... plot of water surface in 2D simulation of Exp-1 …………….95 Figure 6.35 Carpet plot of water surface in 2D simulation of Exp-2…………… 95 Figure 6.36 Carpet plot of water surface in 2D simulation of ... plot of water surface in 2D simulation of Exp-4…………… 96 Figure 6.38 Carpet plot of water surface in 2D simulation of Exp-5…………… 97 Figure 6.39 Carpet plot of water surface in 2D simulation of ... surface profile analysis of open channel flows over a circular surface. Journal of Applied Mechanics, JSCE, Vol. 8, pp 847-854, 2005. 5. Anh T. N. and Hosoda T.: Free surface profile analysis of flows...
... measure of goodness -of- fit. When the third item suggests that the modelis an unlikely match to the data, then items (i) and (ii) are probably worthless.Unfortunately, many practitioners of parameter ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission ... based on? The answerto these questions takes us into the subject of maximum likelihood estimators.Given a particular data set of xi’s and yi’s, we have the intuitive feeling thatsome parameter...
... distribution is often rather poorly realized, if it isrealized at all. We are often taught, rather casually, that, on average, measurementswill fall within ±σ of the true value 68 percent of the time, ... time, within ±2σ 95 percent of the time, and within ±3σ 99.7 percent of the time. Extending this, one wouldexpect a measurement to be off by ±20σ only one time out of 2 × 1088.Weallknow that ... that thestandard deviations σ of these normal distributionsare the same for all points. Thenthe probability of the data set is the product of the probabilities of each point,P ∝Ni=1exp−12yi−...
... problem of fitting a set of N data points (xi,yi)toa straight-line modely(x)=y(x;a, b)=a+bx (15.2.1)666Chapter 15. Modelingof DataSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... variance of the linear combination yi− a − bxi of two randomvariables xiand yi,Var( yi− a − bxi)=Var( yi)+b2Var( xi)=σ2yi+b2σ2xi≡1/wi(15.3.3)The sum of the square of N ... =1, ,M (15.1.7)Equation (15.1.7) is, in general, a set of M nonlinear equations for the M unknownak. Various of the procedures described subsequently in this chapter derive from(15.1.7) and...
... float *xx,*yy,*sx,*sy,*ww,aa,offs;float chixy(float bang)Captive function of fitexy, returns the value of (χ2−offs) for the slopeb=tan(bang).Scaled data andoffsare communicated via the ... variance of the linear combination yi− a − bxi of two randomvariables xiand yi,Var(yi− a − bxi)=Var(yi)+b2Var(xi)=σ2yi+b2σ2xi≡1/wi(15.3.3)The sum of the square of N ... of varying b, successful numerical root-findingleads to a value of ∆a that minimizes χ2for the value of ∆b that gives ∆χ2=1.This(seeFigure 15.3.1) directly gives the tangent projection of...
... combinations of the columns of V, with coefficients obtained by forming dot products of the columns672Chapter 15. Modelingof DataSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... thestandard deviationsof themeasureddependentvariable.The measured values of the dependent variable do not enter the design matrix.Solution by Use of the Normal EquationsThe minimum of (15.4.3) occurs ... vector of length M.The equations (15.4.6) or (15.4.7) are called the normal equations of the least-squares problem. They can be solved for the vector of parameters a by the standardmethods of...
... andalamdade-creases by a factor of 10. If a step failsalamdagrows by a factor of 10. You must call this684Chapter 15. Modelingof DataSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... even when they must be offered without proof.Figure 15.6.1 shows the conceptual scheme of an experiment that “measures”a set of parameters. There is some underlying true set of parameters atruethat ... name of the game is to find some way of estimating or approximatingthe probability distributionof a(i)−atruewithout knowing atrueand withouthavingavailable to us an infinite universe of hypothetical...
... even when they must be offered without proof.Figure 15.6.1 shows the conceptual scheme of an experiment that “measures”a set of parameters. There is some underlying true set of parameters atruethat ... name of the game is to find some way of estimating or approximatingthe probability distributionof a(i)−atruewithout knowing atrueand withouthavingavailable to us an infinite universe of hypothetical ... techniques, and seek to minimize the uncertainty of the result accordingto any desired criteria. Offered the choice between mastery of a five-foot shelf of analytical statistics books and middling...
... function of a single variable z ≡ [yi− y(xi)]/σi.If we now define the derivative of ρ(z) to be a function ψ(z),ψ(z) ≡dρ(z)dz(15.7.4)then the generalization of (15.1.7) to the case of a ... on the following fact: The median cM of a set of numbers ciis also that value which minimizes the sum of the absolute deviationsi|ci− cM|(Proof: Differentiate the above expression ... example, the equality orinequality of two distributions can be estimated by the Wilcoxon test of computingthe mean rank of one distribution in a combined sample of both distributions.The Kolmogorov-Smirnov...
... convergence of embedded minimalsurfaces of a fixed genus in a fixed 3-manifold. This is due to the next theoremwhich loosely speaking asserts that any sequence of embedded minimal surfaces of fixed ... application of Rado’s theorem; cf. the proof of theorem 1.22in [CM4]), Σ0,tis itself a graph, giving the lemma.The next lemma bounds the radius of the intrinsic tubular neighborhood of B2∩ ... The fourth paper of this series will deal with howthe multi-valued graphs fit together and, in particular, prove regularity of theset of points of large curvature – the axis of the double spiral...
... price of steel in 2004 and 2005 and the near doubling of the price of titanium between January 2005 and April 2006.Data and Price Trends 15Ideally, we would next explore the sources of escalation ... Naval Air Systems CommandNGC Northrop Grumman CorporationOPNAV Office of the Chief of Naval OperationsOSD Office of the Secretary of DefenseOSHA Occupational Safety and Health AdministrationPA&E ... Labor RatesAs described above, labor costs are a significant portion of the total system cost. us, changes in labor rates are of importance to under-standing the influence of the economic factors....
... Od1 of the side chain of Asn87 in the S4 strand, withdistances of 2.82 A˚and 2.97 A˚, respectively. Theplane of the base of A20 and the end plane of thecarbamoyl group of Asn87 are out of ... tRNAArgICG.Distances between C1¢ of C72 of the 1Æ72 pair in theacceptor stem, C1¢ of G18 forming a hydrogen bondto U55 of the T-loop and C1¢ of C31 or G39 of 31Æ39 of the anticodon stem of tRNA were compared ... formation of the hydrogenbond between O2 of the carboxyl group of the cognateamino acid and the 2¢-OH group of the ribose moiety of A76 maintains the appropriate orientation of thecarboxyl group of...
... particular,phosphorylation of dGuo was reduced markedly.Structure of Dm-dNK-E52D in complex with dTTPThe structure of one of the mutants, E52D, was solvedusing X-ray crystallography at a resolution of 2.5 A˚incomplex ... levels of phosphorylation expressed in rela-tion to percentage dThd phosphorylation of the wild-type. Relativephosphorylation expressed as a relation of the percentage of dThdphosphorylation of ... quantitation of microgram quantities of protein util-izing the principle of protein–dye binding. Anal Biochem72, 248–254.22 Knecht W, Bergjohann U, Gonski S, Kirschbaum B &Loffler M (1996)...