Spatial Analysis 1st Edition John T. Kent, Kanti V. Mardia - Solutions

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A glass bottle manufacturing company has recorded data on the average number of defects per 10,000 bottles due to stones (small pieces of rock embedded in the bottle wall) and the number of weeks since the last furnace overhaul. The data are shown below.Defects per 10,000 Weeks Defects per 10,000
Consider the data shown below. Construct a scatter diagram and suggest an appropriate form for the regression model. Fit this model to the data and conduct the standard tests of model adequacy x 10 15 18 12 9 8 11 6 y 0.17 0.13 0.09 0.15 0.20 0.21 0.18 0.24
The data shown below present the average number of surviving bacteria in a canned food product and the minutes of exposure to 300 ° F heat.Number of Bacteria Minutes of Exposure 175 1 108 2 95 3 82 4 71 5 50 6 49 7 31 8 28 9 17 10 16 11 11 12a. Plot a scatter diagram. Does it seem likely that a
The following table gives the vapor pressure of water for various temperatures.Temperature ( ° K)Vapor Pressure (mm Hg)273 4.6 283 9.2 293 17.5 303 31.8 313 55.3 323 92.5.333 149.4 343 233.7 353 355.1 363 525.8 373 760.0a. Plot a scatter diagram. Does it seem likely that a straight - line model
Byers and Williams ( “ Viscosities of Binary and Ternary Mixtures of Polyaromatic Hydrocarbons, ” Journal of Chemical and Engineering Data , 32 , 349 – 354, 1987) studied the impact of temperature (the regressor) on the viscosity (the response) of toluene - tetralin blends. The following
Consider the wine quality of young red wines data in Table B.19 . For the purposes of this exercise, ignore regressor x1 . Perform a thorough residual analysis of these data. What conclusions do you draw from this analysis?
Consider the life expectancy data given in Problem 3.16 and Table B.16 .a. For each model construct a normal probability plot of the residuals from the full model. Does there seem to be any problem with the normality assumption?b. For each model construct and interpret a plot of the residuals
Consider the air pollution and mortality data given in Problem 3.15 and Table B.15 .a. Construct a normal probability plot of the residuals from the full model.Does there seem to be any problem with normality assumption?b. Construct and interpret a plot of the residuals versus the predicted
Consider the advertising data given in Problem 2.18.a. Construct a normal probability plot of the residuals from the full model.Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.
Table B.14 contains data on the transient points of an electronic inverter.Using only the regressors x1 , . . . , x4 , fi t a multiple regression model to these data.a. Investigate the adequacy of the model.b. Suppose that observation 2 was recorded incorrectly. Delete this observation, refi t the
Consider the test for lack of fi t. Find E ( MSPE ) and E ( MSLOF ).
Myers Montgomery and Anderson - Cook ( Response Surface Methodology 3 rd edition, Wiley, New York, 2009) discuss an experiment to determine the infl uence of fi ve factors:x1 — acid bath temperature x2 — cascade acid concentration x3 — water temperature x4 — sulfi de concentration x5 —
Derringer and Suich ( “ Simultaneous Optimization of Several Response Variables, ” Journal of Quality Technology , 1980) studied the relationship of an abrasion index for a tire tread compound in terms of three factors: x1 , hydrated silica level; x2 , silane coupling agent level; and x3 ,
Coteron, Sanchez, Martinez, and Aracil ( “ Optimization of the Synthesis of an Analogue of Jojoba Oil Using a Fully Central Composite Design, ” Canadian Journal of Chemical Engineering , 1993) studied the relationship of reaction temperature x1 , initial amount of catalyst x2 , and pressure x3
In Problem 3.14, you were asked to fi t a model to the kinematic viscosity data in Table B.10 .a. Construct a normality plot of the residuals from the full model. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted
In Problem 3.12, you were asked to fi t a model to the clathrate formation data in Table B.8 .a. Construct a normality plot of the residuals from the full model. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted
In Problem 3.9, you were asked to fi t a model to the tube - fl ow reactor data in Table B.6 .a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.c.
Problems 2.4 and 3.5 asked you to fi t two different models to the gasoline mileage data in Table B.3 . Calculate the PRESS statistic for these two models.Based on this statistic, which model is most likely to provide better predictions of new data?
Problem 3.8 asked you to fi t two different models to the chemical process data in Table B.5 . Perform appropriate residual analyses for both models.Discuss the results of these analyses. Calculate the PRESS statistic for both models. Do the residual plots and PRESS provide any insight regarding
Consider the simple linear regression model fi t to the tank pressure and volume data in Problem 2.16.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted
Consider the simple linear regression model fi t to the toluene – tetralin viscosity data in Problem 2.15.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the
Consider the simple linear regression model fi t to the copolyester viscosity data in Problem 2.14.a. Construct a normal probability plot of the unscaled residuals. Does there seem to be any problem with the normality assumption?b. Repeat part a using the studentized residuals. Is there any
Consider the simple linear regression model fi t to the ozone data in Problem 2.13.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.c. Plot the
Consider the simple linear regression model fi t to the steam plant data in Problem 2.12.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.c.
Consider the simple linear regression model fi t to the weight and blood pressure data in Problem 2.10.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted
Consider the simple linear regression model fi t to the oxygen purity data in Problem 2.7.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.
Consider the multiple regression model fi t to the house price data in Problem 3.7.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.c. Construct
Consider the multiple regression model fi t to the gasoline mileage data in Problem 3.5.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.c.
Consider the simple linear regression model fi t to the solar energy data in Problem 2.3.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the predicted response.
Consider the multiple regression model fi t to the National Football League team performance data in Problem 3.1.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the
Consider the simple regression model fi t to the National Football League team performance data in Problem 2.1.a. Construct a normal probability plot of the residuals. Does there seem to be any problem with the normality assumption?b. Construct and interpret a plot of the residuals versus the
Consider the hypotheses for the general linear model, which are of the form H H 0 1 : ,: Tc Tc b b = ≠where T is a q × p matrix of rank q. Derive the appropriate F statistic under both the null and alternative hypothesis.
Let Rj 2be the coeffi cient of determination when we regress the j th regressor on the other k − 1 regressors. Show that the j th variance infl ation factor may be expressed as 11 2 − Rj
Consider a correctly specifi ed regression model with p terms, including the intercept. Make the usual assumptions about ε . Prove that Var y p ˆi in( ) ==∑1 2 σ
Suppose we fi t the model y = X1β2 + ε when the true model is actually given by y = X1β2 + X2β2 + ε . For both models, assume E ( ε ) = 0 and Var( ε ) = σ2 I .Find the expected value and variance of the ordinary least - squares estimate,ˆb1. Under what conditions is this estimate unbiased?
Consider the following two models where E ( ε ) = 0 and Var( ε ) = σ2 I :Model A: y = X 1 β 1 + εModel B: yX X = 11 22 ′b be + +Show that R R A B 2 2 ≤ .
Let xj be the j th row of X , and X−j be the X matrix with the j th row removed.Show that Var ˆβ σ j j j jj jj j j ⎡⎣ ⎤⎦ = ⎡ ′ − ′ ′ ( ) ′ ⎣ ⎤⎦ − −−−− 2 1 xx xX X X X x
Constrained least squares . Suppose we wish to fi nd the least - squares estimator of β in the model y = Xβ + ε subject to a set of equality constraints on β , say Tβ = c . Show that the estimator is b b = + ( ) ′ ′ ′ ⎡ ( ) b ⎣ ⎤⎦ ( − ) − − − ˆ ˆ XX T T XX T c T 1 1 1
Prove that R2 is the square of the correlation between y and yˆ .
For the multiple linear regression model, show that SSR ( β ) = y ′ Hy .
Show that the residuals from a linear regression model can be expressed as e = ( I − H ) ε . [Hint: Refer to Eq. (3.15b) .]
Consider the multiple linear regression model y = Xβ + ε . Show that the least -squares estimator can be written asˆb b = + e = ( ) ′ ′ − R R XX X where 1
For the simple linear regression model, show that the elements of the hat matrix areDiscuss the behavior of these quantities as x i moves farther from x, SEX 1(x-x)(x-x) and h 1, (x,-x) + hij n SEX n
Prove that the matrices H and I − H are idempotent, that is, HH = H and( I − H )( I − H ) = I − H .
Show that Var( ) y H ˆ = σ2 .
Suppose that we have two independent samples, sayTwo models can be fi t to these samples, y xi n i ii =+ + = ββ ε 0 1 1 2 2 , ,, , …y x in n n n i ii =+ + =+ + + γγ ε 0 1 , , ,, 1 1 12 1 2 …a. Show how these two separate models can be written as a single model.b. Using the result in
Consider the multiple linear regression model y xxxx =+ + ββ β β β ε 0 11 22 33 44 + + +Using the procedure for testing a general linear hypothesis, show how to testa. H0 : β1 = β2 = β3 = β4 = βb. H0 : β1 = β2 , β3 = β4c. H01 2 3 1 2 2 4 2 0
Show that an alternate computing formula for the regression sum of squares in a linear regression model is SS y ny i in R = −=∑ ˆ 2 2 1
Suppose that a linear regression model with k = 2 regressors has been fi t to n = 25 observations and R2= 0.90.a. Test for signifi cance of regression at α = 0.05. Use the results of the previous problem.b. What is the smallest value of R2 that would lead to the conclusion of a signifi cant
Show that an equivalent way to perform the test for signifi cance of regression in multiple linear regression is to base the test on R2 as follows: To test H0 :β1 = β2 = . . . = βk versus H1 : at least one βj ≠ 0, calculate F Rn p k R 0 22 1 = ( ) −( ) −and to reject H0 if the computed
A chemical engineer is investigating how the amount of conversion of a product from a raw material ( y ) depends on reaction temperature ( x1 ) and reaction time ( x2 ). He has developed the following regression models:1. y x ˆ = + 100 0 2 4 . 1 2 + x 2. y x ˆ =+ + + 95 0 15 3 1 . 1 2 12 x x x
Rossman [ 1994 ] presents an interesting study of average life expectancy of 40 countries. Table B.16 gives the data. The study has three responses: LifeExp is the overall average life expectancy. LifeExpMale is the average life expectancy for males, and LifeExpFemale is the average life expectancy
McDonald and Ayers [ 1978 ] present data from an early study that examined the possible link between air pollution and mortality. Table B.15 summarizes the data. The response MORT is the total age - adjusted mortality from all causes, in deaths per 100,000 population. The regressor PRECIP is the
The kinematic viscosity of a certain solvent system depends on the ratio of the two solvents and the temperature. Table B.10 summarizes a set of experimental results.a. Fit a multiple linear regression model relating the viscosity to the two regressors.b. Test for signifi cance of regression. What
An engineer studied the effect of four variables on a dimensionless factor used to describe pressure drops in a screen - plate bubble column. Table B.9 summarizes the experimental results.a. Fit a multiple linear regression model relating this dimensionless number to these regressors.b. Test for
A chemical engineer studied the effect of the amount of surfactant and time on clathrate formation. Clathrates are used as cool storage media. Table B.8 summarizes the experimental results.a. Fit a multiple linear regression model relating clathrate formation to these regressors.b. Test for signifi
An engineer performed an experiment to determine the effect of CO 2 pressure, CO 2 temperature, peanut moisture, CO 2 fl ow rate, and peanut particle size on the total yield of oil per batch of peanuts. Table B.7 summarizes the experimental results.a. Fit a multiple linear regression model relating
The quality of Pinot Noir wine is thought to be related to the properties of clarity, aroma, body, fl avor, and oakiness. Data for 38 wines are given in Table B.11.a. Fit a multiple linear regression model relating wine quality to these regressors.b. Test for signifi cance of regression. What
The concentration of NbOCl 3 in a tube - fl ow reactor as a function of several controllable variables is shown in Table B.6.a. Fit a multiple regression model relating concentration of NbOCl 3 ( y ) to concentration of COCl 2 , ( x1 ) and mole fraction ( x4 ).b. Test for signifi cance of
The data in Table B.5 present the performance of a chemical process as a function of sever controllable process variables.a. Fit a multiple regression model relating CO 2 product ( y ) to total solvent( x6 ) and hydrogen consumption ( x7 ).b. Test for signifi cance of regression. Calculate R2 and
Consider the house price data in Table B.4.a. Fit a multiple regression model relating selling price to all nine regressors.b. Test for signifi cance of regression. What conclusions can you draw?c. Use t tests to assess the contribution of each regressor to the model.Discuss your fi ndings.d. What
In Problem 2.4 you were asked to compute a 95% CI on mean gasoline prediction interval on mileage when the engine displacement x1 = 275 in. 3 Compare the lengths of these intervals to the lengths of the confi dence and prediction intervals from Problem 3.5 above. Does this tell you anything about
Consider the gasoline mileage data in Table B.3.a. Fit a multiple linear regression model relatmg gasoline mileage y (miles per gallon) to engine displacement x1 and the number of carburetor barrels x6 .b. Construct the analysis - of - variance table and test for signifi cance of regression.c.
Reconsider the National Football League data from Problem 3.1. Fit a model to these data using only x7 and x8 as the regressors.a. Test for signifi cance of regression.b. Calculate R2 and RAdj 2 . How do these quantities compare to the values computed for the model in Problem 3.1, which included an
Refer to Problem 3.1.a. Find a 95% CI on β7 .b. Find a 95% CI on the mean number of games won by a team when x2 = 2300, x7 = 56.0, and x8 = 2100.
Using the results of Problem 3.1, show numerically that the square of the simple correlation coeffi cient between the observed values y i and the fi tted values yˆi equals R2.
Consider the National Football League data in Table B.1.a. Fit a multiple linear regression model relating the number of games won to the team ’ s passing yardage ( x2 ), the percentage of rushing plays ( x7 ), and the opponents ’ yards rushing ( x8 ).b. Construct the analysis - of - variance
Consider the least - squares residuals e yy i ii = − ˆ , i = 1, 2, . . . , n , from the simple linear regression model. Find the variance of the residuals Var( e i ). Is the variance of the residuals a constant? Discuss.
Consider the simple linear regression model y x =+ + ββ ε 0 1 where the intercept β0 is known.a. Find the least - squares estimator of β1 for this model. Does this answer seem reasonable?b. What is the variance of the slope ˆ (β1 ) for the least - squares estimator found in part a?c. Find a
Consider the data in Problem 2.12 and assume that steam usage and average temperature are jointly normally distributed.a. Find the correlation between steam usage and monthly average ambient temperature.b. Test the hypothesis that ρ = 0.c. Test the hypothesis that ρ = 0.5.d. Find a 99% CI for ρ .
Suppose that we are fi tting a straight line and wish to make the standard error of the slope as small as possible. Suppose that the “ region of interest ” for x is − 1 ≤ x ≤ 1. Where should the observations x1 , x2 , . . . , x n be taken? Discuss the practical aspects of this data
Consider the maximum - likelihood estimator σ 2 of σ2 in the simple linear regression model. We know that σ 2 is a biased estimator for σ2.a. Show the amount of bias in σ 2.b. What happens to the bias as the sample size n becomes large?
Suppose that we have fi t the straight - line regression model ˆ ˆ ˆ y x = + β β 0 11 but the response is affected by a second variable x2 such that the true regression function is Ey x x ( ) =+ + ββ β 0 11 22a. Is the least - squares estimator of the slope in the original simple linear
Consider the simple linear regression model y = β0 + β1x + ε , with E ( ε ) = 0, Var( ε ) = σ2, and ε uncorrelated.a. Show that E MS S ( ) R = + σ β xx 2 1 2 .b. Show that E ( MSRes ) = σ2.
Consider the simple linear regression model y = β0 + β1x + ε , with E ( ε ) = 0, Var( ε ) = σ2, and ε uncorrelated.a. Show that Cov ˆ , ˆ ββ σ 0 1 2 ( ) = −x Sxx .b. Show that Cov(y, β1 ) = 0.
Repeat Problem 2.20 using only 10 observations in each samle, drawing one observation from each level x = 1, 2, 3, . . . , 10. What impact does using n = 10 have on the questions asked in Problem 2.17? Compare the lengths of the CIs and the appearance of the histograms.
Consider the simple linear regression model y = 50 + 10 x + ε where ε is NID(0, 16). Suppose that n = 20 pairs of observations are used to fi t this model.Generate 500 samples of 20 observations, drawing one observation for each level of x = 1, 1.5, 2, . . . , 10 for each sample.a. For each
On March 1, 1984, the Wall Street Journal published a survey of television advertisements conducted by Video Board Tests, Inc., a New York ad - testing company that interviewed 4000 adults. These people were regular product users who were asked to cite a commercial they had seen for that product
Atkinson ( Plots, Transformations, and Regression , Clarendon Press, Oxford, 1985 ) presents the following data on the boiling point of water ( ° F) and barometric pressure (inches of mercury). Construct a scatterplot of the data and propose a model that relates a model that relates boiling point
Carroll and Spiegelman ( “ The Effects of Ignoring Small Measurement Errors in Precision Instrument Calibration, ” Journal of Quality Technology , 18 , 170 –173, 1986) look at the relationship between the pressure in a tank and the volume of liquid. The following table gives the data. Use an
Byers and Williams ( “ Viscosities of Binary and Ternary Mixtures of Polynomatic Hydrocarbons, ” Journal of Chemical and Engineering Data , 32 , 349 – 354, 1987) studied the impact of temperature on the viscosity of toluene –tetralin blends. The following table gives the data for blends
Hsuie, Ma, and Tsai ( “ Separation and Characterizations of Thermotropic Copolyesters of p - Hydroxybenzoic Acid, Sebacic Acid, and Hydroquinone, ”Journal of Applied Polymer Science , 56 , 471 – 476, 1995) study the effect of the molar ratio of sebacic acid (the regressor) on the intrinsic
Davidson ( “ Update on Ozone Trends in California ’ s South Coast Air Basin, ” Air and Waste , 43 , 226, 1993) studied the ozone levels in the South Coast Air Basin of California for the years 1976 – 1991. He believes that the number of days the ozone levels exceeded 0.20 ppm (the
The number of pounds of steam used per month at a plant is thought to be related to the average monthly ambient temperature. The past year ’ s usages and temperatures follow.Month Temperature Usage/l000 Month Temperature Usage/l000 Jan. 21 185.79 Jul. 68 621.55 Feb. 24 214.47 Aug. 74 675.06 Mar.
Consider the weight and blood pressure data in Problem 2.10. Fit a no -intercept model to the data and compare it to the model obtained in Problem 2.10. Which model would you conclude is superior?
The weight and systolic blood pressure of 26 randomly selected males in the age group 25 – 30 are shown below. Assume that weight and blood pressure(BP) are jointly normally distributed.a. Find a regression line relating systolic blood pressure to weight.b. Estimate the correlatiou coeffi
Consider the soft drink delivery time data in Table 2.9. After examining the original regression model (Example 2.9), one analyst claimed that the model was invalid because the intercept was not zero. He argued that if zero cases were delivered, the time to stock and service the machine would be
Consider the oxygen plant data in Problem 2.7 and assume that purity and hydrocarbon percentage are jointly normally distributed random variables.a. What is the correlation between oxygen purity and hydrocarbon percentage?b. Test the hypothesis that ρ = 0.c. Construct a 95% CI for ρ .
The purity of oxygen produced by a fractional distillation process is thought to be related to the percentage of hydrocarbons in the main condensor of the processing unit. Twenty samples are shown below.Purily (%)Hydrocarbon(%) Purily (%) Hydrocarbon (%)86.91 1.02 96.73 1.46 89.85 1.12 99.42 1.55
Table B.4 presents data for 27 houses sold in Erie, Pennsylvania.a. Fit a simple linear regression model relating selling price of the house to the current taxes ( x1 ).b. Test for signifi cance of regression.c. What percent of the total variability in selling price is explained by this model?d.
Consider the gasoline mileage data in Table B.3 . Repeat Problem 2.4(partsa,b, andc) using vehicle weight x10 as the regressor variable. Based on a comparison of the two models, can you conclude that x1 is a better choice of regressor than x10 ?
Table B.3 presents data on the gasoline mileage performance of 32 different automobiles.a. Fit a simple linear regression model relating gasoline mileage y (miles per gallon) to engine displacement xl (cubic inches).b. Construct the analysis - of - variance table and test for signifi cance of
Table B.2 presents data collected during a solar energy project at Georgia Tech.a. Fit a simple linear regression model relating total heat fl ux y (kilowatts)to the radial defl ection of the defl ected rays x4 (milliradians).b. Construct the analysis - of - variance table and test for signifi
Suppose we would like to use the model developed in Problem 2.1 to predict the number of games a team will win if it can limit opponents ’ yards rushing to 1800 yards. Find a point estimate of the number of games won when x8 = 1800. Find a 90% prediction interval on the number of games won.
Table B.1 gives data concerning the performance of the 26 National Football League teams in 1976. It is suspected that the number of yards gained rushing by opponents ( x8 ) has an effect on the number of games won by a team ( y ).a. Fit a simple linear regression model relating games won y to
8.8 This exercise generalizes Example 8.1 in two ways. First, the stationary covariance function ????(h) is replaced by a limiting intrinsic covariance function. If the range parameter ???? tends to ∞ and ????(h) is multiplied by ????, then a limiting IRF-0 random field is obtained with an
8.7 Suppose the mean of the random field in Example 8.1 is assumed known.Show that the kriging variance takes the same form as in (8.15), but without the final term. Deduce that the kriging variance is still maximized at t = 0, so that the optimal location of the new site is unchanged.
8.6 Consider a tensor product covariance model for a zero-mean bivariate stationary spatial process on ℝ1 where cov(Xu(t), X????(t + h)) = cu???? exp(−0.5|h|), u, ???? = 1, 2, h ∈ ℝ1, and C =[1 b b 1], b = 0.7.Suppose the following observations are available, x1(0), x1(1), x2(0), and that
8.5 (Mardia and Goodall, 1993). Using the notation from Section 8.5, assume the tensor product model (8.4) for processes X1(t), ..., Xq(t). Given data on each process at sites, t1, ..., tn, confirm that the covariance matrix of the data can be written in the form Σ∗∗ = A ⊗ Σ. Show that in
8.4 Suppose the signal S in Exercise 8.2 comes from a stationary Gaussian process with mean ???? and covariance function ????(h), observed at sites t1, ..., tn, with ????(0) = ????2. Show that the elements of the observation vector Z have a constant mean and covariances, which can be expressed in

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Spatial Analysis 1st Edition John T. Kent, Kanti V. Mardia - Solutions

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