A Review of Uncertainty Quantification of Estimation of Frequency Response Functions
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
School of Dynamic Systems
College of Engineering and Applied Science
July 17, 2012
by
Christopher Majba
Bachelor of Science in Mechanical Engineering
University of Cincinnati, Cincinnati, Ohio, USA, 45221-0072.
Thesis Advisor and Committee Chair: Dr. Randall Allemang
Committee Members: Dr. David L. Brown, Dr. Allyn Phillips,
Abstract
This thesis investigates methods for calculating the statistical reduction of variance in
FRF measurements.
Traditional analyses will be presented and the links to the analysis
presented herein will be established. As the method for determining the variance, or confidence
bounds, has changed since its first introduction nearly 40 years ago, both the original and more
modern analysis will be discussed with descriptions of the pros and cons of each.
After
weighing these pros and cons and evaluating the assumptions made, the appropriate analysis is
chosen for the testing at hand. The subsequent application in a data acquisition and analysis
software package is detailed, including modifying the equations for use in the MATLAB
language and some of the current drawbacks of the analysis. Conclusions from this work and
areas for future work will be discussed to aid future iterations of this analysis.
ii
iii
Acknowledgements
I would like to thank Dr. Randall Allemang for his constant guidance, motivation, and
opportunity to work for SDRL and be exposed to the materials which ultimately lead to this
thesis. I would also like to thank Dr. Allyn Phillips for his tutelage in MATLAB programming
and his patience and guidance in developing the software ultimately used to complete the
contained analysis. And of course, a special thanks to my family for supporting me and my 8
years of education.
iv
Table of Contents
Abstract................................................................................................................................................... ii
Acknowledgements ................................................................................................................................ iv
List of Tables ......................................................................................................................................... vi
List of Figures ........................................................................................................................................ vi
Chapter 1.
Motivation, Introduction, and Literature Review ............................................................... 1
Chapter 2.
FRF Confidence Bounds: Theory and Background........................................................... 3
2.1.
FRF Estimation ........................................................................................................................ 3
2.2
Use of Confidence Bounds ....................................................................................................... 6
Chapter 3: Confidence Bounds Derivation ............................................................................................... 8
3.1
Introduction to Confidence Interval Derivation ......................................................................... 8
3.2
Modern Bendat Analysis .......................................................................................................... 8
3.2.1
PHASE .................................................................................................................................. 11
3.2.2
FRF ESTIMATORS............................................................................................................... 12
3.3
Classical Bendat Analysis ...................................................................................................... 12
3.3.1
PHASE .................................................................................................................................. 18
3.4.
Application in X-Modal ......................................................................................................... 26
Chapter 4: Results ................................................................................................................................. 30
4.1
Section 1: Simulated Data ...................................................................................................... 30
Chapter 5.
Conclusions and Future Work ......................................................................................... 40
Bibliography ......................................................................................................................................... 43
v
List of Tables
Table 1: Normal Distribution Confidence Coefficient (Bendat & Piersol, 1971) ..................................... 10
Table 2: F-Distribution Confidence Coefficient (90% CI) ...................................................................... 15
Table 3: Classical vs. Modern Bendat Analysis ...................................................................................... 20
Table 4: HP Table ................................................................................................................................. 21
Table 5: HP Table vs. 90% Normal Distribution .................................................................................... 22
Table 6: HP Table vs. 90% F-Distribution (H1) ...................................................................................... 23
Table 7: Confidence Interval Comparison .............................................................................................. 35
List of Figures
Figure 1: Confidence Interval Diagram (Bendat & Piersol, 1971)........................................................... 11
Figure 2: F-Distribution Discontinuity ................................................................................................... 25
Figure 3: Confidence Bounds for Test 1, 10 Ensemble, 90% Confidence Interval ................................... 33
Figure 4: Confidence Bounds for Test 1, 10 Ensemble, 99% Confidence Interval ................................... 34
Figure 5: Confidence Bounds Trend, H1................................................................................................. 36
Figure 6: Confidence Bounds Trends, Hv ............................................................................................... 37
Figure 7: Confidence Bounds Trends, H2 ............................................................................................... 38
vi
Chapter 1.
Motivation, Introduction, and Literature Review
The intent of this thesis is to allow greater insight into the random errors, or variance,
associated with the FRF of an experimentally measured system. While most data acquisition
software packages allow varying FRF estimation techniques (H1, H2, Hv, etc.) which aim to
eliminate bias errors, the variance of the FRF calculation must be reduced using an appropriate
number of data ensembles or cyclic averages, if available. However, one cannot always be
certain that the number of ensembles acquired has removed the suitable amount of random noise.
This is particularly true if the technician is inexperienced or working on a new structure or with
new hardware. Therefore, a confidence bounds approach is presented to statistically determine
how much of the random noise has been eliminated after a particular number of ensembles for a
particular test setup. In addition to giving an indication of the amount of random noise that has
been removed, this analysis will provide an actual range wherein the expected FRF will lie.
The first analytical attempt at this analysis is from the literature of Bendat and Piersol
(Bendat & Piersol, 1971) based on the works of Goodman and Enochson (Enochson &
Goodman, 1965). This approach was later modified by Bendat using several assumptions to
make the analysis easier. This modified approach is later used in Farrar (Farrar, Doebling, &
Cornwell), where the idea for this current thesis first originated. Originally, it seemed that the
Farrar work grossly misused the referenced Bendat analysis, as it differed significantly from the
original Bendat equations (Bendat & Piersol, 1971). As more references on the subject were
revealed, it became clear that Farrar simply made use of Bendat’s more modern analysis (Bendat
& Piersol, 1993), although without making note of the necessary assumptions to make it work.
This gap in understanding and the assumptions and drawbacks of the method used by Farrar will
be addressed in this thesis.
1 / 56
Bendat (1971) is one of the early writings on FRF analysis and also includes sections on
statistical principles. A good portion of this early book drew upon work that was funded at the
USAF Research Laboratories in the 1960s, documented in research reports (Enochson &
Goodman, 1965). The principles presented therein are then used in applying confidence bounds
to an FRF and corresponding phase measurement. Bendat’s later analysis, however, differs
greatly from his original works. Further complicating matters, the Bendat work was written
before most conventional, common nomenclature for modal data acquisition and analysis had
been introduced. It was also believed, at the time of its writing, that tests such as multiple-input
multiple-output would not be practical or even possible due to the unwanted correlation of the
multiple inputs. This can make interpreting Bendat’s work a challenge, particularly when trying
to bridge the gap between his original analysis and the more modern interpretations. This
confusion will be addressed and attempts will be made to clarify the larger discontinuities
discovered as part of the development of this thesis.
2 / 56
Chapter 2.
2.1.
FRF Confidence Bounds: Theory and Background
FRF Estimation
The most common application of frequency response functions is the measurement of the input-
output relationships of a structure or system. Once the data is collected (in the scope of this paper, the
default input will be force and the default output acceleration), a Fourier transform is performed on the
data to show the data in the frequency domain. However, the integral Fourier transform requires time
histories from negative to positive infinity. Obviously, this is not practically possible as experimental
data is taken over finite time intervals and the data is sampled digitally at discrete time intervals.
Therefore, a fast Fourier transform, or FFT, algorithm is used so that the transform from the time domain
to the frequency domain can be carried out computationally. This causes potential errors due to the
random error in the measured data as well as the bias error resulting from the limited frequencies involved
in the FFT. In order to reduce these errors, digital signal processing methods must be understood as well
as the issue of the intrinsic uniqueness of the measured FRF (Allemang, 2007)
Once the time domain data has been transformed into the frequency domain, an estimation of the
actual FRF is calculated. This estimation is used in order to limit the effect of noise on the response, the
input, or both. The estimation algorithms typically used are referred to as H1, H2, and Hv, each with their
own pros and cons. Each of these techniques produces an FRF that differs from the expected value by
using the auto (Gff and Gxx) and cross (Gxf) power values to determine the FRF rather than the standard
definition of
, or the response measurement relative to the input. The differences are most apparent
at the system resonances and anti-resonances. Simply, the H1 technique will underestimate the expected
result at the resonances, H2 will overestimate, and Hv will have a value at resonance bounded by the
results found using H1 and H2. Furthermore, some of these techniques have practical drawbacks. H2, for
example, will fail if the Gxf matrix has a zero in it or, in the case of MIMO testing, if the number of inputs
does not equal the number of outputs or the inputs are not sufficiently uncorrelated. Hv is also used with
hesitation as it requires much more computation than either the H 1 or H2 methods and does not result in a
3 / 56
significant improvement in the phase calculations of the FRF measurement. The additional computation
also has a noticeable impact on testing time. For these reasons, H1 is the most commonly used FRF
estimator.
It is also worth noting that, in the absence of any bias noise, any of the FRF estimation
techniques will yield the expected FRF of the system (H1 = H2 = Hv = H). As one of these techniques is
usually selected at the beginning of a test and left unchanged, it is uncommon to be able to see each of
these estimators compared side by side for common FRF measurement making it difficult to see which
provides the most suitable result. For this reason, a separate analysis tool was implemented at the same
time as the development of the confidence bounds analysis within X-Modal, a modal data acquisition and
processing software developed by UC-SDRL. This additional tool allows the engineer to compare the H1,
H2, and Hv estimation techniques. These comparisons are beyond the scope of this thesis, but offer a good
glimpse of the effects that these different estimation techniques on the experimental data.
In addition to the benefits allotted by the different FRF estimation techniques, the auto and cross
power measurements also lend themselves to the calculation of coherence. Coherence is used as a
measure of linearity, sometimes referred to as causality between an input and output at a given frequency.
At each frequency, the coherence is given a value between zero and one, with one representing perfect
input-output linearity. Any coherence value less than one indicates that the measured output is not solely
dependent on the measured input. Often, this is the result of extraneous noise and simply means that
more averages will be required to obtain a reliable FRF measurement. The reliability of the FRF
measurement can be measured by a reduction of variance of the coherence function ensemble to ensemble
(e.g. the coherence approaches some expected value with increased ensembles). However, if the low
coherence is the result of system nonlinearities, non-measured inputs, or leakage, a larger number of
ensembles will not reduce this variance and the coherence will continue to deteriorate. This is important
to note as much of the analysis conducted in this thesis makes use of multiple-input, multiple output
measurements. One must also be aware that, assuming no change in the test setup between ensembles,
the coherence itself will not improve by taking more ensembles.
4 / 56
While this thesis will focus primarily on MIMO testing, it should be noted that there are
different coherence functions for multiple input and single input testing. Ordinary coherence, used in
single input cases, can be explained using the description mentioned earlier in this chapter. Multiple
coherence provides a measure of the linear dependence between all of the inputs into the system and an
output. This relationship is easily understood mathematically: as long as no correlation exists between
different output records and a particular desired output record, the multiple coherence function between
an output of interest and all the inputs is the result of the summation of the ordinary coherence functions
between each input and the output. If, after removing the effects of the input records, there appear
correlations between the output records then the summation must be extended to include these
contributions (Bendat & Piersol, 1993). The implications of MIMO or SIMO on the confidence bounds
equations testing will also be discussed later during the derivation of Bendat’s equations for use in data
acquisition and analysis.
While the use of FRF estimators can help to reduce the noise from the data and variance in the
coherence function can help determine if enough ensembles have been collected to produce a stable FRF,
it is often unclear exactly how many averages are required to sufficiently eliminate the variance in the
measurements. The easiest way is, of course, to take more ensembles of data until the variance reaches an
acceptable level. However, as mentioned earlier, this is extremely difficult for impact tests carried out by
a human being. Taking a large number of averages is much more useful when MIMO testing is being
conducted. Even in this analysis, there is currently no real measure of how much of the variance has been
eliminated which requires the technician to make an educated guess as to how many ensembles will be
required. Guessing too low may mean that all the data may need retaken, extending test time. Starting
with more than enough ensembles can have a similar effect in that the test runs longer than necessary and
requires more resources to store the additional data. To better gauge the reduction of variance, the
confidence bounds equations can be applied.
5 / 56
2.2
Use of Confidence Bounds
Confidence intervals are frequently used when a test technician or data analyst wants to inspect
experimentally measured data to determine how close it comes to the expected values. Often there are a
few outliers or other data points that may have been influenced more by random noise than by the test
itself. Therefore, confidence bounds offer a way of determining how close the averaged measured value
is to the expected value. This is where the nomenclature quantification of uncertainty comes from. From
this definition, it stands to reason that if the same test is repeated many times and the data from each test
is averaged then the confidence bounds should steadily close in around the averaged measurement data as
it approaches the expected value. It is important to note that the confidence bounds give no indication of
any bias errors associated with an FRF measurement. These errors include things like signal noise,
resolution errors, nonlinear systems, and unmeasured inputs that are correlated with the measured input
and contribute to the output (Bendat & Piersol, 1993). Confidence bounds only help to quantify the
random errors of measured system.
How quickly the confidence interval closes in is a function of the confidence level which can be
chosen by the technician or analyst.
This value, often represented as a percentage, indicates the
probability that the expected value will lie within the confidence interval. Therefore, the higher the
confidence level, the larger the confidence interval and the more averages or ensembles it will take the
interval to close in on the measured data. Some typical confidence levels, and the ones used in this thesis,
are 90%, 95%, 97.5%, and 99%. The use of these values in the FRF confidence bounds equations will be
discussed in greater depth later in this thesis.
Original FRF confidence bounds equations, which will be presented in detail in the next
section, made use of the assumption that the random noise on the FRF has an F-Distribution. In
simple terms, this distribution is used when the measurement is influence by two statistically
independent variables which each have a Chi-Square distribution. A detailed explanation of this
derivation and meaning of these terms is available in Bendat (1971). Bendat’s original analysis,
6 / 56
based on that of Goodman and Enochson, makes it clear that this would be the appropriate
method to use. Farrar’s work, which uses the confidence bounds primarily as a method to
determine theoretical bounds to apply random noise, uses a normal distribution. This method is
detailed in Bendat’s later work and works under the assumption that the error (or variance) is
very small (ε≤0.10). The advantages and disadvantages of both of these methods will be a
primary focus of this paper.
7 / 56
Chapter 3: Confidence Bounds Derivation
3.1
Introduction to Confidence Interval Derivation
This chapter will outline the derivations of the various FRF confidence bounds techniques. First
will be the method used by Farrar in the analysis of the Alamosa Canyon Bridge, designated as Modern
Bendat Analysis. This was the first technique used in the formation of this thesis and thus provides a
good starting point of the progression of this thesis. This will then be followed by the classical Bendat
analysis.
This method was discovered later as most modal analysis involving confidence bounds,
however uncommon such analysis is, simply uses the more modern analysis for its simplicity. Also,
when the modern Bendat analysis did not agree with some results published in documentation found from
the 1970s, it was clear that some sort of further study/analysis was required. Once the two methods are
discussed, the analytical benefits and disadvantages will be discussed and the proper technique chosen for
use in the rest of this thesis.
3.2
Modern Bendat Analysis
As explained in the preceding section, the first experience with the confidence bounds analysis
was from Farrar’s work. This analysis was later found to actually be based off of a much later Bendat
analysis which provides a simpler analysis under the assumption that the variance is small and uses the
normal distribution assumption. Farrar’s use of this method and some of the details of the analysis will
be discussed in this section.
According to Bendat’s later work, the error term for the confidence bounds calculation can be
found in the equation below.
(| ̂ ( )|)
√
( )
( )√
| ̂ ( )|
(1)
8 / 56
Where
(| ̂ ( )|) is the statistical random error,
frequency
, and
( ) is the value of coherence at a particular
is the total number of collected data ensembles. For a full explanation of the
derivation of this equation, please consult Bendat’s later work (Bendat & Piersol, 1993)
One can quickly tell that this analysis is actually quite simple given modern modal data
acquisition systems. All that is required from the measured data for the confidence bounds equations is
the coherence and FRF at each Δf and the current number of ensembles. Statistically all that is required is
the selection of a confidence interval. Assuming a small error and, thus, a normal variance distribution,
this confidence interval will yield a specific confidence coefficient. In other words, the confidence
coefficient will not need recalculated after each ensemble. The significance of this will become clearer
during the discussions of Bendat’s earlier work.
Using the above equation to solve for the variance, the FRF confidence bounds can be defined
thusly:
| ̂ ( )|
(| ̂ ( )|)
In the above equation,
| ( )|
| ̂ ( )|
(| ̂ ( )|)
(2)
is the confidence coefficient for a particular confidence interval, | ̂ ( )| is the
estimated value of the FRF at a particular frequency
, and | ( )| is the actual value of the FRF. The C f
term is obtained from Table 1 below, which gives specific confidence coefficients (combination of x- and
y-axis values) for normally distributed variance based on the desired confidence interval (the values
inside the table multiplied by 2 and then subtracted from 1). In this table, the values for the confidence
interval are actually representative of the one-sided, tail-end probability. Therefore, a value of 0.4013 in
the table does not represent a 40% confidence interval, rather a 20% confidence interval. For example,
for a 90% confidence interval, the confidence coefficient will be equal to 1.65.
9 / 56
za
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.5000
0.4960
0.4920
0.4880
0.4840
0.4801
0.4761
0.4721
0.4681
0.4641
0.4602
0.4562
0.4522
0.4483
0.4443
0.4404
0.4364
0.4325
0.4286
0.4247
0.4207
0.4168
0.4129
0.4090
0.4052
0.4013
0.3974
0.3936
0.3897
0.3859
0.3821
0.3783
0.3745
0.3707
0.3669
0.3632
0.3594
0.3557
0.3520
0.3483
0.3446
0.3409
0.3372
0.3336
0.3300
0.3264
0.3228
0.3192
0.3156
0.3121
0.3085
0.3050
0.3015
0.2981
0.2946
0.2912
0.2877
0.2843
0.2810
0.2776
0.2743
0.2709
0.2676
0.2643
0.2611
0.2578
0.2546
0.2514
0.2483
0.2451
0.2420
0.2389
0.2358
0.2327
0.2297
0.2266
0.2236
0.2207
0.2177
0.2148
0.2119
0.2090
0.2061
0.2033
0.2005
0.1977
0.1949
0.1922
0.1894
0.1867
0.1841
0.1814
0.1788
0.1762
0.1736
0.1711
0.1685
0.1660
0.1635
0.1611
0.1587
0.1563
0.1539
0.1515
0.1492
0.1469
0.1446
0.1423
0.1401
0.1379
0.1357
0.1335
0.1314
0.1292
0.1271
0.1251
0.1230
0.1210
0.1190
0.1170
0.1151
0.1131
0.1112
0.1094
0.1075
0.1057
0.1038
0.1020
0.1003
0.0985
0.0968
0.0951
0.0934
0.0918
0.0901
0.0885
0.0869
0.0853
0.0838
0.0823
0.0808
0.0793
0.0778
0.0764
0.0749
0.0735
0.0721
0.0708
0.0694
0.0681
0.0668
0.0655
0.0643
0.0630
0.0618
0.0606
0.0594
0.0582
0.0571
0.0559
0.0548
0.0537
0.0526
0.0516
0.0505
0.0495
0.0485
0.0475
0.0465
0.0455
0.0446
0.0436
0.0427
0.0418
0.0409
0.0401
0.0392
0.0384
0.0375
0.0367
0.0359
0.0352
0.0344
0.0336
0.0329
0.0322
0.0314
0.0307
0.0301
0.0294
0.0287
0.0281
0.0274
0.0268
0.0262
0.0256
0.0250
0.0244
0.0239
0.0233
0.0228
0.0222
0.0217
0.0212
0.0207
0.0202
0.0197
0.0192
0.0188
0.0183
0.0179
0.0174
0.0170
0.0166
0.0162
0.0158
0.0154
0.0150
0.0146
0.0143
0.0139
0.0136
0.0132
0.0129
0.0126
0.0122
0.0119
0.0116
0.0113
0.0110
0.0107
0.0104
0.0102
0.00990
0.00964
0.00939
0.00914
0.00889
0.00866
0.00842
0.00820
0.00798
0.00776
0.00755
0.00734
0.00714
0.00695
0.00676
0.00657
0.00639
0.00621
0.00604
0.00587
0.00570
0.00554
0.00539
0.00523
0.00508
0.00494
0.00480
0.00466
0.00453
0.00440
0.00427
0.00415
0.00402
0.00391
0.00379
0.00368
0.00357
0.00347
0.00336
0.00326
0.00317
0.00307
0.00298
0.00289
0.00280
0.00272
0.00264
0.00256
0.00248
0.00240
0.00233
0.00226
0.00219
0.00212
0.00205
0.00199
0.00193
0.00187
0.00181
0.00175
0.00169
0.00164
0.00159
0.00154
0.00149
0.00144
0.00139
Table 1: Normal Distribution Confidence Coefficient (Bendat & Piersol, 1971)
The above equation states that the expected value of | ( )| will lie between an upper and lower limit of
the FRF defined by the calculated FRF and its associated variance. This agrees with comments made
earlier in this thesis.
10 / 56
3.2.1 PHASE
When the input-output data is collected from the structure, multiple measurements are actually
calculated using the hardware and software. Aside from the often logarithmically plotted FRF magnitude
data, there is also coherence and phase. Coherence was discussed earlier in this paper as one possible
indicator for the reduction of variance. In fact, methods for applying confidence bounds to coherence do
exist. However, in the scope of this thesis, the phase confidence interval is of greater interest.
Below is a “confidence diagram” which helps to pictorially describe the confidence interval
application to the FRF data collected (Bendat & Piersol, 1971).
Figure 1: Confidence Interval Diagram (Bendat & Piersol, 1971)
Inspection of this diagram gives direct insight into how the phase variance can be calculated
once the FRF and its confidence interval are determined. As has already been discussed, when using the
more modern Bendat analysis one assumes small random error. From this assumption it can be said that
the statistical phase variance
̂ ( ) is also small. Therefore, from the small angle theorem:
11 / 56
̂( )
(| ̂ ( )|)
| ̂ ( )|
√
( )
(3)
( )√
The variable names defined in this section represent the more modern terminology currently used. This
will continue to be used throughout this thesis even where the classical analysis uses more antiquated
terms in order to keep the explanations more clear.
3.2.2 FRF ESTIMATORS
From the reading of Bendat’s more modern work, it is made clear that the above analysis is
meant for use with H1 FRF estimator (Bendat & Piersol, 1993). Recalling what was discussed early in
this thesis, this makes practical sense. Not only is the H 1 estimator more robust, it is also more realistic
than the H2 or Hv methods. These alternative methods aim to reduce noise on the input or noise on both
the input and output, respectively. In most modern data acquisition situations, the significant random
noise can be thought to be confined to the output alone, with the input signals being free from most
random noise.
For the small variance case, Gaussian distributed confidence bounds equations, the H1-specific
analysis will be the only one presented here. This is done as this simplified analysis is a direct derivation
of the more in-depth classical analysis about to be presented. As the classical work already assumed an
H1 estimation, it may have a direct impact on the simplified analysis presented above. Potential analysis
methods for the different estimators will be discussed for the classical work.
3.3
Classical Bendat Analysis
After working with what was discovered to be the more modern Bendat analysis, the original
methods were found while investigating references. There existed stark differences in the analyses that
prompted further investigation to see which provided the best results. This was particularly true seeing as
how the Gaussian analysis only works for small random errors and that the results from the Gaussian
analysis did not coincide with commercially available data (Hewlett Packard, 1978). It is for these
reasons that the classical techniques are presented here.
12 / 56
As was mentioned earlier in this thesis, Bendat’s classical work is somewhat difficult to follow.
The greatest difficulty comes from nomenclature differences, particularly in the use of the term “degrees
of freedom.” It seems to change from meaning the number of input and outputs in one area to being used
to express block size in the next. This mismatching does lead to some confusion as to how to properly
apply the techniques in practical use, particularly as degrees of freedom are most normally identified as
the number of input/output relationships on a test structure. One of the goals of this section is to sort out
this confusion.
The equation for the FRF confidence bounds as presented in Bendat’s classical work is presented
below. For the exact derivation of this equation, please consult Bendat (1971). This approach makes use
of the F-distribution following that the error term is influenced by two statistically independent inputs: the
estimation of the frequency response function (real and imaginary parts) and estimation of the of the
power spectral density of the estimated error. Both of these terms are distributed as chi-square (Bendat &
Piersol, 1971). Bendat actually presents two such equations with the other meant for use in multi-input,
single-output cases. This was presented because, at the time of its writing, it wasn’t believed it was
possible to properly correlate the input-output combinations to obtain individual FRFs. This analysis will
be disregarded here as most modern data acquisition hardware and software can properly separate the
input-output combinations. If not, there would be input-input correlations and the multiple-input analysis
should be used. Again, please consult Bendat (1971) for more information on this analysis.
(| ̂ ( )|)
The
(
)
[
( )]
̂
( )
̂
( )
(4)
term represents the confidence coefficient for the desired confidence interval and associated
particular degrees of freedom, which will change on each ensemble. These values, similar to the values
used for the small-error/normally distributed case explained earlier are obtained from tables. In the
following table are various values for a F-confidence coefficient for a 90% confidence interval. Different
tables are used for different confidence intervals.
13 / 56
Once the F-Distribution confidence interval and appropriate table is chosen, the confidence
coefficient is determined from the values of the two degrees of freedom. In the case of this analysis, the
two degrees of freedom are the number of ensembles minus 2 (df2) and the number inputs to a particular
FRF multiplied by 2 (df1) (real and imaginary parts). By finding the point on the table where these two
degrees of freedom intersect, one can obtain the confidence coefficient.
df1
5
6
1
39.8635
49.5
53.5932
55.833
57.2401
58.2044
58.906
59.439
59.8576
2
8.52632
9
9.16179
9.24342
9.29263
9.32553
9.34908
9.36677
9.38054
3
5.53832
5.46238
5.39077
5.34264
5.30916
5.28473
5.26619
5.25167
5.24
4
4.54477
4.32456
4.19086
4.10725
4.05058
4.00975
3.97897
3.95494
3.93567
5
4.06042
3.77972
3.61948
3.5202
3.45298
3.40451
3.3679
3.33928
3.31628
6
3.77595
3.4633
3.28876
3.18076
3.10751
3.05455
3.01446
2.98304
2.95774
7
3.58943
3.25744
3.07407
2.96053
2.88334
2.82739
2.78493
2.75158
2.72468
8
3.45792
3.11312
2.9238
2.80643
2.72645
2.66833
2.62413
2.58935
2.56124
df2
9
1
2
3
4
7
8
9
3.3603
3.00645
2.81286
2.69268
2.61061
2.55086
2.50531
2.46941
2.44034
10
3.28502
2.92447
2.72767
2.60534
2.52164
2.46058
2.41397
2.37715
2.34731
11
3.2252
2.85951
2.66023
2.53619
2.45118
2.38907
2.34157
2.304
2.2735
12
3.17655
2.8068
2.60552
2.4801
2.39402
2.33102
2.28278
2.24457
2.21352
13
3.13621
2.76317
2.56027
2.43371
2.34672
2.28298
2.2341
2.19535
2.16382
14
3.10221
2.72647
2.52222
2.39469
2.30694
2.24256
2.19313
2.1539
2.12195
15
3.07319
2.69517
2.48979
2.36143
2.27302
2.20808
2.15818
2.11853
2.08621
16
3.04811
2.66817
2.46181
2.33274
2.24376
2.17833
2.128
2.08798
2.05533
17
3.02623
2.64464
2.43743
2.30775
2.21825
2.15239
2.10169
2.06134
2.02839
18
3.00698
2.62395
2.41601
2.28577
2.19583
2.12958
2.07854
2.03789
2.00467
19
2.9899
2.60561
2.39702
2.2663
2.17596
2.10936
2.05802
2.0171
1.98364
20
2.97465
2.58925
2.38009
2.24893
2.15823
2.09132
2.0397
1.99853
1.96485
21
2.96096
2.57457
2.36489
2.23334
2.14231
2.07512
2.02325
1.98186
1.94797
22
2.94858
2.56131
2.35117
2.21927
2.12794
2.0605
2.0084
1.9668
1.93273
23
2.93736
2.54929
2.33873
2.20651
2.11491
2.04723
1.99492
1.95312
1.91888
24
2.92712
2.53833
2.32739
2.19488
2.10303
2.03513
1.98263
1.94066
1.90625
25
2.91774
2.52831
2.31702
2.18424
2.09216
2.02406
1.97138
1.92925
1.89469
26
2.90913
2.5191
2.30749
2.17447
2.08218
2.01389
1.96104
1.91876
1.88407
27
2.90119
2.51061
2.29871
2.16546
2.07298
2.00452
1.95151
1.90909
1.87427
28
2.89385
2.50276
2.2906
2.15714
2.06447
1.99585
1.9427
1.90014
1.8652
29
2.88703
2.49548
2.28307
2.14941
2.05658
1.98781
1.93452
1.89184
1.85679
30
2.88069
2.48872
2.27607
2.14223
2.04925
1.98033
1.92692
1.88412
1.84896
40
2.83535
2.44037
2.22609
2.09095
1.99682
1.92688
1.87252
1.82886
1.7929
60
2.79107
2.39325
2.17741
2.04099
1.94571
1.87472
1.81939
1.77483
1.73802
120
2.74781
2.34734
2.12999
1.9923
1.89587
1.82381
1.76748
1.72196
1.68425
inf
2.70554
2.30259
2.0838
1.94486
1.84727
1.77411
1.71672
1.6702
1.63152
14 / 56
10
12
15
20
24
30
40
60
120
INF
60.195
60.7052
61.2203
61.7403
62.0021
62.265
62.5291
62.7943
63.0606
63.3281
9.39157
9.40813
9.42471
9.44131
9.44962
9.45793
9.46624
9.47456
9.48289
9.49122
5.23041
5.21562
5.20031
5.18448
5.17636
5.16811
5.15972
5.15119
5.14251
5.1337
3.91988
3.89553
3.87036
3.84434
3.83099
3.81742
3.80361
3.78957
3.77527
3.76073
3.2974
3.26824
3.23801
3.20665
3.19052
3.17408
3.15732
3.14023
3.12279
3.105
2.93693
2.90472
2.87122
2.83634
2.81834
2.79996
2.78117
2.76195
2.74229
2.72216
2.70251
2.66811
2.63223
2.59473
2.57533
2.55546
2.5351
2.51422
2.49279
2.47079
2.53804
2.50196
2.46422
2.42464
2.4041
2.38302
2.36136
2.3391
2.31618
2.29257
2.41632
2.37888
2.33962
2.29832
2.27683
2.25472
2.23196
2.20849
2.18427
2.15923
2.3226
2.28405
2.24351
2.20074
2.17843
2.15543
2.13169
2.10716
2.08176
2.05542
2.24823
2.20873
2.16709
2.12305
2.10001
2.07621
2.05161
2.02612
1.99965
1.97211
2.18776
2.14744
2.10485
2.05968
2.03599
2.01149
1.9861
1.95973
1.93228
1.90361
2.13763
2.09659
2.05316
2.00698
1.98272
1.95757
1.93147
1.90429
1.87591
1.8462
2.0954
2.05371
2.00953
1.96245
1.93766
1.91193
1.88516
1.85723
1.828
1.79728
2.05932
2.01707
1.97222
1.92431
1.89904
1.87277
1.84539
1.81676
1.78672
1.75505
2.02815
1.98539
1.93992
1.89127
1.86556
1.83879
1.81084
1.78156
1.75075
1.71817
2.00094
1.95772
1.91169
1.86236
1.83624
1.80901
1.78053
1.75063
1.71909
1.68564
1.97698
1.93334
1.88681
1.83685
1.81035
1.78269
1.75371
1.72322
1.69099
1.65671
1.95573
1.9117
1.86471
1.81416
1.78731
1.75924
1.72979
1.69876
1.66587
1.63077
1.93674
1.89236
1.84494
1.79384
1.76667
1.73822
1.70833
1.67678
1.64326
1.60738
1.91967
1.87497
1.82715
1.77555
1.74807
1.71927
1.68896
1.65691
1.62278
1.58615
1.90425
1.85925
1.81106
1.75899
1.73122
1.70208
1.67138
1.63885
1.60415
1.56678
1.89025
1.84497
1.79643
1.74392
1.71588
1.68643
1.65535
1.62237
1.58711
1.54903
1.87748
1.83194
1.78308
1.73015
1.70185
1.6721
1.64067
1.60726
1.57146
1.5327
1.86578
1.82
1.77083
1.71752
1.68898
1.65895
1.62718
1.59335
1.55703
1.5176
1.85503
1.80902
1.75957
1.70589
1.67712
1.64682
1.61472
1.5805
1.54368
1.5036
1.84511
1.79889
1.74917
1.69514
1.66616
1.6356
1.6032
1.56859
1.53129
1.49057
1.83593
1.78951
1.73954
1.68519
1.656
1.62519
1.5925
1.55753
1.51976
1.47841
1.82741
1.78081
1.7306
1.67593
1.64655
1.61551
1.58253
1.54721
1.50899
1.46704
1.81949
1.7727
1.72227
1.66731
1.63774
1.60648
1.57323
1.53757
1.49891
1.45636
1.76269
1.71456
1.66241
1.60515
1.57411
1.54108
1.50562
1.46716
1.42476
1.37691
1.70701
1.65743
1.60337
1.54349
1.51072
1.47554
1.43734
1.3952
1.34757
1.29146
1.65238
1.6012
1.545
1.48207
1.44723
1.40938
1.3676
1.32034
1.26457
1.19256
1.59872
1.54578
1.48714
1.4206
1.38318
1.34187
1.29513
1.23995
1.1686
1
Table 2: F-Distribution Confidence Coefficient (90% CI) (Dinov, 2011)
A quick look at the above equation reveals that it will not readily work with conventional data
acquisition software, in a post processing implementation, since the ̂ ( ) term or the ̂ ( ) term is
15 / 56
not always saved. In order to simplify the integration of this equation into data acquisition software, the
output power/input power term must be converted to something obtained during an FRF measurement.
H1: Noise on Response Model
To get the above Bendat equation into a form that can use the common H1 estimator and ordinary
coherence, the following relationships will be used.
̂( )
̂
( )
̂
( )
̂ ( )
̂
( )
̂
( )
(6)
̂ ( )̂ ( )
( )
(5)
̂
( ) ̂
( )
̂
( ) ̂
( )
̂
( )
̂
( )
̂
( )
̂
( )
(7)
(8)
Therefore,
̂
( )
̂
( )
̂ ( )̂ ( )
(9)
( )
̂( )
(| ̂ ( )|)
(| ̂ ( )|)
(
)
[
̂ ( ) and
( )]
̂ ( )̂ ( )
( )
(10)
Again, the F-term can be determined from supplied tables (see Table 2 for reference, or the appendix for
more detail).
With this equation the classical Bendat analysis can now use data that is already calculated
within data acquisition software. By looking at the above equation, one can come to the conclusion that it
only works with the H1 FRF estimator. This was presented first as 1) it is the most common FRF
estimator used and 2) it starts to resemble Bendat’s more modern analysis with a coherence term in the
denominator. As the more modern Bendat analysis states specifically that it is used with the H 1 estimator,
this makes practical sense. Since modifying the classical Bendat analysis for use with different FRF
estimators proved relatively easy, use with other estimators will be discussed below.
16 / 56
H2: Noise on Input Model
While H1 is a commonly used FRF estimator, it may be technician’s prerogative to use other
estimation techniques, such as H2 or Hv. From the preceding derivation, it becomes quite simple to
arrange the terms to appropriately estimate the confidence bounds when the H2 estimator is used.
̂( )
̂ ( )
̂
( )
̂
( )
̂
( )
̂
( )
(12)
̂
̂ ( )̂ ( )
( )
(11)
̂
( ) ̂
( ) ̂
̂
( )
̂
( )
̂
( )
̂
( )
( )
(13)
( )
(14)
Therefore,
̂
( )
̂
( )
̂ ( )̂ ( )
( )
(15)
̂( )
(| ̂ ( )|)
(| ̂ ( )|)
(
)
[
̂ ( ) and
( )] ̂ ( )̂ ( )
( )
(16)
Notice how the coherence term is now in the numerator as opposed to the denominator when using the H 1
technique. The implications of this will be discussed later.
Hv: Noise on Response and Input Model
In the case of the Hv estimate, a conversion does not seem as simple as those outlined above.
Knowing that, in theory, the H v estimate should be bounded by the H 1 and H2 estimates, it is hypothesized
that the confidence bounds can be applied straight away, without any manipulation of the confidence
bounds equation. This approach also makes sense in that the Hv estimator is meant to reduce noise on
both the input and output, meaning it should theoretically be the closest to the actual FRF. Therefore, the
proposed equation for such an analysis using Bendat’s original methods is as follows.
(| ̂ ( )|)
(
)
[
( )] ̂ ( )
17 / 56
(17)
No work has been done to verify the validity of this equation. As of this writing it remains a hypothetical
approach to the application of the confidence bounds to the H v FRF estimation.
One interesting thing to note is the clear difference in the confidence bounds between the FRF
estimators. Knowing that the coherence term
( ) must be less than or equal to one, it is obvious that
for a given FRF measurement at a specific Δf, the size of the confidence bounds will be as follows:
(| ̂ ( )|)
(| ̂ ( )|)
(| ̂ ( )|)
(18)
Again, this is assuming that there is noise on the input and/or output. If not, all FRF estimators yield the
same result and the coherence will be unity.
3.3.1 PHASE
Just as in the modern analysis, the F-Distribution confidence interval analysis makes provisions
for measurements of phase as well as magnitude.
Also similar to the modern analysis, the phase
confidence interval is calculated using essentially the same error term as for the FRF itself. However,
since the F-Distribution does not assume small error, the small angle theorem no longer applies.
Keeping the (| ̂ ( )|) term from above the same, the upper and lower limits of the phase can be
computed by the following.
̂( )
[
(| ̂ ( )|)
]
| ̂ ( )|
(19)
Another look at the confidence diagram shows the validity of the above equation. Just as in the analysis
mentioned earlier the
̂ ( ) represents the angle formed between the measured and calculated FRF and
the confidence bounds. The key difference is that now the value of the phase variance is being estimated
without the small angle assumption.
Other difficulties in making the original Bendat phase calculations work within a data acquistion
framework are discussed in the “Application” section of this chapter.
Classical Bendat vs. Modern Bendat
Now that the actual methods used in obtaining the FRF confidence bounds have been established,
the primary differences and similarities will be investigated. The advantages and disadvantages of each
18 / 56
analysis will be presented and the methods used for the remainder of this thesis will be selected.
Justification for the decision will be offered.
The key analytical difference between Bendat’s classical equation and the more modern and
simplified method is the choice of statistical distribution. The original analysis makes use of the Fdistribution, which is a one-sided asymmetrical distribution. Use of the F-Distribution is not an arbitrary
choice but rather the result of the variables affecting the FRF (Bendat & Piersol, 1971). The analysis was
simplified later on by assuming small errors and forcing the distribution to appear Gaussian. However,
this assumption is only valid at high coherence, high ensemble count, or both. While final test results
after many averages with an acceptable coherence may yield a Gaussian distributed random noise, it is
unlikely that such small errors will present when the test is first started (i.e. <20 ensembles). Therefore,
for an accurate analysis that will hold throughout the test procedure, the F-Distribution technique is
preferred.
Actually calculating the confidence coefficients required for the different analyses is important as
well. Here, the Gaussian distributed method’s simplicity is most evident as once a confidence interval is
selected (90%, 95%, 99%, etc.), a single coefficient can be assigned for the entire test.
The F-
Distribution, on the other hand, has full tables for each confidence interval with values that correspond to
the appropriate degrees of freedom. Recalling from before, this means that for a certain confidence
interval the confidence coefficient will constantly be changing based on the number of ensembles
acquired. This does make this analysis a bit more cumbersome.
The primary issue with the assumption of a normal distribution is that it is only applicable in
cases where the error term is very small. This is because the F-Distribution will actually approach
Normal for sufficiently random variance (<0.10). This assumption does hold some weight, as with
modern data acquisition hardware the amount of noise variance can be kept low using MIMO setups.
However, a quick look at the following table gives some indication how valid an assumption this actually
is. In it, the confidence coefficient as calculated assuming an F-Distribution are displayed. Values for
19 / 56
which a Gaussian approximation is valid (ε<0.10) are highlighted in green. Yellow, orange and red
highlight as the approximation becomes less valid, with red being the worst (ε>0.30).
γxy
2
nd
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
1.22
0.81
0.62
0.50
0.41
0.33
0.27
0.20
0.14
32
0.83
0.56
0.42
0.34
0.28
0.23
0.18
0.14
0.09
64
0.57
0.38
0.29
0.23
0.19
0.16
0.13
0.10
0.06
128
0.40
0.27
0.21
0.16
0.13
0.11
0.09
0.07
0.04
256
0.29
0.19
0.15
0.12
0.10
0.08
0.06
0.05
0.03
Table 3: Classical vs. Modern Bendat Analysis
Calculating the “true” values for the error term using the original Bendat analysis at various
coherences and ensemble numbers one can see the relatively rare cases in which this assumption holds
true. As most MIMO tests are run to an average of 100 ensembles, the coherence would have to be quite
high (>0.60) in order for the error to be sufficiently small enough for the Gaussian assumption. While
this can happen at certain Δf’s, it is often unlikely at the resonances where analysis is so important and
bias errors can be dominant. It can also skew the variance calculation at anti-resonances and other areas
in the frequency domain that may be of interest.
Also, it is important to note that for lightly damped structures, the dominant error in the vicinity
of the resonance (peak) or anti-resonance (valley) is due to the bias error known as leakage. In such a
structure, the peak should be relatively sharp compared to a wider peak evidenced in more heavily
damped structures. As a result, it is more likely that the peak will be missed as a result of poor resolution,
causing a leakage error. The Bendat statistical analysis presented in this thesis assumes a random error
and does not discuss the aspects of bias errors.
20 / 56
Another method used for choosing the statistical method for analysis in this thesis is comparison
with available literature. In this case, a table is presented by HP for the confidence bounds similarly as
presented above (Hewlett Packard, 1978). This table is presented below.
nd
γxy2
16
32
64
128
256
5.2
14.6
54
4.2
-8.4
38
3.5
-6
30
3
-4.5
24
2.5
-3.5
19
2.1
-2.7
15
1.6
-2
12
1.1
-1.3
8
3.8
7.1
34
3.1
-4.8
25
2.6
-3.6
20
2.1
-2.8
16
1.8
-2.2
13
1.5
-1.7
10
1.1
-1.3
8
0.8
-0.8
5
2.8
4.2
23
2.2
-3
17
1.8
-2.3
14
1.5
-1.9
11
1.3
-1.5
9
1
-1.2
7
0.8
-0.9
6
0.5
-0.6
4
2.1
2.7
16
1.6
-2
10
1.3
-1.6
10
1.1
-1.3
8
0.9
-1
6
0.7
-0.8
5
0.6
-0.6
4
0.4
-0.4
3
1.5
1.8
11
1.2
-1.4
7
1
-1.1
7
0.8
-0.9
5
0.7
-0.7
4
0.5
-0.6
4
0.4
-0.4
3
0.3
-0.3
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Table 4: HP Table
This table is commonly presented in classrooms when discussing the amount of random error
associated with a particular FRF. However, the methods used for determining the bounds presented
within are often left out. To see if this table corresponds to either the classical or modern Bendat analysis,
21 / 56
the differences of each with respect to the HP table were investigated. The results of that investigation
are presented below.
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
-5.44 -4.16 -3.14 -2.33 -1.71 Upper H
-17.7 -8.29 -4.96 -3.19 -2.13 Lower H
-0.9 -0.6 -0.4 -0.4 -0.2 Phase
1.23 0.82 0.59 0.48 0.32 Upper H
7.07 2.52 1.23 0.71 0.44 Lower H
53.4 33.6 22.7 15.8 10.9 Phase
1.02 0.73 0.46 0.34 0.29 Upper H
3.32 1.54 0.83 0.52 0.38 Lower H
37.6 24.7 16.8
9.8
6.9 Phase
0.86 0.65 0.38 0.27 0.26 Upper H
2.19 1.09
0.6 0.43 0.29 Lower H
29.6 19.7 13.8
9.9
6.9 Phase
0.79 0.48 0.32 0.25 0.19 Upper H
1.53 0.81 0.54 0.36 0.25 Lower H
23.7 15.8 10.9
7.9
4.9 Phase
0.65 0.46 0.33
0.2
0.2 Upper H
1.15 0.61 0.41 0.24 0.17 Lower H
18.8 12.8
8.9
5.9
3.9 Phase
0.59 0.41 0.21 0.14
0.1 Upper H
0.87 0.45 0.33
0.2 0.18 Lower H
14.8
9.9
6.9
4.9
4 Phase
0.42 0.25 0.19 0.17 0.09 Upper H
0.64 0.36 0.25 0.14 0.08 Lower H
11.9
7.9
5.9
3.9
3 Phase
0.3 0.23 0.09 0.11 0.09 Upper H
0.42 0.19 0.17
0.1 0.09 Lower H
7.9
4.9
4
3
2 Phase
Table 5: HP Table vs. 90% Normal Distribution
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γxy2
nd
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
-6.93 -5.26 -3.94 -2.95 -2.18 Upper H
∞
-15.56 -7.4 -4.49 -2.91 Lower H
-90 -56.4
-35 -23.8 -16.6 Phase
0.02 -0.04 -0.01 0.03 -0.01 Upper H
-0.04
0.06 0.01 -0.03 -0.03 Lower H
-0.6
0.3
0.5
0.4
0 Phase
0
0.03 -0.03 -0.02 0.02 Upper H
-0.05
0
0
0 0.04 Lower H
-0.5
-0.1
0 -1.9 -1.3 Phase
-0.02
0.06 -0.03 -0.03 0.04 Upper H
0 -0.01 -0.02 0.03 0.03 Lower H
0.1
0.1
0.5
0.5
0.3 Phase
0.03 -0.03 -0.02
0 0.01 Upper H
-0.04 -0.03 0.06 0.04 0.03 Lower H
0
-0.1
0
0.3 -0.5 Phase
0.01
0.02 0.04 -0.01 0.05 Upper H
-0.01 -0.03 0.03 -0.01
0 Lower H
-0.4
-0.1
0 -0.3 -0.4 Phase
0.05
0.05 -0.02 -0.03 -0.02 Upper H
0.01 -0.04 0.04
0 0.04 Lower H
-0.5
-0.5 -0.2 -0.1
0.4 Phase
-0.01 -0.03 0.01 0.03
0 Upper H
0.02
0 0.03 -0.01 -0.02 Lower H
0.2
0
0.5
0.1
0.3 Phase
-0.01
0.03 -0.04 0.02 0.03 Upper H
0.03 -0.04 0.03
0 0.02 Lower H
0.2
-0.3
0.3
0.4
0.2 Phase
Table 6: HP Table vs. 90% F-Distribution (H 1 )
Looking at the preceding comparisons, it is easy to see that the F-Distribution with the H 1
technique presented in this thesis yields the closest comparison with the HP table. This comparison was
also performed with different confidence intervals for both the Gaussian and F-Distribution techniques, as
well the different FRF estimation techniques presented in this thesis. These tables can be found in the
Appendix for reference.
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The absence of the two statistically independent variables from the normally distributed method
can be explained simply by the small-error assumption. As the simplified analysis assumes a small error,
the error term can be assumed to be zero. When this term drops out there remains only a single variable
influencing the confidence interval (the number of ensembles), thus allowing the normally distributed
model to be used.
Another unresolved discontinuity between the classical and modern Bendat analysis is the
presence of the √ term in the denominator in the modern analysis. See the equation below.
(| ̂ ( )|)
( )
√
( )√
| ̂ ( )|
This term could be a result of the reduction in terms from the classical analysis. It could also be a result
of an assumption that the modern analysis would only be applied to RMS values. Regardless, the
classical analysis as implemented for this thesis does not have an additional √ term in the denominator.
The resolution of the significance of this term is still unknown.
The most important area where the difference between these two analyses is evident is in the
generic confidence bounds themselves. From the table for the normally distributed values, there appear
no discontinuities or strange values. This was also seen in the results, where the confidence bounds
calculated assuming a normal distribution were continuous and behaved as expected. When one looks at
the values for the F-Distribution values, it is immediately clear that there is a discontinuity for low
coherence and low ensemble count.
24 / 56
Figure 2: F-Distribution Discontinuity
This indicates that, in these areas, the error term is greater than unity and the statistical variance is
greater than the measurement value itself. While this does not seem to make much sense practically, one
must remember that the confidence bounds in this thesis are measuring the possible variance from a
statistical point of view. In such circumstances, it is possible for the measured value to be due more to
variance than the user-controlled inputs to the system. One must be wary however, as when such areas
are plotted on the decibel or logarithmic scale (as FRF’s often are) such large variances appear as breaks
in the plot. For a more complete picture, one may have to look at the real and imaginary plots or the nonlogarithmic FRF magnitude plots to see the true value of the expected variance.
As most modern data acquisition software packages do not offer much in the way of variance
measurement and/or graphical display, both of the confidence bounds analyses described above are
beneficial. Both offer a unique insight into the behavior of the measurement and allow the technician to
statistically determine how much variance has been removed from a measurement. However, the modern
Bendat analysis, while easy to implement, makes certain assumptions that simply aren’t necessary with
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the computer tools available today. Despite its added complexity, the F-distribution technique will be
used for this thesis as it is more useable under a wider range of test scenarios. The additional complexity
is largely remedied by correct implementation with the software which will be discussed in the next
section.
3.4.
Application in X-Modal
The initial application of the FRF confidence bounds in a software package was in the V-
Acquisition module of the X-Modal software package developed by the University of Cincinnati’s
Structural Dynamics Research Lab (UC-SDRL). This software was developed for use in graduate level
vibrations courses in order to have complete control over the acquisition and analysis techniques from a
software point of view. Most commercially available software packages do not reveal the exact methods
used in calculations or assumptions made. This leads to “black box” analyses which don’t necessary help
ones understanding of the data. Depending on the analysis, this could affect the results and skew the
understanding of the structure(s) under analysis. Having control allows UC-SDRL to better understand
the results of the various tests performed and provide a better learning tool for students. The ability to see
exactly how the program is carrying out commands and calculations also make it an ideal test bed for the
confidence interval calculations in an actual acquisition and analysis package.
As has already been discussed, the modern Bendat analysis is computationally very
straightforward whereas the classical approach requires additional considerations. From the last section,
it is known that changing the confidence interval is not as simple as changing a confidence coefficient as
it was for the normally distributed error. Each confidence interval has its own table for the F-values
required for the confidence bounds calculation. On top of that, the coefficient changes depending on the
two degrees of freedom: the number of ensembles and the number of correlated input autopowers to an
output autopower. With modern data acquisition software and hardware and for the purposes of this
thesis, it will be assumed that each FRF will be the result of a single input, thus the number of input
autopowers to this analysis can be held constant at one.
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For the number of ensembles, it is necessary to look up a new value each time the FRF is
recalculated.
Therefore, in the V-Acquisition module there is script which will check the desired
confidence interval, call the corresponding table and find the number of ensembles and pull the
appropriate F-value for use in the calculation. This allows the correct analysis to be carried out with little
additional input from the user. While this allows the correct implementation of Bendat’s earlier analysis,
it does present certain problems within the V-Acquisition environment itself. These are, namely, time
delay related issues that will be discussed in greater detail in the Conclusions and Future Work section.
Actually writing the lines of code for the equations required was not difficult. The MATLAB
environment makes it particularly easy to enter the algebraic equations quickly. Certain care had to be
taken to ensure that the correct values of the FRF and coherence were fed into the analysis. This was
accomplished by proper indexing of both the input variables and the confidence bounds values
themselves. Furthermore, since the use of multiple FRF estimators was discussed earlier in this thesis it
was desired to have separate analysis techniques for the various estimators. This necessitated the use of a
gateway to see which estimator had been chosen and apply the appropriate confidence bound technique.
The greater programming difficulties actually arose when trying to design the graphical user interface
(GUI) to allow the confidence interval analysis to be easy to use.
The calculations and plotting were originally carried out on V-Acquisition’s “Measure” screen.
This was done so that the confidence bounds calculations could be done “live” as the measurement was
taking place and allow the technician to see in real time the decreasing variance. Applying the analysis in
this way presented a significant drawback as the confidence bounds calculations had to then be carried out
“online”, or as the data was being acquired by the software. This lead to issues where data may be lost or
not stored properly when the user would select different channels for viewing during the acquisition
cycle. Therefore, the analysis was moved to the “Display” screen, where post-acquisition inspection and
brief analysis is often carried out. In this way, the confidence bounds were calculated at a different time
from the data acquisition. This enabled the confidence bounds calculations to be more robust and limited
MATLAB errors sometimes caused by multiple calculations and plotting of data.
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One issue that arose in actually implementing the math in X-Modal was in how the phase
calculations were carried out. For starters, V-Acquisition displays the phase plot by wrapping the phase
by default, where the plots are bounded by ±180°. This is also how the phase confidence interval
calculations were applied. Brief efforts were made to allow the confidence bounds calculations to also
work with “unwrapped” phase plots (a selectable option in V-Acquisition), but this proved problematic
and did not provide significant additional insight. In fact, often times the phase confidence bounds values
would “blow up” and not even display all the data on the screen. For scope of this thesis, the phase
confidence bounds are only applied to the wrapped phase plots.
The fact that the FRF bounds were applied to wrapped phase plots may at first seem trivial.
However, even when the initial application was carried out using Bendat’s classical techniques it was
discovered that the upper and lower bounds would switch signs around the zero degree phase line. This
was especially apparent as the phase will range between -180°to +180° depending on the resonances and
anti-resonances. In this case, since the value of the phase at each Δf was multiplied across all terms in the
confidence bounds equation, the lower bounds would then be added to the measured phase and the upper
bounds subtracted from the measured phase at certain Δf’s. In order to prevent this problem an extra line
of code was used in the phase confidence bounds equation. This was to determine the sign of the phase
calculation at each Δf and apply the appropriate sign in the confidence bounds equation. This prevents
the “flipping” of the confidence bounds and allowed the upper bounds to always be above and the lower
bounds to always be below the measured phase. For further details on this fix, please consult the
appendix.
Once the use of the Bendat equations was finalized (both for the error term and its application in
the phase confidence bounds) there were fewer issues in the application in V-Acquisition. Once the phase
confidence interval term is determined as explained earlier, it was simply added or subtracted, point by
point, to the phase. This was done for each input-output FRF acquired and processed by the hardware and
software, just as for the FRF confidence bounds.
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For actually viewing the FRF confidence bounds, the analysis was originally disabled for
displaying multiple input-output FRFs on the screen at the same time. This was done to limit the amount
of information on the screen and reduce confusion when visually analyzing the data. However, in order
to properly disable the analysis plus all the GUI controls which control the analysis, a call must be sent
back through to the appropriate sub-functions.
Using the existing structure, this call propagated
throughout various parts of the code, including sections that tend to be memory and time intensive. This
added a significant amount of time to the selection of FRFs, even when the confidence bounds analysis
itself is disabled and no calculations are performed. As this program is primarily meant as a dataacquisition tool and this new analysis technique is still much in its infancy, preference was given to the
existing programming.
As a result, the analysis does not disable itself for multiple input-output
combinations. This can lead to rather cluttered plots when looking at the confidence bounds for multiple
FRFs, but the program does run much faster regardless of the number of FRFs selected. For the time
being this is how the confidence bounds will be implemented until a better method for disabling the
analysis is developed.
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Chapter 4: Results
4.1
Section 1: Simulated Data
For the initial analysis of the Bendat method for confidence bounds, simulated data was used.
This was data taken earlier that is used in debugging the X-Modal and V-Acquisition software without
having to connect to actual data acquisition hardware or run tests on physical structures. This allows
quicker inspection and debugging of the confidence bounds equation as well as removing all time
variance from the analysis. Complete removal of time variance enables different analysis procedures to
be used on the same data and the results compared back-to-back. This proved most useful as different
iterations of both of Bendat’s techniques were investigated throughout the course of this thesis. It also
sped up the validation of different programming techniques for their robustness and speed.
As this is just an initial application of this technique, the test technician is currently unable to save
the confidence bounds values through V-Acquisition module itself. The values of the confidence bounds
are simply used to graphically display the confidence interval. Therefore, the actual data was acquired
while MATLAB was placed into debug mode. Once the calculations for the bounds were completed, the
program was halted before the desired variables were overwritten. Then, the appropriate data was saved
into a *.mat file using keystroke commands in the MATLAB command window. This allowed for closer
inspection of the data after the test is completed without requiring the V-Acquisition or X-Modal software
itself. This is also how the plots used in this thesis were generated.
The test setup will vary through the different parts of this section, but a few general notes will be
addressed here. Firstly, the maximum number of ensembles gathered in any single test will be limited to
100. This is primarily a result of the amount of data available for processing using the front-end
simulator. This should not negatively affect the actual results of the analysis as most modal tests are
thought to converge at or below 100 ensembles.
For the simulated data, the test setup is as follows:
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DSP
Avg
Span Freq
Nr Lines
FRF Estimator
Online FRF
Pretrigger
Delay
Window
800 Hz Avg Type
800
Nr Ensembles
H1
Nr Cyclic Blocks per Ensemble
On
Display Update Rate
0
%
Enable Ensemble Preview
Ovld Mode
RMS
w/Cyclic
20
4
1
Off
accept
rng
Hanning
Window P110
-
w/
This is intended to mirror the actual test performed to acquire the data in the first place. As the raw data
is already acquired, it is unnecessary to modify the source setup for correct data interpretation.
The first tests were run using an H1 estimation technique since that is the technique suggested
from Bendat’s later analysis, although here it will be applied to the classical analysis. The goal was to see
the effect that both the FRF estimator and the specific confidence bounds equation had on the data. As
has been established, for the confidence bounds techniques used herein, the type of FRF estimator will
affect the final result. The final confidence bounds analysis for data processed using the H1 FRF
estimator will be presented first. Data will be presented in this section for the confidence bounds analysis
of data processed using the H2 and Hv methods as well.
Before the results of these tests are presented, there is one important thought worth mentioning.
When this analysis was first started, it was hypothesized that the different FRF estimators would yield
asymmetric confidence bounds. For example, the confidence bounds on an FRF determined using a H 1
technique, which underestimates the values at the peaks, would have a larger upper confidence bound
than a lower confidence bound. This hypothesis comes from the thought that the actual FRF is much
more likely to be above the H 1 estimate than below. A quick look at the equations used will show that the
FRF will be equally bounded regardless of the estimator used. Also, the effect of the different FRF
estimators is a result of bias (leakage) error, not random error. Therefore, this error is beyond the scope
of this thesis and will not be addressed here.
31 / 56
The initial test setup described above yields very “clean” FRF data, even with a relatively low
number of ensembles for a MIMO test. A large reason for this is the use of cyclic averaging to help
reduce bias and variance without the need for a large number of actual ensembles. While this data is
helpful in the validation of the confidence bounds, it lends itself to data with relatively small variance.
For the purposes of this thesis, less polished FRFs will also be determined from the raw data to see how
much the variance increases with these noisier measurements and how well the confidence bounds
analysis reflects the difference.
The first thing to do is inspect that the measured FRF and phase are in fact bounded by the
confidence bounds as calculated by the script. This was checked at various frequencies to see how the
script handled noisy data as well as the data at resonances and anti-resonances. Quick examples of the
FRF and phase confidence bounds are presented here.
32 / 56
Confidence Bounds
150
100
Magnitude (db)
50
0
-50
-100
-150
0
100
200
300
400
500
600
700
800
900
Frequency (Hz)
Figure 3: Confidence Bounds for Test 1, 10 Ensemble, 90% Confidence Interval
A quick glance at this data shows what one would expect from the confidence interval
calculation. The confidence bounds encapsulate the measured and calculated FRF, with the black line
indicating the upper bounds and the red line indicating the lower bounds. The blue line is the plot of the
estimate of the actual measured FRF. Also, as one increases the confidence interval, it becomes clear that
the confidence bounds also increase.
33 / 56
Confidence Bounds
150
100
Magnitude (db)
50
0
-50
-100
-150
0
100
200
300
400
500
600
700
800
900
Frequency (Hz)
Figure 4: Confidence Bounds for Test 1, 10 Ensemble, 99% Confidence Interval
The complete FRF is not shown here because the confidence interval is only noticeable at the
peaks. For the phase measurements, the difference between the two confidence intervals is most apparent
between the phase shifts, i.e. 400-550 Hz. For clarity, the numerical differences with respect to the
measured FRF are presented below.
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Upper Bounds
Lower Bounds
90%
1.3891
-2
99%
2.2055
-2.9635
Table 7: Confidence Interval Comparison
This verifies that the script is performing its calculations correctly and shows that the analysis
presented herein follows the theory of confidence bounds logically. From an analytical point of view, this
also indicates that for a higher confidence interval more ensembles will be required to reach the same
upper and lower bounds as for a lower confidence interval. The applications of this observation will be
discussed in greater length in the Conclusions and Future Work section of this thesis.
Next, it is important to verify that both the FRF and phase confidence bounds do in fact become
smaller as more and more ensembles are gathered. To do this, a test will be conducted and the resultant
FRF and confidence bounds will be saved at specified number of ensembles. Then, the upper and lower
bounds at various frequencies will be inspected at each number of ensembles to see if the bounds are in
fact approaching the measured FRF. As ensembles increase, results from this inspection will then be
graphed to more easily identify trends in the confidence bounds as a function of ensemble number. This
will represent how well the measured data is approaching the expected value as the variance is removed
from the data. The plot of these trends is plotted below.
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Difference from Measured FRF (dB)
2.5
2
1.5
Upper Bound
1
Lower Bound
0.5
0
10
20
30
40
50
60
70
80
90 100
# of Ensembles
Figure 5: Confidence Bounds Trend, H 1
The preceding plot does in fact show how the increasing number of ensembles removes variance
from the measurement. Of particular note is how the variance drop off is very steep initially and then
starts to level out. This is what one would expect from these results: the larger the sample size (number
of ensembles in this case), the smaller the variance should become as the error is averaged out in the
calculation of the mean FRF.
Hv FRF Estimator
For comparison purposes, the confidence bounds were applied to the different kinds of FRF
estimators.
In the case of the Hv estimation technique, one can see that the variance trends in much the same
way as for the H1 estimator. It can also be seen that the two confidence limits for the two differ primarily
on scaling. For example, the highest value of the lower bounds using the H 1 technique is around 2 db,
whereas it is closer to 1.1 db in the case of the H v FRF estimator. This behavior supports what one would
expect. The FRF estimators are used to help eliminate noise from the input or output signals. Therefore,
if the estimator removes noise from both the input and the output, the variance itself should be smaller.
36 / 56
This will be evidenced by the scaling of the plot of the upper and lower confidence bounds as a function
of ensemble.
Difference from Measured FRF (db)
1.2
1
0.8
0.6
Upper Bounds
0.4
Lower Bounds
0.2
0
10
20
30
40
50
60
70
80
90 100
# of Ensembles
Figure 6: Confidence Bounds Trends, H v
It is also an interesting observation that, theoretically anyway, the H v technique would be the best
at finding the expected FRF amplitude, not including bias errors. Thus, it wouldn’t be too far of a stretch
to imagine that the unaltered Bendat equations work best with this particular analysis.
One drawback of the Hv analysis is that it does take much longer than the H1 method. This is
likely due to the more intensive mathematical manipulation required for the total least squares solution.
H2
For comparison purposes, the confidence bounds were also applied to FRFs determined using the
H2 FRF estimator. The program automatically selects the appropriate confidence interval calculations
based on the FRF estimator currently selected.
One can easily see from the equations determined earlier in this thesis that any confidence bounds
determined this way will actually be the smallest. This will be a result of the mathematical relationships
used. The H1 estimator has a coherence term in the denominator whereas the theoretical H v estimator has
37 / 56
no additional coherence term. The H2 estimator has this additional coherence term in the numerator
which then causes this term to be the smallest of the three estimators discussed here; remember,
coherence always has a value between zero and one. This goes against the theory that the Hv estimator
should be the best at eliminating the most noise from the calculations.
The confidence interval trend for the H2 estimator is presented below.
Difference from Measured FRF (db)
1.2
1
0.8
0.6
Upper Bounds
0.4
Lower Bounds
0.2
0
10
20
30
40
50
60
70
80
90 100
# of Ensembles
Figure 7: Confidence Bounds Trends, H 2
After plotting the convergence characteristics of each of the confidence intervals for the different
FRF estimators, it is noticed that they do not converge to the same level of variance for each measurement
(H1 converges noticeably higher than the other two methods). This leads to the conclusion that the errors
associated with this measurement are largely related to bias, not random variance. A check of the
simulated data and test structure confirms this. Further investigation into this will be discussed in the
Conclusions and Future work section of this Thesis.
GENERAL OBSERVATIONS
Some interesting behavior exhibited during earlier analysis of the confidence bounds was at lower
frequencies. Looking at the data presented, the lower bounds often appeared above the measured FRF.
While this could be indicative of some odd behavior of the Bendat equations at low frequency or low
38 / 56
coherence resulting from the use of AC accelerometers at low frequency, it was most likely the result of a
sign issue in the confidence bounds script. While performing other maintenance on the program (not
related to the confidence bounds calculations themselves) this anomaly disappeared and has not
resurfaced since the final Bendat equations have been applied. This may be something to be cautious of
when using this analysis in the future when expanding the script to more accurately portray Bendat’s
methods.
As far as the hypothesis on how the FRF estimator affects the confidence bounds the data and
equations allow a clear picture. With any of the analyses discussed in this thesis, either the smallerror/normally distributed variance or when using the F-distribution model, the upper and lower bounds
will be equal. In other words, while the different FRF estimators may affect the actual confidence
interval, the upper and lower bounds will always be equal. This is because either the underestimation or
overestimation of the FRF as resulted from the FRF estimators is actual a bias error and is not accounted
for in the variance calculation. Therefore, this phenomenon is outside the scope of this analysis and
thesis.
39 / 56
Chapter 5.
Conclusions and Future Work
The purpose of this thesis was to investigate methods for determining a confidence
interval for an FRF measurement.
Original methods were presented alongside more
contemporary analyses in the attempt to resolve most, if not all, of the inconsistencies and offer a
reasonable and diverse method for FRF confidence bounds analysis.
While the data presented herein does offer a way to quantitatively measure the amount of
variance of an FRF measurement, it does not take into account the bias errors associated with
such a measurement (Bendat & Piersol, 1971). These errors would be most evident at the
resonances and anti-resonances and would be significantly larger in lightly damped systems.
The analysis as currently presented cannot quantify these errors as they are constant throughout
the test. As a result, any analysis which measures the variance of the measurement would serve
no use in finding these errors. The only way to determine such fallacies would be double check
test and channel setup and/or to compare the acquired data with known benchmark data.
The concern over bias error on the measurements is demonstrated in the convergence
data presented in this thesis.
While the confidence bounds do converge with increasing
ensembles, they converge about different values for the different FRF estimators. This is a result
of the convergence data being taken from one of the peaks in the FRF magnitude plot where the
coherence was quite low, indicating that bias errors are dominate. Further validation should be
conducted on a system where random errors dominate here. This can be achieved on a MIMO
setup by attaching an additional shaker to the structure and powering it with an external power
source and random excitation. This creates a condition where random noise dominates the
measurement and, if the confidence bounds analysis is correct, the upper and lower bounds
should converge to a the same number for each FRF estimators.
40 / 56
One possible application of this work in future iterations may include a form of
convergence measurement. This can be particularly useful in shaker tests which, while requiring
more complicated setups than impact testing, allow the test technician to take an almost
unlimited number of ensembles. Of course, this can be limited by the number of data channels,
block size, and system memory. Such a program would not be appropriate for impact testing as
the number of averages required to eliminate significant variance would likely be very large.
This is quite impractical for impact testing. There is also much more random error inherent in
impact testing, such as being able to impact the same spot consecutively.
By taking a greater number of ensembles, the technician is able to reduce the amount of
variance in the FRF calculation.
This will be visibly and numerically illustrated by the
confidence bounds. Therefore, an option could be implemented in future software packages
where, instead of a user defined number of ensembles, the program itself could determine an
acceptable number of averages. The technician could simply setup the hardware and digital
signal processing and run the test. When the confidence interval became small enough, as
determined either by other experimental data or by user preference, the program would
automatically terminate the test. This would help eliminate the need for the technician to preselect a number of ensembles which may or may not eliminate enough variance. Furthermore, if
the number of ensembles is not enough to sufficiently remove the variance the technician
typically would have to start the test all over again. Having a general “convergence” criterion
could help eliminate this problem for less experienced technicians. Again, this would work best
as an aid. More experienced technicians would likely know the test structure and particular test
objectives and, thus, be able to choose a more appropriate number of ensembles.
41 / 56
While this analysis is geared predominantly toward MIMO shaker testing, its usefulness
does expand to testing and even operational data, which is very common in NVH testing
performed at commercial facilities. In impact testing, the confidence bounds analysis could help
determine if the technician is hitting consistently enough, albeit in the relative absence of bias
errors. Also, when impact testing is being conducted in the field with many potential noise
sources, it will provide a clearer indicator of how many impacts are required to sufficiently
minimize the variance. The same benefit could be realized in operational testing where taking
multiple runs over time could yield varied results. The various runs could be averaged and the
confidence bounds determined and then the individual runs compared with the confidence
interval. Any runs outside the confidence interval can be considered outliers and discarded.
They key thing to note when performing this sort of analysis is that, in order to develop an FRF
calculation from operational data, one must chose a certain point as the input. This often means
developing A/A FRFs rather than the A/F ratios used in this thesis.
42 / 56
Bibliography
Allemang, R. J. (2007). Vibrations: Experimental Modal Analysis. Cincinnati: UC-SDRL.
Bendat, J. (1978). Statistical Errors in Measurement of Coherence Functions and Input/Output
Quantities. Journal of Sound and Vibration, 405-421.
Bendat, J. S., & Piersol, A. G. (1971). Random Data: Analysis and Measurement Procedures.
John Wiley & Sons, Inc.
Bendat, J. S., & Piersol, A. G. (1993). Engineering Applications of Correlation and Spectral
Analysis: Second Edition. New York City: John Wiley & Sons, Inc.
Enochson, L., & Goodman, N. (1965). Gaussian Approximations to the Distribution of Sample
Coherence. Wright Patterson Air Force Base: Air Force Flight Dynamics Laboratory:
Research and Technology Division: Air Force Systems Command.
Farrar, C. R., Doebling, S. W., & Cornwell, P. J. (n.d.). A Comparison Study of Modal
Parameter Confidence Intervals Computed Using the Monte Carlo and Bootstrap
Techniques.
Hewlett Packard. (1978). Measuring the Coherence Function with the HP 3582A Spectrum
Analyzer. Palo Alto, California, U.S.A: Hewlett Packard.
Majba, C., Allemang, R., & Phillips, A. (2012). A Review of Uncertainty Quantification
Estimation of Frequency Response Functions. IMAC XXX.
43 / 56
Appendix
Appendix 1: Error Term Calculations
Error Term (HP Table vs. 90% F-Distribution (Hv Technique))
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
-2.84 -2.03 -1.45 -1.04 -0.75 Upper H
-4.24 -2.66 -1.74 -1.19 -0.82 Lower H
-22.73 -15.28 -10.45 -7.34 -5.17 Phase
2.50
1.87
1.43 1.11 0.79 Upper H
10.66
4.62
2.57 1.59 1.03 Lower H
32.63 19.61 13.15 9.08 6.12 Phase
1.65
1.29
0.91 0.67 0.54 Upper H
4.78
2.50
1.49 0.96 0.68 Lower H
18.07 11.56
7.80 3.53 2.44 Phase
1.12
0.91
0.60 0.44 0.38 Upper H
2.71
1.50
0.91 0.64 0.44 Lower H
11.61
7.58
5.48 4.01 2.78 Phase
0.80
0.54
0.40 0.31 0.24 Upper H
1.55
0.90
0.64 0.43 0.30 Lower H
7.26
4.67
3.23 2.54 1.15 Phase
0.51
0.39
0.31 0.19 0.19 Upper H
0.91
0.52
0.38 0.23 0.16 Lower H
4.07
2.88
2.06 1.11 0.55 Phase
0.35
0.27
0.14 0.08 0.06 Upper H
0.51
0.27
0.24 0.13 0.14 Lower H
2.11
1.25
0.99 0.77 1.02 Phase
0.15
0.08
0.09 0.09 0.04 Upper H
0.25
0.15
0.12 0.06 0.02 Lower H
1.50
0.86
1.10 0.55 0.56 Phase
0.05
0.07 -0.01 0.04 0.04 Upper H
0.10
0.00
0.06 0.02 0.03 Lower H
0.60 -0.04
0.53 0.56 0.28 Phase
44 / 56
Error Term (HP Table) (Hewlett Packard, 1978)
nd
γxy2
16
32
64
128
256
5.2
14.6
54
4.2
-8.4
38
3.5
-6
30
3
-4.5
24
2.5
-3.5
19
2.1
-2.7
15
1.6
-2
12
1.1
-1.3
8
3.8
7.1
34
3.1
-4.8
25
2.6
-3.6
20
2.1
-2.8
16
1.8
-2.2
13
1.5
-1.7
10
1.1
-1.3
8
0.8
-0.8
5
2.8
4.2
23
2.2
-3
17
1.8
-2.3
14
1.5
-1.9
11
1.3
-1.5
9
1
-1.2
7
0.8
-0.9
6
0.5
-0.6
4
2.1
2.7
16
1.6
-2
10
1.3
-1.6
10
1.1
-1.3
8
0.9
-1
6
0.7
-0.8
5
0.6
-0.6
4
0.4
-0.4
3
1.5
1.8
11
1.2
-1.4
7
1
-1.1
7
0.8
-0.9
5
0.7
-0.7
4
0.5
-0.6
4
0.4
-0.4
3
0.3
-0.3
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
45 / 56
Error Term (F-Distribution 90% (H1 Technique))
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
6.93
5.26 3.94 2.95 2.18 Upper H
-Inf
-15.56 -7.4 -4.49 -2.91 Lower H
90
56.4
35 23.8 16.6 Phase
5.18
3.84 2.81 2.07 1.51 Upper H
-14.64
-7.04 -4.19 -2.73 -1.83 Lower H
54.6
33.7 22.5 15.6
11 Phase
4.2
3.07 2.23 1.62 1.18 Upper H
-8.45
-4.8
-3
-2 -1.36 Lower H
38.5
25.1
17 11.9
8.3 Phase
3.52
2.54 1.83 1.33 0.96 Upper H
-6 -3.61 -2.32 -1.57 -1.07 Lower H
29.9
19.9 13.5
9.5
6.7 Phase
2.97
2.13 1.52
1.1 0.79 Upper H
-4.54 -2.83 -1.84 -1.26 -0.87 Lower H
24
16.1
11
7.7
5.5 Phase
2.49
1.78 1.26 0.91 0.65 Upper H
-3.51 -2.23 -1.47 -1.01 -0.7 Lower H
19.4
13.1
9
6.3
4.4 Phase
2.05
1.45 1.02 0.73 0.52 Upper H
-2.69 -1.74 -1.16 -0.8 -0.56 Lower H
15.5
10.5
7.2
5.1
3.6 Phase
1.61
1.13 0.79 0.57
0.4 Upper H
-1.98
-1.3 -0.87 -0.61 -0.42 Lower H
11.8
8
5.5
3.9
2.7 Phase
1.11
0.77 0.54 0.38 0.27 Upper H
-1.27 -0.84 -0.57 -0.4 -0.28 Lower H
7.8
5.3
3.7
2.6
1.8 Phase
46 / 56
Error Term (F-Distribution 90% (H2 Technique))
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
1.00
-1.13
7.02
1.31
-1.54
9.38
1.49
-1.79
10.76
1.58
-1.93
11.51
1.61
-1.98
11.75
1.58
-1.93
11.51
1.49
-1.79
10.76
1.31
-1.54
9.38
1.00
-1.13
7.02
32
0.70
-0.76
4.78
0.92
-1.02
6.38
1.04
-1.18
7.31
1.11
-1.27
7.82
1.13
-1.30
7.98
1.11
-1.27
7.82
1.04
-1.18
7.31
0.92
-1.02
6.38
0.70
-0.76
4.78
64
0.48
-0.51
3.29
0.64
-0.69
4.39
0.73
-0.80
5.03
0.78
-0.85
5.37
0.79
-0.87
5.49
0.78
-0.85
5.37
0.73
-0.80
5.03
0.64
-0.69
4.39
0.48
-0.51
3.29
128
0.34
-0.36
2.32
0.46
-0.48
3.09
0.52
-0.55
3.54
0.55
-0.59
3.78
0.57
-0.61
3.86
0.55
-0.59
3.78
0.52
-0.55
3.54
0.46
-0.48
3.09
0.34
-0.36
2.32
256
0.24
-0.25
1.63
0.32
-0.34
2.18
0.37
-0.39
2.50
0.40
-0.41
2.67
0.40
-0.42
2.72
0.40
-0.41
2.67
0.37
-0.39
2.50
0.32
-0.34
2.18
0.24
-0.25
1.63
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
47 / 56
Error Term (F-Distribution 90% (Hv Technique))
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
2.84
-4.24
22.73
2.70
-3.94
21.37
2.55
-3.62
19.93
2.38
-3.29
18.39
2.20
-2.95
16.74
1.99
-2.59
14.93
1.75
-2.19
12.89
1.45
-1.75
10.50
1.05
-1.20
7.40
32
64 128
256
2.03 1.45 1.04 0.75 Upper H
-2.66 -1.74 -1.19 -0.82 Lower H
15.28 10.45 7.34 5.17 Phase
1.93 1.37 0.99 0.71 Upper H
-2.48 -1.63 -1.11 -0.77 Lower H
14.39 9.85 6.92 4.88 Phase
1.81 1.29 0.93 0.66 Upper H
-2.30 -1.51 -1.04 -0.72 Lower H
13.44 9.20 6.47 4.56 Phase
1.69 1.20 0.86 0.62 Upper H
-2.10 -1.39 -0.96 -0.66 Lower H
12.42 8.52 5.99 4.22 Phase
1.56 1.10 0.79 0.56 Upper H
-1.90 -1.26 -0.87 -0.60 Lower H
11.33 7.77 5.46 3.85 Phase
1.41 0.99 0.71 0.51 Upper H
-1.68 -1.12 -0.77 -0.54 Lower H
10.12 6.94 4.89 3.45 Phase
1.23 0.86 0.62 0.44 Upper H
-1.43 -0.96 -0.67 -0.46 Lower H
8.75 6.01 4.23 2.98 Phase
1.02 0.71 0.51 0.36 Upper H
-1.15 -0.78 -0.54 -0.38 Lower H
7.14 4.90 3.45 2.44 Phase
0.73 0.51 0.36 0.26 Upper H
-0.80 -0.54 -0.38 -0.27 Lower H
5.04 3.47 2.44 1.72 Phase
48 / 56
Error Term (Normal Distribution 90%)
nd
γxy2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
5.44
-17.7
0.9
3.97
-7.53
0.6
3.18
-5.08
0.4
2.64
-3.81
0.4
2.21
-2.97
0.3
1.85
-2.35
0.2
1.51
-1.83
0.2
1.18
-1.36
0.1
0.8
-0.88
0.1
32
4.16
-8.29
0.6
2.98
-4.58
0.4
2.37
-3.26
0.3
1.95
-2.51
0.3
1.62
-1.99
0.2
1.34
-1.59
0.2
1.09
-1.25
0.1
0.85
-0.94
0.1
0.57
-0.61
0.1
64
3.14
-4.96
0.4
2.21
-2.97
0.3
1.74
-2.17
0.2
1.42
-1.7
0.2
1.18
-1.36
0.1
0.97
-1.09
0.1
0.79
-0.87
0.1
0.61
-0.65
0.1
0.41
-0.43
0
128
2.33
-3.19
0.4
1.62
-1.99
0.2
1.26
-1.48
0.2
1.03
-1.17
0.1
0.85
-0.94
0.1
0.7
-0.76
0.1
0.56
-0.6
0.1
0.43
-0.46
0.1
0.29
-0.3
0
256
1.71
-2.13
0.2
1.18
-1.36
0.1
0.91
-1.02
0.1
0.74
-0.81
0.1
0.61
-0.65
0.1
0.5
-0.53
0.1
0.4
-0.42
0
0.31
-0.32
0
0.21
-0.21
0
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
Upper H
Lower H
Phase
49 / 56
Appendix 2: F-Distribution Confidence Coefficient
F-Distribution Confidence Coefficient Table (95.0% CI): (Dinov, 2011)
df1
df2
1
2
3
4
5
6
7
8
9
1
161.448
199.5
215.707
224.583
230.162
233.986
236.768
238.883
240.543
2
18.5128
19
19.1643
19.2468
19.2964
19.3295
19.3532
19.371
19.3848
3
10.128
9.5521
9.2766
9.1172
9.0135
8.9406
8.8867
8.8452
8.8123
4
7.7086
6.9443
6.5914
6.3882
6.2561
6.1631
6.0942
6.041
5.9988
5
6.6079
5.7861
5.4095
5.1922
5.0503
4.9503
4.8759
4.8183
4.7725
6
5.9874
5.1433
4.7571
4.5337
4.3874
4.2839
4.2067
4.1468
4.099
7
5.5914
4.7374
4.3468
4.1203
3.9715
3.866
3.787
3.7257
3.6767
8
5.3177
4.459
4.0662
3.8379
3.6875
3.5806
3.5005
3.4381
3.3881
9
5.1174
4.2565
3.8625
3.6331
3.4817
3.3738
3.2927
3.2296
3.1789
10
4.9646
4.1028
3.7083
3.478
3.3258
3.2172
3.1355
3.0717
3.0204
11
4.8443
3.9823
3.5874
3.3567
3.2039
3.0946
3.0123
2.948
2.8962
12
4.7472
3.8853
3.4903
3.2592
3.1059
2.9961
2.9134
2.8486
2.7964
13
4.6672
3.8056
3.4105
3.1791
3.0254
2.9153
2.8321
2.7669
2.7144
14
4.6001
3.7389
3.3439
3.1122
2.9582
2.8477
2.7642
2.6987
2.6458
15
4.5431
3.6823
3.2874
3.0556
2.9013
2.7905
2.7066
2.6408
2.5876
16
4.494
3.6337
3.2389
3.0069
2.8524
2.7413
2.6572
2.5911
2.5377
17
4.4513
3.5915
3.1968
2.9647
2.81
2.6987
2.6143
2.548
2.4943
18
4.4139
3.5546
3.1599
2.9277
2.7729
2.6613
2.5767
2.5102
2.4563
19
4.3807
3.5219
3.1274
2.8951
2.7401
2.6283
2.5435
2.4768
2.4227
20
4.3512
3.4928
3.0984
2.8661
2.7109
2.599
2.514
2.4471
2.3928
21
4.3248
3.4668
3.0725
2.8401
2.6848
2.5727
2.4876
2.4205
2.366
22
4.3009
3.4434
3.0491
2.8167
2.6613
2.5491
2.4638
2.3965
2.3419
23
4.2793
3.4221
3.028
2.7955
2.64
2.5277
2.4422
2.3748
2.3201
24
4.2597
3.4028
3.0088
2.7763
2.6207
2.5082
2.4226
2.3551
2.3002
25
4.2417
3.3852
2.9912
2.7587
2.603
2.4904
2.4047
2.3371
2.2821
26
4.2252
3.369
2.9752
2.7426
2.5868
2.4741
2.3883
2.3205
2.2655
27
4.21
3.3541
2.9604
2.7278
2.5719
2.4591
2.3732
2.3053
2.2501
28
4.196
3.3404
2.9467
2.7141
2.5581
2.4453
2.3593
2.2913
2.236
29
4.183
3.3277
2.934
2.7014
2.5454
2.4324
2.3463
2.2783
2.2229
30
4.1709
3.3158
2.9223
2.6896
2.5336
2.4205
2.3343
2.2662
2.2107
40
4.0847
3.2317
2.8387
2.606
2.4495
2.3359
2.249
2.1802
2.124
60
4.0012
3.1504
2.7581
2.5252
2.3683
2.2541
2.1665
2.097
2.0401
120
3.9201
3.0718
2.6802
2.4472
2.2899
2.175
2.0868
2.0164
1.9588
inf
3.8415
2.9957
2.6049
2.3719
2.2141
2.0986
2.0096
1.9384
1.8799
50 / 56
10
12
241.882
243.906
19.3959
15
20
24
30
40
60
120
INF
245.95
248.013
249.052
250.095
251.143
252.196
253.253
254.314
19.4125
19.4291
19.4458
19.4541
19.4624
19.4707
19.4791
19.4874
19.4957
8.7855
8.7446
8.7029
8.6602
8.6385
8.6166
8.5944
8.572
8.5494
8.5264
5.9644
5.9117
5.8578
5.8025
5.7744
5.7459
5.717
5.6877
5.6581
5.6281
4.7351
4.6777
4.6188
4.5581
4.5272
4.4957
4.4638
4.4314
4.3985
4.365
4.06
3.9999
3.9381
3.8742
3.8415
3.8082
3.7743
3.7398
3.7047
3.6689
3.6365
3.5747
3.5107
3.4445
3.4105
3.3758
3.3404
3.3043
3.2674
3.2298
3.3472
3.2839
3.2184
3.1503
3.1152
3.0794
3.0428
3.0053
2.9669
2.9276
3.1373
3.0729
3.0061
2.9365
2.9005
2.8637
2.8259
2.7872
2.7475
2.7067
2.9782
2.913
2.845
2.774
2.7372
2.6996
2.6609
2.6211
2.5801
2.5379
2.8536
2.7876
2.7186
2.6464
2.609
2.5705
2.5309
2.4901
2.448
2.4045
2.7534
2.6866
2.6169
2.5436
2.5055
2.4663
2.4259
2.3842
2.341
2.2962
2.671
2.6037
2.5331
2.4589
2.4202
2.3803
2.3392
2.2966
2.2524
2.2064
2.6022
2.5342
2.463
2.3879
2.3487
2.3082
2.2664
2.2229
2.1778
2.1307
2.5437
2.4753
2.4034
2.3275
2.2878
2.2468
2.2043
2.1601
2.1141
2.0658
2.4935
2.4247
2.3522
2.2756
2.2354
2.1938
2.1507
2.1058
2.0589
2.0096
2.4499
2.3807
2.3077
2.2304
2.1898
2.1477
2.104
2.0584
2.0107
1.9604
2.4117
2.3421
2.2686
2.1906
2.1497
2.1071
2.0629
2.0166
1.9681
1.9168
2.3779
2.308
2.2341
2.1555
2.1141
2.0712
2.0264
1.9795
1.9302
1.878
2.3479
2.2776
2.2033
2.1242
2.0825
2.0391
1.9938
1.9464
1.8963
1.8432
2.321
2.2504
2.1757
2.096
2.054
2.0102
1.9645
1.9165
1.8657
1.8117
2.2967
2.2258
2.1508
2.0707
2.0283
1.9842
1.938
1.8894
1.838
1.7831
2.2747
2.2036
2.1282
2.0476
2.005
1.9605
1.9139
1.8648
1.8128
1.757
2.2547
2.1834
2.1077
2.0267
1.9838
1.939
1.892
1.8424
1.7896
1.733
2.2365
2.1649
2.0889
2.0075
1.9643
1.9192
1.8718
1.8217
1.7684
1.711
2.2197
2.1479
2.0716
1.9898
1.9464
1.901
1.8533
1.8027
1.7488
1.6906
2.2043
2.1323
2.0558
1.9736
1.9299
1.8842
1.8361
1.7851
1.7306
1.6717
2.19
2.1179
2.0411
1.9586
1.9147
1.8687
1.8203
1.7689
1.7138
1.6541
2.1768
2.1045
2.0275
1.9446
1.9005
1.8543
1.8055
1.7537
1.6981
1.6376
2.1646
2.0921
2.0148
1.9317
1.8874
1.8409
1.7918
1.7396
1.6835
1.6223
2.0772
2.0035
1.9245
1.8389
1.7929
1.7444
1.6928
1.6373
1.5766
1.5089
1.9926
1.9174
1.8364
1.748
1.7001
1.6491
1.5943
1.5343
1.4673
1.3893
1.9105
1.8337
1.7505
1.6587
1.6084
1.5543
1.4952
1.429
1.3519
1.2539
1.8307
1.7522
1.6664
1.5705
1.5173
1.4591
1.394
1.318
1.2214
1
51 / 56
F-Distribution Confidence Coefficient Table (97.5% CI): (Dinov, 2011)
df1
df2
1
2
3
4
5
6
7
8
9
1
647.789
799.5
864.163
899.583
921.848
937.111
948.217
956.656
963.285
2
38.5063
39
39.1655
39.2484
39.2982
39.3315
39.3552
39.373
39.3869
3
17.4434
16.0441
15.4392
15.101
14.8848
14.7347
14.6244
14.5399
14.4731
4
12.2179
10.6491
9.9792
9.6045
9.3645
9.1973
9.0741
8.9796
8.9047
5
10.007
8.4336
7.7636
7.3879
7.1464
6.9777
6.8531
6.7572
6.6811
6
8.8131
7.2599
6.5988
6.2272
5.9876
5.8198
5.6955
5.5996
5.5234
7
8.0727
6.5415
5.8898
5.5226
5.2852
5.1186
4.9949
4.8993
4.8232
8
7.5709
6.0595
5.416
5.0526
4.8173
4.6517
4.5286
4.4333
4.3572
9
7.2093
5.7147
5.0781
4.7181
4.4844
4.3197
4.197
4.102
4.026
10
6.9367
5.4564
4.8256
4.4683
4.2361
4.0721
3.9498
3.8549
3.779
11
6.7241
5.2559
4.63
4.2751
4.044
3.8807
3.7586
3.6638
3.5879
12
6.5538
5.0959
4.4742
4.1212
3.8911
3.7283
3.6065
3.5118
3.4358
13
6.4143
4.9653
4.3472
3.9959
3.7667
3.6043
3.4827
3.388
3.312
14
6.2979
4.8567
4.2417
3.8919
3.6634
3.5014
3.3799
3.2853
3.2093
15
6.1995
4.765
4.1528
3.8043
3.5764
3.4147
3.2934
3.1987
3.1227
16
6.1151
4.6867
4.0768
3.7294
3.5021
3.3406
3.2194
3.1248
3.0488
17
6.042
4.6189
4.0112
3.6648
3.4379
3.2767
3.1556
3.061
2.9849
18
5.9781
4.5597
3.9539
3.6083
3.382
3.2209
3.0999
3.0053
2.9291
19
5.9216
4.5075
3.9034
3.5587
3.3327
3.1718
3.0509
2.9563
2.8801
20
5.8715
4.4613
3.8587
3.5147
3.2891
3.1283
3.0074
2.9128
2.8365
21
5.8266
4.4199
3.8188
3.4754
3.2501
3.0895
2.9686
2.874
2.7977
22
5.7863
4.3828
3.7829
3.4401
3.2151
3.0546
2.9338
2.8392
2.7628
23
5.7498
4.3492
3.7505
3.4083
3.1835
3.0232
2.9023
2.8077
2.7313
24
5.7166
4.3187
3.7211
3.3794
3.1548
2.9946
2.8738
2.7791
2.7027
25
5.6864
4.2909
3.6943
3.353
3.1287
2.9685
2.8478
2.7531
2.6766
26
5.6586
4.2655
3.6697
3.3289
3.1048
2.9447
2.824
2.7293
2.6528
27
5.6331
4.2421
3.6472
3.3067
3.0828
2.9228
2.8021
2.7074
2.6309
28
5.6096
4.2205
3.6264
3.2863
3.0626
2.9027
2.782
2.6872
2.6106
29
5.5878
4.2006
3.6072
3.2674
3.0438
2.884
2.7633
2.6686
2.5919
30
5.5675
4.1821
3.5894
3.2499
3.0265
2.8667
2.746
2.6513
2.5746
40
5.4239
4.051
3.4633
3.1261
2.9037
2.7444
2.6238
2.5289
2.4519
60
5.2856
3.9253
3.3425
3.0077
2.7863
2.6274
2.5068
2.4117
2.3344
120
5.1523
3.8046
3.2269
2.8943
2.674
2.5154
2.3948
2.2994
2.2217
inf
5.0239
3.6889
3.1161
2.7858
2.5665
2.4082
2.2875
2.1918
2.1136
52 / 56
10
12
15
20
24
30
40
60
120
968.627
976.708
984.867
993.103
997.249
1001.41
1005.6
1009.8
1014.02
1018.26
39.398
39.4146
39.4313
39.4479
39.4562
39.465
39.473
39.481
39.49
39.498
14.4189
14.3366
14.2527
14.1674
14.1241
14.081
14.037
13.992
13.947
13.902
8.8439
8.7512
8.6565
8.5599
8.5109
8.461
8.411
8.36
8.309
8.257
6.6192
6.5245
6.4277
6.3286
6.278
6.227
6.175
6.123
6.069
6.015
5.4613
5.3662
5.2687
5.1684
5.1172
5.065
5.012
4.959
4.904
4.849
4.7611
4.6658
4.5678
4.4667
4.415
4.362
4.309
4.254
4.199
4.142
4.2951
4.1997
4.1012
3.9995
3.9472
3.894
3.84
3.784
3.728
3.67
3.9639
3.8682
3.7694
3.6669
3.6142
3.56
3.505
3.449
3.392
3.333
3.7168
3.6209
3.5217
3.4185
3.3654
3.311
3.255
3.198
3.14
3.08
3.5257
3.4296
3.3299
3.2261
3.1725
3.118
3.061
3.004
2.944
2.883
3.3736
3.2773
3.1772
3.0728
3.0187
2.963
2.906
2.848
2.787
2.725
3.2497
3.1532
3.0527
2.9477
2.8932
2.837
2.78
2.72
2.659
2.595
3.1469
3.0502
2.9493
2.8437
2.7888
2.732
2.674
2.614
2.552
2.487
3.0602
2.9633
2.8621
2.7559
2.7006
2.644
2.585
2.524
2.461
2.395
2.9862
2.889
2.7875
2.6808
2.6252
2.568
2.509
2.447
2.383
2.316
2.9222
2.8249
2.723
2.6158
2.5598
2.502
2.442
2.38
2.315
2.247
2.8664
2.7689
2.6667
2.559
2.5027
2.445
2.384
2.321
2.256
2.187
2.8172
2.7196
2.6171
2.5089
2.4523
2.394
2.333
2.27
2.203
2.133
2.7737
2.6758
2.5731
2.4645
2.4076
2.349
2.287
2.223
2.156
2.085
2.7348
2.6368
2.5338
2.4247
2.3675
2.308
2.246
2.182
2.114
2.042
2.6998
2.6017
2.4984
2.389
2.3315
2.272
2.21
2.145
2.076
2.003
2.6682
2.5699
2.4665
2.3567
2.2989
2.239
2.176
2.111
2.041
1.968
2.6396
2.5411
2.4374
2.3273
2.2693
2.209
2.146
2.08
2.01
1.935
2.6135
2.5149
2.411
2.3005
2.2422
2.182
2.118
2.052
1.981
1.906
2.5896
2.4908
2.3867
2.2759
2.2174
2.157
2.093
2.026
1.954
1.878
2.5676
2.4688
2.3644
2.2533
2.1946
2.133
2.069
2.002
1.93
1.853
2.5473
2.4484
2.3438
2.2324
2.1735
2.112
2.048
1.98
1.907
1.829
2.5286
2.4295
2.3248
2.2131
2.154
2.092
2.028
1.959
1.886
1.807
2.5112
2.412
2.3072
2.1952
2.1359
2.074
2.009
1.94
1.866
1.787
2.3882
2.2882
2.1819
2.0677
2.0069
1.943
1.875
1.803
1.724
1.637
2.2702
2.1692
2.0613
1.9445
1.8817
1.815
1.744
1.667
1.581
1.482
2.157
2.0548
1.945
1.8249
1.7597
1.69
1.614
1.53
1.433
1.31
2.0483
1.9447
1.8326
1.7085
1.6402
1.566
1.484
1.388
1.268
1
53 / 56
INF
F-Distribution Confidence Coefficient Table (99.0% CI): (Dinov, 2011)
df1
1
2
3
4
5
6
7
8
9
1
4052.18
4999.5
5403.35
5624.58
5763.65
5858.99
5928.36
5981.07
6022.47
2
98.503
99
99.166
99.249
99.299
99.333
99.356
99.374
99.388
3
34.116
30.817
29.457
28.71
28.237
27.911
27.672
27.489
27.345
4
21.198
18
16.694
15.977
15.522
15.207
14.976
14.799
14.659
5
16.258
13.274
12.06
11.392
10.967
10.672
10.456
10.289
10.158
6
13.745
10.925
9.78
9.148
8.746
8.466
8.26
8.102
7.976
7
12.246
9.547
8.451
7.847
7.46
7.191
6.993
6.84
6.719
8
11.259
8.649
7.591
7.006
6.632
6.371
6.178
6.029
5.911
9
10.561
8.022
6.992
6.422
6.057
5.802
5.613
5.467
5.351
10
10.044
7.559
6.552
5.994
5.636
5.386
5.2
5.057
4.942
11
9.646
7.206
6.217
5.668
5.316
5.069
4.886
4.744
4.632
12
9.33
6.927
5.953
5.412
5.064
4.821
4.64
4.499
4.388
13
9.074
6.701
5.739
5.205
4.862
4.62
4.441
4.302
4.191
14
8.862
6.515
5.564
5.035
4.695
4.456
4.278
4.14
4.03
15
8.683
6.359
5.417
4.893
4.556
4.318
4.142
4.004
3.895
16
8.531
6.226
5.292
4.773
4.437
4.202
4.026
3.89
3.78
17
8.4
6.112
5.185
4.669
4.336
4.102
3.927
3.791
3.682
18
8.285
6.013
5.092
4.579
4.248
4.015
3.841
3.705
3.597
19
8.185
5.926
5.01
4.5
4.171
3.939
3.765
3.631
3.523
20
8.096
5.849
4.938
4.431
4.103
3.871
3.699
3.564
3.457
21
8.017
5.78
4.874
4.369
4.042
3.812
3.64
3.506
3.398
22
7.945
5.719
4.817
4.313
3.988
3.758
3.587
3.453
3.346
23
7.881
5.664
4.765
4.264
3.939
3.71
3.539
3.406
3.299
24
7.823
5.614
4.718
4.218
3.895
3.667
3.496
3.363
3.256
25
7.77
5.568
4.675
4.177
3.855
3.627
3.457
3.324
3.217
26
7.721
5.526
4.637
4.14
3.818
3.591
3.421
3.288
3.182
27
7.677
5.488
4.601
4.106
3.785
3.558
3.388
3.256
3.149
28
7.636
5.453
4.568
4.074
3.754
3.528
3.358
3.226
3.12
29
7.598
5.42
4.538
4.045
3.725
3.499
3.33
3.198
3.092
30
7.562
5.39
4.51
4.018
3.699
3.473
3.304
3.173
3.067
40
7.314
5.179
4.313
3.828
3.514
3.291
3.124
2.993
2.888
df2
60
7.077
4.977
4.126
3.649
3.339
3.119
2.953
2.823
2.718
120
6.851
4.787
3.949
3.48
3.174
2.956
2.792
2.663
2.559
inf
6.635
4.605
3.782
3.319
3.017
2.802
2.639
2.511
2.407
54 / 56
10
12
15
20
24
30
40
60
120
INF
6055.85
6106.32
6157.29
6208.73
6234.63
6260.65
6286.78
6313.03
6339.39
6365.86
99.399
99.416
99.433
99.449
99.458
99.466
99.474
99.482
99.491
99.499
27.229
27.052
26.872
26.69
26.598
26.505
26.411
26.316
26.221
26.125
14.546
14.374
14.198
14.02
13.929
13.838
13.745
13.652
13.558
13.463
10.051
9.888
9.722
9.553
9.466
9.379
9.291
9.202
9.112
9.02
7.874
7.718
7.559
7.396
7.313
7.229
7.143
7.057
6.969
6.88
6.62
6.469
6.314
6.155
6.074
5.992
5.908
5.824
5.737
5.65
5.814
5.667
5.515
5.359
5.279
5.198
5.116
5.032
4.946
4.859
5.257
5.111
4.962
4.808
4.729
4.649
4.567
4.483
4.398
4.311
4.849
4.706
4.558
4.405
4.327
4.247
4.165
4.082
3.996
3.909
4.539
4.397
4.251
4.099
4.021
3.941
3.86
3.776
3.69
3.602
4.296
4.155
4.01
3.858
3.78
3.701
3.619
3.535
3.449
3.361
4.1
3.96
3.815
3.665
3.587
3.507
3.425
3.341
3.255
3.165
3.939
3.8
3.656
3.505
3.427
3.348
3.266
3.181
3.094
3.004
3.805
3.666
3.522
3.372
3.294
3.214
3.132
3.047
2.959
2.868
3.691
3.553
3.409
3.259
3.181
3.101
3.018
2.933
2.845
2.753
3.593
3.455
3.312
3.162
3.084
3.003
2.92
2.835
2.746
2.653
3.508
3.371
3.227
3.077
2.999
2.919
2.835
2.749
2.66
2.566
3.434
3.297
3.153
3.003
2.925
2.844
2.761
2.674
2.584
2.489
3.368
3.231
3.088
2.938
2.859
2.778
2.695
2.608
2.517
2.421
3.31
3.173
3.03
2.88
2.801
2.72
2.636
2.548
2.457
2.36
3.258
3.121
2.978
2.827
2.749
2.667
2.583
2.495
2.403
2.305
3.211
3.074
2.931
2.781
2.702
2.62
2.535
2.447
2.354
2.256
3.168
3.032
2.889
2.738
2.659
2.577
2.492
2.403
2.31
2.211
3.129
2.993
2.85
2.699
2.62
2.538
2.453
2.364
2.27
2.169
3.094
2.958
2.815
2.664
2.585
2.503
2.417
2.327
2.233
2.131
3.062
2.926
2.783
2.632
2.552
2.47
2.384
2.294
2.198
2.097
3.032
2.896
2.753
2.602
2.522
2.44
2.354
2.263
2.167
2.064
3.005
2.868
2.726
2.574
2.495
2.412
2.325
2.234
2.138
2.034
2.979
2.843
2.7
2.549
2.469
2.386
2.299
2.208
2.111
2.006
2.801
2.665
2.522
2.369
2.288
2.203
2.114
2.019
1.917
1.805
2.632
2.496
2.352
2.198
2.115
2.028
1.936
1.836
1.726
1.601
2.472
2.336
2.192
2.035
1.95
1.86
1.763
1.656
1.533
1.381
2.321
2.185
2.039
1.878
1.791
1.696
1.592
1.473
1.325
1
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Appendix 3: Phase Calculation Corrections
Initial Calculation, to prevent “flipping” of upper and lower bounds:
mySigns = sign(myPhase); %Determines current sign of phase in block
%Calculate upper and lower bounds using error term and applying correct sign
upperPhase(odof,idof,:)=myPhase(odof,idof,:).*(1+Cf*(mySigns(odof,idof,:).*
sigmaPhase(odof,1,:)));
lowerPhase(odof,idof,:)=myPhase(odof,idof,:).*(1-Cf*(mySigns(odof,idof,:).*
sigmaPhase(odof,1,:)));
Final calculation used when “flipping” was no longer a concern:
upperPhase(odof,idof,:) = myPhase(odof,idof,:)+(sigmaPhase(odof,1,:));
lowerPhase(odof,idof,:) = myPhase(odof,idof,:)-(sigmaPhase(odof,1,:));
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