# Signal Reconstruction from Noisy Random Projections

Save this PDF as:

Size: px
Start display at page:

## Transcription

5 5 Let f m) deote the best m-term aroximatio of f i the basis. That is, if f has a reresetatio f = i= θ iψ i i the basis {ψ i }, the f m) = m i= θ i)ψ i), where coefficiets ad basis fuctios are re-ordered such that θ ) θ ) θ ). Assume that the average squared error f f m) / i= f i f m) i ) satisfies for some α 0 ad some costat C A f f m) C A m α > 0. Power-law decays like this arise quite commoly i alicatios. For examle, smooth ad iecewise smooth sigals ad sigals of bouded variatio exhibit this sort of behavior [, [3. It is also uecessary to restrict oe s attetio orthoormal basis exasios. Much more geeral aroximatio strategies ca be accomodated [3, but to kee the resetatio as simle as ossible we will ot delve further ito such extesios. Now let us take F to be a suitably quatized collectio of fuctios, rereseted i terms of the basis {ψ i } the costructio of F is discussed i Sectio IV). We have the followig error boud. Theorem If cf) = log) {# o-zero coefficiets of f} the there exists a costat C = C B, σ, C A ) > 0 such that [ E f k f ) k α/α+) C C, log where C is as give i Theorem. Note that the exoet α/α + ) is the usual exoet goverig the rate of covergece i oarametric fuctio estimatio. A stroger result is obtaied if the sigal is sarse, as stated i the followig Corollary. Corollary Suose that f has at most m ozero coefficiets. The there exists a costat C = C B, σ) > 0 such that [ E f k f C C ) k, m log where C is as give i Theorem.

8 8 where k)!! )3)... k ). Thus we ca boud the variace of U j by ) varu j ) g g + 4σ. Sice g satisfies g 4B ad rf, f ) = f f / = g /, the boud becomes varu j ) 8B + 4σ ) rf, f ). So, we ca relace the term i the Craig-Berstei iequality that deeds o the variace by k var k U j = k varu j ) 8B + 4σ ) rf, f ). k k j= For a give fuctio f, we have P rf, f ) rf, f ) > t 8B + + 4σ ) ) ɛrf, f ) < e t ζ) or, by lettig δ = e t P j= rf, f ) rf, f ) > log δ ) 8B + 4σ ) ) ɛrf, f ) + < δ. ζ) Now assig to each f F a o-egative ealty term cf) such that the ealties satisfy the Kraft Iequality [5 cf) f F ad let δf) = cf) δ. The by alyig the uio boud we have for all f F ad for all δ > 0 rf, f ) rf, f ) cf) log + log ) δ 8B + 4σ ) ɛrf, f ) + ζ) with robability at least δ. Now set ζ = ɛh, defie 8B + 4σ ) ɛ a, ζ) ad choose ɛ < 0B + 8 Bσ + σ. Notice that a < ad ζ < by choice of ɛ. The a)rf, f ) rf, f ) + cf) log + log ) δ holds with robability at least δ for all f F ad ay δ > 0.

9 9 For the give traiig samles, we ca miimize the uer boud by choosig { } f k = arg mi rf, f cf) log ) + f F which is equivalet to { f k = arg mi Rf) + f F f k arg mi f F } cf) log sice we ca igore Rf ) whe erformig the otimizatio. If we defie { } Rf) + cf) log the with robability at least δ a)r f k, f ) r f k, f ) + c f k ) log + log ) δ rf k, f ) + cf k ) log + log ) δ sice f k miimizes the comlexity-regularized emirical risk criterio. Usig the Craig-Berstei iequality agai to boud rf k, f ) rf k, f ) with the same variace boud as before) we get that with robability at least δ rf k, f ) rfk, f ) a rfk, f ) + log ) δ. 3) We wat both ) ad 3) to hold simultaeously, so we use the uio boud to obtai ) r f + a k, f ) rfk a, f ) + cf k ) log + log )) δ a holdig with robability at least δ. Let δ = e t a)/ to obtai P r f k, f ) + a a Itegratig this relatio gives [ E r f k, f ) ) rfk, f ) cf k ) log ) a) t e t a)/. ) + a a rfk, f ) + cf k ) log + 4. a) )

10 0 Now, sice a is ositive, [ E f k f [ = E r f k, f ) where C = + a)/ a). = + a a + a a + a a ) + a mi a f F { f f + = C mi f F ) rfk, f ) + cf k ) log + 4 a) ) rfk, f ) + + a) cf k ) log + 4 a) ) { Rfk ) Rf ) + cf k ) log + 4 { Rf) Rf ) + Tyical values of C ca be determied by aroximatig the costat ɛ < 0B + 8 Bσ + σ. cf) log + 4 } } cf) log + 4 }, Uer boudig the deomiator guaratees that the coditio is satisfied, so let ɛ = / B + σ) ). Now 8B + 4σ ) ɛ a = = ζ) If we deote the sigal to oise ratio by S = B /σ the 4B + σ 5B )Bσ + σ. 4S + a = 5S )S + for which the extremes are a mi = / ad a max = 4/5, givig costats C i the rage of [3/9, 9. IV. ERROR BOUNDS FOR COMPRESSIBLE SIGNALS I this sectio we rove Theorem ad Corollary. Suose that f is comressible i a certai orthoormal basis {ψ i } i=. Secifically, let f m) deote the best m-term aroximatio of f i terms of {ψ i }, ad assume that the error of the aroximatio obeys for some α 0 ad a costat C A > 0. f f m) C A m α Let us use the basis {ψ i } for the recostructio rocess. Ay vector f F ca be exressed i terms of the basis {ψ i } as f = i= θ iψ i, where θ = {θ i } are the coefficiets of f i this basis. Let T deote the trasform that mas coefficiets to fuctios, so that f = T θ ad defie Θ = {θ : T θ B, θ i quatized to levels} to be the set of cadidate solutios i the basis {ψ i }. The ealty term

11 cf) writte i terms of the basis {ψ i } is cf) = cθ) = + ) log) i= I θ i 0 = + ) log θ 0. It is easily verified that f F cf) by otig that each θ Θ ca be uiquely ecoded via a refix code cosistig of + ) log bits er o-zero coefficiet log bits for the locatios ad log bits for the quatized values) i which case the codelegths cf) must satisfy the Kraft iequality [6. The oracle iequality [ E f k f { f f C mi + f F } cf) log + 4 ca be writte i terms of the ew class of cadidate recostructios as [ E f k f { θ θ } cθ) log + 4 C mi + θ Θ where f = T θ. For each iteger m, let θ m) deote the coefficiets corresodig to the best m-term aroximatio of f ad let θ m) q deote the earest elemet i Θ. The maximum ossible dyamic rage for the coefficiet magitudes, ± B, is quatized to levels, givig θ q m) θ m) 4B / = C Q /. Thus, θ m) q θ = θ m) q θ m) + θ m) θ θ m) q θ m) + θ m) q θ m) θ m) θ + θ m) θ C Q CA C Q + m α + C A m α. Now isert θ q m) i lace of θ i the oracle boud, ad ote that cθ q m) ) = + )m log to obtai [ E f { } k f C Q m α CA C Q C mi + θ Θ / + C A m α + )m log log Choosig large eough makes the first two terms egligible. Balacig the third ad fourth terms gives ) ) + ) log α+ k α+ m = ɛc A log so ad sice ) α ) α + ) log C A m α α+ k α+ = C A, ɛc A log k < log k ) α α+

12 whe k > ad > e, [ E f k f ) α k α+ C C, log as claimed i the Theorem, where C = { C A + ) log ɛc A } α ɛ ) α Suose ow that f has oly m ozero coefficiets. I this case, θ m) q θ C Q sice θ m) θ = 0. Now the ealty term domiates i the oracle boud ad [ E f k f ) k C C, m log where { } + ) log + 4 C =. ɛ V. OPTIMIZATION SCHEME Although our otimizatio is o-covex, it does ermit a simle, iterative otimizatio strategy that roduces a sequece of recostructios for which the corresodig sequece of comlexity-regularized emirical risk values is o-decreasig. This algorithm, which is described below, has demostrated itself to be quite effective i similar deoisig ad recostructio roblems [7 [9. A ossible alterative strategy might etail covexifyig the roblem by relacig the l 0 ealty with a l ealty. Recet results show that ofte the solutio to this covex roblem coicides with or aroximates the solutio to the origial o-covex roblem [0. Let us assume that we wish to recostruct our sigal i terms of the basis {ψ i }. Usig the defiitios itroduced i the revious sectio, the recostructio is equivalet to f k = T θ k where f k = arg mi f F θ k = arg mi θ Θ { Rf) + { RT θ) + Thus, the otimizatio roblem ca the be writte as { y P T θ + θ k = arg mi θ Θ } cf) log } cθ) log } log) log) θ 0 ɛ

13 3 where P = Φ T, the trasose of the k rojectio matrix Φ, y is a colum vector of the k observatios, ad θ 0 = i= I {θ i 0}. To solve this, we use a iterative boud-otimizatio rocedure, as roosed i [7 [9. This rocedure etails a two-ste iterative rocess that begis with a iitializatio θ 0) ad comutes:. ϕ t) = θ t) + λ P T )T y P T θ t) ). θt+) i = ϕ t) i if ϕ t) log) log) i λɛ 0 otherwise where λ is the largest eigevalue of P P. This rocedure is desirable sice the secod ste, i which the comlexity term lays its role, ivolves a simle coordiate-wise thresholdig oeratio. It is easy to verify that the iteratios roduce a mootoically o-icreasig sequece of comlexity-regularized emirical risk values [9. Thus, this rocedure rovides a simle iteratio that teds to miimize the origial objective fuctio, ad aears to give good results i ractice [7. The iteratios ca be termiated whe the etries uiformly satisfy θ t+) i θ t) i δ, for a small ositive tolerace δ. The comutatioal comlexity of the above rocedure is quite aealig. Each iteratio requires oly Ok) oeratios, assumig that the trasform T ca be comuted i O) oeratios. For examle, the discrete wavelet of Fourier trasforms ca be comuted i O) ad O log ) oeratios, resectively. Multilicatio by P is the most itesive oeratio, requirig Ok) oeratios. The thresholdig ste is carried out ideedetly i each coordiate, ad this ste requires O) oeratios as well. Of course, the umber of iteratios required is roblem deedet ad difficult to redict, but i our exeriece i this alicatio ad others [7, [9 algorithms of this sort ted to coverge i a reasoably small umber of iteratios, eve i very high dimesioal cases. Oe oit worthy of metio relates to the factor /ɛ = B +σ) i the ealty. As is ofte the case with coservative bouds of this tye, the theoretical ealty is larger tha what is eeded i ractice to achieve good results. Also, a otetial hurdle to calibratig the algorithm is that it deeds o kowledge of B ad σ, either of which may be kow a riori. Strictly seakig, these values do ot eed to be kow ideedetly but rather we eed oly estimate B +σ). To that ed, otice that each observatio is a radom variable with variace equal to f / + σ. Let B = f /, which is the miimum B satisfyig the stated boud f B. The the variace of each observatio is B + σ. Further, it is easy to verify that B +σ ) B +σ). So, a scheme could be develoed whereby the samle variace is used as a surrogate for the ukow quatities i the form i which they aear i the arameter ɛ.

18 8 VIII. ACKNOWLEDGEMENTS The authors would like to thak Rui Castro for his assistace i all stages of the work reseted here. A. The Craig-Berstei Momet Coditio APPENDIX The cetral requiremet of the Craig-Berstei iequality is the satisfactio of the momet coditio E [ X E[X!varX)h for itegers with some ositive costat h that does ot deed o, a coditio that is automatically verified for = with ay fiite value of h. For higher owers this coditio ca be very difficult to verify for several reasos, ot the least of which is the absolute value reset i the momets. Previous work that made use of the Craig-Berstei iequality assumed that the observatios were bouded, forcig a ossibly urealistic case of bouded oise [3. This assumtio is ot sufficiet for the rates of covergece we claim. Ideed, the aïve boud o the observatios is y i B yieldig a costat h that would grow roortioally to. With that motivatio, we develo a framework uder which a boudig costat h ca be determied more directly. First, observe that the momet coditio is usually easier to verify for the eve owers because the absolute value eed ot be dealt with directly. This is sufficiet to guaratee the momet coditio is satisfied for all itegers, as roved i the followig lemma. Lemma Suose the Craig-Berstei momet coditio holds for all eve itegers greater tha or equal to, that is E [ X E[X k k)!varx)hk, k sice the k = case is satisfied trivially for ay h. The the coditio holds also for the odd absolute momets with h = h. Thus A ad B, E [ X E[X k k )!varx) h k 3, k E [ X E[X!varX)h),. Proof: For ease of otatio, let Z = X E[X. Hölder s Iequality states, for ay radom variables E[ AB E[ A / E[ B q /q

19 9 where <, q < ad / + /q =. Take A = Z, B = Z k, ad = q = to get E[ Z k E[Z E[Z 4k 4 where the absolute values iside the square root have bee droed because the exoets are eve. Now by assumtio so E[Z 4k 4 4k 4)!E[Z h 4k 6 E[ Z k We wat to satisfy the followig iequality by choice of h which meas h must satisfy 4k 4)!E[Z ) h 4k 6 4k 4)! E[Z h k 3 E[ Z k k )!E[Z h k 3 If we choose k )! 4k 4)! h k 3 h k 3. h max k 4k 4)! k )! ) k 3 h the the momet coditio will be satisfied for the odd exoets k. A uer boud for the term i brackets is, as show here. For k, the boud k)! k k!) holds ad ca be verified by iductio o k. This imlies ) 4k 4)! k 3 ) k 3. 4) k )! k Now, the term i aretheses o the right had side of 4) is always less tha for k. The fial ste is to show that lim k k ) k 3 =, which is verified by otig that lim log k k ) k 3 = lim log ) ) log k ) = 0. k k 3

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### 3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2

MECH 5 Egieerig Sciece 3 Eergy 3.3. No-Flow Eergy Equatio (NFEE) You may have oticed that the term system kees croig u. It is ecessary, therefore, that before we start ay aalysis we defie the system that

### MATH 140A - HW 5 SOLUTIONS

MATH 40A - HW 5 SOLUTIONS Problem WR Ch 3 #8. If a coverges, ad if {b } is mootoic ad bouded, rove that a b coverges. Solutio. Theorem 3.4 states that if a the artial sums of a form a bouded sequece; b

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### 1 n. n > dt. t < n 1 + n=1

Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

### Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms.

Lectures # 7: The Class Number Formula For Positive efiite Biary uadratic Forms. Noah Syder July 17, 00 1 efiitios efiitio 1.1. A biary quadratic form (BF) is a fuctio (x, y) = ax +bxy+cy (with a, b, c

### Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

Divergece of / Adria Dudek adria.dudek[at]au.edu.au Whe I was i high school, my maths teacher cheekily told me that it s ossible to add u ifiitely may umbers ad get a fiite umber. She the illustrated this

### Section IV.5: Recurrence Relations from Algorithms

Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### 13 Fast Fourier Transform (FFT)

13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform.

### Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS

Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater

### RF Engineering Continuing Education Introduction to Traffic Planning

RF Egieerig otiuig Educatio Itroductio to Traffic Plaig Queuig Systems Figure. shows a schematic reresetatio of a queuig system. This reresetatio is a mathematical abstractio suitable for may differet

### 7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### Regression with a Binary Dependent Variable (SW Ch. 11)

Regressio with a Biary Deedet Variable (SW Ch. 11) So far the deedet variable (Y) has bee cotiuous: district-wide average test score traffic fatality rate But we might wat to uderstad the effect of X o

### 8.5 Alternating infinite series

65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

### 3. Covariance and Correlation

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

### Capacity of Wireless Networks with Heterogeneous Traffic

Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata

### B1. Fourier Analysis of Discrete Time Signals

B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### 1 Introduction to reducing variance in Monte Carlo simulations

Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

### Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

### Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

### Normal Distribution.

Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

### Modified Line Search Method for Global Optimization

Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

### 7) an = 7 n 7n. Solve the problem. Answer the question. n=1. Solve the problem. Answer the question. 16) an =

Eam Name MULTIPLE CHOICE. Choose the oe alterative that best comletes the statemet or aswers the questio. ) Use series to estimate the itegral's value to withi a error of magitude less tha -.. l( + )d..79.9.77

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### 3. Continuous Random Variables

Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay

### An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

### The second difference is the sequence of differences of the first difference sequence, 2

Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

### Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

### Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

### Lecture 7: Borel Sets and Lebesgue Measure

EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

### On the L p -conjecture for locally compact groups

Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### Confidence Intervals for One Mean with Tolerance Probability

Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

### Irreducible polynomials with consecutive zero coefficients

Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

### 5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

### 1 The Binomial Theorem: Another Approach

The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

### represented by 4! different arrangements of boxes, divide by 4! to get ways

Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,

### 1 The Gaussian channel

ECE 77 Lecture 0 The Gaussia chael Objective: I this lecture we will lear about commuicatio over a chael of practical iterest, i which the trasmitted sigal is subjected to additive white Gaussia oise.

### Systems Design Project: Indoor Location of Wireless Devices

Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

### Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### Lesson 12. Sequences and Series

Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### Section 7-3 Estimating a Population. Requirements

Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

### PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

### Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network

This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. Major Coefficiets Recovery: a Compressed Data Gatherig

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### Maximum Likelihood Estimators.

Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

### A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

### Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio

### Chapter Gaussian Elimination

Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio

### Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

### Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

### π d i (b i z) (n 1)π )... sin(θ + )

SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive

### Sequences II. Chapter 3. 3.1 Convergent Sequences

Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

O the Throughut Scalig of Cogitive Radio Ad Hoc Networks Chegzhi Li ad Huaiyu Dai Deartmet of Electrical ad Comuter Egieerig North Carolia State Uiversity, Raleigh, NC email: {cli3, hdai}@csu.edu Abstract

### Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

### Universal coding for classes of sources

Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)

BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet

### 4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### Measurable Functions

Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

### Nuclear and Particle Physics - Lecture 18 The nuclear force

1 Itroductio Nuclear ad Particle Physics - Lecture 18 The uclear force Protos ad eutros ca actually bid together through the strog force strogly eough to form boud states. These ca alteratively be thought

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### A Gentle Introduction to Algorithms: Part II

A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic bouds

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

### Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

### Differential Quartet, A Novel Circuit Building Block for High Slew Rate Differential Amplification

4th WSEAS Iteratioal Coferece o ELECTRONICS, CONTROL ad SIGNAL PROCESSING, Miami, Florida, USA, 7-9 November, 5 (.6-66 Differetial Quartet, A Novel Circuit Buildig Block for High Slew Rate Differetial

### Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

### Research Article Sign Data Derivative Recovery

Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

### Estimating Probability Distributions by Observing Betting Practices

5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,

### ARITHMETIC AND GEOMETRIC PROGRESSIONS

Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

### Hypothesis Tests Applied to Means

The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

### The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

### Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:

Math 355 - Discrete Math 4.1-4.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let

### Methods of Evaluating Estimators

Math 541: Statistical Theory II Istructor: Sogfeg Zheg Methods of Evaluatig Estimators Let X 1, X 2,, X be i.i.d. radom variables, i.e., a radom sample from f(x θ), where θ is ukow. A estimator of θ is