linear Regression Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uversty Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal

More information

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall

More information

Study on a Fire Detection System Based on Support Vector Machine

Study on a Fire Detection System Based on Support Vector Machine Sesors & Trasducers, Vol. 8, Issue, November 04, pp. 57-6 Sesors & Trasducers 04 by IFSA Publshg, S. L. http://www.sesorsportal.com Study o a Fre Detecto System Based o Support Vector Mache Ye Xaotg, Wu

More information

Lecture 8: Linear Regression

Lecture 8: Linear Regression Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE

More information

Descriptive Statistics

Descriptive Statistics Page Techcal Math II Descrptve Statstcs Descrptve Statstcs Descrptve statstcs s the body of methods used to represet ad summarze sets of data. A descrpto of how a set of measuremets (for eample, people

More information

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Generalization of the Dissimilarity Measure of Fuzzy Sets

Generalization of the Dissimilarity Measure of Fuzzy Sets Iteratoal Mathematcal Forum 2 2007 o. 68 3395-3400 Geeralzato of the Dssmlarty Measure of Fuzzy Sets Faramarz Faghh Boformatcs Laboratory Naobotechology Research Ceter vesa Research Isttute CECR Tehra

More information

CS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x

CS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x CS 75 Mache Learg Lecture 8 Lear regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Lear regresso Fucto f : X Y s a lear combato of put compoets f + + + K d d K k - parameters

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Support vector machines

Support vector machines CS 75 Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Outle Outle: Algorthms for lear decso boudary Support vector maches Mamum marg hyperplae.

More information

Analyzing Fuzzy System Reliability Using Vague Set Theory

Analyzing Fuzzy System Reliability Using Vague Set Theory Iteratoal Joural of Appled Scece ad Egeerg 2003., : 82-88 Aalyzg Fuzzy System Relablty sg Vague Set Theory Shy-Mg Che Departmet of Computer Scece ad Iformato Egeerg, Natoal Tawa versty of Scece ad Techology,

More information

An Improved Support Vector Machine Using Class-Median Vectors *

An Improved Support Vector Machine Using Class-Median Vectors * A Improved Support Vector Mache Usg Class-Meda Vectors Zhezhe Kou, Jahua Xu, Xuegog Zhag ad Lag J State Ke Laborator of Itellget Techolog ad Sstems Departmet of Automato, Tsghua Uverst, Bejg 100084, P.R.C.

More information

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?

More information

COV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.

COV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs,

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 3423 Simple Linear Regression Page 12-01 ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

A Method for Damping Estimation Based On Least Square Fit

A Method for Damping Estimation Based On Least Square Fit Amerca Joural of Egeerg Research (AJER) 5 Amerca Joural of Egeerg Research (AJER) e-issn: 3-847 p-issn : 3-936 Volume-4, Issue-7, pp-5-9 www.ajer.org Research Paper Ope Access A Method for Dampg Estmato

More information

Beam Warming Second-Order Upwind Method

Beam Warming Second-Order Upwind Method Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet

More information

ABOUT ONE APPROACH TO APPROXIMATION OF CONTINUOUS FUNCTION BY THREE-LAYERED NEURAL NETWORK

ABOUT ONE APPROACH TO APPROXIMATION OF CONTINUOUS FUNCTION BY THREE-LAYERED NEURAL NETWORK ABOUT ONE APPROACH TO APPROXIMATION OF CONTINUOUS FUNCTION BY THREE-LAYERED NEURAL NETWORK Ram Rzayev Cyberetc Isttute of the Natoal Scece Academy of Azerbaa Republc ramrza@yahoo.com Aygu Alasgarova Khazar

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure

More information

Unsupervised Learning and Other Neural Networks

Unsupervised Learning and Other Neural Networks CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all

More information

An Improved Differential Evolution Algorithm Based on Statistical Log-linear Model

An Improved Differential Evolution Algorithm Based on Statistical Log-linear Model Sesors & Trasducers, Vol. 59, Issue, November, pp. 77-8 Sesors & Trasducers by IFSA http://www.sesorsportal.com A Improved Dfferetal Evoluto Algorthm Based o Statstcal Log-lear Model Zhehuag Huag School

More information

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture) CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.

More information

Arithmetic Mean and Geometric Mean

Arithmetic Mean and Geometric Mean Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,

More information

Chapter Two. An Introduction to Regression ( )

Chapter Two. An Introduction to Regression ( ) ubject: A Itroducto to Regresso Frst tage Chapter Two A Itroducto to Regresso (018-019) 1 pg. ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the

More information

ANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK

ANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

Median as a Weighted Arithmetic Mean of All Sample Observations

Median as a Weighted Arithmetic Mean of All Sample Observations Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of

More information

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted

More information

Lecture 07: Poles and Zeros

Lecture 07: Poles and Zeros Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto

More information

Lecture 7: Linear and quadratic classifiers

Lecture 7: Linear and quadratic classifiers Lecture 7: Lear ad quadratc classfers Bayes classfers for ormally dstrbuted classes Case : Σ σ I Case : Σ Σ (Σ daoal Case : Σ Σ (Σ o-daoal Case 4: Σ σ I Case 5: Σ Σ j eeral case Lear ad quadratc classfers:

More information

Applications of Multiple Biological Signals

Applications of Multiple Biological Signals Applcatos of Multple Bologcal Sgals I the Hosptal of Natoal Tawa Uversty, curatve gastrectomy could be performed o patets of gastrc cacers who are udergoe the curatve resecto to acqure sgal resposes from

More information

Generative classification models

Generative classification models CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato

More information

Newton s Power Flow algorithm

Newton s Power Flow algorithm Power Egeerg - Egll Beedt Hresso ewto s Power Flow algorthm Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 2 Calculatos. For the referece bus #, we set : V = p.u. ad δ = 0 For all other

More information

Nonparametric Regression with Trapezoidal Fuzzy Data

Nonparametric Regression with Trapezoidal Fuzzy Data Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: 3-869 Volume: 3 Issue: 6 386-383 Noparametrc Regresso wt rapezodal Fuzzy Data. Razzaga Departmet of Statstcs Roudee Brac Islamc Azad Uversty

More information

Dimensionality Reduction and Learning

Dimensionality Reduction and Learning CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that

More information

Gender Classification from ECG Signal Analysis using Least Square Support Vector Machine

Gender Classification from ECG Signal Analysis using Least Square Support Vector Machine Amerca Joural of Sgal Processg, (5): 45-49 DOI:.593/.asp.5.8 Geder Classfcato from ECG Sgal Aalyss usg Least Square Support Vector Mache Raesh Ku. rpathy,*, Ashutosh Acharya, Sumt Kumar Choudhary Departmet

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

Statistics MINITAB - Lab 5

Statistics MINITAB - Lab 5 Statstcs 10010 MINITAB - Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of

More information

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura (agakura@z7.keo.jp) 1 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad

More information

Probability and. Lecture 13: and Correlation

Probability and. Lecture 13: and Correlation 933 Probablty ad Statstcs for Software ad Kowledge Egeers Lecture 3: Smple Lear Regresso ad Correlato Mocha Soptkamo, Ph.D. Outle The Smple Lear Regresso Model (.) Fttg the Regresso Le (.) The Aalyss of

More information

Non-uniform Turán-type problems

Non-uniform Turán-type problems Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at

More information

Comparison of SVMs in Number Plate Recognition

Comparison of SVMs in Number Plate Recognition Comparso of SVMs Number Plate Recogto Lhog Zheg, Xaga He ad om Htz Uversty of echology, Sydey, Departmet of Computer Systems, {lzheg, sea, htz}@t.uts.edu.au Abstract. Hgh accuracy ad hgh speed are two

More information

Rademacher Complexity. Examples

Rademacher Complexity. Examples Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed

More information

MAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE

MAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE merca Jr of Mathematcs ad Sceces Vol, No,(Jauary 0) Copyrght Md Reader Publcatos wwwjouralshubcom MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES OF FUZZY DIVERGENCE RKTul Departmet of Mathematcs SSM College

More information

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers

More information

A Comparison of Neural Network, Rough Sets and Support Vector Machine on Remote Sensing Image Classification

A Comparison of Neural Network, Rough Sets and Support Vector Machine on Remote Sensing Image Classification A Comparso of Neural Network, Rough Sets ad Support Vector Mache o Remote Sesg Image Classfcato Hag XIAO 1, Xub ZHANG 1, Yume DU 1: School of Electroc, Iformato ad Electrcal Egeerg Shagha Jaotog Uversty

More information

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions.

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions. Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos

More information

Source-Channel Prediction in Error Resilient Video Coding

Source-Channel Prediction in Error Resilient Video Coding Source-Chael Predcto Error Reslet Vdeo Codg Hua Yag ad Keeth Rose Sgal Compresso Laboratory ECE Departmet Uversty of Calfora, Sata Barbara Outle Itroducto Source-chael predcto Smulato results Coclusos

More information

Statistical characteristics of the normalized Stokes parameters

Statistical characteristics of the normalized Stokes parameters Scece Cha Seres F: Iformato Sceces 008 SCIENCE IN CHINA PRESS Sprger www.sccha.com fo.sccha.com www.sprgerlk.com Statstcal characterstcs of the ormalzed Stokes parameters LIU Tao 1 WANG XueSog 1 & XIAO

More information

Bootstrap Method for Testing of Equality of Several Coefficients of Variation

Bootstrap Method for Testing of Equality of Several Coefficients of Variation Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. -6, Artcle ID Sc- Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee

More information

= 1. UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Parameters and Statistics. Measures of Centrality

= 1. UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Parameters and Statistics. Measures of Centrality UCLA STAT Itroducto to Statstcal Methods for the Lfe ad Health Sceces Istructor: Ivo Dov, Asst. Prof. of Statstcs ad Neurology Teachg Assstats: Fred Phoa, Krste Johso, Mg Zheg & Matlda Hseh Uversty of

More information

Section l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58

Section l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58 Secto.. 6l 34 6h 667899 7l 44 7h Stem=Tes 8l 344 Leaf=Oes 8h 5557899 9l 3 9h 58 Ths dsplay brgs out the gap the data: There are o scores the hgh 7's. 6. a. beams cylders 9 5 8 88533 6 6 98877643 7 488

More information

Statistics: Unlocking the Power of Data Lock 5

Statistics: Unlocking the Power of Data Lock 5 STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I- Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re- grades Re-

More information

An Acoustic Method for Condition Classification in Live Sewer Networks

An Acoustic Method for Condition Classification in Live Sewer Networks 18th World Coferece o Nodestructve Testg, 16-2 Aprl 212, Durba, South Afrca A Acoustc Method for Codto Classfcato Lve Sewer Networks Zao FENG, Krll V. HOROSHENKOV, M. Tareq BIN ALI, Smo J. TAIT School

More information

Assignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)

Assignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class) Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

MOLECULAR VIBRATIONS

MOLECULAR VIBRATIONS MOLECULAR VIBRATIONS Here we wsh to vestgate molecular vbratos ad draw a smlarty betwee the theory of molecular vbratos ad Hückel theory. 1. Smple Harmoc Oscllator Recall that the eergy of a oe-dmesoal

More information

EECE 301 Signals & Systems

EECE 301 Signals & Systems EECE 01 Sgals & Systems Prof. Mark Fowler Note Set #9 Computg D-T Covoluto Readg Assgmet: Secto. of Kame ad Heck 1/ Course Flow Dagram The arrows here show coceptual flow betwee deas. Note the parallel

More information

About a Fuzzy Distance between Two Fuzzy Partitions and Application in Attribute Reduction Problem

About a Fuzzy Distance between Two Fuzzy Partitions and Application in Attribute Reduction Problem BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND IORMATION TECHNOLOGIES Volume 6, No 4 Sofa 206 Prt ISSN: 3-9702; Ole ISSN: 34-408 DOI: 0.55/cat-206-0064 About a Fuzzy Dstace betwee Two Fuzzy Parttos ad Applcato

More information

Naïve Bayes MIT Course Notes Cynthia Rudin

Naïve Bayes MIT Course Notes Cynthia Rudin Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.

More information

The equation is sometimes presented in form Y = a + b x. This is reasonable, but it s not the notation we use.

The equation is sometimes presented in form Y = a + b x. This is reasonable, but it s not the notation we use. INTRODUCTORY NOTE ON LINEAR REGREION We have data of the form (x y ) (x y ) (x y ) These wll most ofte be preseted to us as two colum of a spreadsheet As the topc develops we wll see both upper case ad

More information