Particle Size Distributions from SANS Data Using the Maximum Entropy Method. By J. A. POTTON, G. J. DANIELL AND B. D. RAINFORD
|
|
- Noel Evans
- 5 years ago
- Views:
Transcription
1 3 J. Appl. Cryst. (1988). 21,3-8 Particle Size Distributins frm SANS Data Using the Maximum Entrpy Methd By J. A. PTTN, G. J. DANIELL AND B. D. RAINFRD Physics Department, The University, Suthamptn S9 5NH, England (Received 2 ctber 1987; accepted 2 April 1988) Abstract The extractin f particle size distributins frm small-angle neutrn scattering data is an example f a practical linear inverse prblem. Additinal assumptins are necessary t btain a unique slutin. The applicatin f the maximum entrpy methd t select a realistic size distributin is discussed. Principal features f the methd include a prper treatment f experimental errrs, n interplatin r smthing f data, n fitting t empirical mdels, and guaranteed psitivity f the slutin everywhere in spite f statistical nise in the data. The resulting slutin is the mst unifrm cnsistent with the data. Mdel data results are presented t shw that the maximum entrpy criterin prves very useful in prblems f this type. 1. Intrductin Small-angle scattering f either X-rays r neutrns is ne f the mst direct ways f extracting particle size infrmatin. The scattering intensities are given by It= ~ St(r)p(r ) dr, j = 1,..., m (1) where St(r) is the scattering functin fr a single particle f linear dimensin r, calculated fr the jth scattering vectr qt, and p(r) is the unknwn prbability distributin sught. There are M bservatins. We assume that there are n interparticle interference effects; hwever, any knwn effects may be included in St(r). The determinatin f p(r) frm the measured scattering pattern is an example f a linear inverse prblem. In practice it is impssible t extract a unique slutin since nt nly are the bservatins subject t experimental errr but, mre imprtantly, nly a finite number M f bservatins ver a limited range f scattering vectrs is available. Bth f these limitatins restrict the infrmatin cntent in the data. An imprvement t existing analyses t extract a feasible p(r) slutin will include n interplatin r smthing f data, explicit use f experimental errrs and n need fr a mdel functin, which wuld prejudice the slutin. Facilities t decnvlve instrumental reslutin and t crrect fr pssible systematic errrs intrduced via backgrund subtractin wuld be a benefit. The maximum entrpy methd (MEM) determines a unique slutin f the inverse prblem, which pssesses the useful prperty f being everywhere psitive. The resulting histgram f p(r) has the maximum cnfiguratinal entrpy cmpatible with the available data, and is als the mst unifrm. This apprach has been widely used in gephysical prblems and image recnstructin (Parker, 1977) and mre recently in the analysis f particle sizes by the technique f magnetic granulmetry (Pttn, Daniell & Melville, 1984). 2. The maximum entrpy methd The prblem is t determine a set f numbers xi, i= 1,..., N, which are values f an intrinsically psitive physical quantity, given the bservatins d t, j = 1,..., M. The bservatins are measurements f anther quantity Yt, which is linearly related t xi by N Yt = ~ tixi, J= 1,..., M. (2) i=1 Since the number f unknwns is greater than the number f data pints, and the matrix ji may have n inverse, a unique slutin is impssible. Even where a slutin frmally exists it is frequently unstable and large amplificatin f the errrs in the data may ccur in cmputing xi, leading t physically impssible negative values. Additinal assumptins are necessary t btain sensible values fr xi, and the maximum entrpy criterin has prved a very useful ne. The entrpy f a set f numbers x1 is defined as N S = - ~ xi lg xi/b i (3) i=1 where b1 are a prescribed set f values which give a scale t the magnitude f xi. The MEM selects the set x1 fr which S is a maximum subject t the values yj being cnsistent with the available data dj. It is dangerus t cnsider slutins that fit the data exactly. Such slutins will have features that are /88/ Internatinal Unin f Crystallgraphy
2 4 CNFERENCE PRCEEDINGS necessary nly t accmmdate bservatinal errrs. A measure f the misfit between experiment (d j) and Yi predicted by (2) is given by the X z statistic M )~2= ~ (dj- yj)zfiry (4) j=l where ej is the standard deviatin f the measurement dj. Each pint cntributes unity n average, and thus the mean f the distributin is M, the number f data pints. The MEM selects the set f xi fr which the cnfiguratinal entrpy S is a maximum, subject t the cnstraint that X2= M. Details f the search prcedure are given by Skilling & Bryan (1984). The matrix j~ specifies the particular prblem under cnsideratin. Fr the small-angle prblem it takes the frm f a pretabulated matrix f scattering functins Sj(r) calculated fr cmbinatins f linear dimensin r~ and scattering vectr qj. bviusly we must define the scattering functin fr a particular shape. If the particles can be assumed t be spherical then the spherical particle scattering functin is applicable, S(q, r) = 12rc(p - p3 2 (sin qr- qr cs qr)z/q (5) where p and Ps are the scattering-length densities f the particles and the substrate respectively. If the errrs f all the bservatins were infinite, the data wuld in n way cnstrain the slutin. The uncnstrained maximum entrpy slutin is then simply x~ = bile = Bi where e is the number As the errr bars are reduced frm infinity t their true values, a pint in the maximum entrpy map x~ wuld deviate frm its default value B~ prvided there is infrmatin in the data abut this particular x~. Departures frm the default are necessary in rder t fit the experimental data and cannt be artefacts intrduced by experimental errrs r bad data prcessing. We may chse the default B~ t reflect knwn physical reasns why the slutin must take particular values in particular ranges f i, even thugh the data may cntain little infrmatin t supprt this. The physical limits f p(r) are p(r)~ as r~ p(r)~ asr~. () Since we are t determine a prbability distributin p(r), each particle size is equally likely in the absence f data. We culd therefre chse Bi = 1/N as the starting pint fr the iterative prcedure. Hwever, we knw that p() = and p(rmax) ~ S in practice the default level is chsen t be as small as pssible (greater than zer) withut hindering cnvergence. Cnvergence can be slw fr very small defaults since the initial value f )~z is enrmus. Since the MEM appraches the frward prblem Pi--* I~ by generating trial distributins which are then cmpared with bservatin, it is straightfrward t take int accunt the effects f instrumental gemetry and reslutin where these are imprtant. These refinements require nly mdificatin f the matrix ~i. In sme cases we must deal with data which include systematic errrs. Fr small-angle neutrn scattering measurements the magnitude f the backgrund is uncertain. We may minimize )~2 and iterate t btain a gd estimate f the backgrund crrectin. In the next sectin we present mdel data results t shw the applicability f the MEM t small-angle scattering data. 3. Investigatin f mdel data Small-angle scattering was simulated fr nninteracting spherical particles with varius distributins f particle sizes. A mdel experiment cnsisted f 5 intensity measurements, in equal steps in full scattering angle frm 1 t a maximum f 1. This is equivalent t a minimum scattering vectr f.183 and a maximum f "1825 A-1 if we assume a neutrn wavelength typically f A [the standard definitin fr q is 4n (sin /2)/2 where is the full scattering angle]. The minimum value was chsen t reflect beamstp practicalities. Statistical errrs were included. Randm errrs with a Gaussian distributin f zer mean and chsen standard deviatin were added t the scattering data. The maximum entrpy methd was used t analyse the resulting data. A default f.1 was used thrughut. We present a selectin f the results t illustrate the behaviur f resultant slutins. All the fllwing figures depict particle size distributins. The input distributin is cntinuus and the maximum entrpy slutin is shwn as discrete. The size f the errrs f the crrespnding scattering data is indicated at the bttm right f each figure. Initially we investigated unimdal size distributins f similar Gaussian shape but varying in peak psitin up t a maximum f 1.~. The errrs in the scattering data were fixed at 1%. All peaks in the distributins were reprduced well, bth in respect f psitin in radius and height. Figs. l and shw the results fr small particles (1 A peak) and large particles (7 A peak) respectively. Next we examined the results as a functin f the size f the errrs in the simulated data. A Gaussian particle size distributin with peak at 2 A radius was chsen. Statistical errrs in the data varied frm as little as.2% up t a maximum f 1%. Figs. 2 and shw the results fr 1% and 5% errrs respectively. Fr errrs better than 2%, the distributin was reprduced almst perfectly. As the errrs are increased we bserve a gradual reductin in the peak height, althugh its psitin in radius is crrect.
3 .. CNFERENCE PRCEEDINGS 5 Fr 5% errrs the discrepancy between input and slutin is 13%. In reality we wuld hpe t measure intensities t better than 5%, s this effect is acceptable. Indeed it is reassuring that we can reprduce the distributin s well fr data with errrs as large as 5 r even 1%. The width f the Gaussian distributin functin was fund nt t affect the resultant slutin. A wide distributin was reprduced as well as a delta-like functin. Fig. 3 shws the result fr a distributin with a half-peak width f -,~ 5 A, twice that f Fig. 2. A serius questin arises when errr size is being cnsidered. The MEM relies n the experimenter knwing the standard deviatin f his data. Althugh this is nt an unreasnable request, ften errr estimatin is inaccurate. The maximum entrpy algrithm selects that slutin with maximum cnfiguratinal entrpy which cnfrms t the cnstraint that ~2- M, the number f data pints. This ptimum value f Z 2 relies n the errrs being knwn de EL precisely. We addressed the prblem f ver- and underestimated errrs. Again a Gaussian particle size distributin with a peak at 2 A radius was chsen. Simulated scattering data were generated with 1% errrs included. The data were subsequently analysed fr varius assumed errr bars using the MEM. bservatinal errrs were verestimated by as much as a factr f 1 and underestimated by as much as a factr f 1. Fig. 2 shws that the distributin is reprduced well fr crrectly predicted errrs (1%). Fr verestimated errrs, we bserve an increasing discrepancy in the peak height as the discrepancy in the errrs increases. c. ~ --- "J n I 1 I I i 1 2 RADIUS (/~) 17= I.L I I I 1 I I I I RADIUS (~,) [3_. ry EL_ r, ~ (D 4 A A A A A A A 7= I I --I -- --I ~ I J RAD I US (h) Fig. 1. This and all the fllwing figures depict particle size distributins. The input distributin is cntinuus and the maximum entrpy slutin is shwn as discrete. The size f the errrs in the crrespnding scattering data is indicated at the bttm right f each figure. Here is shwn the agreement between input and maximum entrpy slutin f unimdal distributins with peaks at 1 A and 7 A. The errrs in the scattering data were 1%. ~. ~.5% A AAAAAAAA A AAAII, IA.A t i ~ I J T I I RADIUS Fig. 2. Agreement between input and maximum entrpy slutin f a unimdal distributin with a peak at 2 A fr errrs in the scattering data f 1% and 5%. cy v f7 AAAA A ~..r~a ~ ~ k A A A A A A A A AA A I I -- I I I I RADIUS (~,) Fig. 3. Agreement between input and maximum entrpy slutin fr a brad distributin (t be cmpared with Fig. 2 which has a half-peak width a factr f tw smaller).
4 CNFERENCE PRCEEDINGS The psitin f the peak in radius remains crrect. verestimated errrs result in a maximum entrpy slutin with an increased value f final Z 2. We have necessarily lst infrmatin and wuld expect a brader distributin than reality t result. ur bservatins agree with cmmn sense. Fig. 4 shws the slutin btained fr errr bars f 5%, a factr f five larger than reality. We bserve a small reductin in the peak height. The questin f underestimated errrs can be mre serius. Fr significant underestimatin f errrs, cnvergence f 2 t M is nt attained. Instead )(2 cnverges t a value greater than M. The resultant slutin will have maximum cnfiguratinal entrpy subject t )~2 being as small as pssible (but greater than M). It is pssible that the algrithm can reduce )~2 by interpreting the randm nise in the data. We may therefre bserve small instabilities in the slutin. Fig. 4 shws the slutin attained fr errr bars f.2%, a factr f five smaller than reality. We still bserve very gd agreement between input and slutin. Hwever, we bserve an indicatin f a discrepancy at ~. The value f )~2 has been reduced by including this feature. t 13. ~-'~ 13_,=; 1 2 RADI US (#) d p,-)-- AAakkAkAA~kAA 5% 1% I I % The MEM des nt fail fr inaccurate errrs. verestimated errrs lead t underfitting f the data and a cnsequent lss f particle size infrmatin (brader distributin than reality). Underestimated errrs can lead t verfitting f the data and sme randm nise interpretatin. A mre stern test f the technique wuld be t attempt t reprduce bimdal distributins f particle sizes. Such distributins were investigated bth as a functin f peak psitin and errrs in the scattering data. Fig. 5 shws the result fr peaks at 1 and 2/k, and errrs f 1% in the intensities. We bserve a slight reductin in the height f the 1 A peak, thugh bth maxima are in the crrect psitins in radius. Fig. 5 shws hw the slutin may be imprved by reducing the errrs in the data t.2%. Fig. 5(c) illustrates the results fr peaks at significantly larger radii, at and 7 A, and errrs f 1% in the simulated data. We bserve gd agreement between input and slutin fr bth peaks. The agreement is marginally better than Fig. 5. It appears t be easier t reprduce bimdal distributins at higher radii. Indeed, Fig. 5(d) shws that data with as much as 5% errrs can result in reasnable peak reprductin. Finally, scattering data representing a trimdal particle size distributin were investigated. The initial distributin cnsisted f peaks at 1, 2 and 3/k, with errrs f 1% included in the crrespnding mdel data. Fig. illustrates the results. We bserve best agreement fr the "3 A peak, reasnable agreement fr the 2 A peak and sme agreement fr the 1 A peak. The maximum f each peak appears at the crrect radius value. Again, verall agreement can be imprved by reducing the size f the errrs in the bservatins. Fig. shws the result fr errrs f.2%. Reprductin is very gd. Fig. (c) shws the result at crrespndingly larger radii. Peaks are psitined at 5, and 7 A. Fr 1% data all three peaks are reprduced well, bth in respect f psitin in radius and height. Data with errrs as large as 5% can prduce three reasnable peaks, as illustrated in Fig. (d). t t"t". "--s 13_ AAAAAAA& AAA AA <~ I i i I 1 2 RADIUS ' k.... 1% ~AAAAAAAA I I I 3 4 Fig. 4. Agreement between input and maximum entrpy slutin fr ver- and underestimated errrs f 5% and.2%. The true errrs in the scattering data are 1%. Fig. 2 shws the result fr crrect errr estimatin. Cncluding remarks We have shwn hw the apprach f maximizing cnfiguratinal entrpy t chse a unique slutin f a practical linear inverse prblem may be applied t small-angle neutrn scattering. It must be stressed that the MEM is n substitute fr gd data; it simply makes the best f available data. The algrithm successively generates trial slutins fr the prbability functin p(r) and cmpares these with bservatin using X 2 statistics. Cnvergence is attained when Z 2 is equal t the number f data pints and the cnfiguratinal entrpy is a maximum. The
5 CNFERENCE PRCEEDINGS 7 ~d iv-)-. t CE "--' _ k_ (3._ t g t LI ~' 1% I I I I 1 I I RADIUS 2Z AAAA A A A A A A A A A A A A A ~ I I 1 I I 1 I I RADI US A ry CL CL It:: "-~(9 CL t {D J IV')" c I I I I I I RADI US..2Z AA~AA A A AAAA I I i I T I I I RADI US 4 k AAAAAAAAAAA,~ A A ~ ~ ~AAA A I I I I ~ I I RADIUS (c) 4 I 1 I I 3" I I RADIUS (/~) (c) cy 13_ g.sz AA AAAAAAAA &A I k" I ~AAA I I --I I RADIUS (/~) (c/) Fig. 5. Agreement between input and maximum entrpy slutin fr large- and small-radius bimdal distributins fr varying errrs in the scattering data. Peaks at 1 and 2/~ with errrs in the data f 1% then "2%. Peaks at and 7/~, with errrs f(c) 1% in the data and then (d) 5%. ry CL AA*AA A * i I 1 m I T 4 Z 5 7 RADI US I 8 Fig... Agreement between input and maximum entrpy slutin fr large- and small-radius trimdal distributins fr varying errrs in the intensity data. Peaks at 1, 2 and 3,~ with errrs in the data f 1% and "2%. Peaks at 5, and 7,~ with errrs in the data f(c) 1% and (d) 5%.
6 8 CNFERENCE PRCEEDINGS resultant p(r) will be the mst unifrm slutin cnsistent with the data, and has the attractive prperty! being everywhere psitive. The methd may be extended t include crrectins fr any reslutin r slit smearing since these effects enter as a cnvlutin f a reslutin functin with the calculated scattering results, which are then cmpared with bservatin. This is mre straightfrward than attempting t decnvlve the reslutin functin frm the data befre appraching the inverse prblem t extract a particle size distributin. The mdel results are very encuraging. Applicatin f the MEM t experimental small-angle scattering data has been carried ut successfully (Pttn, Daniell & Rainfrd, 198). Mre results tgether with a full descriptin f a cmplementary technique which enables errr bars t be assigned t each pint n a maximum entrpy slutin are published in this vlume (Pttn, Daniell & Rainfrd, 1988). References PARKER, R. L. (1977). Rev. Earth Planet. Sci. 5, PTTN, J. A., DANIELL, G. J. & MELVILLE, D. (1984). J. Phys. D, 17, PTTN, J. A., DANIELL, G. J. & RAINFRD, B. D. (198). Inst. Phys. Cnf. Set. 8, PTTN, J. A., DANIELL, G. J. & RAINFRD, B. D. (1988). J. Appl. Cryst. 21, SKILLING, J. BRYAN, R. K. (1984). Mn. Nt. R. Astrn. Sc. 211,
Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationApplication of ILIUM to the estimation of the T eff [Fe/H] pair from BP/RP
Applicatin f ILIUM t the estimatin f the T eff [Fe/H] pair frm BP/RP prepared by: apprved by: reference: issue: 1 revisin: 1 date: 2009-02-10 status: Issued Cryn A.L. Bailer-Jnes Max Planck Institute fr
More informationA New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation
III-l III. A New Evaluatin Measure J. Jiner and L. Werner Abstract The prblems f evaluatin and the needed criteria f evaluatin measures in the SMART system f infrmatin retrieval are reviewed and discussed.
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationChapter 3: Cluster Analysis
Chapter 3: Cluster Analysis } 3.1 Basic Cncepts f Clustering 3.1.1 Cluster Analysis 3.1. Clustering Categries } 3. Partitining Methds 3..1 The principle 3.. K-Means Methd 3..3 K-Medids Methd 3..4 CLARA
More informationResampling Methods. Chapter 5. Chapter 5 1 / 52
Resampling Methds Chapter 5 Chapter 5 1 / 52 1 51 Validatin set apprach 2 52 Crss validatin 3 53 Btstrap Chapter 5 2 / 52 Abut Resampling An imprtant statistical tl Pretending the data as ppulatin and
More informationSUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis
SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical mdel fr micrarray data analysis David Rssell Department f Bistatistics M.D. Andersn Cancer Center, Hustn, TX 77030, USA rsselldavid@gmail.cm
More informationKinetic Model Completeness
5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More informationDetermining the Accuracy of Modal Parameter Estimation Methods
Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system
More informationBiplots in Practice MICHAEL GREENACRE. Professor of Statistics at the Pompeu Fabra University. Chapter 13 Offprint
Biplts in Practice MICHAEL GREENACRE Prfessr f Statistics at the Pmpeu Fabra University Chapter 13 Offprint CASE STUDY BIOMEDICINE Cmparing Cancer Types Accrding t Gene Epressin Arrays First published:
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationAP Statistics Notes Unit Two: The Normal Distributions
AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres).
More informationInternal vs. external validity. External validity. This section is based on Stock and Watson s Chapter 9.
Sectin 7 Mdel Assessment This sectin is based n Stck and Watsn s Chapter 9. Internal vs. external validity Internal validity refers t whether the analysis is valid fr the ppulatin and sample being studied.
More information3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression
3.3.4 Prstate Cancer Data Example (Cntinued) 3.4 Shrinkage Methds 61 Table 3.3 shws the cefficients frm a number f different selectin and shrinkage methds. They are best-subset selectin using an all-subsets
More informationand the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are:
Algrithm fr Estimating R and R - (David Sandwell, SIO, August 4, 2006) Azimith cmpressin invlves the alignment f successive eches t be fcused n a pint target Let s be the slw time alng the satellite track
More informationLeast Squares Optimal Filtering with Multirate Observations
Prc. 36th Asilmar Cnf. n Signals, Systems, and Cmputers, Pacific Grve, CA, Nvember 2002 Least Squares Optimal Filtering with Multirate Observatins Charles W. herrien and Anthny H. Hawes Department f Electrical
More informationAdmin. MDP Search Trees. Optimal Quantities. Reinforcement Learning
Admin Reinfrcement Learning Cntent adapted frm Berkeley CS188 MDP Search Trees Each MDP state prjects an expectimax-like search tree Optimal Quantities The value (utility) f a state s: V*(s) = expected
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationWhat is Statistical Learning?
What is Statistical Learning? Sales 5 10 15 20 25 Sales 5 10 15 20 25 Sales 5 10 15 20 25 0 50 100 200 300 TV 0 10 20 30 40 50 Radi 0 20 40 60 80 100 Newspaper Shwn are Sales vs TV, Radi and Newspaper,
More informationName: Block: Date: Science 10: The Great Geyser Experiment A controlled experiment
Science 10: The Great Geyser Experiment A cntrlled experiment Yu will prduce a GEYSER by drpping Ments int a bttle f diet pp Sme questins t think abut are: What are yu ging t test? What are yu ging t measure?
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationYou need to be able to define the following terms and answer basic questions about them:
CS440/ECE448 Sectin Q Fall 2017 Midterm Review Yu need t be able t define the fllwing terms and answer basic questins abut them: Intr t AI, agents and envirnments Pssible definitins f AI, prs and cns f
More informationk-nearest Neighbor How to choose k Average of k points more reliable when: Large k: noise in attributes +o o noise in class labels
Mtivating Example Memry-Based Learning Instance-Based Learning K-earest eighbr Inductive Assumptin Similar inputs map t similar utputs If nt true => learning is impssible If true => learning reduces t
More informationthe results to larger systems due to prop'erties of the projection algorithm. First, the number of hidden nodes must
M.E. Aggune, M.J. Dambrg, M.A. El-Sharkawi, R.J. Marks II and L.E. Atlas, "Dynamic and static security assessment f pwer systems using artificial neural netwrks", Prceedings f the NSF Wrkshp n Applicatins
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More informationDistributions, spatial statistics and a Bayesian perspective
Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics
More informationCOMP 551 Applied Machine Learning Lecture 11: Support Vector Machines
COMP 551 Applied Machine Learning Lecture 11: Supprt Vectr Machines Instructr: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted fr this curse
More informationAPPLICATION OF THE BRATSETH SCHEME FOR HIGH LATITUDE INTERMITTENT DATA ASSIMILATION USING THE PSU/NCAR MM5 MESOSCALE MODEL
JP2.11 APPLICATION OF THE BRATSETH SCHEME FOR HIGH LATITUDE INTERMITTENT DATA ASSIMILATION USING THE PSU/NCAR MM5 MESOSCALE MODEL Xingang Fan * and Jeffrey S. Tilley University f Alaska Fairbanks, Fairbanks,
More informationComparison of two variable parameter Muskingum methods
Extreme Hydrlgical Events: Precipitatin, Flds and Drughts (Prceedings f the Ykhama Sympsium, July 1993). IAHS Publ. n. 213, 1993. 129 Cmparisn f tw variable parameter Muskingum methds M. PERUMAL Department
More informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More informationFive Whys How To Do It Better
Five Whys Definitin. As explained in the previus article, we define rt cause as simply the uncvering f hw the current prblem came int being. Fr a simple causal chain, it is the entire chain. Fr a cmplex
More informationSequential Allocation with Minimal Switching
In Cmputing Science and Statistics 28 (1996), pp. 567 572 Sequential Allcatin with Minimal Switching Quentin F. Stut 1 Janis Hardwick 1 EECS Dept., University f Michigan Statistics Dept., Purdue University
More informationHiding in plain sight
Hiding in plain sight Principles f stegangraphy CS349 Cryptgraphy Department f Cmputer Science Wellesley Cllege The prisners prblem Stegangraphy 1-2 1 Secret writing Lemn juice is very nearly clear s it
More informationLead/Lag Compensator Frequency Domain Properties and Design Methods
Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin
More informationModelling of Clock Behaviour. Don Percival. Applied Physics Laboratory University of Washington Seattle, Washington, USA
Mdelling f Clck Behaviur Dn Percival Applied Physics Labratry University f Washingtn Seattle, Washingtn, USA verheads and paper fr talk available at http://faculty.washingtn.edu/dbp/talks.html 1 Overview
More informationSection 6-2: Simplex Method: Maximization with Problem Constraints of the Form ~
Sectin 6-2: Simplex Methd: Maximizatin with Prblem Cnstraints f the Frm ~ Nte: This methd was develped by Gerge B. Dantzig in 1947 while n assignment t the U.S. Department f the Air Frce. Definitin: Standard
More information2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS
2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 6. An electrchemical cell is cnstructed with an pen switch, as shwn in the diagram abve. A strip f Sn and a strip f an unknwn metal, X, are used as electrdes.
More informationThe blessing of dimensionality for kernel methods
fr kernel methds Building classifiers in high dimensinal space Pierre Dupnt Pierre.Dupnt@ucluvain.be Classifiers define decisin surfaces in sme feature space where the data is either initially represented
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationElements of Machine Intelligence - I
ECE-175A Elements f Machine Intelligence - I Ken Kreutz-Delgad Nun Vascncels ECE Department, UCSD Winter 2011 The curse The curse will cver basic, but imprtant, aspects f machine learning and pattern recgnitin
More informationA study on GPS PDOP and its impact on position error
IndianJurnalfRadi& SpacePhysics V1.26,April1997,pp. 107-111 A study n GPS and its impact n psitin errr P Banerjee,AnindyaBse& B SMathur TimeandFrequencySectin,NatinalPhysicalLabratry,NewDelhi110012 Received19June
More informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationWRITING THE REPORT. Organizing the report. Title Page. Table of Contents
WRITING THE REPORT Organizing the reprt Mst reprts shuld be rganized in the fllwing manner. Smetime there is a valid reasn t include extra chapters in within the bdy f the reprt. 1. Title page 2. Executive
More informationFloating Point Method for Solving Transportation. Problems with Additional Constraints
Internatinal Mathematical Frum, Vl. 6, 20, n. 40, 983-992 Flating Pint Methd fr Slving Transprtatin Prblems with Additinal Cnstraints P. Pandian and D. Anuradha Department f Mathematics, Schl f Advanced
More informationCOMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification
COMP 551 Applied Machine Learning Lecture 5: Generative mdels fr linear classificatin Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Jelle Pineau Class web page: www.cs.mcgill.ca/~hvanh2/cmp551
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationChapter 3 Digital Transmission Fundamentals
Chapter 3 Digital Transmissin Fundamentals Errr Detectin and Crrectin CSE 3213, Winter 2010 Instructr: Frhar Frzan Mdul-2 Arithmetic Mdul 2 arithmetic is perfrmed digit y digit n inary numers. Each digit
More informationCHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS
CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS 1 Influential bservatins are bservatins whse presence in the data can have a distrting effect n the parameter estimates and pssibly the entire analysis,
More informationTree Structured Classifier
Tree Structured Classifier Reference: Classificatin and Regressin Trees by L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stne, Chapman & Hall, 98. A Medical Eample (CART): Predict high risk patients
More informationSlide04 (supplemental) Haykin Chapter 4 (both 2nd and 3rd ed): Multi-Layer Perceptrons
Slide04 supplemental) Haykin Chapter 4 bth 2nd and 3rd ed): Multi-Layer Perceptrns CPSC 636-600 Instructr: Ynsuck Che Heuristic fr Making Backprp Perfrm Better 1. Sequential vs. batch update: fr large
More informationTuring Machines. Human-aware Robotics. 2017/10/17 & 19 Chapter 3.2 & 3.3 in Sipser Ø Announcement:
Turing Machines Human-aware Rbtics 2017/10/17 & 19 Chapter 3.2 & 3.3 in Sipser Ø Annuncement: q q q q Slides fr this lecture are here: http://www.public.asu.edu/~yzhan442/teaching/cse355/lectures/tm-ii.pdf
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 1-3 weeks 6-10 Majr Learning Indicatrs Identify situatins relevant
More informationWriting Guidelines. (Updated: November 25, 2009) Forwards
Writing Guidelines (Updated: Nvember 25, 2009) Frwards I have fund in my review f the manuscripts frm ur students and research assciates, as well as thse submitted t varius jurnals by thers that the majr
More informationComparing Several Means: ANOVA. Group Means and Grand Mean
STAT 511 ANOVA and Regressin 1 Cmparing Several Means: ANOVA Slide 1 Blue Lake snap beans were grwn in 12 pen-tp chambers which are subject t 4 treatments 3 each with O 3 and SO 2 present/absent. The ttal
More informationSAMPLING DYNAMICAL SYSTEMS
SAMPLING DYNAMICAL SYSTEMS Melvin J. Hinich Applied Research Labratries The University f Texas at Austin Austin, TX 78713-8029, USA (512) 835-3278 (Vice) 835-3259 (Fax) hinich@mail.la.utexas.edu ABSTRACT
More informationHypothesis Tests for One Population Mean
Hypthesis Tests fr One Ppulatin Mean Chapter 9 Ala Abdelbaki Objective Objective: T estimate the value f ne ppulatin mean Inferential statistics using statistics in rder t estimate parameters We will be
More informationPSU GISPOPSCI June 2011 Ordinary Least Squares & Spatial Linear Regression in GeoDa
There are tw parts t this lab. The first is intended t demnstrate hw t request and interpret the spatial diagnstics f a standard OLS regressin mdel using GeDa. The diagnstics prvide infrmatin abut the
More informationPure adaptive search for finite global optimization*
Mathematical Prgramming 69 (1995) 443-448 Pure adaptive search fr finite glbal ptimizatin* Z.B. Zabinskya.*, G.R. Wd b, M.A. Steel c, W.P. Baritmpa c a Industrial Engineering Prgram, FU-20. University
More informationIntroduction: A Generalized approach for computing the trajectories associated with the Newtonian N Body Problem
A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering
More informationFigure 1a. A planar mechanism.
ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,
More informationo o IMPORTANT REMINDERS Reports will be graded largely on their ability to clearly communicate results and important conclusions.
BASD High Schl Frmal Lab Reprt GENERAL INFORMATION 12 pt Times New Rman fnt Duble-spaced, if required by yur teacher 1 inch margins n all sides (tp, bttm, left, and right) Always write in third persn (avid
More informationPerfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Key Wrds: Autregressive, Mving Average, Runs Tests, Shewhart Cntrl Chart
Perfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Sandy D. Balkin Dennis K. J. Lin y Pennsylvania State University, University Park, PA 16802 Sandy Balkin is a graduate student
More informationLecture 23: Lattice Models of Materials; Modeling Polymer Solutions
Lecture 23: 12.05.05 Lattice Mdels f Materials; Mdeling Plymer Slutins Tday: LAST TIME...2 The Bltzmann Factr and Partitin Functin: systems at cnstant temperature...2 A better mdel: The Debye slid...3
More informationEric Klein and Ning Sa
Week 12. Statistical Appraches t Netwrks: p1 and p* Wasserman and Faust Chapter 15: Statistical Analysis f Single Relatinal Netwrks There are fur tasks in psitinal analysis: 1) Define Equivalence 2) Measure
More informationChemistry 20 Lesson 11 Electronegativity, Polarity and Shapes
Chemistry 20 Lessn 11 Electrnegativity, Plarity and Shapes In ur previus wrk we learned why atms frm cvalent bnds and hw t draw the resulting rganizatin f atms. In this lessn we will learn (a) hw the cmbinatin
More informationSOLUTION OF THREE-CONSTRAINT ENTROPY-BASED VELOCITY DISTRIBUTION
SOLUTION OF THREECONSTRAINT ENTROPYBASED VELOCITY DISTRIBUTION By D. E. Barbe,' J. F. Cruise, 2 and V. P. Singh, 3 Members, ASCE ABSTRACT: A twdimensinal velcity prfile based upn the principle f maximum
More informationinitially lcated away frm the data set never win the cmpetitin, resulting in a nnptimal nal cdebk, [2] [3] [4] and [5]. Khnen's Self Organizing Featur
Cdewrd Distributin fr Frequency Sensitive Cmpetitive Learning with One Dimensinal Input Data Aristides S. Galanpuls and Stanley C. Ahalt Department f Electrical Engineering The Ohi State University Abstract
More informationInference in the Multiple-Regression
Sectin 5 Mdel Inference in the Multiple-Regressin Kinds f hypthesis tests in a multiple regressin There are several distinct kinds f hypthesis tests we can run in a multiple regressin. Suppse that amng
More informationMATCHING TECHNIQUES. Technical Track Session VI. Emanuela Galasso. The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Emanuela Galass The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Emanuela Galass fr the purpse f this wrkshp When can we use
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationBASD HIGH SCHOOL FORMAL LAB REPORT
BASD HIGH SCHOOL FORMAL LAB REPORT *WARNING: After an explanatin f what t include in each sectin, there is an example f hw the sectin might lk using a sample experiment Keep in mind, the sample lab used
More informationChapter 8: The Binomial and Geometric Distributions
Sectin 8.1: The Binmial Distributins Chapter 8: The Binmial and Gemetric Distributins A randm variable X is called a BINOMIAL RANDOM VARIABLE if it meets ALL the fllwing cnditins: 1) 2) 3) 4) The MOST
More informationLab 1 The Scientific Method
INTRODUCTION The fllwing labratry exercise is designed t give yu, the student, an pprtunity t explre unknwn systems, r universes, and hypthesize pssible rules which may gvern the behavir within them. Scientific
More informationMedium Scale Integrated (MSI) devices [Sections 2.9 and 2.10]
EECS 270, Winter 2017, Lecture 3 Page 1 f 6 Medium Scale Integrated (MSI) devices [Sectins 2.9 and 2.10] As we ve seen, it s smetimes nt reasnable t d all the design wrk at the gate-level smetimes we just
More informationPhysical Layer: Outline
18-: Intrductin t Telecmmunicatin Netwrks Lectures : Physical Layer Peter Steenkiste Spring 01 www.cs.cmu.edu/~prs/nets-ece Physical Layer: Outline Digital Representatin f Infrmatin Characterizatin f Cmmunicatin
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationMethods for Determination of Mean Speckle Size in Simulated Speckle Pattern
0.478/msr-04-004 MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 Methds fr Determinatin f Mean Speckle Size in Simulated Speckle Pattern. Hamarvá, P. Šmíd, P. Hrváth, M. Hrabvský nstitute f Physics f the Academy
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationMATCHING TECHNIQUES Technical Track Session VI Céline Ferré The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Céline Ferré The Wrld Bank When can we use matching? What if the assignment t the treatment is nt dne randmly r based n an eligibility index, but n the basis
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More information^YawataR&D Laboratory, Nippon Steel Corporation, Tobata, Kitakyushu, Japan
Detectin f fatigue crack initiatin frm a ntch under a randm lad C. Makabe," S. Nishida^C. Urashima,' H. Kaneshir* "Department f Mechanical Systems Engineering, University f the Ryukyus, Nishihara, kinawa,
More informationStatistical Learning. 2.1 What Is Statistical Learning?
2 Statistical Learning 2.1 What Is Statistical Learning? In rder t mtivate ur study f statistical learning, we begin with a simple example. Suppse that we are statistical cnsultants hired by a client t
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationCAUSAL INFERENCE. Technical Track Session I. Phillippe Leite. The World Bank
CAUSAL INFERENCE Technical Track Sessin I Phillippe Leite The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Phillippe Leite fr the purpse f this wrkshp Plicy questins are causal
More informationCells though to send feedback signals from the medulla back to the lamina o L: Lamina Monopolar cells
Classificatin Rules (and Exceptins) Name: Cell type fllwed by either a clumn ID (determined by the visual lcatin f the cell) r a numeric identifier t separate ut different examples f a given cell type
More informationVideo Encoder Control
Vide Encder Cntrl Thmas Wiegand Digital Image Cmmunicatin 1 / 41 Outline Intrductin Encder Cntrl using Lagrange multipliers Lagrangian ptimizatin Lagrangian bit allcatin Lagrangian Optimizatin in Hybrid
More informationA Simple Set of Test Matrices for Eigenvalue Programs*
Simple Set f Test Matrices fr Eigenvalue Prgrams* By C. W. Gear** bstract. Sets f simple matrices f rder N are given, tgether with all f their eigenvalues and right eigenvectrs, and simple rules fr generating
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationHow do scientists measure trees? What is DBH?
Hw d scientists measure trees? What is DBH? Purpse Students develp an understanding f tree size and hw scientists measure trees. Students bserve and measure tree ckies and explre the relatinship between
More informationT Algorithmic methods for data mining. Slide set 6: dimensionality reduction
T-61.5060 Algrithmic methds fr data mining Slide set 6: dimensinality reductin reading assignment LRU bk: 11.1 11.3 PCA tutrial in mycurses (ptinal) ptinal: An Elementary Prf f a Therem f Jhnsn and Lindenstrauss,
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More information