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
Analysis of Nonideal Data 303
where b
k
and c
k
are constants accounting for the contributions of the higher-
order derivatives. Again, for 0 < x < 1 the sum of the higher powers will
likely be negligible, and for this reason, empirical models are excellent inter-
polators. Nonetheless, a good theoretical model would eliminate this bias
and would require fewer terms for an adequate fit to the data.
4.2.2 Design Bias
We have seen from the previous section that an improper model specification
is a problem if we extrapolate beyond the bounds of the experimental design.
The proper model derived from theoretical considerations ameliorates this

problem. We have also seen that a purely empirical model will do a very
good job within the design boundaries even if it is wrong. However, even
with the proper model, an improper experimental design may still bias the
coefficients. We refer to errors introduced by a less than ideal X matrix as
design bias. Conversely, proper experimental design can eliminate this bias.
Consider a classical one-factor-at-a-time design given in Table 4.1. Here,
x
1
is the excess oxygen concentration in the furnace, x
2
is the air preheat
temperature (APH) of the combustion air, and x
3
is the furnace temperature,
measured at the bridgewall of the furnace (BWT). Let us represent this factor
space by S:
(4.10a)
Then
(4.10b)
TABLE 4.1
A Classical Design in Three Factors
y = ln(NOx) x
1
= O
2
, % x
2
= APH, °F x
3
= BWT, °F

2.19 1 25 1000
2.70 5 25 1000
2.95 1 500 1000
2.83 1 25 2000
S =
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
1 1 25 1000
1 5 25 1000
1 1 500 1000
1 1 25 2000
⎟⎟
⎟
⎟
⎟
SS
T
=
4 8 575 5000
8 28 675 9000
575 675 251 875 600 000,,
55000 251 875 600 000 7 000 000,,,,
⎛
⎝

⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
© 2006 by Taylor & Francis Group, LLC
304 Modeling of Combustion Systems: A Practical Approach
This is not a promising start, as S
T
S contains not a single zero value;
everything mutually biases everything else. Coding will zero some of the
off-diagonal values. Using the coding transforms, we have
(4.11a)
(4.11b)
(We show a merely to give the coefficient references.) These coded data are
0 1 3
depicts the classical design.
It forms a right-angled tetrahedron in factor space. Since it is neither scaled
nor centered, the edges are not equal lengths, nor does the design center
(centroid of the tetrahedron) coincide with the center of the factor space
(centroid of the cubic region).
Since it is not centered, the design center is not coincident with the center
of the factor space.
centers are now coincident. However, the design is still not orthogonal
because it is not balanced about the coordinate center.

scaled, and since it is balanced about the origin, it also gives an orthogonal
matrix. Let us represent this factor space by T. Then,
(4.12a)
(4.12b)
X =
−−−
−−
−−
−
1050505
1150505
1051505
10



.550515−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟


XXa
T
=
−−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
4000
0311
0131
0113
0
1
a
a
aa
a
2
3

⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
T =
−−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1111
1111

1111
1111
TT
T
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
4
4
4
4
© 2006 by Taylor & Francis Group, LLC
better. At least a is unbiased, but a to a still bias one another. Figure 4.3a
Figure 4.3b is the same design scaled to 0/1 coordinates, but not centered.
Figure 4.3c shows the design in ±1 coordinates. The design and coordinate
Figure 4.3d is an example of a fractional factorial design. It is centered and
Analysis of Nonideal Data 305
If we could transform our design to these coordinates, we would have an
orthogonal design. In fact, we can.
4.3 Ways to Make Designs Orthogonal

We have two basic remedies to make designs orthogonal: we can either
change the design or morph the factor space. Changing the design means
that before we begin the experiment, we think about what factors are impor-
tant 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 inde-
pendent factors with clear meaning and perform a number of statistical tests,
etc. The only “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.”
FIGURE 4.3
Graphical representation of various experimental designs. (a) The classical design in the original
coordinates. The coordinate center does not coincide with the center of the design. (b) The
design coded to 0/1 coordinates. This conformally shrinks the factor directions to uniform
dimension. (c) The design in ±1 coordinates. The design and coordinate center are now coin-
cident (X). (d) A design that is orthogonal and centered in the new coordinates.
(c) Design in ±1 Coordinates
(centered right-angle tetrahedron)
(d) Orthogonal Design in
±1 Coordinates
(regular tetrahedron)
(a) Design in Original Coordinates
(distorted right-angle tetrahedron)
(b) Design in 0/1 Coordinates
(right-angle tetrahedron)
APH: 25 – 500 F
Coordinate
Center
x

O
2
:1– 5%
Design
Center
BWT: 1000 – 2000 F
A Design
Point (1 of 4)
x
s
2
t
3
t
2
s
3
s
t
t
1
x
© 2006 by Taylor & Francis Group, LLC
306 Modeling of Combustion Systems: A Practical Approach
4.3.1 Source and Target Matrices: Morphing Factor Space
Suppose we want to convert a source matrix (S) that is nonorthogonal but
full rank and square, such as Matrix 4.10a, into an orthogonal target matrix
(T), such as Matrix 4.12a. We could postmultiply by some transformation
(F) matrix:
SF = T (4.13)

Then, solving for F we have
F = S
–1
SF = S
–1
T (4.14)
So long as the source matrix is full rank, it will have an inverse. It does
not matter whether we use the original matrix or first transform the matrix
using a linear transform. Accordingly, let us first scale the numbers by coding
the high and low values to 0 and 1, respectively, using the following trans-
form:
(4.15)
Again, we could just have easily used the original matrix, the above 0/1
coding, or the traditional ±1 coding. But as this is a classical design, one-
factor-at-a-time investigations usually proceed from some origin, which is
more conveniently coded as the coordinate center.
(4.16a)
(4.16b)
w
k
kk
kk
=
−
−
−
+−
ξξ
ξξ
S =

⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1000
1100
1010
1001
SS
T
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟

⎟
4111
1100
1010
1001
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 307
We would like to transform S in Matrix 4.16a into T of Matrix 4.12a. We
will do this with a transformation matrix:
(4.17)
To find F, we apply Equation 4.14 and obtain
(4.18)
We now observe that indeed
(4.19)
Before the transformation, we have something like in
s
1
·s
2
·s
3
factor space. After the transformation, we have
in t
1
·t
2
·t
3
factor space. This latter function is orthogonal in t
1

, t
2
, and t
3
. In
other words, if and SF = T, (where F maps s
1
, s
2
, and s
3
onto t
1
,
t
2
, and t
3
) then . So, on the one hand, we have gained independent
coefficients. On the other hand, we are not sure what they mean. In other
words, we are trading a non-orthogonal design in orthogonal s
1
·s
2
·s
3
factor
space for an orthogonal design in distorted t
1
·t

2
·t
3
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 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.
TSF F==
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1000
1100
1010
1001
F =
−−−

⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1111
02 0 2
00 2 2
02 2 0
TSF==
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−−−1000

1100
1010
1001
1111
02022
0022
0220
1111
1111
1111
111
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
−−−
−
−
−−
⎛
⎝

⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1
yb bs bs bs=+ + +
011223
3
ya at at at=+ + +
0112233
yTaSb==
SFa Sb=
© 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 of
collision frequency (Z). But it is not possible to directly observe molecular
collisions and hence Z. However, Z is related to the temperature (T), pressure
(P), and concentration (C) — all increase the collision frequency. Suppose,
for the sake of argument, that the actual production rate of an important
species, y = f(ζ), were actually a function of the log of the collision frequency,
ζ = ln(Z), and that Z is given by Equation 4.20:
(4.20)
Then

where a
0
= ln(b
0
), x
1
= ln(P), , and x
3
= ln(C). So for y = φ(ζ), the most
parsimonious model would actually be a linear combination of x
1
, x
2
, and
x
3
. In such a case, orthogonal components may be useful to spot such rela-
tions in the data. However, we do not want to distort the original factors.
We seek only to rotate the axes to expose these relations. Eigenvectors and
eigenvalues can do this for us.
4.3.2 Eigenvalues and Eigenvectors
One may use eigenvalues and eigenvectors to decompose a matrix into
orthogonal components, and they are the best alternative for that purpose
because they do not distort the factor space as the source–target method may
do. Eigenvalues (
ΛΛ
ΛΛ
) and eigenvectors (K) are defined for a square matrix
(M) of full rank as follows:
KΛΛ

ΛΛ
= MK (4.21)
where K is the eigenvector matrix of M, and
ΛΛ
ΛΛ
is the diagonal eigenvalue
matrix for M. ΛΛ
ΛΛ
is a diagonal matrix of the eigenvalues (λ) of M:
For real symmetric matrices, we shall derive eigenvectors that are orthog-
onal in the strictest sense. That is,
ZbPe C
a
a
T
a
=
−
0
1
2
3
ζ= + + +aaxaxax
0112233
xT
2
1=−
ΛΛ=
⎛
⎝

⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
λ
λ
λ
1
2
'
n
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 309
(4.22)
Theoretically, eigenvalues are solutions of an n
th
-order polynomial (charac-
teristic) equation, where n is the number of rows in the starting matrix,
presuming it is nonsingular and square. Matrix algebra texts give the pro-
cedure.
3
However, the mechanics can become unwieldy and dedicated soft-
ware is really a must for this procedure. Regrettably, 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 spread-
sheet, 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.
In the above equation, we are relying on context to obviate any equivoca-
tion for n, (for M
n
the superscript is an authentic exponent).
Thus, M
2
= MM. However, for t
n
and m
n
kk
, the superscript n is mere nomen-
clature. Once we have t
n
, the characteristic equation and its solutions follow:
(characteristic equation) (4.23)
(coefficient solutions) (4.24)
where λ are the latent roots (also called eigenvalues, proper values, or char-
acteristic values). To clarify these concepts, we illustrate with an example.
KK K K I
T
==
−1
ttr m
nn

kk
n
k
n
=
()
=
=
∑
M
1
MMMMM
n
k
n
==
()()()
=
∏
1
$
c
k
k
k
n
λ
=
∑
=

0
0
c
nj
ct
jn
j
nk
k
nj
njk
=
−
−
⎧
⎨
⎪
⎪
⎩
⎪
⎪
=
−
=
−−
−−
∑
1
1
0

1
if
othherwise
© 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.
Solution: Matrix 4.16b is a full-rank (nonsingular) matrix having
four rows (n = 4). We solve for t
n
in the following manner: Let
Then
Now the traces of each matrix become
M =
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
4111
1100

1010
1001
M
2
4111
1100
1010
1001
4111
1100
1010
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
11001
19555
5211
5121
5112
⎛

⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟⎟
⎟
⎟
⎟
M
3
91 24 24 24
24 7 6 6
24 6 7 6
24 6 6 7

=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
M
4
436 115 115 115
115 31 30 30
115 30 31 30
115 30 30 31
=
⎛
⎝⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟

⎟
⎟
ttr=
()
=+++=M 41117
ttr
22
25=
()
=M
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 311
Solving for the coefficients of the characteristic matrix according
to Equation 4.24, we have
(the n
th
coefficient of the characteristic equation is always 1)
and the characteristic equation is 1 – 7λ + 12λ
2
– 7λ
3
+ λ
4
. Fortu-
nately, this equation factors as (λ
2
– 5λ + 1)(λ- 1)
2
= 0 with the solu-
tions

λ =
Since these are solutions for a single variable, one may also use
numerical procedures such as the goal seek algorithm in Excel to
solve for them. Also, the rational roots (if they exist) will always
be factors of the constant. In our case, the constant is 1, so we
would try ±1, finding 1 to be a double root, as shown above. This
rational roots procedure can often help to factor the equation and
reduce the order of the remainder, simplifying the final solution.
t
3
112=
t
4
529=
c
4
1=
cct
34
7=− =−
cctct
234
2
1
2
1
2
77 125 12=− +
()
=− −

()()
+
()( )
⎡
⎣
⎤
⎦
=
c ctctct
123
2
4
3
1
3
1
3
12 7 7 25 1=− + +
()
=−
()()
+−
()()
+
(()( )
⎡
⎣
⎤
⎦
=−112 7

c ctctctct
012
2
3
3
4
4
1
4
1
4
7 7 12 2=− + + +
()
=− −
()()
+
()
55 7 112 1 529 1
()
+−
()( )
+
()( )
⎡
⎣
⎤
⎦
=
11
521

2
521
2
,, ,
−+
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
© 2006 by Taylor & Francis Group, 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 eigenvector such that
(M – λI)k = 0 (4.25)
where k is an eigenvector. The eigenvectors are not unique in the sense that
any scalar multiple of an eigenvector will itself be an eigenvector. To resolve
this problem, we shall reduce the eigenvectors to unit magnitude, i.e.,
For real, symmetric matrices (the only kind we need to consider in this
text), the eigenvectors are always orthogonal. That is,
j
T
k = 0 (4.26)
where j and k are any two different vectors in the K matrix.

For the case at hand, Equation 4.25 reduces to
(4.27)
We illustrate the procedure for one of the eigenvalues in the next example.
Example 4.3 Finding an Eigenvector from an Eigenvalue
Problem statement: For the Matrix 4.27, we have shown that the
characteristic equation is (λ
2
– 5λ + 1)(λ- 1)
2
= 0, having solutions
λ =
Find the eigenvector associated with the eigenvalue = .
* Any standard mathematical text will have solutions for up to fourth-order polynomials. See,
for example, Gellert, W. et al., Eds., The VNR Concise Encyclopedia of Mathematics, American Edi-
tion, Van Nostrand Reinhold Company, New York, 1977, pp. 80-101. General equations of fifth
order and higher have been proven impossible to solve, though many special equations of arbi-
trary order are solvable; e.g., the triquadratic equation ax
6
+ bx
3
+ c = 0 may be reduced to a
quadratic equation with the substitution u = x
3
.
k
j
j
n
2
0

1
1
=
−
∑
=
MIb−
()
=
−
−
−
−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
λ
λ
λ
λ
λ
4 111

11 0 0
101 0
1001
⎟⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
k

k
k
k
0
1
2
3
0
0
0
0
11
521
2
521
2
,, ,
−+
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
521
2

0 2087
−
≈ .
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 313
Solution: We can find the eigenvector numerically using a spread-
sheet.
Step 1: First, we substitute a selected eigenvalue, e.g., λ = 0.2097:
Step 2: Now we arbitrarily set k
3
= 1,
and reduce the matrix by one column and the eigenvector and so-
lution vector by one row, so that the system becomes soluble.
Step 3: Premultiplying by the inverse of the matrix we have
But we had arbitrarily set k
3
= 1, so the full vector is
and this is an eigenvector associated with λ = 0.2097.
MIk−
()
=λ
3 7913 1 1 1
1 0 7913 0 0
1 0 0 7913 0
1000791
.
.
.
.33
0

0
0
0
0
1
2
3
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟

⎟
=
⎛
⎝
⎜
⎜
k
k
k
k
⎜⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
3 7913 1 1 1
10791300
1 0 0 7913 0
0
1
2
.
.
.
⎛
⎝
⎜

⎜
⎜
⎞
⎠
⎟
⎟
⎟
k
k
k
11
0
0
0
0
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝

⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
3 7913 1 1
1 0 7913 0
1 0 0 7913
0
1
2
.
.
.
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛

⎝
⎜
k
k
k
⎜⎜
⎜
⎞
⎠
⎟
⎟
⎟
=−
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
1
0
0
b
b
b
0

1
2
3 7913 1 1
1 0 7913 0
1 0 0 7913
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
=−
⎛
.
.
.
⎝⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛

⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
=
−
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
−1
1
0
0
0 791
1
1
.
⎟⎟
k

k
k
k
0
1
2
3
0 791
1
1
1
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
−
⎛
⎝
⎜
⎜
⎜

⎜
⎞
⎠
⎟
⎟
⎟
⎟
.
© 2006 by Taylor & Francis Group, LLC
314 Modeling of Combustion Systems: A Practical Approach
Step 4: Normalizing this by root of the sum of squares,
, we obtain the unit eigenvector
associated with λ = 0.2097:
where the subscript denotes the column of the column vector in the
eigenvectors’ matrix K.
So long as the eigenvectors are distinct, this method will lead to the associated
eigenvectors. The major advantage of this method is that spreadsheets can do
all the calculations. However, if the eigenvectors are not distinct (e.g., multiple
roots), we will end up with a problem — two different eigenvectors associated
with two identically valued eigenvalues. We can continue without problem to
obtain an eigenvector associated with .
But we run into trouble almost immediately, solving for the eigenvectors
associated with the double root, λ = {1, 1}, generating the matrix
It reduces to the following equations: 3k
0
+ k
1
+ k
2
= –1 and k

0
= 0. Substituting
one into the other, we obtain k
1
+ k
2
= –1, from which we may evaluate the
remaining two eigenvectors.
1 1 1 0 7913 1 9042
222 2
+++ =
k
3
0 416
0 525
0 525
0 525
=
−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟

⎟
.
.
.
.
().5 21 2 4 7913+≈
k
4
0 910
0 240
0 240
0 240
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
.
.
.
.
3111

1000
1000
10001
0
1
2
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
k
k
k
⎟⎟

⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
0
0
0
0
′
=
−
−+ −+
K
0 0 0 416 0 910
0 525 0 240
11
05



() ()
.
ab
ab
225 0 240
11
0 525 0 240
.

⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 315
Here a and b are undetermined coefficients. We use a and b because the
remaining two eigenvectors cannot be the same; eigenvectors for real sym-
metric matrices are always mutually orthogonal. Note that we have not yet
normalized the first two vectors in K to unit magnitude, so for now we label
the eigenvector matrix as K
′′

′′
rather than K. Now if the first two column
vectors in K′′
′′
(let us call them k
0
and k
1
) are mutually orthogonal, then k
0
T
k
1
= 0, giving ab + (a + 1)(b + 1) + 1 = 0, which reduces to .
Arbitrarily choosing b = 1 gives a = –1. Substituting into the matrix gives
Normalizing the first two vectors to unit magnitude gives
MathCAD gives the following solution, which the reader may verify is
equally correct, yielding the relations given in Equations 4.21, 4.22, and 4.25.
(Multiple roots do not have unique associated eigenvectors.)
(4.28)
At any rate, once we obtain the eigenvalues and eigenvectors, we can move
on to making real symmetric matrices orthogonal. Least squares solutions
always generate real symmetric matrices; thus, they are amenable to this
treatment.
Recall that for real symmetric matrices, eigenvectors are orthogonal in the
strictest sense. And so it follows for our example that
ab b=− + +()( )22 1
′
=
−

−
−
K
0 0 0 416 0 910
11
0 525 0 240
2
0 0 525 0 240
1



11
0 525 0 240
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
K =
−
−
−

0 0 0 416 0 910
0 408 0 707 0 525 0 240
0 816 0


.00 525 0 240
0 408 0 707 0 525 0 240


⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
K =
−
−
−
0 0 0 416 0 910
0 272 0 770 0 525 0 240
0 803 0





149 0 525 0 240
0 531 0 620 0 525 0 240−−
⎛
⎝
⎜
⎜
⎜
⎜
⎞⎞
⎠
⎟
⎟
⎟
⎟
© 2006 by Taylor & Francis Group, LLC
316 Modeling of Combustion Systems: A Practical Approach
or
4.3.3 Using Eigenvectors to Make Matrices Orthogonal
Premultiplying Equation 4.21 by K
T
gives
K
T
KΛΛ
ΛΛ


≡ ΛΛ

ΛΛ
≡ K
T
MK (4.29)
Given y = Xa, we seek another system of factors giving linear combinations
of X such that y = Ub, and also where U
T
U = D, a diagonal matrix. Here is
the procedure:
Step 1: Express the first specification mathematically:
y = Xa = Ub (4.30)
Step 2: Define the following:
M = X
T
X (4.31)
U = XK (4.32)
b = K
T
a (4.33)
0 000 0 408 0 816 0 408
0 000 0 707 0 000 0 707


−
−
−00 416 0 525 0 525 0 525
0 910 0 240 0 240 0 240


⎛

⎝
⎜⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−
−
0 000 0 000 0 416 0 910
0 408 0 707

00 525 0 240
0 816 0 000 0 525 0 240
0 408 0 707 0



−
.525 0 240
1
1
1
1
⎛
⎝

⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
0 000 0 272 0 803 0 531
0 000 0 770 0 149 0 62


−−
−−00
0 416 0 525 0 525 0 525

0 910 0 240 0 240 0 240
−

⎛⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−
−
0 000 0 000 0 416 0 910
0 272 0 7

770 0 525 0 240
0 803 0 149 0 525 0 240
0 531 0



−
−−6620 0 525 0 240
1
1

1
1

⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
© 2006 by Taylor & Francis Group, LLC
Analysis of Nonideal Data 317

Step 3: The above equations complete the transformation: substitution
of these relations into Equation 4.30 gives Ub = XKK
T
a = Xa because
KK
T
= I; therefore, y = Ub represents an alternate system of factors
and coefficients for y = Xa.
To see this, consider the necessary properties of U
T
U. Premultiplying y =
Ub by U
T
gives U
T
y = U
T
Ub. Substituting in terms of X gives U
T
U =
(xK)
T
XK = K
T
X
T
XK. But M = X
T
X, and in light of Equation 4.29, this substi-
tution gives U

T
U = K
T
MK = ΛΛ
ΛΛ
. Therefore, U is a diagonal matrix — the
eigenvector matrix of X
T
X to be exact. Collecting these equations:
y = Xa = Ub
K = eigenvectors (X
T
X) (4.34)
U = XK
b = K
T
a
U
T
U = ΛΛ
ΛΛ
(4.35)
X
T
X = KΛΛ
ΛΛ
K
T
(4.36)
K

T
X
T
XK = ΛΛ
ΛΛ
(4.37)
Equations 4.28 and 4.30 amount to the following in light of Equation 4.32:
u
1
= –0.272 x
1
+ 0.803 x
2
– 0.531 x
3
u
2
= 0.770 x
1
– 0.149 x
2
– 0.620 x
3
u
3
= –0.416 + 0.525 (x
1
+ x
2
+ x

3
)
u
4
= 0.910 + 0.240 (x
1
+ x
2
+ x
3
)
Thus, the u
k
represent linear combinations of x
k
, and either system will give
identical values for y: y = a
0
+ a
1
x
1
+ a
2
x
2
+ a
3
x
3

= b
1
u
1
+ b
2
u
2
+ b
3
u
3
+ b
4
u
4
.
The advantage of the eigenvalue procedure over the source–target matrix
procedure is that:
1. We have merely rotated axes, not distorted factors.
2. Both the original and new coordinate axes are orthogonal.
3. We may apply the procedure to any nonsingular matrix, even when
X is nonsquare, because M = X
T
X will always be square.
© 2006 by Taylor & Francis Group, LLC