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T p 1 ) T p 2 / ( 1 − T p 2 ) give 95% confidence limits for the odds ratio in the absence of covariates. 3.2. The Common Odds RatioConsider K independent studies (or strata from the same study), where from the kth study, we have observations for two independent binomial random variables X 1 k and X 2 k with respective success probabilities p 1 k and p 2 k , and respective sample sizes n 1 k and n 2 k , k = 1, 2, ...., K. Thus, the odds ratio from the kth study is δ k = p 1 k / ( 1 − p 1 k ) p 2 k / ( 1 − p 2 k ) , k = 1, 2, ...., K. Assuming that the odds ratio is the same across the K studies, we have δ 1 = δ 2 = . . . . = δ K = δ (say). 3.2.1. An Approximate GPQ for the Common Odds RatioAn approximate GPQ for each δ k , to be denoted by T δ k , can be constructed from the kth study, proceeding as mentioned in Section 3.1. We now combine these GPQs in order to obtain an approximate GPQ for the common odds ratio δ. For this, we propose a weighted average of the study-specific GPQs on the log scale. The weights that we shall use are motivated as follows. For i = 1, 2, if p ^ i k denote sample proportions from the kth study, and if δ ^ k = p ^ 1 k / ( 1 − p ^ 1 k ) p ^ 2 k / ( 1 − p ^ 2 k ) , k = 1, 2, ...., K, then using the delta method, an approximate variance of log ( δ ^ k ) , say 1 / w k , is given by: 1 / w k = 1 n 1 k p 1 k + 0.5 + 1 n 1 k ( 1 − p 1 k ) + 0.5 + 1 n 2 k p 2 k + 0.5 + 1 n 2 k ( 1 − p 2 k ) + 0.5 , where we have also used a continuity correction. Noting that log ( δ ) = ∑ k = 1 K w k log ( δ k ) / ∑ k = 1 K w k , an approximate GPQ T δ for the common odds ratio can be obtained from log ( T δ ) = ∑ k = 1 K T w k log ( T δ k ) / ∑

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Interval for the NNT can be obtained from a confidence interval for p 2 − p 1 . An approximate GPQ as well as fiducial quantities can be used for computing confidence intervals for p 2 − p 1 . In fact, fiducial quantities for these parameters based on Equation (2) are given in [12]. 3.4. Epidemiological Measures under the Logistic and Log-Binomial ModelsSo far, our adaptation of the methodology based on GPQs and fiducial quantities has been for situations where covariates are absent. Clearly, the odds ratio, as well as the other epidemiological measures, have extensive practical applications in the context of binomial responses that depend on covariates. The logistic model is very often used to model the response probability. The log-binomial model is sometimes used to estimate the risk ratio in the presence of covariates, when the outcome is not rare. As noted in Section 2.4, under the log-binomial model, the binomial success probabilities p satisfies ln ( p ) = x ′ β , where x is a covariate vector. Writing x = ( x 1 , x 2 , . . . . , x s ) ′ and β = ( β 1 , β 2 , . . . . , β s ) ′ , where s is the number of covariates, the parameter β 1 is the prevalence ratio (PR) for a one unit increase in x 1 , adjusted for the other covariates. We recall that GPQs and fiducial quantities for β are given in Section 2.3 and Section 2.4 for the logistic model and the log-binomial model, respectively. From this, GPQs and fiducial quantities can be constructed for any function of β ; in particular, for the various epidemiological measures, including the prevalence ratio. 4. Numerical ResultsThe accuracy of the proposed procedures based on GPQs and fiducial quantities is assessed using simulations. Here, we have presented the results for only two scenarios: interval estimation of a common odds ratio (under binomial distributions without covariates), and the interval estimation of a prevalence ratio (under the log-binomial model). We refer to [12] for numerical results on the performance of fiducial intervals for a few other parameters, including that for the difference between binomial proportions. Note that coverage probability for the latter is equivalent to that for the NNT. 4.1. Common Odds RatioTable 1 gives the coverage probabilities of the confidence intervals based on different approaches for a common odds ratio from K = 5 studies, for a 95% nominal level. We also assume that, for the different studies, n 1 k = n 1 , and n 2 k = n 2 (k = 1, 2, …, 5), where we have used

Of the behavior of an untrained network across different classes. This analysis serves as a critical indicator of the local linear operators’ capability to distinguish between various class characteristics.To facilitate the comparison among the various individual correlation matrices, which may exhibit varying sizes due to the differing number of data points per class, each matrix is evaluated separately: E C = ∑ i = 0 N ∑ j = 0 N log ( | ( P J C ) i , j | + ε ) , if C constant ∑ i = 0 N ∑ j = 0 N log ( | ( P J C ) i , j | + ε ) | | P J C | | , otherwise where ε is a small constant. We denote | | P J C | | as the number of elements of the set P J C .Finally, we use S to evaluate the discriminability of neural networks: S = ∑ t = 0 C | E t | , if C constant ∑ i = 0 C ∑ j = i C | E i − E j | | | E | | , otherwise where E is the vector containing all correlation matrices’ scores. 3.3.3. Evaluation Method with Three IndicatorsWith an additional indicator included, we have three indicators: trainability K N , expressivity R ^ N , and discriminability S N .We define the performance of three different neural networks, N 1 , N 2 , and N 3 , as follows: F 1 = λ K · Δ K N 1 + λ R · Δ R ^ N 1 + λ S · Δ S N 1 F 2 = λ K · Δ K N 2 + λ R · Δ R ^ N 2 + λ S · Δ S N 2 F 3 = λ K · Δ K N 3 + λ R · Δ R ^ N 3 + λ S · Δ S N 3 We let the three neural networks compete with each other. An overview of the TrueSkill algorithm with the three indicators is shown in Figure 4.Once two neural networks, N 1 and N 2 , have competed, we fix one of the indicators and update the other two indicators.Combined with the TrueSkill algorithm, we can update the equations of these three indicators:Let μ 1 = λ K ( Δ K N 1 − Δ K N 2 ) , μ 2 = − [ λ R ( Δ R ^ N 1 − Δ R ^ N 2 ) + C S ] , C S = λ S ( Δ S

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Browse Presentation Creator Pro Upload Apr 04, 2019 370 likes | 432 Views Lead Controller Design. p lead. z lead. 20log(Kz lead /p lead ). Goal: select z and p so that max phase lead is at desired wgc and max phase lead = PM defficiency !. Lead Design . From specs => PM d and w gcd From plant, draw Bode plot Download Presentation Lead Controller Design An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher. Presentation Transcript Lead Controller Designplead zlead 20log(Kzlead/plead) Goal: select z and p so that max phase lead is at desired wgc and max phase lead = PM defficiency!Lead Design • From specs => PMd and wgcd • From plant, draw Bode plot • Find PMhave = 180 + angle(G(jwgcd) • DPM = PMd - PMhave + a few degrees • Choose a=plead/zlead so that fmax = DPM and it happens at wgcdLead Design Variation • There is ess and Mp requirement • From ess_d, compute K = ess/ess_d • Draw Bode plot of KG, find PM & wgc • From Mp specs, find PMd • fmax = DPM = PMd - PMhave + extraextraLead design example • Plant transfer function is given by: • n=[50]; d=[1/5 1 0]; • Desired design specifications are: • Step response overshoot n=[50]; d=[1/5 1 0]; figure(1); clf; margin(n,d); grid; hold on; ess2ramp= 1/200; Kvd=1/ess2ramp; Kva = n(end)/d(end-1); Kzp = Kvd/Kva; figure(2); margin(Kzp*n,d); grid; [GM,PM,wpc,wgc]=margin(Kzp*n,d); w_gcd=wgc; Mp = 20/100; zeta =sqrt((log(Mp))^2/(pi^2+(log(Mp))^2)); PMd = zeta * 100 + 10; phimax = (PMd-PM)*pi/180; alpha=(1+sin(phimax))/(1-sin(phimax)); z=w_gcd/sqrt(alpha); p=w_gcd*sqrt(alpha); ngc = conv(n, alpha*Kzp*[1 z]); dgc = conv(d, [1 p]); figure(3); margin(tf(ngc,dgc)); grid; [ncl,dcl]=feedback(ngc,dgc,1,1); figure(4); step(ncl,dcl); grid; figure(5); margin(ncl*1.414,dcl); grid;Problem • wgc moved to a new location • At a higher frequency • Phi_max was not at actual wgc • PM at actual wgc is too small • Solution: • Anticipate this upward wgc change • Place both z_lead and p_lead at a higher frequency than original wgcn=[50]; d=[1/5 1 0]; figure(1); clf; margin(n,d); grid; hold on; ess2ramp= 1/200; Kvd=1/ess2ramp; Kva = n(end)/d(end-1); Kzp = Kvd/Kva; figure(2); margin(Kzp*n,d); grid; [GM,PM,wpc,wgc]=margin(Kzp*n,d); w_gcd=wgc; Mp = 20/100; zeta =sqrt((log(Mp))^2/(pi^2+(log(Mp))^2)); PMd