Multivariate EWMA Control Chart

Size: px
Start display at page:

Download "Multivariate EWMA Control Chart"

Transcription

1 Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant when the varables are hghly correlated, snce jont out-of-control condtons can occur wthout any ndvdual varable volatng ts control lmts when plotted separately. The EWMA chart s smlar to the T-Squared chart, except that the ponts plotted on the chart are a weghted average of current and past observatons. Sample StatFolo: mvewma.sgp Sample Data: The fle grt.sgd contans measurements made on n = 56 batches of grt, from Holmes and Mergen (1993). The data represent the percentages of large, medum, and small partcles n the grt. The table below shows a partal lst of the data n that fle: Large Medum Small Means Covarances , Snce the percentages n the frst three columns sum to 100% n each row, t s only necessary to create a chart based on the frst 2 columns. In addton to the data, the fle contans columns wth the standard means and covarances for the percentage of large and medum partcles, establshed when montorng of the process was frst begun by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 1

2 Data Input The data to be analyzed consst of p = 2 or more numerc columns contanng the varables of nterest, wth optonal entry of the varable means and varance-covarance matrx. Data: 2 or more numerc columns contanng the n samples, one sample per row. Subgroup numbers or sze: If the data were obtaned as ndvduals, leave ths feld blank or enter 1. If the data were collected n subgroups, each of sze m, enter the sngle value m. In such a case, each consecutve m rows n the fle wll be consdered to form a subgroup. If the subgroup szes are not equal, enter the name of an addtonal numerc or non-numerc column contanng group dentfers. The program wll scan ths column and place all sequental rows wth dentcal codes nto the same group. Standard Means: For an ntal study or Phase I analyss where the data wll be used to determne the control lmts, leave ths feld blank. For a control-to-standard or Phase II analyss, enter the name of a column contan p means. Standard Covarances: For an ntal study or Phase I analyss, leave ths feld blank. For a control-to-standard or Phase II analyss, enter the name of a column contan the p 2 varances and covarances. In enterng the values n a covarance matrx, enter the values n the frst row of the matrx, then the values n the second row, and so forth. Note: f you select Save Results after performng a Phase I analyss, the covarance matrx wll be saved n ths exact format by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 2

3 Labels: optonal labels for each subgroup. The labels wll be appled n sequence to the subgroups when plottng the control charts. Select: subset selecton. Analyss Summary The Analyss Summary shows the number of observatons ncluded n the analyss and the locaton of the control lmts on the control charts. Multvarate EWMA Charts Data varables: Large Medum Number of observatons ncluded = 56 Number of observatons excluded = 0 Smoothng parameter lambda: 0.2 Intalzaton: centerlne Phase 2 - covarance specfed based on standard Chart Alpha LCL UCL Beyond lmts T-Squared Included n the table are: Smoothng parameter lambda: the value of the EWMA parameter, specfed on the Analyss Optons dalog box. The default value of s determned from the settngs on the Control Charts tab of the Preferences dalog box, accessble from the Edt menu. Phase: If Phase 1, the method for estmatng the covarance matrx s dsplayed. If Phase 2, the assumptons about the nput covarance matrx are shown. Chart: the type of chart. For ndvduals data, only a T-Squared chart s created. For grouped data, a generalzed varance chart s ncluded f the subgroup sze exceeds the number of varables. Alpha: the false alarm probablty of the chart, specfed usng Analyss Optons. For standard 3-sgma control charts, = LCL: the lower control lmt. UCL: the upper control lmt. Beyond lmts: the number of ponts on the control chart that are beyond the control lmts. In the example, the process generated 15 out-of-control sgnals on the EWMA T-Squared chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 3

4 Analyss Optons Alpha: the false alarm probablty for postonng the control lmts. For the equvalent of a standard 3-sgma control chart, set = 0.27%. Covarance Matrx: detals about the covarance matrx. There are 4 possbltes: 1. If the Standard Covarances feld was left blank on the data nput dalog box and the data are n subgroups, no entry s necessary snce the covarances wll be estmated from the pooled wthn-group varablty. 2. If the Standard Covarances feld was left blank on the data nput dalog box and the data are ndvduals, select Pooled estmator to estmate the covarance between varables j and k usng the usual estmator s jk 1 n 1 n 1 x x x k xk j j (1) Select Successve dfferences to estmate the covarance usng n 1 xj x 1, j x k x 1 k (2) s jk, 2( n 1) 1 The second estmator s a more local estmator, n the sense that t captures only shortterm varablty, n a smlar manner to the way n whch a movng range s used to estmate varablty for a standard ndvduals chart. 3. If an entry was made n the Standard Covarances feld and the estmates provded were obtaned from a prevous sample, enter the sze of that prevous sample (f grouped, the number of subgroups) n the Standard Sample Sze feld. 4. If an entry was made n the Standard Covarances feld and the covarances are assumed to be known, leave the Standard Sample Sze feld empty. Lambda: a value for the EWMA parameter 0 < < 1. The value of controls the amount of weght gven to the past hstory of the process. The smaller the value, the more weght gven to older observatons or subgroups. Ths also mpacts the average run length of the chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 4

5 EWMA Chart The EWMA Chart shows the exponentally weghted value of T 2 for each data value or subgroup: T-Squared Multvarate EWMA Control Chart UCL = 11.83, lambda = Observaton The EWMA procedure begns by smoothng the observed data vector at tme by EWMA x (3) ( 1 ) EWMA 1 for ndvduals data and by EWMA x (4) ( 1 ) EWMA 1 for grouped data, wth EWMA 0 set equal to the mean vector or x. The -th value of T-squared s then calculated from 1 EWMA EWMA 2 T (5) Z where s the covarance matrx of the nput data, and 2 Z (6) For the sample data, the chart starts out well below the control lmt but then rses durng the latter secton. 15 out-of-control sgnals are generated by the chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 5

6 Pane Optons STATGRAPHICS Rev. 7/24/2009 Pont Symbols: select Beyond Lmts to draw specal pont symbols only for ponts fallng above the control lmt. Select Largest Contrbutor to dentfy the varable that contrbutes most to each value of T 2. Decmal Places for Lmts: number of decmal places for dsplayng the control lmt. Color Zones: check ths box to dsplay green and red zones. Example: Identfyng Largest Contrbutor If Largest Contrbutor s selected, the chart wll take the followng form: T-Squared Multvarate EWMA Control Chart UCL = 11.83, lambda = 0.2 Largest Large Medum Observaton Each pont on the chart s coded accordng to the varable that contrbutes the most to the value of T 2. In the plot above, the bggest contrbutor to the frst rse appears to be the percentage of Medum partcles, whle the bggest contrbutor to the second rse appears to be the percentage of Large partcles by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 6

7 Multvarate Control Chart Report Ths pane tabulates the ponts on the control chart: Multvarate Chart Report Observaton T-Squared Large Medum 27 * * * * * * * * * * * * * * * An astersk ndcates any value beyond the control lmts. Any ponts excluded from the analyss usng the Exclude button are ndcated wth an X. Pane Optons Dsplay All Subgroups f checked, all observatons or subgroups wll be tabulated. Otherwse, only ponts beyond the control lmts wll be ncluded n the table. Dsplay Orgnal Data f checked, the values of each varable wll be dsplayed. Otherwse, only the control chart values wll be tabulated by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 7

8 Generalzed Varance Chart A T-Squared chart s desgned to montor the means of p varables. To montor the varance, a Generalzed Varance Chart by sometmes be dsplayed: Generalzed Varance Chart Gen. Varance UCL = CL = LCL = Subgroup Ths chart s created only for data arranged n subgroups, and only f the subgroup sze s at least p + 1. The generalzed varance S for the -th subgroup s defned as the determnant of ts varance-covarance matrx. The above chart shows the grt data grouped n subgroups of 4 consecutve observatons each. Any pont beyond the upper control lmt would ndcate an unusually large amount of varablty wthn that subgroup. In ths case, no such ponts are present. Control Ellpse If out-of-control sgnals are shown on the control chart, t s useful to examne those values n detal. A good chart to use n the case of p = 2 varables s the Control Ellpse: Control Ellpse Medum Large The upper control lmt on the T-Squared chart corresponds to an ellptcal regon n the space of any two of the varables, wth the other varables held at a fxed value. For p = 2, any out-ofcontrol sgnals wll show up as ponts outsde the ellpse by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 8

9 In the sample data, t wll be notced that some out-of-control sngles correspond to a hgh percentage of Large partcles whle others correspond to a low percentage of Medum partcles. Pane Optons Select 2 varables: select any 2 varables to defne the control ellpse. The ellpse wll be plotted assumng that all other varables at held at ther mean levels. Care should be taken n nterpretng the plot f p > 2, snce the true ellptcal control regon for each pont depends on the value of the varables that are not plotted by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 9

10 EWMA Decomposton The T 2 statstc can be decomposed nto components attrbutable to each of the varables. One method for measurng the contrbuton of the j-th varable to an out-of-control T 2 value s by lookng at how much smaller T 2 would be f the j-th varable was not ncluded n the analyss. The T-Squared Decomposton pane does such a decomposton for each out-of-control sgnal on the T-Squared chart: T-Squared Decomposton Observaton Large Medum Followng Runger, Alt and Montgomery (1996), the table shows d j 2 2 T T( j) (7) where 2 T ( j) s the value of the statstc usng all varables except the j-th. For the current data, Medum appears to be the domnant varable for the early out-of-control sgnals, whle Large appears to be the domnant varable for the later sgnals. 3-D Control Chart When the data consst of p = 3 varables, a 3-D control chart can be very helpful, snce the control regon s then an ellpsod by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 10

11 Control Ellpsod X X2 X1 The above plot shows the outlne of a typcal control ellpsod for 3 varables. Note: to explore ths plot, t s very helpful to use the dynamc rotaton button on the analyss toolbar. Pane Optons Select 3 varables: select any 3 varables to defne the control ellpsod. The ellpsod wll be plotted assumng that all other varables at held at ther mean levels. Care should be taken n nterpretng the plot f p > 3, snce the true control regon for each pont depends on the value of the varables that are not plotted by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 11

12 Save Results The followng results can be saved to the datasheet: 1. T-Squared the calculated T 2 values for each observaton or subgroup. 2. Means the p varable means. 3. Covarances the p 2 varances and covarances n rowwse order. 4. Labels the labels correspondng to each value of T Generalzed Varance - f calculated, the values of S. Calculatons T-Squared Control Lmt f Covarances are Known UCL X (8) 2, p T-Squared Control Lmt f Covarances are Estmated from k Prevous Samples UCL p( k 1)( k 1) F (9) k( k p), p, k p T-Squared Control Lmt f Covarances are Estmated from Current Data 2 ( n 1) UCL Beta, p / 2,( n p1) / 2 (10) n Generalzed Varance Control Lmts where UCL b 1 3 b 2 (11) CL b 1 (12) LCL b 1 3 b 2 (13) b 1 p ( n 1 p ( n 1) ) 1 p p p 1 b2 ( n ) ( n j 2) ( n j) 2 p ( n 1) 1 j1 j1 (14) (15) If s not known, t s replaced by the estmate S /b by StatPont Technologes, Inc. Multvarate EWMA Control Chart - 12

Control Charts for Means (Simulation)

Control Charts for Means (Simulation) Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Analysis of Covariance

Analysis of Covariance Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Imperial College London

Imperial College London F. Fang 1, C.C. Pan 1, I.M. Navon 2, M.D. Pggott 1, G.J. Gorman 1, P.A. Allson 1 and A.J.H. Goddard 1 1 Appled Modellng and Computaton Group Department of Earth Scence and Engneerng Imperal College London,

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

Faraday's Law of Induction

Faraday's Law of Induction Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy

More information

Multivariate Statistical Process Control Charts and the Problem of Interpretation: A Short Overview and Some Applications in Industry

Multivariate Statistical Process Control Charts and the Problem of Interpretation: A Short Overview and Some Applications in Industry Multvarate Statstcal Process Control Charts and the Problem of Interpretaton: A Short Overvew and Some Applcatons n Industry S. Bersms 1 J. Panaretos and S. Psaraks Abstract- Woodall and Montgomery [35]

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15

The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15 The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6 PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

Nonlinear data mapping by neural networks

Nonlinear data mapping by neural networks Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Control Charts with Supplementary Runs Rules for Monitoring Bivariate Processes

Control Charts with Supplementary Runs Rules for Monitoring Bivariate Processes Control Charts wth Supplementary Runs Rules for Montorng varate Processes Marcela. G. Machado *, ntono F.. Costa * * Producton Department, Sao Paulo State Unversty, Campus of Guaratnguetá, 56-4 Guaratnguetá,

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

An Interest-Oriented Network Evolution Mechanism for Online Communities

An Interest-Oriented Network Evolution Mechanism for Online Communities An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

Meta-Analysis of Hazard Ratios

Meta-Analysis of Hazard Ratios NCSS Statstcal Softare Chapter 458 Meta-Analyss of Hazard Ratos Introducton Ths module performs a meta-analyss on a set of to-group, tme to event (survval), studes n hch some data may be censored. These

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

MULTIPLE LINEAR REGRESSION IN MINITAB

MULTIPLE LINEAR REGRESSION IN MINITAB MULTIPLE LINEAR REGRESSION IN MINITAB Ths document shows a complcated Mntab multple regresson. It ncludes descrptons of the Mntab commands, and the Mntab output s heavly annotated. Comments n { } are used

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.

RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo. ICSV4 Carns Australa 9- July, 007 RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM

IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span alespe@sa.upv.es The dentfcaton and control of a

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1.

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1. HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher

More information

Quantization Effects in Digital Filters

Quantization Effects in Digital Filters Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

Fixed income risk attribution

Fixed income risk attribution 5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two

More information

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching) Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

Enterprise Master Patient Index

Enterprise Master Patient Index Enterprse Master Patent Index Healthcare data are captured n many dfferent settngs such as hosptals, clncs, labs, and physcan offces. Accordng to a report by the CDC, patents n the Unted States made an

More information

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika.

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika. VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths user-frendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

More information

A Multi-mode Image Tracking System Based on Distributed Fusion

A Multi-mode Image Tracking System Based on Distributed Fusion A Mult-mode Image Tracng System Based on Dstrbuted Fuson Ln zheng Chongzhao Han Dongguang Zuo Hongsen Yan School of Electroncs & nformaton engneerng, X an Jaotong Unversty X an, Shaanx, Chna Lnzheng@malst.xjtu.edu.cn

More information

Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Conversion between the vector and raster data structures using Fuzzy Geographical Entities Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

More information

Demographic and Health Surveys Methodology

Demographic and Health Surveys Methodology samplng and household lstng manual Demographc and Health Surveys Methodology Ths document s part of the Demographc and Health Survey s DHS Toolkt of methodology for the MEASURE DHS Phase III project, mplemented

More information

FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES

FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES Zuzanna BRO EK-MUCHA, Grzegorz ZADORA, 2 Insttute of Forensc Research, Cracow, Poland 2 Faculty of Chemstry, Jagellonan

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

1 De nitions and Censoring

1 De nitions and Censoring De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks Jont Resource Allocaton and Base-Staton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Section C2: BJT Structure and Operational Modes

Section C2: BJT Structure and Operational Modes Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Mean Molecular Weight

Mean Molecular Weight Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of

More information

Texas Instruments 30Xa Calculator

Texas Instruments 30Xa Calculator Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

More information

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

Measuring Ad Effectiveness Using Geo Experiments

Measuring Ad Effectiveness Using Geo Experiments Measurng Ad Effectveness Usng Geo Experments Jon Vaver, Jm Koehler Google Inc Abstract Advertsers have a fundamental need to quantfy the effectveness of ther advertsng For search ad spend, ths nformaton

More information

Damage detection in composite laminates using coin-tap method

Damage detection in composite laminates using coin-tap method Damage detecton n composte lamnates usng con-tap method S.J. Km Korea Aerospace Research Insttute, 45 Eoeun-Dong, Youseong-Gu, 35-333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The con-tap test has the

More information

RequIn, a tool for fast web traffic inference

RequIn, a tool for fast web traffic inference RequIn, a tool for fast web traffc nference Olver aul, Jean Etenne Kba GET/INT, LOR Department 9 rue Charles Fourer 90 Evry, France Olver.aul@nt-evry.fr, Jean-Etenne.Kba@nt-evry.fr Abstract As networked

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

The Current Employment Statistics (CES) survey,

The Current Employment Statistics (CES) survey, Busness Brths and Deaths Impact of busness brths and deaths n the payroll survey The CES probablty-based sample redesgn accounts for most busness brth employment through the mputaton of busness deaths,

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

HALL EFFECT SENSORS AND COMMUTATION

HALL EFFECT SENSORS AND COMMUTATION OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth three-phase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are: polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

More information

Instructions for Analyzing Data from CAHPS Surveys:

Instructions for Analyzing Data from CAHPS Surveys: Instructons for Analyzng Data from CAHPS Surveys: Usng the CAHPS Analyss Program Verson 4.1 Purpose of ths Document...1 The CAHPS Analyss Program...1 Computng Requrements...1 Pre-Analyss Decsons...2 What

More information

Solutions to the exam in SF2862, June 2009

Solutions to the exam in SF2862, June 2009 Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

Characterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University

Characterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Influence and Correlation in Social Networks

Influence and Correlation in Social Networks Influence and Correlaton n Socal Networks Ars Anagnostopoulos Rav Kumar Mohammad Mahdan Yahoo! Research 701 Frst Ave. Sunnyvale, CA 94089. {ars,ravkumar,mahdan}@yahoo-nc.com ABSTRACT In many onlne socal

More information

Regression Models for a Binary Response Using EXCEL and JMP

Regression Models for a Binary Response Using EXCEL and JMP SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

Daily O-D Matrix Estimation using Cellular Probe Data

Daily O-D Matrix Estimation using Cellular Probe Data Zhang, Qn, Dong and Ran Daly O-D Matrx Estmaton usng Cellular Probe Data 0 0 Y Zhang* Department of Cvl and Envronmental Engneerng, Unversty of Wsconsn-Madson, Madson, WI 0 Phone: -0-- E-mal: zhang@wsc.edu

More information

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation Exhaustve Regresson An Exploraton of Regresson-Based Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The

More information

Clustering Gene Expression Data. (Slides thanks to Dr. Mark Craven)

Clustering Gene Expression Data. (Slides thanks to Dr. Mark Craven) Clusterng Gene Epresson Data Sldes thanks to Dr. Mark Craven Gene Epresson Proles we ll assume we have a D matr o gene epresson measurements rows represent genes columns represent derent eperments tme

More information