Fundamentals Of Linear State Space Systems 1st Edition John S Bay - Solutions

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Table 8.3 shows the three-point shooting, by game, of Ray Allen of the Boston Celtics during the 2010 NBA (basketball) playoffs (e.g., he made 0 of 4 shots in game 1). Commentators remarked that his shooting varied dramatically from game to game. In game i, suppose that niyi = number of three-point
Verify formula (9.1) for the effects of correlation on between-cluster and within-cluster effects.
How does positive correlation affect the SE for between-cluster effects with binary data? Let y11,…, y1d be Bernoulli trials with E(y1j) = ????1 and let y21,…, y2d be Bernoulli trials with E(y2j) = ????2. Suppose corr(yij, yik) = ???? for i = 1, 2 and corr(y1j, y2k) = 0 for all j and k. Find
Suppose y1,…yn have E(yi) = ????, var(yi) = ????2, and corr(yi, yj) = ???? for i ≠ j.Show that E(s2) = ????2(1 − ????).
Formulate a normal random-effects model that generates the within-cluster and between-cluster effects described in Section 9.1.1.
In the analysis of covariance model, observation j in group i has ????ij = ????0 +????1xij + ????i for a quantitative variable xij and qualitative {????1,…, ????c}. Describe an application in which the qualitative variable would naturally be treated as a random effect. Show then how to express
A crossover study comparing d = 2 drugs observes a continuous response(yi1, yi2) for each subject for each drug. Let ????1 = E(yi1) and ????2 = E(yi2) and consider H0: ????1 = ????2.a. Construct the normal linear mixed model that generates a paired-difference t test (with test statistic t =
For the normal linear mixed model (9.6), derive expression (9.7) for var(yi).
For the extension of the random-intercept linear mixed model (9.8) that assumes cov(????ij, ????ik) = ????2???? ????|j−k|, show that corr(yij, yik) = (????2 u + ????|j−k|????2????)∕(????2 u + ????2????).
Consider the model discussed in Section 9.2.4 having a random intercept and a random slope. Is the fit for subject i any different than using least squares to fit a line using only the data for subject i? Explain.
For the linear mixed model, show that ????̃ (for known {Vi}) is unbiased, and derive its variance. Show how ????̃ and var(????̃) simplify when the model does not contain random effects.
When Xi and Vi in the linear mixed model are the same for each subject, show that the generalized least squares solution (9.10) can be expressed in terms of ȳ = (1∕n)∑i yi.
For the balanced random-intercept linear mixed model (9.8) based on conditional independence given the random effect (Section 9.2.1), show that????̃ = (XTV−1X)−1 XTV−1y simplifies to the ordinary least squares estimator, (XTX)−1 XTy; i.e., with compound symmetry, observations do not need
Using the joint normal density of y and u, derive Henderson’s mixed-model equations.
When R = ????2???? I, show that as ????−1 u tends to the zero matrix, the mixed model equations for a linear mixed model tend to the ordinary normal equations that treat both ???? and u as fixed effects.
For the random-effects one-way layout model (Section 9.3.2), show that ????̂0 and ũi are as stated there (i.e., with the known variances). Show that ũi is a weighted average of 0 and the least squares estimate based on treating u as fixed effects. Give an application for which you would prefer
BLUPs are unbiased. Explain why this does not imply that E(ũ ∣ u) = u. Illustrate by a simulation using a simple linear mixed model. For your simulation, show how ũ tends to shrink toward 0 relative to u.
Show how fitting and prediction results of Section 9.3 for linear mixed models simplify when var(????i) = ????2???? I instead of R.
For the REML approach for the normal null model described in Section 9.3.3, find L, derive the distribution of Ly, and find the REML estimator of ????2.
For the binary matched-pairs model (9.5), consider a strictly fixed effects approach, replacing ????0 + ui in the model by ????0i. Assume independence of responses between and within subjects.a. Show that the joint probability mass function is proportional to exp[∑n i=1????0i(yi1 + yi2) +
Refer to the previous exercise. Unlike the conditional ML estimator of ????1, the unconditional ML estimator is inconsistent (Andersen 1980, pp. 244–245).a. Averaging over the population, explain why????21 = E[ 1 1 + exp(????0i)exp(????0i + ????1)1 + exp(????0i + ????1)], where the expectation
A binary response yij = 1 or 0 for observation j on subject i, i = 1,…, n, j = 1,…,d. Let ȳ.j = ∑i yij∕n, ȳi. = ∑j yij∕d, and ȳ = ∑i∑j yij∕nd. Regard{yi+} as fixed, and suppose each way to allocate the yi+ “successes” to the d observations is equally likely. Show that
For the GLMM for binary data using probit link function,Φ−1[P(yij = 1 ∣ ui)] = xij???? + zijui show that the corresponding marginal model is Φ−1[P(yij = 1] = xij????[1 +zij????uzT ij]−1∕2. Compare effects in the GLMM and marginal model.
From Exercise 7.25, with any link and a factor predictor in the one-way layout, the negative binomial ML fitted means equal the sample means. Show this is not true for the Poisson GLMM.
For the Poisson GLMM (9.12) with random intercept, use the normal mgf to show that for j ≠ k, cov(yij, yik) = exp[(xij + xik)????][exp (????2 u) (exp (????2 u)− 1)]Find corr(yij, yik). Explain why, as in binary GLMMs, corr(yij, yik) ≥ 0.
For recent US Presidential elections, in each state wealthier voters tend to be more likely to vote Republican, yet states that are wealthier in an aggregate sense are more likely to have more Democrat than Republican votes (Gelman and Hill 2007, Section 14.2.). Sketch a plot that illustrates how
Construct a marginal model that is a multivariate analog of the normal linear model for a balanced two-way layout, assuming an absence of interaction.Show how to express the two main effect hypotheses for that model in terms of model parameters.
For subject i, yi = (yi1,…, yid)T are d repeated measures on a response variable. Consider a model by which yi ∼ N(????,????????) with ????1 = ⋯ = ????d and???? = corr(yij, yik) for all j ≠ k. The hypothesis of interest is H0: ????1 = ⋯ = ????d.a. Express this scenario as a marginal model
When y ∼ N(0, ????????), in terms of elements kij from the concentration matrix K = ????−1???? , show that the pdf f for y has the form log f(y) = constant − 1 2∑i kiiy2 i −∑i∑j kijyiyj, and the elements of K are natural parameters in the exponential family representation. Explain why
For the GEE (9.15) with R(????) = I, show that the equations simplify to(1∕????)∑n i=1 XT i ????i(yi − ????i) = 0, where ????i is the diagonal matrix with elements ????????ij∕????????ij on the main diagonal for j = 1,…,d, for natural parameter ????ij and linear predictor ????ij. Show
Generalizing the heuristic argument in Section 8.3.2, justify why formula(9.16) is valid for the sandwich covariance matrix (Liang and Zeger 1986, Appendix).
For both linear mixed models and marginal models, the generalized least squares estimator ????̂ = (XTV−1X)−1 XTV−1y is useful. Suppose V is not the true var(y) but we can consistently estimate var(y) empirically by S. Describe a way to do this. Explain why it is then sensible to estimate
For the smoking prevention and cessation study (Section 9.2.3), fit multilevel models to analyze whether it helps to add any interaction terms. Interpret fixed and random effects for the model that has a SC × TV interaction.
Using the R output shown for the simple analyses of the FEV data in Section 9.2.5, show that the estimated values of corr(yi1, yi2) and corr(yi1, yi8) are 0.74 for the random intercept model and 0.86 and 0.62 for the model that also permits autoregressive within-patient errors.
Refer to Exercise 1.21 and the longitudinal analysis in Section 9.2.5. Analyze the data in file FEV2.dat at www.stat.ufl.edu/~aa/glm/data , investigating the correlation structure for the eight FEV responses and modeling how FEV depends on the hour and the drug, adjusting for the baseline
A field study11 analyzed associations between a food web response measure and plant invasion and tidal restriction in salt marsh habitats. The response, observed in a species of small marsh fish (Fundulus heteroclitus) that swim in and out of the salt marshes with the tides, was a stable carbon
For Table 7.5 on counts of victims of homicide, specify and fit a Poisson GLMM. Interpret estimates. Show that the deviance decreases by 116.6 compared with the Poisson GLM, and interpret.
The data file Maculatum.dat at the text website is from a study13 of salamander embryo development. These data refer to the spotted salamander(Ambystoma maculatum). One purpose of the study was to compare four rearing environments (very humid air, and water with low, medium, and saturated dissolved
Refer to the previous exercise. For all the embryos, use a logistic GLMM to model the probability of hatching in terms of the treatment. Interpret results, and compare to those obtained with an ordinary logistic GLM that ignores the clustering.
Download the file Rats.dat at the text website for the teratology study in Table 8.1.a. Use the GEE approach to fit the logistic model, assuming an exchangeable working correlation structure for observations within a litter. Show how the empirical sandwich adjustment increases the SE values
Refer to Exercise 9.34. Analyze the FEV data with marginal models, and compare results to those obtained with linear mixed models.
A crossover study analyzed by B. Jones and M. Kenward (1987, Stat. Med. 6:555–564) compared three drugs on a binary outcome (success = 1, failure =0). Counts for the eight possible response patterns for drugs (A, B, C) were 6 for 000, 9 for 001, 4 for 010, 45 for 011, 3 for 100, 7 for 101, 4 for
Suppose y1,…, yc are independent from a Poisson distribution with mean ????.Conditional on ∑ yi = n, are y1,…, yc exchangeable? Independent? Explain.
Independent observations y = (y1,…, yn) come from the N(????, ????2) distribution, with ????2 known, and ???? has a N(????, ????2) prior. Show that the posterior predictive distribution for a future y∗ is normal with mean equal to the posterior mean of ???? and variance equal to the posterior
Find the posterior mean and variance for ???? in the null model with a N(????, ????2) response for unknown ????2, using the improper-priors approach of Section 10.2.5.
Suppose y1,…, yn are independent from a N(????,????) distribution, and ???? has a N(????0,????0) prior distribution. With ???? known, derive the posterior distribution.Explain how the posterior mean is a weighted average of the prior mean and the sample mean and the posterior precision is the sum
For the Bayesian ordinary normal linear model, using a flat improper prior for ???? but a proper inverse-gamma distribution (10.5) for ????2, find the posterior Bayes estimate of ????2 and express it as a weighted average of s2 and the prior mean of ????2.
Suppose we assume the Bayesian normal linear model for a one-way layout, but the actual conditional distribution of y is highly skewed to the right (e.g., y = annual income). For large {ni}, would you expect Bayesian inference about {????i} to be relatively robust? Would you expect Bayesian
You regard m potential models, M1,…, Mm, to be (a priori) equally likely.Use Bayes’ theorem to conduct Bayesian model averaging by finding an expression for the posterior P(Mi ∣ y) in terms of the marginal {P(y ∣ Mj)}for the models (integrating out the parameters). Extend the result to
With a beta(????1, ????2) prior for the binomial parameter ???? and sample proportion y, if n is large relative to ????1 + ????2, show that the posterior distribution of ????has approximate mean y and approximate variance y(1 − y)∕n. Interpret.
With a beta prior for the binomial parameter ???? having ???? = ????1∕(????1 + ????2)and letting n∗ = ????1 + ????2, find E( ̃???? − ????)2 and express it as a weighted average of [????(1 − ????)]∕n and (???? − ????)2. Compare this with E( ̂???? − ????)2, for ̂???? = y.Evaluate
A beta prior for the binomial parameter ???? has ????1 = ????2 = ????.a. When y = 0, for what values of ???? is the posterior density of ???? monotone decreasing and hence the HPD posterior interval of the form (0, U)? For such ????, report U as the quantile of a beta distribution.b. Show that the
In Exercise 4.11 for the binomial probability ???? of being a vegetarian, the proportion y = 0 of n = 25 students were vegetarians.a. Report the ML estimate. Find the 95% confidence interval based on inverting the likelihood-ratio chi-squared test.b. Using a uniform prior distribution for ????,
This exercise is based on an example in the keynote lecture by Carl Morris(see www.youtube.com/watch?v=JOovvj_SKOg) at a symposium held in his honor in October 2012. Before a Presidential election, polls are taken in two states that are usually swing states. In State A, the proportion y1 = 0.590 of
For a binomial distribution with beta prior, show that the marginal distribution of s = ny is the beta-binomial. State its mean and variance.
For a binomial distribution with beta prior, show how to conduct Bayesian estimation of {????i} for c groups in the one-way layout.
In a diagnostic test for a disease, let D denote the event of having the disease, and let + (−) denote a positive (negative) diagnosis by the test. Let????1 = P(+ ∣ D) (the sensitivity), ????2 = P(+ ∣ Dc) (the false positive rate), and???? = P(D) (the prevalence). More relevant to a patient
Consider independent binary observations from two groups with ????i =P(yi = 1) = 1 − P(yi = 0), and a binary predictor x. For the 2×2 contingency table summarizing the two binomials, let logit(????i) = ????0 + ????1xi, where xi = 1 or xi = 0 according to a subject’s group classification. Also,
In the previous exercise, suppose ????̂ = 0 and the HPD interval for ???? is (0, U).Is (1∕U, ∞) the HPD interval or the equal-tail posterior interval for ????∗?Do you think it is a sensible interval? Explain, and give implications for forming posterior intervals for parameters in logistic
Re-do the analysis with prior ???? = 1 for the example in Section 10.3.2, but use (0, 1) coding for the indicator variable x1 instead of −0.5 and 0.5. Why is the posterior mean for ????1 so different?
For a Poisson random variable y with mean ????, show that the Jeffreys prior distribution for ???? is improper. Using it, find the posterior distribution and indicate whether it is improper.
For iid Poisson variates y1,…, yn with parameter ????, suppose ???? ∼ gamma(????, k) (Recall Section 4.7.2).a. Show that the posterior distribution of ???? is gamma (i.e., the prior is conjugate), with posterior mean that is a weighted average of the sample mean and the prior mean. Explain how
Show how the results in (a) in the previous exercise generalize to estimating Poisson parameters for c groups in the one-way layout. For fixed gamma hyperparameters, show that the estimate of ????i for group i is a weighted average of ȳi and the mean of the gamma prior distribution. When those
Show how results in Section 10.1.3 generalize for n independent multinomial trials and a Dirichlet prior for the multinomial probabilities. (Good (1965)gave one of the first Bayesian analyses with this model, using empirical Bayes and hierarchical approaches. See also Section 11.2.2.)
Two independent multinomial variates have ordered response categories.Using independent Dirichlet priors, explain how to simulate to find the posterior probability that those two distributions are stochastically ordered.(Altham (1969) found an exact expression for this probability.)
Refer to the Presidential election data in Section 10.4.3.a. Use a Bayesian approach to fit model (10.7), with independent N(0, 100)priors for {????i}. Find corresponding estimates of {????i}, and evaluate their performance by finding the MSE and comparing with the ordinary sample proportions.b.
In the previous exercise, repeat (b), but using an empirical Bayes approach for a N(logit(qi), ????2) prior distribution for ????i.
Conduct Bayesian analogs of the frequentist modeling for the FEV clinical trial data from Exercise 3.31. Compare Bayesian and frequentist results.
Conduct Bayesian modeling for the smoking prevention data of Section 9.2.3.Compare results to the frequentist results presented there.
Refer to Exercise 5.34 on horseshoe crabs. Repeat this exercise using Bayesian methods.
For the endometrial cancer example in Section 10.3.2, fit the logistic model using a hierarchical Bayesian approach with a diffuse inverse chi-squared distribution for the ????2 hyperparameter. Interpret results.
Show that M-estimation with ????(ei) = |ei| gives the ML solution assuming a Laplace distribution for the response.
The breakdown point of an estimator is the proportion of observations that must be moved toward infinity in order for the estimator to also become infinite. The higher the breakdown point, the more robust the estimator. For estimating the center of a symmetric distribution, explain why the
Refer to the equations solved to obtain ????̂ for M-estimation and the expression for var(????̂). Show how they simplify for least squares.
In M-estimation, let ????(x) = 2(√1 + x2∕2 − 1). Find the influence function, and explain why this gives a compromise between least squares and ????(x) =|x|, having a bounded influence function with a smooth derivative at 0.
Since the Gauss-Markov theorem says that least squares estimates are “best,”are not estimates obtained using M-estimation necessarily poorer? Explain.
For the saturated model, E(yi) = ????i, i = 1,…, n, find the ridge-regression estimate of ????i and interpret the impact of ????.
For the normal linear model, explain how (a) the ridge-regression estimates relate to Bayesian posterior means when {????j} have independent N(0, ????2)distributions, (b) the lasso estimates relate to Bayesian posterior modes when {????j} have independent Laplace (double-exponential) distributions
Consider the linear model yi = ????1xi1 + ????2xi2 + ⋯ + ????40xi,40 + ????i with ????1 = 1 and ????2 = ⋯ = ????40 = 0, where xij = ui + vj with {ui}, {vj}, and{????i} being iid N(0, 1) random variables.a. Find the correlation between y and x1, y and xj for j ≠ 1, and xj and xk for j ≠ k,
Refer to the Dirichlet prior distribution introduced for multinomial parameters in Section 11.2.2. Explain why a multivariate normal prior for multinomial logits provides greater flexibility.
Refer to Copas’s kernel smoother (11.1) for binary regression, with ????(u) =exp(−u2∕2).a. To describe how close this estimator falls at a particular x value to a corresponding smoothing in the population, use the delta method to show that an estimated asymptotic variance is̃????(x)[1 −
When p > n, why is backward elimination not a potential method for selecting a subset of explanatory variables?
Sometimes nonlinear regression models can be converted to ordinary GLMs by employing a link function for the mean response and/or transforming the explanatory variables. Explain how this could be done for the normalresponse models (a) E(yi) = ????0 exp(????1xi) (an exponential growth model),(b)
Refer to the form of iterations for the Gauss–Newton algorithm described in Section 11.3.7. Show that an analogous formula????̂ − ???? = (XTX)−1XT????holds for the ordinary linear model, where ???? = y − ???? is a “true residual.”
Randomly generate nine observations satisfying a normal linear model by taking xi ∼ N(50, 20) and yi = 45.0 + 0.1xi + ????i with ????i ∼ N(0, 1). Now add to the dataset a contaminated outlying observation 10 having x10 = 100, y10 = 100. Fit the normal linear model to the 10 observations using
For the data analyzed in Section 11.1.8 on risk factors for endometrial cancer, compare the results shown there with those you obtain using the lasso.
For the horseshoe crab data introduced in Section 1.5.1 and modeled in Section 7.5, suppose you use graphics to investigate how a female crab’s number of male satellites depends on the width (in centimeters) of the carapace shell of the crab. If you plot the response counts of satellites against
For the horseshoe crab dataset, Exercise 5.32 used logistic regression to model the probability that a female crab has at least one male satellite. Plot these binary response data against the crab’s carapace width. Also plot a curve based on smoothing the data using a kernel smoother or a
For the Housing.dat file analyzed in Sections 3.4 and 4.7, use methods of this chapter to describe how the house selling price depends on its size.
Continue the previous exercise, now using all the explanatory variables.
2.4. Given the following FIR filter impulse responses what are their H(z) and H(z-1)? What are the zeros of the filters? Express the transfer functions in terms of zeros and poles? Prove that these filters have linear phases.a. h[n] = {0.5, 0.5}b. h[n] = {0.5, 0.0, -0.5}c. h[n] = {0.5, 0.0, 0.5}d.
2.3. Which of the FIR filters defined by the following impulse responses are linear phase filters?a. h[n] = {0.2, 0.3, 0.3, 0.2}b. h[n] = {0.1, 0.2, 0.2, 0.1, 0.2, 0.2}c. h[n] = {0.2, 0.2, 0.1, 0.1, 0.2, 0.2}d. h[n] = {0.05, 0.15, 0.3, -0.15,-0.15}e. h[n] = {0.05, 0.3, 0.0, -0.3, -0.05}
2.2. Show how the z-Transform can become the DFT.
2.5. Which of the following vector pairs are orthogonal or orthonormal?a. [1, -3, 5]T and [-1, -2, -1]Tb. [0.6, 0.8]T and [4, -3]Tc. [0.8, 0.6]T and [0.6, -0.8]Td. [1, 2, 3]T and [4, 5, 6]T
2.6. Which of the following matrices are Toeplitz?What is special about matricesc, d,e, and f? 3 2 1 2 3 3 2 1 [1 2 3] a. 4 3 2, b. 12 3, c. 111, d. 2 12,e. 2 12 543 1 2 3 (1+j) (1-j) f. (1-j) 1 [(1+j) (1-j) 1 (1+j) 1 1 1 1 2 3 3 2 1
2.7. Which of the following matrices are orthogonal? Compute the matrix inverses of those that are orthogonal. 0 10] 1 1 0 0 [12 3 a. 0 0 1 , b. 0 1 0 00 . C. , d. 0 20 , e. 21 2 0 0 0 0 3 3 2
2.8. Why are ergodic processes important?
2.9. Find the eigenvalues of the following 2 x 2 Toeplitz matrix,Find the eigenvectors for a = 4 and b = 1. A a b b a
7.8. Compute the MV spectral estimate of a random phase complex exponential process in white noise, defined by,[ ][ ] has a variance of .is a random variable uniformly distributed over , .where :[ ] [ ]2 11 wj jn w n x n A e e w nσφ π πφ θ−= +Use Woodbury’s identity to find the required
2.10. Compute the rounding quantization error variance for an Analogue to Digital Converter (ADC) with a quantization interval equal to Δ. Assume that the signal distribution is uniform and that the noise is stationary white noise.v2.11. What is the mean and autocorrelation of the random phase

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Fundamentals Of Linear State Space Systems 1st Edition John S Bay - Solutions

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