3.2.3. Detection of 5-HMF, Furfural and pH Value

The negative influence of 5-HMF and furfural as main carbohydrate degradation products on enzymatic hydrolysis and subsequent fermentation is well described [41–44]. Therefore, 5-HMF and furfural were analyzed. As these compounds are unstable, the detection was performed directly in the extract obtained after the steam refining of the corn stover. The results are shown in Figure 5.

**Figure 5.** Effect of steaming at different severities on pH and the amounts of 5-HMF and furfural in the extract fraction.

Figure 5 shows that both, the content of furfural and 5-HMF in the extract fraction, increased steadily with increasing severity factors. This has particular importance for the approach to saccharify the fibers and the extract simultaneously. However, it is also important for subsequent processes such as EH of the fibers to fermentable carbohydrates or the production of biogas from the extract. The degradation of hexoses and pentoses to 5-HMF and furfural and the lowering of the pH value during steam explosion and steam refining is also well-described in the literature [11,45–49]. Ruiz et al. [49] report almost similar results for 5-HMF and furfural after steam explosion of sunflower stalks in comparison with the presented results for corn stover. Therefore, it can be assumed that the lower furan contents are typical for agricultural residues. Jacquet et al. [47] report results for really high severities up to 5.56 at extreme temperatures around 260 ◦C. For severities below severities of 4.5, they report similar data to the present data, although they used microcrystalline cellulose. For further results, a strong increase of 5-HMF, but not for furfural, is reported. In this context Um and van Walsum [50] report about the formation of furfural with increasing severity by a simultaneous decrease of the xylose contents in a corn stover dilute acid pretreatment. However, for severities above 4.43, they report a decrease and degradation of furfural contents and an increase of formate. In contrast to the present data, they received high severities by time, not by temperature.

However, the measured values for 5-HMF and furfural after steam refining of corn stover are much lower than the measured values for the furans after steam refining of poplar wood [11].

Figure 5 depicts as well the pH value in the extracts, which decreases with increasing severity. This is due to the formation of formic and acetic acid. Formic acid concentrations ranging from 0.61% at a severity factor of 3.65% up to 1.53% at a severity factor of 4.54%, all values based on raw material. The corresponding data for acetic are 2.02% and 2.89%, based on raw material.

Due to the formation of these organic acids, mainly caused by hemicellulose degradation, the pH value is reduced and autohydrolysis is intensified.


6.2.1. 18F-FDG-PET/CT

The most commonly used PET tracer is 18F-fluorodeoxyglucose (FDG). Uptake of this tracer is relatively higher in cells that are metabolically active, such as inflammatory cells. FDG is indicated in several infectious and inflammatory diseases and is useful for the diagnosis and follow-up of various autoimmune diseases [4,101,102]. In pSS patients, several case reports showed abnormal FDG uptake in salivary glands [103–105] (Figure 5). However, physiological FDG accumulation in salivary glands is highly variable, and subjects without known head and neck pathology frequently show increased FDG uptake in the salivary glands [106,107]. A possible explanation for the wide range in physiological uptake is that salivary glands of different subjects utilize different amounts of glucose for metabolism [108,109]. Which human salivary gland cells have the highest glucose uptake is not known, but in rodent salivary glands, both acinar and ductal cells seem to play a role in glucose uptake [110,111].

**Figure 5.** FDG-PET uptake patterns within the parotid glands. FDG-PET/CT showing (**A**) no increased uptake in the parotid glands of a subject without pathology in the head/neck region, (**B**) increased uptake in both parotid glands in a pSS patient, and (**C**) intense uptake in both parotid glands of a pSS patients with histologically confirmed parotid salivary gland MALT lymphoma.

Whether pSS patients show increased FDG uptake in inflamed salivary and lacrimal glands is not yet clear. Although Cohen et al. [112] reported increased FDG uptake in salivary glands of pSS patients compared to a control group of patients who underwent PET for an isolated pulmonary nodule, another study could not confirm this finding [36]. However, both studies used different scoring methods and camera settings, and standardized guidelines and EANM Research Ltd. (EARL) reconstructions were not applied in these studies.

Besides visualizing salivary gland inflammation, several authors reported the ability of FDG-PET/CT to detect systemic disease activity in pSS [104,105,112–115]. Abnormal FDG uptake was observed in 75–85% of pSS patients, mainly within salivary glands, lymph nodes, and lungs [112,113]. However, not all organ involvement in pSS can be visualized by FDG-PET/CT, such as neuropathies, cutaneous vasculitis, and other skin abnormalities, which is mainly due to limited spatial resolution of PET cameras. Since the optimal spatial resolution is around 3-4 mm in newer PET/CT systems, it would be of interest to use these newer systems to study the effectiveness of FDG-PET/CT in assessing systemic disease activity in representative pSS cohorts.

The applicability of FDG-PET/CT in MALT lymphomas is still controversial due to variable FDG avidity at different MALT locations. Pulmonary and head/neck MALT lymphomas seem to be most FDG avid, and FDG-PET/CT shows higher sensitivities at these regions [116,117]. Since pSS-associated MALT lymphomas frequently develop in the salivary glands and/or lungs, there may be a role for this technique in the detection of lymphomas associated with pSS [113,118,119] (Figure 3). Cohen et al. [112] reported a higher maximum standardized uptake value (SUVmax) in lymphoma patients compared to pSS patients without lymphoma. Since only four lymphoma patients were included in this retrospective cohort, the study was not powered to affirm the usefulness of FDG-PET/CT in the diagnosis of pSS-associated MALT lymphoma. In another retrospective study by Kerean et al. [113], 8 out of 15 pSS patients had confirmed salivary gland or pulmonary MALT lymphoma. They found that a SUVmax of ≥4.7 in the parotid glands and presence of focal lung lesions were associated with lymphoma. An important advantage of FDG-PET/CT is the possibility to detect extraglandular lymphoma locations by using whole-body imaging. In addition, Kerean et al. [113] showed that FDG-PET/CT is useful for biopsy guiding and for treatment response monitoring [113]. However, the mentioned SUVmax threshold found in this study cannot directly be used by others, as different nonstandardized camera systems were used in this multicenter study. Importantly, both retrospective studies state that the frequent presence of benign lymph node uptake in pSS patients can be misleading and should be taken into account when using FDG-PET/CT for the diagnosis of lymphoma in pSS patients [112,113].


**2. Separately coherent upper conditional previsions**

Given a metric space (Ω, *d*) the Borel *σ*-field is the *σ*-field generated by the open sets of Ω. Let **B** be a Borel-measurable partition of Ω, i.e. all sets of the partition are Borel sets.

For every *B* ∈ **B** let us denote by *X*|*B* the restriction to *B* of a random variable defined on Ω and by sup(*X*|*B*) the supremum value that *X* assumes on *B*.

Separately coherent upper conditional previsions *P*(·|*B*) are functionals, defined on a linear space of bounded random variables, i.e. bounded real-valued functions, satisfying the axioms of separate coherence [19].

**Definiton 1.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear space of bounded random variables on *B*. Separately coherent upper conditional previsions are functionals *P*(·|*B*) defined on **K**(B), such that the following conditions hold for every *X* and *Y* in **K**(*B*) and every strictly positive constant *λ*:

2 Stochastic Control

outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension *s*. Otherwise if the conditioning event has Hausdorff outer measure in its dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability. Coherent upper and lower conditional probabilities are obtained ([6])

If the conditioning event B has positive and finite Hausdorff outer measure in its Hausdorff dimension then the given upper conditional prevision defined on a linear lattice of bounded random variables is proven to be a functional, which is monotone, submodular, comonotonically additive and continuous from below. Moreover all these properties are proven to be a sufficient condition under which the upper conditional probability defined by Hausdorff outer measure is the unique monotone set function, which represent a coherent upper conditional prevision as Choquet integral. The given model of coherent upper

Many complex systems are strongly dependent on the initial conditions, that is small differences on the initial conditions lead the system to entirely different states. These systems are called *chaotic systems*. Thus uncertainty in the initial conditions produces uncertainty in the final state of the system. Often the final state of the system, called *strange attractor* is represented by a fractal set, i.e., a set with non-integer Hausdorff dimension. The model of coherent upper prevision, introduced in this chapter, can be proposed to forecast in a chaotic system when the conditional prevision of a random variable is conditioned to the attractor of

In Section 2 The notion of separately coherent conditional previsions and their properties are

In Section 3 separately coherent upper conditional previsions are defined in a metric space by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension. Otherwise they are defined by

In Section 4 results are given such that a coherent upper conditional prevision, defined on a linear lattice of bounded random variables containing all constants, is uniquely represented as the Choquet integral with respect to its associated Hausdorff outer measure if and only if it

Given a metric space (Ω, *d*) the Borel *σ*-field is the *σ*-field generated by the open sets of Ω. Let

For every *B* ∈ **B** let us denote by *X*|*B* the restriction to *B* of a random variable defined on Ω

Separately coherent upper conditional previsions *P*(·|*B*) are functionals, defined on a linear space of bounded random variables, i.e. bounded real-valued functions, satisfying the axioms

**B** be a Borel-measurable partition of Ω, i.e. all sets of the partition are Borel sets.

conditional prevision can be applied to make prevision in chaotic systems.

when only 0-1 valued random variables are considered.

the chaotic system.

of separate coherence [19].

recalled.

The outline of the chapter is the following.

a 0-1 valued finitely, but not countably, additive probability.

**2. Separately coherent upper conditional previsions**

and by sup(*X*|*B*) the supremum value that *X* assumes on *B*.

is monotone, submodular and continuous from below.

Coherent upper conditional previsions can always be extended to coherent upper previsions on the class **L**(*B*) of all bounded random variables defined on *B*. If coherent upper conditional previsions are defined on the class **L**(*B*) no measurability condition is required for the sets *B* of the partition **B**.

Suppose that *P*(*X*|*B*) is a coherent upper conditional prevision on a linear space **K**(*B*) then its conjugate coherent lower conditional prevision is defined by *P*(*X*|*B*) = −*P*(−*X*|*B*). If for every *X* belonging to **K**(*B*) we have *P*(*X*|*B*) = *P*(*X*|*B*) = *P*(*X*|*B*) then *P*(*X*|*B*) is called a coherent *linear* conditional prevision (de Finetti [**?** ]) and it is a linear positive functional on **K**(*B*).

**Definition 2.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear space of bounded random variables on *B*. Then linear coherent conditional previsions are functionals *P*(·|*B*) defined on **K**(*B*), such that the following conditions hold for every *X* and *Y* in **K**(*B*) and every strictly positive constant *λ*:

1') if *X* ≥ 0 then *P*(*X*|*B*) ≥ 0 (positivity); 2') *P*(*λX*|*B*) = *λP*(*X*|*B*) (positive homogeneity); 3') *P*(*X* + *Y*)|*B*) = *P*(*X*|*B*) + *P*(*Y*|*B*) (linearity); 4') *P*(*B*|*B*) = 1.

A class of bounded random variables is called a *lattice* if it is closed under point-wise maximum ∨ and point-wise minimum ∧.

Two random variables *X* and *Y* defined on *B* are *comonotonic* if, (*X*(*ω*1) − *X*(*ω*2))(*Y*(*ω*1) − *Y*(*ω*2)) ≥ 0 ∀*ω*1, *ω*2 ∈ *B*.

**Definition 3.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear lattice of bounded random variables defined on *B* and let *P*(·|*B*) be a coherent upper conditional prevision defined on **K**(*B*) then for every X, Y, *Xn* in **K**(B) *P*(·|*B*) is

A bounded random variable is called *B-measurable* or measurable with respect to the partition **B** [19, p.291] if it is constant on the atoms *B* of the partition. Let *G*(**B**) be the class of all **B**-measurable random variables.

The diameter of a non empty set *U* of Ω is defined as |*U*| = sup {*d*(*x*, *y*) : *x*, *y* ∈ *U*} and if a

*hs*(*A*) = lim*δ*→0 *hs*,*δ*(*A*).

This limit exists, but may be infinite, since *hs*,*δ*(*A*) increases as *δ* decreases because less *δ*-covers are available. The *Hausdorff dimension* of a set *A*, *dimH*(*A*), is defined as the unique

(*A*)=+∞ if 0 ≤ s < dimH(A),

(*A*) = 0 if dimH(A) < s < +∞.

*hs*(*E* ∪ *F*) = *hs*(*E*) + *hs*(*F*) whenever *d*(*E*, *F*) = inf {*d*(*x*, *y*) : *x* ∈ *E*, *y* ∈ *F*} > 0.

A subset *A* of Ω is called *measurable* with respect to the outer measure *hs* if it decomposes

*hs*(*E*) = *hs*(*A* ∩ *E*) + *hs*(*E* − *A*) for all sets *E* ⊆ Ω.

All Borel subsets of Ω are measurable with respect to any metric outer measure [11, Theorem 1.5]. So every Borel subset of Ω is measurable with respect to every Hausdorff outer measure

The restriction of *hs* to the *σ*-field of *hs*-measurable sets, containing the *σ*-field of the Borel sets, is called Hausdorff s-dimensional measure. In particular the Hausdorff 0-dimensional measure is the counting measure and the Hausdorff 1-dimensional measure is the Lebesgue

*hs*(*A* ∪ *B*) + *hs*(*A* ∩ *B*) = *hs*(*A*) + *hs*(*B*)

for every pair of Borelian sets *A* and *B*; so that [5, Proposition 2.4] the Hausdorff outer

*hs*(*A* ∪ *B*) + *hs*(*A* ∩ *B*) ≤ *hs*(*A*) + *hs*(*B*).

In [15, p.50] and [11, Theorem 1.6 (a)] it has been proven that if *A* is any subset of Ω there is a *Gσ*-set *G* containing *A* with *hs*(*A*) = *hs*(*G*). In particular *hs* is an *outer regular* measure.

Moreover Hausdorff outer measures are *continuous from below* [11, Lemma 1.3], that is for any

The Hausdorff *s*-dimensional measures are *modular* on the Borel *σ*-field, that is

We can observe that if 0 < *hs*(*A*) < +∞ then *dimH*(*A*) = *s*, but the converse is not true.

Let *s* be a non-negative number. For *δ* >0 we define *hs*,*δ* (*A*) = inf ∑+∞

The *Hausdorff s-dimensional outer measure* of *A*, denoted by *hs*(*A*), is defined as

*hs*

*hs*

Hausdorff outer measures are *metric* outer measures:

every subset of Ω additively, that is if

*hs* since Hausdorff outer measures are metric.

increasing sequence of sets {*Ai*} we have

*i Ui* and 0 < |*Ui*| < *δ* for each i, the class {*Ui*} is called a

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

ß=1 |*Ui*| *s*

Coherent Upper Conditional Previsions De ned by

, where the

55

subset *A* of Ω is such that *A* ⊂

infimum is over all *δ*-covers {*Ui*}.

*δ*-cover of *A*.

value, such that

measure.

measures are *submodular*

Denote by *P*(*X*|**B**) the random variable equal to *P*(*X*|*B*) if *ω* ∈ *B*.

Separately coherent upper conditional previsions *P*(*X*|*B*) can be extended to a common domain **H** so that the function *P*(·|**B**) can be defined from **H** to *G*(**B**) to summarize the collection of *P*(*X*|*B*) with *B* ∈ **B**.

*P*(·|**B**) is assumed to be separately coherent if all the *P*(·|*B*) are separately coherent. In Theorem 1 [9] the function *P*(*X*|**B**) is compared with the Radon-Nikodym derivative.

It is proven that, every time that the *σ*-field of the conditioning events is properly contained in the *σ*-field of the probability space and it contains all singletons, the Radon-Nikodym derivative cannot be used as a tool to define coherent conditional previsions. This is due to the fact that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the *σ*-field of the conditioning events, contradicts a necessary condition for the coherence.

Analysis done points out the necessity to introduce a different tool to define coherent conditional previsions.


**Author details**

Afsheen Raza Hamad Medical Corporation, Qatar

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Introductory Chapter: Introduction to New Insights and Recent Progress in Immune Checkpoint… DOI: http://dx.doi.org/10.5772/intechopen.107901*


**Development of Real-Time Hardware in the Loop Based MPC for Small-Scale Helicopter**

Zahari Taha2, Abdelhakim Deboucha1,

Azeddein Kinsheel1 and Raja Ariffin Bin Raja Ghazilla1

*1Centre for Product Design and Manufacturing Department of Engineering Design, Manufacture Faculty of Engineering- University of Malaya, Kuala Lumpur, 2Department of Manufacturing Engineering, University Malaysia Pahang, Gambang, Pahang, Malaysia*


**2. Methods**

Multiple variables create greater challenges. The concept is to create a distribution with shape and location parameters for each variable, then create distributions to define the mean and shape values (priors). Each of those distributions can be solved with MCMC methods to create a predictive distribution. That predictive distribution can be sampled and multiplied (or added) to the results of all other variables solved and sampled similarly. The assignment of distributions need not stop with the priors for shape location and location. Those parameters also can be assigned a distribution, and likewise those (and so on). Model parameters create a structure or a series of levels that look like a hierarchy, whereby the priors of a given solution are dependent on priors to those variable distributions and integrate that information across levels simultaneously [12], thereby separating the observed variability into parts attributable to both random and true differences [13–15]. Each investigation will have a different series of variables, each with different associated variables, priors, and priors of the priors.

Allenby et al. [16] stated that hierarchical Bayesian models are really "the combination of two things: i) a model written in hierarchical form that is ii) estimated using Bayesian methods." Shaddick et al. [17] consider there are to be at least three levels: (1) the observation or measurement level, (2) the underlying process level, and (3) the parameter level. Kruschke and Vanpaemel [18] noted that hierarchical Bayesian data analysis involves "describing data by meaningful mathematical models and allocating credibility to parameter values that are consistent with the data and with prior knowledge." Using Bayesian methods, hierarchical Bayesian models can yield estimates of the true effects at each level of the hierarchy [14, 19]. By considering the results across all levels, hierarchical Bayesian models can be used to rigorously integrate information with a complex underlying structure [14], resulting in a tendency to shrink differences when multiple variables are incorporated [20]. An important aspect of the hierarchical approach is that the model is usually a flexible version of a base model [21], and if needed, the models allow for adding extra levels depending on the hyperparameters [22].

Applying Bayesian prediction and weighting in a unified approach to Bayesian regression models can account for complex design features under the framework of multilevel regression and poststratification [23–25]. Weighting in a hierarchical model can be used as an extension of linear or logistic regression models. Methods for hierarchical functional data typically require that all curves are observed over or standardized to fall in the same region [26–28]. While classical weighting usually relies on many user-defined choices for regression that is difficult to codify [29], the Bayesian approach allows prior information to be incorporated and the distributions automatically adjusted [30, 31].

MCMC allows the user to approximate aspects of posterior distributions that cannot be directly calculated (e.g., random samples from the posterior, posterior means, etc.). There are examples of current applications of this approach. Draper [32] considered Bayesian hierarchical Poisson regression models, Wang et al. [33] created hierarchical Bayesian model developed for predicting monthly residential per capita electricity consumption at the state level across the United States, and Maddala et al. [34] studied the relationship of income elasticity on energy demand in the United States by applying a dynamic linear regression model under Bayesian framework. Roman et al. [35] and Neil and Fenton [36] used hierarchical Bayesian model for evaluation of treatments for Covid-19.

There are limits to hierarchical Bayesian model. The first is the underlying resulting model assumption may be wrong [14, 31]. Thais et al. [37] noted that "ill behaved likelihoods" at the lower levels in the hierarchy may create either excessive concentration about a mean or noninformative results. Rouder et al. [38] note that although hierarchical linear models are suitable in several domains, they rarely make good models of psychological process. Heller and Gharamani [39] note the algorithm provides no guide to choosing the "correct" number of clusters.

As a result, Shaddick et al. [18] note that Bayesian hierarchical models are an extremely useful and flexible framework in which to model complex relationships and dependencies in data, while Kruschke and Vanpaemel [17] suggest they provide flexibility in designing models that are appropriate for describing the data at hand that can provide a complete representation of parameter uncertainty (i.e., the posterior distribution) that can be directly interpreted. Some examples on how to create a hierarchical Bayesian model are helpful to demonstrate the process.


Reactive Oxygen Species - Advances and Developments

*Edited by Rizwan Ahmad*

Published in London, United Kingdom

Reactive Oxygen Species - Advances and Developments http://dx.doi.org/10.5772/intechopen.107774 Edited by Rizwan Ahmad

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