This section is devoted to the numerical validation of the aforementioned quadratic form approximation. Two strategies recalled in this section validate theχ2l approximation from Theorem 5.26 and Theorem 5.28, and are tested on the cases of MAC11 and MAC12. The procedures proposed in this section closely follow Section 4.3.2.
Recall that the estimates of MAC computed from each realization ofmMonte Carlo simulations are collected in MACM C ∈Rm×1, where each estimate approaches the theoretical boundary of the MAC domain, 0 or 1. Subsequently
QMACM C =N(MACM C−gmac(ϕ∗, ψ∗)) , (5.29) which, based on Theorems 5.26 and 5.28, is a quadratic form that can be approximated with theχ2l distribution. Notice thatQMACM C yields
QMACM C =N(1−MACM C) , (5.30)
for collinear and
QMACM C =N(MACM C) , (5.31)
for orthogonal cases respectively. The number of degrees of freedomlM C of MACM C
can be derived from the empirical cumulants of the Monte Carlo histogram as described in Section 5.3.1. Note that 1≤lM C ≤r−1, after Lemma 5.5. The density of the χ2 approximation can be analytically expressed as in (5.21); letQMAC
M C be a scaled and shiftedQMACM C such that
QMACM C = (QMACM C−β)/ss α , (5.32)
whereαs =sσQ/sσχ2 andβs=µsQ−(µsχ2sσQ)/σsχ2. The scalarsσsQ,µsQ,σsχ2 andµsχ2
are computed from the cumulants ofQMACM C as in Section 5.3.1. As such, knowing the Monte Carlo histograms, the characterization of theχ2 approximation is easily performed. That can be achieved by defining a vectorv1∈Rm×1 whosej-th entry is drawn from a χ2lM C distribution such that vj ∼ χ2lM C. As such, each entry of v1 is drawn from a corresponding realization ofχ2lM C and the resulting aggregated histogram of all components ofv1 should correspond to the theoretical histogram of χ2lM C. A comparison of the cumulative distribution functions of bothQMACM C and v1 is shown in the left parts of Figure 5.14 for the collinear and Figure 5.16 for the orthogonal mode shape cases.
Prior to analyzing the figures, it is illustrated that the perturbation approach is also capable of such approximation. For that define a vectorQMAC
P T ∈Rm×1 such that forj= 1. . . mitsj-th entry writes
QMAC
P T ,j = (QMACM C,j−βj)/αj, (5.33)
whereβjandαjare thej-thβandαestimated after Section 5.3.1 for the corresponding realization of the Monte Carlo simulation. The main problem here is that each QMACP T ,j is drawn from some distribution that can be characterized by a different lj. A common baseline is needed. That is achieved by modifying the approximated distributions to fit with a theoreticalχ2pdistribution, wherepwill be constant for all realizations and not equal toljbut 2r−1 as explained below. Contrary to (5.32), each QMACM C,j is normalized with its ownαandβcoming from the perturbation theory and not from averaging over the Monte Carlo histogram. In that sense,QMACM C,j
is properly following aχ2lP T ,j distribution, wherelP T ,jis computed fromβjandαj. Still, each realization follows its own distinct distribution, which is not optimal for a comparison. This comparison is also more difficult when considering that the Monte Carlo histogram is fitted with its ownχ2lM C distribution characterized with ownlM C. To deal with that the additive property of independentχ2distributions is used. Recall from Lemma 5.5 that 1≤lP T ,j ≤r−1 for any j. The bounds forlM C, as stated in Corollary 5.6, are 1≤lM C ≤r−1. As such, it exists some complement for all these numbers e.g. one that is equal to 2r−1, such that both 2r−1−lP T ,j >0 for alljand 2r−1−lM C >0. Now, define random vectorsv2, v3∈Rm×1 such that v2∼χ22r−1−lM C, and forj= 1. . . meach entry ofv3,j∼χ22r−1−lP T ,j. Consequently using additive property of independentχ2 distributions define
ZM C =QMAC
M C+v2, ZM C ∼χ22r−1 , (5.34)
ZP T =QMAC
P T +v3, ZP T ∼χ22r−1 . (5.35)
A comparison between CDFs ofZM C,ZP T evaluated for both MAC11 and MAC12
with a theoreticalχ22r−1CDF is illustrated in the right parts of Figure 5.14 for collinear and Figure 5.16 for orthogonal mode shapes.
In case when a theoretical reference is known, a statistical measure can be used to quantify the dispersion between the aforementioned distributions. That dispersion can be quantified with Pearsonχ2 statistics between the expectedχ22r−1 distributed variable and the observedZP Tj ∈Rm×1 vector. It writes as
ZP Tj = (QMACM C−βj)/αj+v4,j , (5.36)
wherev4,j ∼χ22r−1−lP T ,j. This leads to a collection of histograms that mirror the quality of eachαj, βj, similarly as in the Gaussian case. The histograms of the Pearson χ2statistic for MAC11and MAC12estimates are depicted respectively on the left parts of Figure 5.15 and Figure 5.17. Consequently, a comparison of using the best, median and the worst approximations of MAC11and MAC12estimates, which correspond to respective quantiles of the Pearsonχ2 statistics, is depicted on the respective right parts of Figure 5.15 and Figure 5.17.
5.3.3.1 Collinear mode shapes
Figure 5.14 and Figure 5.15 illustrate the proposed validation scheme applied to the Monte Carlo simulations of MAC11. The left part of the Figure 5.14 illustrates that both CDF coincide meaning that the distribution of the Monte Carlo histogram is well characterized by a quadratic form which distribution is approximated by aχ2 distribution. The right part of Figure 5.14 shows that the correctedZM C for Monte Carlo and combined realizations from the perturbation theoryZP T have similar CDF.
This shows the validity of the perturbation-based approach to compute the variances
5.3 Quadratic approximation ofgmac( ˆϕ, ψ∗) 77
0 2 4 6 8 10 12 14
MAC11 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Probability
Monte Carlo CDF v1 CDF
0 2 4 6 8 10 12 14 16 18 20
ZMAC 11 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Probability
ZMC empirical Monte Carlo distribution median fit CDF
0.975 quantile fit CDF 0.025 quantile fit CDF ZPT first order perturbation theory
2(2r-1) CDF
Figure 5.14: χ2l
M C CDF and CDF of MAC computed between the exact mode shape from the model and estimated one. Theoreticalχ22r−1CDF compared to CDF ofZM C (5.34) and ZP T (5.35) computed with parameters estimated for MAC between exact mode shape from the model and estimated one (right).
needed to characterize the quadratic distribution and yield similar distribution param-eters as the full Monte Carlo histogram. Also, the plots corresponding to the best, median and worst quantiles are all quite similar. They match the Monte Carlo and the perturbation-basedZ plots. This means that almost any of the realization among the Monte Carlo experiments can be used to give such variance information.
0 10 20 30 40 50 60 70
Pearson 2 statistics MAC11 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
errors median 0.975 quantile 0.025 quantile
2 fit mean errors
2 nDOF
0.9975 0.998 0.9985 0.999 0.9995 1
MAC11 0
0.05 0.1 0.15 0.2 0.25 0.3
probability
MAC11 0.025 quantile fit 0.975 quantile fit median fit
Figure 5.15: Histogram of errors inχ2l
P T approximation with best, median and worst fits for MAC between exact mode shape from the model and estimated one (left). Scaled and shifted χ2 fits to empirical probability distribution of MAC between exact mode shape from the model and estimated one based on median, 0.95 and 0.025 quantiles of Pearsonχ2 statistics (right).
Figure 5.15 shows the dispersion of the distance between all the empirical histograms based on the perturbation theory. It can be seen that the empirical mean of the Pearson histogram is close to its theoretical mean. The right part of Figure 5.15 shows the fitting of MAC histogram by the perturbation-based densities corresponding to selected best, median and worst quantiles. It can be seen that the quadratic assumption here is much more adequate for MAC11than the Gaussian approximation depicted on Figure 5.10.
5.3.3.2 Orthogonal mode shapes
Figure 5.16 and Figure 5.17 created for the orthogonal mode shapes case illustrate that both Monte Carlo based histogram of MAC12and the perturbation theory based Z variables are well approximated by aχ2 law, as expected. They exhibit very similar behavior regarding validity ofχ2approximation as the results illustrated on Figure 5.14 and Figure 5.15 made for the collinear case, which shows the versatility of the devised validation schemes.
0 2 4 6 8 10 12 14
MAC12 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Probability
Monte Carlo CDF v1 CDF
0 5 10 15 20 25 30 35
ZMAC 12 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Probability
ZMC empirical Monte Carlo distribution median fit CDF
0.975 quantile fit CDF 0.025 quantile fit CDF ZPT first order perturbation theory
2(2r-1) CDF
Figure 5.16: χ2(lM C) CDF and CDF of MAC computed between the exact mode shape from the model and estimated one. Theoreticalχ22r−1CDF compared to CDF ofZM C (5.34) andZP T (5.35) computed with parameters estimated for MAC between exact mode shape from the model and estimated one (right).
0 5 10 15 20 25 30
Pearson 2 statistics MAC12 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
errors median 0.975 quantile 0.025 quantile
2 fit mean errors
2 nDOF
0 1 2 3
MAC12 10-4
0 0.05 0.1 0.15 0.2 0.25 0.3
probability
MAC12 0.025 quantile fit 0.975 quantile fit median fit
Figure 5.17: Histogram of errors inχ2(lP T) approximation with best, median and worst fits for MAC between exact mode shape from the model and estimated one (left). Scaled and shiftedχ2fits to empirical probability distribution of MAC between exact mode shape from the model and estimated one based on median, 0.95 and 0.025 quantiles of Pearsonχ2 statistics (right).
5.3.4 Influence of sample length on distribution of gmac( ˆϕ, ψ∗): a χ2