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Course: ASTR 10, Fall 2008
School: Maryland
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10. Class Statistics and the K-S Test Statistical Description of Data Cf. NRiC 14. Statistics provides tools for understanding data. In the wrong hands these tools can be dangerous! Heres a typical data analysis cycle: 1. Apply some formula to data to compute a statistic. 2. Find where that value falls in a probability distribution computed on the basis of some null hypothesis. 3. If it falls in an unlikely...

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10. Class Statistics and the K-S Test Statistical Description of Data Cf. NRiC 14. Statistics provides tools for understanding data. In the wrong hands these tools can be dangerous! Heres a typical data analysis cycle: 1. Apply some formula to data to compute a statistic. 2. Find where that value falls in a probability distribution computed on the basis of some null hypothesis. 3. If it falls in an unlikely spot (on distribution tail), conclude null hypothesis is false for your data set. Statistics Statistics and probability theory are closely related. Statistics can never prove things, only disprove them by ruling out hypotheses. Distinguish between model-independent statistics (this class, e.g., mean, median, mode) and model-dependent statistics (next class, e.g., least-squares tting). Will make use of special functions (e.g., gamma function) described in NRiC 6. Moments of a Distribution The mean, median, and mode of distributions are called measures of central tendency. The most common description of data involves its moments, sums of integer powers of the values. The most familiar moment is the mean: x = <x> = Variance The width of the central value is estimated by its second moment, called the variance, N 1 Var = (xi x)2 , N 1 i=1 1 N N xi . i=1 or its square root, the standard deviation, = Var. 1 Why N 1? If the mean is known a priori, i.e., if its not measured from the data, then use N, else N 1. If this matters to you, then N is probably too small! A clever way to minimize round-o error when computing the variance is to use the corrected two-pass algorithm. First compute x, then do: Var = 1 1 (xi x)2 N 1 i=1 N N N i=1 2 (xi x) . The second sum would be zero if x were exact, but otherwise it does a good job of correcting RE in Var. Proof: EFTS (hint: set x x + ). Other moments Higher moments, like skewness (3rd moment) and kurtosis (4th moment) are also sometimes used, but can be unreliable. Cf. NRiC 14.1. Distribution Functions A distribution function (DF) p(x) gives the probability of nding a value between x 2 and x + dx, e.g., the familiar normal (Gaussian) distribution p(x) dx = 1 ex /2 dx. 2 The expected mean data value is: <x> = For a discrete DF: <x> = i x p(x) dx . p(x) dx xi pi . i pi Similar to weighted means, e.g., center of mass. Median The median of a DF is the value xmed for which larger and smaller values of x are equally probable: xmed 1 p(x) dx = = p(x) dx. 2 xmed For discrete values, sort in ascending order (i = 1, 2, ..., N), then: xmed = x(N +1)/2 , if N is odd, 1 (xN/2 + xN/2+1 ), if N is even. 2 2 Mode The mode of a probability DF p(x) is the value of x where the DF takes on a maximum value. Most useful when there is a single, sharp max, in which case it estimates the central value. Sometimes a distribution will be bimodal, with two relative In maxima. this case the mean and median are not very useful since they give only a compromise value between the two peaks. Comparing Distributions Often want to know if two distributions have dierent means or variances (NRiC 14.2): 1. Students t-test for signicantly dierent means. (a) Find number of standard errors /N 1/2 between two means. (b) Compute statistic using nasty formula: probability that the two means are dierent by chance. (c) Small numerical value indicates signicant dierence. 2. F -test for signicantly dierent variances. (a) Compute F = Var1 /Var2 and plug into nasty formula (the distribution of F in the case that the variances are the samethe null hypothesisis related to the incomplete beta function). (b) Small value indicates signicant dierence. Given two sets of data, can generalize to a single question: Are the sets drawn from the same DF? E.g., are stars distributed uniformly in the sky? Do two brands of lightbulbs have the same distribution of burn-out times? Recall can only disprove (to a certain condence level), not prove. May have continuous or binned data. May want to compare one data set with known DF, or two unknown data sets with each other. Popular technique for binned data is the 2 test. For continuous data, use the KS test. Cf. NRiC 14.3. Chi-square (2 ) test Suppose have Ni events in ith bin but expect ni : 2 = i (Ni ni )2 . ni 3 Large value of 2 indicates unlikely match (i.e., Ni s probably not drawn from population represented by ni s). Compute probability Q(2 |) from incomplete gamma function, where is the number of degrees of freedom. Typically = NB , where NB is the number of bins, or NB 1, if the ni s are normalized such that i ni = i Ni . Null hypothes...

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Maryland - ASTR - 415
Class 10. Statistics and the K-S TestStatistical Description of Data Cf. NRiC 14. Statistics provides tools for understanding data. In the wrong hands these tools can be dangerous! Heres a typical data analysis cycle: 1. Apply some formula to da
Maryland - ASTR - 11
Class 11. Modeling of Data NRiC 15. Model depends on adjustable parameters. Can be used for constrained interpolation. Basic approach: 1. Choose gure-of-merit function (e.g., 2 ). 2. Adjust best-t parameters: minimize merit function. 3. Compute g
Maryland - ASTR - 415
Class 11. Modeling of Data NRiC 15. Model depends on adjustable parameters. Can be used for constrained interpolation. Basic approach: 1. Choose gure-of-merit function (e.g., 2 ). 2. Adjust best-t parameters: minimize merit function. 3. Compute g
Maryland - ASTR - 12
Class 12. Random Numbers NRiC 7. Frequently needed to generate initial conditions. Often used to solve problems statistically. How can a computer generate a random number? It cant! Generators are pseudo-random. Generators are deterministic: its
Maryland - ASTR - 415
Class 12. Random Numbers NRiC 7. Frequently needed to generate initial conditions. Often used to solve problems statistically. How can a computer generate a random number? It cant! Generators are pseudo-random. Generators are deterministic: its
Maryland - ASTR - 13
Class 13. Numerical IntegrationSimple Monte Carlo Integration (NRiC 7.6) Can use RNGs to estimate integrals. Suppose we pick n random points x1 , ., xN uniformly in a multi-D volume V . Basic theorem of Monte Carlo integration: f dV 1 NNVV &lt;
Maryland - ASTR - 415
Class 13. Numerical IntegrationSimple Monte Carlo Integration (NRiC 7.6) Can use RNGs to estimate integrals. Suppose we pick n random points x1 , ., xN uniformly in a multi-D volume V . Basic theorem of Monte Carlo integration: f dV 1 NNVV &lt;
Maryland - ASTR - 14
Class 14. Ordinary Dierential Equations NRiC 16. ODEs involve derivatives with respect to one independent variable, e.g., time t. ODEs can always be reduced to a set of rst-order equations (i.e., involving only rst derivatives). E.g., d2 y dy + b(
Maryland - ASTR - 415
Class 14. Ordinary Dierential Equations NRiC 16. ODEs involve derivatives with respect to one independent variable, e.g., time t. ODEs can always be reduced to a set of rst-order equations (i.e., involving only rst derivatives). E.g., d2 y dy + b(
Maryland - ASTR - 15
Class 15. ODEs, Part 2The Leapfrog Integrator Very useful for second-order DEs in which d2 x/dt2 = f(x), e.g., SHM, N-body, etc. NOTE: Now dropping the prime ( ) from f. Suppose x is position, so d2 x/dt2 is acceleration. Procedure: dene v = dx/
Maryland - ASTR - 415
Class 15. ODEs, Part 2The Leapfrog Integrator Very useful for second-order DEs in which d2 x/dt2 = f(x), e.g., SHM, N-body, etc. NOTE: Now dropping the prime ( ) from f. Suppose x is position, so d2 x/dt2 is acceleration. Procedure: dene v = dx/
Maryland - ASTR - 16
Class 16. ODEs, Part 3Sti ODEs A system of more than one ODE is sti if solutions vary on two or more widely disparate lengthscales. E.g., y = 100y. General solution: y = Ae10x + Be+10x . Suppose BCs are y(0) = 1, y (0) = 10. Then B = 0, i.e., pur
Maryland - ASTR - 415
Class 16. ODEs, Part 3Sti ODEs A system of more than one ODE is sti if solutions vary on two or more widely disparate lengthscales. E.g., y = 100y. General solution: y = Ae10x + Be+10x . Suppose BCs are y(0) = 1, y (0) = 10. Then B = 0, i.e., pur
Maryland - ASTR - 17
Class 17. ODEs, Part 42-pt BVP: Shooting MethodProcedure: 1. At x1 , must specify N starting values for yi , i = 1, ., N. n1 values given by BC at x1 . n2 = N n1 values can be freely chosen. 2. Represent the free values as a vector V of dimensi
Maryland - ASTR - 415
Class 17. ODEs, Part 42-pt BVP: Shooting MethodProcedure: 1. At x1 , must specify N starting values for yi , i = 1, ., N. n1 values given by BC at x1 . n2 = N n1 values can be freely chosen. 2. Represent the free values as a vector V of dimensi
Maryland - ASTR - 18
Class 18. N -body Techniques, Part 1The N -body Problem Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physi
Maryland - ASTR - 415
Class 18. N -body Techniques, Part 1The N -body Problem Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physi
Maryland - ASTR - 19
Class 19. N -body Techniques, Part 2Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g.,
Maryland - ASTR - 415
Class 19. N -body Techniques, Part 2Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g.,
Maryland - ASTR - 20
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 415
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 21
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
Maryland - ASTR - 415
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
Maryland - ASTR - 22
Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 415
Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 23
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 415
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 24
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
ASTR415 Survey ResultsSpring 2007 11 respondents 1. Computer familiarity [1=Master, 5=None]: Avg = 2.7 [Skilled], Min = 1 [Master], Max = 4 [Novice] 2. Unix familiarity [1=Master, 5=None]: Avg = 3.6 [Skilled], Min = 2 [Expert], Max = 5 [None] 3. Uni
Maryland - ASTR - 415
ASTR415 Spring 2007Due May 08, 2007Term Project1) For your term project you will install and learn how to use a freely available (opensource) 3D visualization tool based on Open-GL. You will write a short report and present in class the results
Maryland - ASTR - 415
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Maryland - ASTR - 415
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Maryland - ASTR - 415
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Maryland - ASTR - 415
ICadafWWaHH\$CHIaRp\$a\$wraWCawHa 9 i i S x i U bIxpU B @ e F k i 6 d B wP v q D A B FP PT B 8 6 qT Q v D 4 B v 4 @ D B F f D F B A @ A B F 6P k Uif'fbS7)UdRT Y rtuCaCdRR99R7\$17t\$ICdP Y `CaC9ECGR'\$i S
Maryland - ASTR - 415
A Crash Course on UNIXUNIX is an &quot;operating system&quot;.Interface between user and data stored on computer. A Windows-style interface is not required. Many flavors of UNIX (and windows interfaces).Solaris, Mandrake, RedHat (fvwm, Gnome, KDE)
Maryland - ASTR - 415
Data RepresentationsComputers store data as different variable types, e.g. integer, floating point, complex, etc. Different machines have different wordlengths, e.g. 4-byte ints on a 32-bit machine (Pentium), 8-byte ints on a 64-bit machine (Alpha).
Maryland - ASTR - 415
VisualizationVisualization is useful for:1) Data 2) Codeentry (initial conditions) debugging and performance analysis and display of results3) Interpretation Our focus will be #3. The computational astrophysicist can either:1) Develop 2) U
Maryland - ASTR - 415
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra &quot;analytically&quot; (e.g. Mathematica, Maple). Numerical methods needed when:
Maryland - ASTR - 415
Nonlinear EquationsOften (most of the time?) the relevant system of equations is not linear in the unknowns. Then, cannot decompose as Ax = b. Oh well. Instead write as:(1) (2) f(x) = 0 f(x) = 0function of one variable (1-D) x = (x1,x2,.,xn
Maryland - ASTR - 415
Statistical Description of Data Cf. NRiC, Chapter 14. Statistics provides tools for understanding data.In the wrong hands these tools can be dangerous! Apply some formula to data to compute a &quot;statistic&quot;. Find where value falls in a probability
Maryland - ASTR - 415
Modeling of Data NRiC Chapter 15. Model depends on adjustable parameters. Can be used for &quot;constrained interpolation&quot;. Basic approach:1. 2. 3. 4.Choose figure-of-merit function (e.g. 2). Adjust best-fit parameters: minimize merit function. Co
Maryland - ASTR - 415
Random Numbers NRiC Chapter 7. Frequently needed to generate initial conditions. Often used to solve problems statistically. How can a computer generate a random number? It can't! Generators are pseudo-random. Generators are deterministic: i
Maryland - ASTR - 415
Numerical Integration (Quadrature) NRiC Chapter 4. Already seen Monte Carlo integration. Can cast problem as a differential equation (DE): = is equivalent to solving for I y(b) the DE dy/dx = f(x) with the boundary condition (BC) y(a) =
Maryland - ASTR - 415
Ordinary Differential Equations (ODEs) NRiC Chapter 16. ODEs involve derivatives wrt one independent variable, e.g. time t. ODEs can always be reduced to a set of firstorder equations (involving only first derivatives).e.g. = is
Maryland - MATH - 241
MATH 241 CALCULUS III FIRST MIDTERM EXAM Instructions. Answer each question on a separate answer sheet. Show all your work. Be sure your name, section number, and problem number are on each answer sheet, and that you have copied and signed the honor
Maryland - MATH - 241
MATH 241 CALCULUS III FIRST MIDTERM EXAM SOLUTIONS (1) For this problem, u = + k, and v = + 2 + 3 k. ij i j (a) u v = 1 + 2 3 = 2. k ij (b) u v = det 1 1 1 = 5 2 + 3 k. i j 1 2 3 (c) The symmetric form of the equations are: 2 x =
Maryland - ASTR - 601
Astronomy 601 - Fall 2005 Radiative ProcessesInstructor Prof. Massimo Ricotti Oce: CSS 0213 E-mail: ricotti@astro.umd.edu Phone: (301) 405 5097 Oce hours: by appointment Class web page: http:/www.astro.umd.edu/ricotti/NEWWEB/teaching/ASTR601.html Sc
Maryland - ECE - 2003
Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana and Vijay Bharadwaj Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 2
Maryland - ECE - 2003
Proceedings of the DARPA Information Survivability Conference and Exposition (DISCEX03) 0-7695-1897-4/03 \$17.00 2003 IEEEProceedings of the DARPA Information Survivability Conference and Exposition (DISCEX03) 0-7695-1897-4/03 \$17.00 2003 IEEEPr
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Integrated Security Services for Dynamic CoalitionsHimanshu Khurana1, Serban Gavrila1, Rakeshbabu Bobba, Radostina Koleva, Anuja Sonalker, Emilian Dinu, Virgil Gligor, and John Baras Electrical and Computer Engineering Department, University of Mary
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Towards Automated Negotiation of Access Control PoliciesVijay G. Bharadwaj and John S. Baras Institute for Systems Research, University of Maryland, College Park MD 20742, USA. vgb,baras @umd.eduAbstractWe examine the problem of negotiating acces
Maryland - ECE - 2003
DYNAMIC ADAPTATION OF ACCESS CONTROL POLICIESVijay Bharadwaj and John Baras Institute for Systems Research University of Maryland College Park MD 20742ABSTRACTWe describe an architecture and algorithms for deriving an access control policy by com
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Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 20742 DARPA DC PI Mee
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Reasoning about Joint Administration of Access Policies for Coalition ResourcesHimanshu Khurana Virgil Gligor John Linnjlinn@rsasecurity.com{hkhurana, gligor}@eng.umd.eduUniversity of Maryland College Park, MD.RSA Labs Bedford, MA.OutlineC
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Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementVirgil D. Gligor and John S. Baras Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 207
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Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana and Vijay Bharadwaj Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 2
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DK6037 FLPraveen K. MurthyFujitsu Laboratories of America, Sunnyvale, California, USAShuvra S. BhattacharyyaUniversity of Maryland, College Park, USAEffective Strategies for Aggressive Memory OptimizationAlthough programming in memory restri
Maryland - ECE - 2003
about the book. . . Ranging from low-level applicationand architectureoptimizationsto high-level modeling and exploration concerns,this text/reference compiles essentialresearchon various levels of abstractionappearingin embeddedsystemsand softwarede
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Marcel Dekker Catalog: Embedded MultiprocessorsPage 1 of 1Embedded MultiprocessorsScheduling and Synchronization Sundararajan Sriram, Texas Instruments, Inc., Dallas, Texas, and Shuvra S. Bhattacharyya, University of Maryland, College Park serie
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Springer - Signals &amp; CommunicationPage 1 of 1springer.com Springer Berlin Heidelberg New YorkSignals &amp; CommunicationSoftware Synthesis from Dataflow Graphs Series: The International Series in Engineering and Computer Science, Vol. 360 Bhattach