Modeling of Combustion Systems A Practical Approach 12 doc

293 4 Analysis of Nonideal Data Chapter Overview In this chapter, we show how to obtain information from less than ideal data. Thus far, we have studied statistically cognizant exper- imental designs yielding balanced, symmetrical data with ideal statistical properties. Statistical experimental design (SED) has great advantages, and whenever we have an opportunity to use SED, we should. However, there will be many occasions when the data we receive are historical, or from plant operating history, or other nonideal sources with much less desirable statistical properties. But even poorly designed (or nondesigned) experi- ments usually contain recoverable information. On rarer occa- sions, we may not be able to draw firm conclusions, but even this is preferable to concluding falsehoods unawares. We begin our analysis with plant data. With the advent of the distributed control systems (DCSs), plant data are ubiquitous. However, they almost certainly suffer from maladies that lead to correlated rather than independent errors. Also, bias due to an improper experimental design or model can lead to nonrandom errors. In such cases, a mechanical application of ANOVA and statistical tests will mislead; F ratios will be incorrect; coefficients will be biased. Since furnaces behave as integrators, we look briefly at some features of moving average processes and lag plots for serial correlation, as well as other residuals plots. The chapter shows how to orthogonalize certain kinds of data sets using source and target matrices and, more importantly, eigenvalues and eigen- vectors. Additionally, we discuss canonical forms for interpreting multidimensional data and overview a variety of helpful statistics to flag troubles. Such statistics include the coefficient of determina- tion (r 2 ), the adjusted coefficient of determination (r A 2 ), the prediction sum of squares (PRESS) statistic and a derivative, r P 2 , and variance inflation factors (VIFs) for multicollinear data. We also introduce the hat matrix for detecting hidden extrapolation. In other cases, the phenomena are so complex or theory so lacking that we simply cannot formulate a credible theoretical or © 2006 by Taylor & Francis Group, LLC 294 Modeling of Combustion Systems: A Practical Approach even semiempirical model. In such a case, it is preferable to produce some kind of model. For this purpose, we shall use purely empirical models, and we show how to derive them beginning with a Taylor series approximation to the true but unknown function. This chapter also examines categorical factors and shows how to analyze designs with restricted randomization such as nested and split-plot designs. This requires rules for deriving expected mean squares, and we provide them. On occasion, the reader may need to fit parameters for categorical responses, and we touch on this subject as well. The last part of the chapter concerns mixture designs for fuel blends and how to simulate complex fuels with many fewer com- ponents. This requires a brief overview of fuel chemistry, which we present. We conclude by showing how to combine mixture and factorial designs and fractionate them. 4.1 Plant Data Plant data typically exhibit serial correlation, often strongly. Serial correlation indicates errors that correlate with run order rather than the random errors we subsume in our statistical tests. Consider a NOx analyzer attached to a municipal solid waste (MSW) boiler, for example. Suppose it takes 45 min- utes for the MSW to go from trash to ash, after which the ash leaves the Then the natural burning cycle of the unit is roughly 45 minutes or so. If we pull an independent NOx sample every 4 hours, it is unlikely that there will be any correlation among the data. Except in the case of an obvious malfunction, the history of the boiler 4 hours earlier will have no measurable effect on the latest sample. However, let us investigate what will happen by merely increasing the sampling frequency. 4.1.1 Problem 1: Events Too Close in Time DCS units provide a steady stream of continual (and correlated) information. Suppose we analyze NOx with a snapshot every hour. Will one reading be correlated with the next? How about every minute? What about every sec- ond? Surely, if the previous second’s analysis shows high NOx, we would expect the subsequent second to be high as well. In other words, data that are very close in time exhibit positive serial correlation. Negative serial correlation is possible, but rarer in plant environments. However, it can occur in the plant when one effect inhibits another. Nor is this the only cause of serial correlation. © 2006 by Taylor & Francis Group, LLC boiler (Figure 4.1). Analysis of Nonideal Data 295 4.1.2 Problem 2: Lurking Factors Lurking factors are an important cause of serial correlation. For example, O 2 concentration affects both NOx and CO emissions. If we were so naïve as to neglect to measure the O 2 level, we could easily induce a serial correlation. For example, air temperature correlates inversely to airflow, and the former relates to a diurnal cycle. Therefore, we can also expect airflow with fixed damper positions, e.g., most refinery burners, to also show a diurnal cycle. Every effect must have a cause. If we account for all the major sources of fixed variation, then the multiple minor and unknown sources should dis- tribute normally according to the central limit theorem and collect in our error term. Therefore, it behooves us to find every major cause for our response because major fixed effects in the errors can result in correlated rather than normally distributed errors. 4.1.3 Problem 3: Moving Average Processes If we consider the boiler furnace as an integrator, then flue gas emissions and components comprise a moving average process — and moving averages are FIGURE 4.1 A municipal solid waste boiler. It takes roughly 45 minutes for the trash-to-ash cycle. This particular unit is equipped with ammonia injection to reduce NOx. (From Baukal, C.E., Jr., Ed., The John Zink Combustion Handbook, CRC Press, Boca Raton, FL, 2001.) UNDERGRATE COMBUSTION AIR NO x REDUCTION ZONE AMMONIA INJECTION FEED CHUTE COMBUSTION ZONE STOKER GRATE ASH EXHAUST STACK FLUE GAS MUNICIPAL SOLID WASTE © 2006 by Taylor & Francis Group, LLC 296 Modeling of Combustion Systems: A Practical Approach highly and positively correlated. To see this, consider a random distribution — a plot of x k against the next data point in time, x k+1 The first plot shows 100 nearest neighbors from a uniform random distri- bution plotted one against the other. The data were generated with the Excel™ function RAND( )-0.5, representing a uniform distribution with zero mean between –0.5 and 0.5. The nearest-neighbor plot shows no corre- lation to speak of (r 2 = 0.009), the mean is essentially zero ( = 0.04), and the standard deviation is s = 0.28. These are very close to the expected values for these statistics, and it is not so surprising that random data show no trend when plotted against nearest neighbors. formed a moving average using the 10 nearest neighbors: where k indexes each point sequentially. Note that the correlation of ξ k with ξ k+1 in Figure 4.2b has an r 2 of 84.0% despite being drawn from an originally FIGURE 4.2 uniform random number generator, –0.5 < x < 0.5. The graph plots each data point against its neighbor (x k+1 k 2 the same data as 10-point moving averages. Plotting the moving average data in the same fashion gives noticeably less dispersion (s = 0.09 vs. 0.28) and high correlation, despite the fact that the moving averages comprise uniform random data. In the same way, integrating processes such as combustion furnaces can have emissions with serially correlated errors. (a) Nearest Neighbor Plot, Uniform Random Distribution (b) Nearest Neighbor Plot, Moving Average x k+1 ξ x k 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.2 0.1 0.0 -0.1 -0.2 -0.3 R 2 =0.9% s=±0.28 R 2 =84.0% s=0.09 k ξ k ξ k+1 = Σ x k k=1 10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.3 -0.2 -0.1 0.0 0.1 0.2 y ξ kk k n x= ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = = ∑ 1 10 1 10 © 2006 by Taylor & Francis Group, LLC A moving average with random data. Figure 3.7a shows data from 100 points generated by a vs. x ). The correlation is, as expected, nearly zero (r = 0.009). Figure 3.7b shows (Figure 4.2a). But Figure 4.2b tells a different story. To create the second plot, we per- Analysis of Nonideal Data 297 uniform random population with zero mean. Also note that the standard devi- ation of the process has fallen by a factor of three (from 0.28 to 0.094). The deflation of the standard deviation by a factor of three is not a coincidence, for the denominator in the calculation of standard deviation is , or . However, the mean values for both data sets are virtually iden- tical at ~0.0. Since the mean values are unaffected, we may perform regressions and generate accurate values for the coefficients. However, as the moving average process deflates s, our F test will errantly lead us to score insignificant effects as significant ones. That is, failure to account for serial correlation in the data set before analysis will result in inflated F tests. An analysis showing many factors to be statistically significant is a red flag for the deflation of variance from whatever cause. 4.1.4 Some Diagnostics and Remedies Here are a few things we can do to warn of serial correlation and remedy it: 1. Always check for serial correlation as revealed by an x k vs. x k+1 plot and time-ordered residuals. 2. Make sure that the data are sufficiently separate in time and each run condition sufficiently long to ensure that the samples are inde- pendent. 3. Carefully consider the process, not just the data. Since the serially correlated data have both fixed and random components, the prob- lem becomes assessing which are which. One could make an a priori estimate for a moving average process using a well-stirred model of sufficiently large. 4.1.5 Historical Data and Serial Correlation For historical data, we do not have the privilege of changing how the data were collected. Therefore, we must do our best to note serial correlation and deal with it after the fact. Once we recognize serial correlation, the problem becomes recovering independent errors from correlated ones and using only the former in our F tests. As we have noted, most serial correlation will evaporate if we can identify lurking factors or the actual cause for the correlation. We then put that cause into a fixed effect in the model. If there are cyclical trends, an analysis of batch cycles within the plant may lead to the discovery of a lurking factor. Failing this, one may be able to use time series analysis to extract the actual random error term from the correlated n −1 10 1 3−= © 2006 by Taylor & Francis Group, LLC the furnace per the transient mass balance for the boiler in Chapter 2. Using such results, we could adjust the sampling period to be 298 Modeling of Combustion Systems: A Practical Approach one. 1,2 This is not so easy. Such models fall into some subset of an autore- gressive-integrated-moving average (ARIMA) model, with the moving aver- age (MA) model being the most likely. Time series analysis is a dedicated discipline in its own right. Often one will have to do supplemental experi- ments to arrive at reasonable estimates and models. 4.2 Empirical Models The main subject of this text is semiempirical models, i.e., theoretically derived models with some adjustable parameters. These are always prefer- able to purely empirical models for a variety of reasons, including a greater range of prediction, a closer relation to the underlying physics, and a require- ment for the modeler to think about the system being modeled. But in some cases, we know so little about the system that we are at a loss to know how to begin. In such cases, we shall use a purely empirical model. For the time being, let us presume that we have no preferred form for the model. That is, we have sufficient theoretical knowledge to suspect certain factors, but not their exact relationships to the response. For example, sup- pose we know that oxygen (x 1 ), air preheat temperature (x 2 ), and furnace temperature (x 3 ) affect NOx. We may write the following implicit relation: (4.1) where ξ represents the factors in their original metric and φ is the functional notation. Although we do not know the explicit form of the model, we can use a Taylor series to approximate the true but unknown model. Equation 4.2 represents a general Taylor series: (4.2) Here ξ refers to the factors, subscripted to distinguish among them. We reference the Taylor series to some coordinate center in factor space (a 1 , a 2 , … , a p ), where each coordinate is subscripted per its associated factor. The farther we move from the coordinate center, the more Taylor series terms we require to maintain accuracy. For Equation 4.1, the Taylor series of Equation 4.2, truncated to second order, gives the following equation: y =φξ ξ ξ(, ,) 123 yaaa fn k a k k ==+ ∂ ∂ φξ ξ ξ φ φ ξ ξ( , , , ) ( , , , ) 12 12 −− () + ∂ ∂∂ − () − () + ∂ = ∑ a aa k k p jk aa jjkk jk 1 22 φ ξξ ξξ , φφ ξ ξ ∂ − () + ==< − ∑∑∑ k a kk k p k p jk p k a 2 2 11 1 2! © 2006 by Taylor & Francis Group, LLC Analysis of Nonideal Data 299 Now if we code the factors to ±1 with the transforms given earlier, the Taylor series becomes the simpler Maclaurin series, which by definition is centered at the origin (0, 0, 0): Equations 4.3 and 4.4 give heuristics for the infinite Maclaurin and Taylor series terms, respectively. For our purposes, we will usually truncate them at n ≤ 2: Maclaurin series (4.3) Taylor series (4.4) In the above equations, φ( ) is the functional notation; x and ξ are the independent variables (factors), the former being scaled and centered and the latter not — i.e., in their original or customary metrics; k indexes the factors from 1 to f; f is the number of factors in the model; p indexes the order of the series from 1 to n; n is the overall order of the series (for an infinite series n = ∞), and 0 and a are vectors — the former is a vector of f zeros and the latter a vector of f constant terms (a 1 , a 2 , … , a f ) T . For nonlinear models, when n < ∞, the series is no longer exact but approx- imate. In such a case we replace the equality (=) by an approximate equality (≈). We illustrate the use of Equations 4.3 and 4.4 with an example. Example 4.1 Derivation of the Maclaurin Series for Two Factors Problem statement: Use Equations 4.3 and 4.4 to derive the Maclaurin and Taylor series for , truncated to third order. What would the corresponding fitted equation look like? y aaa a a aa ≈ + ∂ ∂ − () + ∂ ∂ − () φ φ ξ ξ φ ξ ξ(, , ) 123 1 11 2 22 12 ++ ∂ ∂ − () + ∂ ∂∂ − () − φ ξ ξ φ ξξ ξξ 3 33 2 12 1122 3 12 a aa a aa , (() + ∂ ∂∂ − () − () + ∂ ∂∂ 2 13 1133 2 23 13 φ ξξ ξξ φ ξξ aa a aa , 223 1 2233 2 1 2 11 2 2 2 , ! a a aa a ξξ φ ξ ξ φ − () − () ∂ ∂ − () + ∂ ∂ ξξ ξ φ ξ ξ 2 2 22 2 2 3 2 33 2 23 22 aa aa− () + ∂ ∂ − () ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ !! ⎩⎩ ⎪ ⎪ ⎪ ⎪ ⎪ + y x x xx xx k k k jk jk ≈+ ∂ ∂ + ∂ ∂∂ + ∂ = ∑ φ φφ (,,) , 000 0 1 3 2 00 22 2 0 2 1 3 1 32 2 φ ∂ ==< ∑∑∑ x x k k kkjk ! yxxx px x k k k f = () =+ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ = ∑ φφ φ φ12 0 1 1 ,, () ! $ 0 T ⎥⎥ ⎥ = =∞ ∑ p p n 1 y p a f k a kk k f k = () =+ ∂ ∂ − () = φξ ξ ξ φ φ ξ ξ 12 1 1 ,,, () ! $ a T ∑∑∑ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = =∞ p p n 1 yxx=φ(, ) 12 © 2006 by Taylor & Francis Group, LLC 300 Modeling of Combustion Systems: A Practical Approach Solution: For f = 2 and n = 3, Equation 4.3 becomes Proceeding step by step, we have the following: If we were to evaluate the above equation numerically from a data set, we could fit the third-order model Here, we have grouped the terms in parentheses by overall order. We may derive the Taylor series in the same manner, replacing x k by ξ k – a k and 0 by a k . y px x k k k f p p n ≈+ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = = = = ∑∑ φ φ (,) ! 00 1 0 1 2 1 3 y x x x x≈+ ∂ ∂ + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∂ ∂ φ φφ φ (,) !! 00 1 1 1 2 1 0 1 1 0 2 1 xx x x x x x x x 1 0 1 1 0 2 2 1 0 1 1 0 2 1 3 + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∂ ∂ + ∂ ∂ φ φφ ! ⎛⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 y x x x x x ≈ + ∂ ∂ + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∂ ∂ φ φφ φ (,) ! 00 1 2 1 0 1 1 0 2 2 1 2 00 1 2 2 12 0 12 2 2 2 0 2 2 2 1 3 x xx xx x x+ ∂ ∂∂ + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + φφ ! ∂∂ ∂ + ∂ ∂∂ + ∂ ∂∂ 3 1 3 0 1 3 3 1 2 2 0 1 2 2 3 12 2 0 33 φφ φ x x xx xx xx xxx x x 12 2 3 2 3 0 2 3 + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ φ y x x x x x x ≈ + ∂ ∂ + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∂ ∂ φ φφ φ (,)00 1 0 1 1 0 2 2 1 2 0 1 222 12 0 12 2 2 2 0 1 2 3 22!! + ∂ ∂∂ + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∂ φφ φ xx xx x x ∂∂ + ∂ ∂∂ + ∂ ∂∂x x xx xx xx x 1 3 0 1 33 1 2 2 0 1 2 2 3 12 2 0 32!! φφ 112 23 2 3 0 2 3 23 x x x !! + ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ φ y aaxax axaxxax a ≈ ++ () ++ + () + 01122 111 2 12 1 2 22 2 2 1111 1 3 112 1 2 2 122 1 2 2 222 2 3 xaxxaxxax+++ () ⎧ ⎨ ⎪ ⎩ ⎪ y bb a b b b b ≈ ++ () ++ + () + 01122 111 2 12 1 2 22 2 2 1 ξξ ξ ξξ ξ 111 1 3 112 1 2 2 122 1 2 2 222 2 3 ξξξξξξ+++ () ⎧ ⎨ ⎪ ⎩ ⎪ bbb © 2006 by Taylor & Francis Group, LLC Analysis of Nonideal Data 301 4.2.1 Model Bias from an Incorrect Model Specification In the previous section, we constructed a model comprising a finite number of terms by truncating an infinite Taylor series; therefore, if higher-order derivatives exist, then they will bias the coefficients. We introduced the explore additional considerations. For example, let us suppose that Equation 4.5 gives the true model for NOx: (4.5) where y is the NOx, A and b are constants, and T is the furnace temperature. Further, suppose that due to our ignorance or out of convenience or what- ever, we fit the following (wrong) model: (4.6) where x is centered and scaled per our usual convention, i.e., Then . The Maclaurin series becomes where We may also write this as (4.7) where So long as the series remains infinite, there is a one-to-one correspondence between the coefficients and the evaluated derivatives. However, once we truncate the model, this is no longer strictly true: higher-order derivatives ln yA b T =− ya aT=+ 01 x TT T = − ˆ TTxT=+ ˆ y d dx x d dx xd dT x =+ + + +φ φφ φ 0 0 2 2 0 23 3 0 3 23!! $ φ() ˆ xe A b Tx T = − + ya axax ax ax=+++++ 01 2 2 3 3 4 4 $ aa d dx a d dx a d dT 0 0 1 0 2 2 2 0 3 3 3 0 1 2 1 3 == = =φ φφφ ,, ! , ! $ © 2006 by Taylor & Francis Group, LLC reader to this concept in Chapter 3 beginning with Section 3.4. Here we 302 Modeling of Combustion Systems: A Practical Approach will bias the lower-order coefficients. Yet, near zero, higher-order terms will vanish more quickly than lower-order ones. So, if x is close to zero then the model has little bias. We refer to the error caused by using an incorrect mathematical expression as model bias. At x = 1 each term is weighted by its Maclaurin series coefficient. As x grows beyond 1, then the higher-order terms exert larger and larger influ- ence; so mild extrapolation leads quickly to erroneous results. This would not be the case if the model were correct. Notwithstanding, even for the incorrect empirical model, this bias may be nil so long as we are within the bounds of our original data set (coded to ±1). For x >> 1, we need to add many additional terms for the empirical model to adequately approximate the true model. As x grows larger and larger, we need more and more empirical terms. This is so, despite the fact that the true model comprises only two terms. This is why it is much more preferable to generate a theoretical or semiempirical form rather than a wholly empir- ical one. Nonetheless, an empirical model of second order at most (and usually less) is sufficient for interpolation. In other words, empirical models are very good interpolators and very poor extrapolators. This is true for all models in the sense that we may never have exactly the right model form, but it is especially so for empirical models. Suppose that we could expand our model to comprise an infinite number of terms (which would require an infinite data set to fit). Then we could evaluate the coefficients for Equation 4.7, generating the following normal equations: (4.8) Because we centered x, the sum of the odd powers is zero, but the sum of the even powers is not. Since our approximate model comprises only two terms — a 0 and a 1 of Equation 4.6 — the higher-order terms will bias them. A careful examination of Equation 4.8 shows that the even terms bias a 0 and the odd terms bias a 1 . We are actually fitting an equation something like (4.9) y xy xy xy xy ∑ ∑ ∑ ∑ ∑ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 2 3 4 % ⎟⎟ = ∑∑ ∑∑ ∑∑∑ ∑∑ ∑∑ Nxx xx xxx xx xxx 24 24 246 46 468 $ $ $ $ ∑∑ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ $ %%%%%' a a a a 0 1 2 3 aa 4 % ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ybc cx cx bc cx cx x= +++ () + +++ () 00 2 2 4 4 11 3 3 5 5 $$ © 2006 by Taylor & Francis Group, LLC [...]... often have no real meaning On the other hand, sometimes a linear combination of factors does have meaning and the linear combination may actually be the penultimate factor © 2006 by Taylor & Francis Group, LLC 308 Modeling of Combustion Systems: A Practical Approach For example, kinetic expressions (those determining the rate of appearance or disappearance of a species like NOx or CO) are really a function... about what factors are important and how we can arrange the test matrix to be orthogonal This generates a balanced design having an equal number of high and low values for each factor equidistant from zero in each factor direction, e.g., factorial designs The advantage of using orthogonal designs is that one can examine independent factors with clear meaning and perform a number of statistical tests,... where λ are the latent roots (also called eigenvalues, proper values, or characteristic values) To clarify these concepts, we illustrate with an example © 2006 by Taylor & Francis Group, LLC 310 Modeling of Combustion Systems: A Practical Approach Example 4.2 The Characteristic Equation Using the Trace Operator Problem statement: Given Matrix 4.16b, find the characteristic equation and the eigenvalues... translating axes, we can always simplify the equation to either of two forms: f y = a0 + ∑ f θ k uk + k =1 ∑λ 2 kk k A canonical form (4.3 9a) u B canonical form (4.4 0a) u k f y= ∑λ 2 kk k k =1 Box and Draper4 call the first the A canonical form and the second the B canonical form The A canonical form represents a rotation of axes The B canonical form represents both a rotation and a translation to a new design... “disadvantage” is that it requires up-front thinking Remember Westheimer’s discovery: a couple of months in the laboratory will save you a couple of hours at the library.” © 2006 by Taylor & Francis Group, LLC 306 4.3.1 Modeling of Combustion Systems: A Practical Approach Source and Target Matrices: Morphing Factor Space Suppose we want to convert a source matrix (S) that is nonorthogonal but full rank... Excel does not have a standard function for this, but software such as MathCAD™ does Dedicated statistical software is the best option The procedure can be done in a spreadsheet, but it is tedious, as we show now We may make use of the trace of the matrix to find the eigenvalues The trace of a matrix is the sum of the diagonal elements We may also define traces for higher-order square matrices n ( ) ∑m... Group, LLC Analysis of Nonideal Data 4.4.2 325 Overfit Now r or r2 will always increase as the number of adjustable parameters in the model increases Continuing to add adjustable model parameters eventually results in a condition known as overfit Overfit is the unjustified addition of model parameters resulting in the fitting of random error to false factor effects This is a statistical no-no, because random... If we are far from the design center, the A canonical form will be more useful If we are close to the design center, we shall prefer the B canonical form 4.3.4.1 Derivation of A Canonical Form We may rewrite Equation 4.38 in matrix form as y = a0 + xTa + xTAx where xT = (x1 x2 … xf), aT = (a1 a2 … af), and © 2006 by Taylor & Francis Group, LLC (4.41) Analysis of Nonideal Data 319 ⎛ 2 a1 1 ⎜ 1 A= ⎜ 2⎜... nonsquare, because M = XTX will always be square © 2006 by Taylor & Francis Group, LLC 318 Modeling of Combustion Systems: A Practical Approach 4 The new system, y = Ub, is orthogonal Therefore, we estimate b without design bias 5 If the linear combinations of factors have meaning, they may represent a more parsimonious model and help to identify an important underlying relationship 4.3.4 Canonical Forms... orthogonal s1·s2·s3 factor space for an orthogonal design in distorted t1·t2·t3 space If the distorted space has no physical meaning, we have gained little We see that after the fact, it may be possible to find combinations of the original factors that represent an orthogonal design However, this is a much weaker approach than conducting a proper design in the first place, because the factor combinations often . LLC 312 Modeling of Combustion Systems: A Practical Approach Analytically, one can always find the solutions for polynomials up to fourth order using various procedures.* Each eigenvalue has an associated. 0 TSF== ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ −−−1000 1100 1010 1001 1111 02022 0022 0220 1111 1111 1111 111 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = −−− − − −− ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 1 yb bs bs bs=+ + + 0 1122 3 3 ya at at at=+ + + 0 1122 33 yTaSb== SFa Sb= © 2006 by Taylor & Francis Group, LLC 308 Modeling of Combustion Systems: A Practical Approach For example, kinetic expressions. CHUTE COMBUSTION ZONE STOKER GRATE ASH EXHAUST STACK FLUE GAS MUNICIPAL SOLID WASTE © 2006 by Taylor & Francis Group, LLC 296 Modeling of Combustion Systems: A Practical Approach highly and positively correlated.