The likelihood formulation is of critical importance in Bayesian inference. To build the likelihood, one needs to assume a probabilistic relation between the model predictions and experimental data in order to account for unavoidable model error as well as experimental or measurement error. There is not just one way to do that, and different likelihood formulations can lead to different results. Therefore, Bayesian inference is subjective in the sense that different likelihood models can be tried using the same data, and the inference results might differ significantly. Prediction error equations, which relate the model predictions with the experimental data probabilistically, are used to formulate the likelihood. Depending on the nature of the data, different prediction error equations can be used for different subsets of the entire data set.

For the modal frequencies, the most common choice is the uncorrelated Gaussian error assumption for each modal frequency (e.g., [

39,

40]). Specifically, the prediction error equation for the

r-th modal frequency is taken as:

where

${\epsilon}_{{\omega}_{r}}$ is the prediction error for the

r-th modal frequency taken to be Gaussian with zero mean and standard deviation

${\sigma}_{{\omega}_{r}}{\widehat{\omega}}_{r}^{2}$. The unknown parameter

${\sigma}_{{\omega}_{r}}$ is included in the parameter set

$\underline{\theta}$ to be estimated from the data. This formulation for the modal frequencies assumes that each modal frequency is uncorrelated with the rest. Then, the likelihood term for the

r-th modal frequency is the probability of observing the measured frequency given specific values of the model parameters

$\underline{\theta}$, derived from Equation (

3) in the form:

where

$N(x;\mu ,{\sigma}^{2})$ denotes the univariate Gaussian PDF evaluated at point

x with mean

$\mu $ and variance

${\sigma}^{2}$.

However, as far as the mode shapes are concerned, the prediction error formulation can be more complex due to the fact that they are vectors with multiple components. Again we make the assumption that all mode shapes are uncorrelated with each other and therefore we can treat each mode shape individually, just like the modal frequencies. Two formulations are presented next. The first one is a review of existing formulations, while the second one is a novel formulation based on features between model predicted and experimentally identified mode shapes.

#### 2.1.1. Formulation Using Probabilistic Models for Mode Shape Vectors

An often-used formulation for the prediction error is to assume that the discrepancy vector between the measured mode shape vector and the model predicted mode shape vector follows a zero-mean multivariable Gaussian distribution with a specified covariance matrix. The prediction error equation for the

r-th mode shape is then

where

${\underline{\epsilon}}_{{\underline{\varphi}}_{r}}$ is the prediction error vector for the

r-th mode shape taken to be Gaussian with zero mean and covariance matrix

${\sigma}_{{\underline{\varphi}}_{r}}^{2}{\mathsf{\Sigma}}_{{\underline{\varphi}}_{r}}$, where the matrix

${\mathsf{\Sigma}}_{{\underline{\varphi}}_{r}}$ specifies the possible correlation structure between the components of the prediction error vector of the

r-th mode shape, the unknown scalar

${\sigma}_{{\underline{\varphi}}_{r}}^{2}$ is included in the parameter set to be estimated, and

is a normalization constant such that the measured mode shape

${\underline{\widehat{\varphi}}}_{r}$ at the

${N}_{0,r}$ measured DOFs is closest to the model mode shape

${\beta}_{r}\left(\underline{\theta}\right){\underline{\varphi}}_{r}\left(\underline{\theta}\right)$ predicted by the particular value of

$\underline{\theta}$, and

${||\underline{z}||}^{2}={\underline{z}}^{\mathrm{T}}\underline{z}$ is the usual Euclidean norm. The scalar

${\beta}_{r}\left(\underline{\theta}\right)$ is introduced in Equation (

6) to account for the fact that the measured modeshape

${\widehat{\underline{\varphi}}}_{r}$ is normalized to have Euclidean norm equal to one, while the model predicted modeshape

${\underline{\varphi}}_{r}\left(\underline{\theta}\right)$ is mass normalized. The scalar

${\beta}_{r}\left(\underline{\theta}\right)$ is derived by minimizing the distance

$||{\widehat{\underline{\varphi}}}_{r}-{\beta}_{r}\left(\underline{\theta}\right){\underline{\varphi}}_{r}\left(\underline{\theta}\right)||$ between the measured mode shape and the scaled version of the model predicted mode shape.

It is important to note in this approach that the number of data points used for each mode shape is equal to the number of measured DOFs ${N}_{0,r}$ for that particular mode. For a spatially uncorrelated model for the prediction error ${\underline{\epsilon}}_{{\underline{\varphi}}_{r}}$ (diagonal ${\mathsf{\Sigma}}_{{\underline{\varphi}}_{r}}$ matrix) each mode shape component counts as a new independent data point in the likelihood. From the Bayesian Central Limit Theorem, the posterior uncertainty is expected to reduce without bounds as the number of mode shape components is increased. However, as the number of measured DOFs increases, the sensors become very close to one another, providing almost the same information content that should not further reduce the posterior uncertainty of the model parameters. The closeness of the sensors depends on the wavelength of the considered measured mode shape. Two sensors are close and are expected to provide redundant information if their distance is a fraction of the wave length of the corresponding mode shape. Therefore, a spatially uncorrelated model for the prediction error vector ${\underline{\epsilon}}_{{\underline{\varphi}}_{r}}$ of the mode shape would not yield the expected behavior regarding posterior uncertainty as the number of mode shape components increases.

A remedy to this is to introduce a correlation model between the components of the prediction error vector of the mode shape, leading to a non-diagonal covariance matrix

${\mathsf{\Sigma}}_{{\underline{\varphi}}_{r}}$. However, a correlation function should be postulated to describe the spatial correlation between two mode shape components (sensors) as a function of their distance, where the closer two sensors get the more they are correlated. Several correlation functions exist in the literature [

41]. The problem is that one cannot know beforehand which correlation function is the proper one for the particular application at hand. This decision of the correlation function might turn out to be extremely difficult to make in practice, because in practical situations one normally has slight to none available information regarding the correlation nature of the prediction error vector. Selecting the proper correlation function might be challenging and failure to do so could easily lead to erroneous results as was demonstrated in [

41]. Finding the proper correlation function is not the goal of this work. More on that issue can be found in [

41,

42,

43]. Herein two cases of correlation models are examined: uncorrelated and exponentially correlated models.

For the simplest case of uncorrelated mode shape prediction error vectors the covariance matrix simplifies to a diagonal matrix:

with

I being the

${N}_{0,r}\text{}\times \text{}{N}_{0,r}$ identity matrix, while for the exponentially correlated model the identity matrix

I is replaced by the correlation matrix

${R}_{r}$ whose

$(i,j)$-th element is given by the exponential correlation function:

where

${x}_{r}(i,j)$ is the Euclidean distance between the

i-th and

j-th mode shape components (sensors) for the

r-th mode, and

${\lambda}_{r}$ is the correlation length for the

r-th mode which is a parameter to be identified.

Using Equation (

5), the likelihood term for the

r-th mode shape is the probability of observing the measured mode shape for given model parameters

$\underline{\theta}$, given by

where

$N(\underline{x};\underline{\mu},\mathsf{\Sigma})$ denotes the multivariate Gaussian PDF evaluated at point

$\underline{x}$ with mean vector

$\underline{\mu}$ and covariance matrix

$\mathsf{\Sigma}$. Following the work of Papadimitriou et al. [

44] which was based on the same prediction error Equation (

5), the likelihood function in Equation (

9) can be expressed in terms of the MAC values between the measured and model predicted mode shapes.

Slightly different prediction error equations for the mode shapes have been proposed in the literature (e.g., [

39,

40]), including versions that do not require the use of the mode correspondence [

4,

6]. In all these alternatives, the likelihood formulation for the mode shapes is based on a probabilistic description of individual components of a vector and thus they fall into the category discussed in this subsection.

#### 2.1.2. Formulation Using Probabilistic Models for MAC Values

The previous formulation uses the mode shapes as full vectors in the likelihood function. Herein we propose a novel formulation for including the mode shapes in the likelihood function which is based on the MAC value between the experimental and model predicted mode shape. The MAC value, defined as $MAC(\underline{u},\underline{v})={\underline{u}}^{T}\underline{v}/(||\underline{u}||\text{}||\underline{v}||)$ between two vectors $\underline{u}$ and $\underline{v}$, is the most common way to measure the similarity between two mode shape vectors. It is a scalar measure which varies from 0 to 1 with a value of 1 indicating a perfect match. The scaling of the mode shapes is not important for the MAC value which means that no normalization is needed for either the experimental or model predicted mode shape.

In the new formulation the experimental mode shape is not compared with the model predicted mode shape in an element-wise fashion, but rather based on its MAC value. This reduces the number of data points used in the likelihood for each mode shape from ${N}_{0,r}$ to just 1. Therefore, instead of calculating the probability of observing the experimental mode shape vector given the model predicted mode shape vector (for some given model parameter values), we calculate the probability of their MAC value taking a value of 1, implying that they match perfectly.

In contrast to the previous vector formulation of the likelihood, the MAC value is a univariate quantity and therefore requires a univariate distribution to model it. Taking into account the fact that the MAC value is strictly bounded in the interval

$[0,\text{}1]$, a Truncated Gaussian distribution is used, although there are many other choices of candidate distributions. The Gaussian distribution is preferred because of its known properties. This leads to the following prediction error equation for the MAC value of the

r-th mode shape:

where

$MA{C}_{r}\left(\underline{\theta}\right)=MAC({\underline{\widehat{\varphi}}}_{r},{\underline{\varphi}}_{r}\left(\underline{\theta}\right))$ is the model-predicted MAC value, defined as the MAC value between the experimental

r-th mode shape and the model predicted

r-th mode shape for the given values of the model parameters

$\underline{\theta}$. The term

${\epsilon}_{{MAC}_{r}}$ is the error in the

r-th MAC value (analogous to the error in the

r-th frequency), assumed to follow a univariate zero-mean Gaussian distribution with standard deviation equal to

${\sigma}_{{MAC}_{r}}$. The standard deviation

${\sigma}_{{MAC}_{r}}$ is a measure of “how far” the observed MAC value

${\widehat{MAC}}_{r}$ can be from the model-predicted MAC value

$MA{C}_{r}\left(\underline{\theta}\right)$ due to model and experimental errors. This can be thought of as completely analogous to the error term for the modal frequencies in (

3). The resulting Gaussian with mean

$MA{C}_{r}\left(\underline{\theta}\right)$ and standard deviation

${\sigma}_{{MAC}_{r}}$ is truncated in 0 and 1 which results in the Truncated Gaussian distribution.

An important issue that should be addressed when using MAC values is the fact that although the MAC is a scalar value, it depends on the number of mode shape components used. This needs to be taken into account in the formulation in order to avoid erroneous results. For example, if only two components of a mode shape are used, there is a chance that the MAC value turns out to be very close to 1 (provided that those two components match well between the two mode shapes). However, if a large number of components is used, due to small errors in each component there is the chance that the MAC value is significantly lower than 1, which would mean that the case with two components would yield a larger MAC value. However, the case of large number of components components is expected to be much more informative than the case of two components since the more components we have the better we know the actual geometry of the mode shape. This naturally leads to the conclusion that the number of mode shape components must be taken into account, assigning higher preference to MAC values calculated with more components than MAC values calculated with less components.

One way to account for this in a Bayesian framework is through manipulation of the MAC value standard deviation parameter

${\sigma}_{{MAC}_{r}}$. We seek a formula through which to define

${\sigma}_{{MAC}_{r}}$ that depends on the number of mode shape components

${N}_{0,r}$. Although there is not only one way to achieve this, the following formula is used:

where

${\sigma}_{{MAC}_{r}}^{\prime}$ is the parameter to be inferred from data. The first term in Equation (

11) describes the uncertainty present in the MAC value that exists independently of the number of sensors. This uncertainty exists even for a large number of sensors and is due to model and experimental errors in the individual components and can not be reduced further. The second term in Equation (

11) depends on the number of sensors and gets smaller as the number of sensors is increased, which reduces the standard deviation of the MAC value. This way more weight (less uncertainty) is given to MAC values calculated with more sensors. These are modeling choices within the Bayesian framework, much like the choice of Gaussian PDFs for the likelihood, independent data, etc. Alternative formulations could also be postulated. In particular, the two terms in Equation (

11) can be weighted differently but this falls outside the scope of the present work.

Then the likelihood term for the MAC value of the

r-th mode shape is the probability of observing a MAC value of 1 for given values of the model parameters

$\underline{\theta}$ (indicating a perfect match between the experimental and model predicted mode shapes), given by the Truncated Gaussian PDF:

where

$TN(x;\mu ,{\sigma}^{2},a,b)$ denotes the Truncated Gaussian PDF evaluated at point

x with mean

$\mu $, variance

${\sigma}^{2}$ and truncation limits

a and

b.