Sufficient diagnostic proposition.
\r\n\t
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Since he graduated he has published several peer-reviewed papers at the national and international levels and he has been a guest researcher and lecturer both at Michigan State University (USA) and at the University of Toronto (Canada) where he has developed part of his Ph.D. research with the Financial support from the Portuguese Foundation for Science and Technology (Ph.D. grant).",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"108118",title:"Dr.",name:"Luis",middleName:null,surname:"Loures",slug:"luis-loures",fullName:"Luis Loures",profilePictureURL:"https://mts.intechopen.com/storage/users/108118/images/system/108118.png",biography:"Luís Loures is a Landscape Architect and Agronomic Engineer, Vice-President of the Polytechnic Institute of Portalegre, who holds a Ph.D. in Planning and a Post-Doc in Agronomy. 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Each node is a random variable. Each arc represents a relationship between two nodes. The strength of a relationship in a graph can be quantified by a number called weight. There are some important relationships such as prerequisite, diagnostic, and aggregation. The difference between BN and normal graph is that the strength of every relationship in BN is represented by a conditional probability table (CPT) whose entries are conditional probabilities of a child node given parent nodes. There are two main approaches to construct a BN, which are as follows
The first approach aims to learn BN from training data by learning machine algorithms.
The second approach is that experts define some graph patterns according to specific relationships and then, BN is constructed based on such patterns along with determined CPTs.
This research focuses on the second approach in which relationships are converted into CPTs. Essentially, relationship conversion aims to determine conditional probabilities based on weights and meanings of relationships. We will have different ways to convert graphic weights into CPTs for different relationships. It is impossible to convert all relationships but some of them such as diagnostic, aggregation, and prerequisite are mandatory ones that we must specify as computable CPTs of BN. Especially, these relationships are adhered to logic X-gates [1] such as AND-gate, OR-gate, and SIGMA-gate. The X-gate inference in this research is derived and inspired from noisy OR-gate described in the book “Learning Bayesian Networks” Neapolitan ([2], pp. 157–159). Díez and Druzdzel [3] also researched OR/MAX, AND/MIN, and noisy XOR inferences but they focused on canonical models, deterministic models, and ICI models whereas I focused on logic gate and graphic relationships. So, their research is different from mine but we share the same result that is AND-gate model. In general, my research focuses on applied probability adhered to Bayesian network, logic gates, and Bayesian user modeling [4]. The scientific results are shared with Millán and Pérez-de-la-Cruz [4].
\nFactor graph [5] represents factorization of a global function into many partial functions. If joint distribution of BN is considered as the global function and CPTs are considered as partial functions, the sumproduct algorithm [6] of factor graph is applied into calculating posterior probabilities of variables in BN. Pearl’s propagation algorithm [7] is very successful in BN inference. The application of factor graph into BN is only realized if all CPT (s) of BN are already determined whereas this research focuses on defining such CPTs firstly. I did not use factor graph for constructing BN. The concept “X-gate inference” only implies how to convert simple graph into BN. However, the arrange sum with a fixed variable mentioned in this research is the “not-sum” ([6], p. 499) of factor graph. Essentially, X-gate probability shown in Eq. (10) is as same as λ message in the Pearl’s algorithm ([6], p. 518) but I use the most basic way to prove the X-gate probability.
\nAs default, the research is applied in learning context in which BN is used to assess students’ knowledge. Evidences are tests, exams, exercises, etc. and hypotheses are learning concepts, knowledge items, etc. Note that diagnostic relationship is very important to Bayesian evaluation in learning context because it is used to evaluate student’s mastery of concepts (knowledge items) over entire BN. Now, we start relationship conversion with a research on diagnostic relationship in the next section.
\nIn some opinions like mine, the diagnostic relationship should be from hypothesis to evidence. For example, disease is hypothesis and symptom is evidence. The symptom must be conditionally dependent on disease. Given a symptom, calculating the posterior probability of disease is essentially to diagnose likelihood of such disease ([8], p. 1666). Inversely, the arc from evidence to hypothesis implies prediction where evidence and hypothesis represent observation and event, respectively. Given an observation, calculating the posterior probability of the event is essentially to predict/assert such event ([8], p. 1666). Figure 1 shows diagnosis and prediction.
\nDiagnosis and prediction with hypothesis X and evidence D.
The weight w of the relationship between X and D is 1. Figure 1 depicts simplest graph with two random variables. We need to convert diagnostic relationship into conditional probabilities in order to construct a simplest BN from the simplest graph. Note that hypothesis is binary but evidence can be numerical. In learning context, evidence D can be test, exam, exercise, etc. The conditional probability of D given X (likelihood function) is P(D|X). The posterior probability of X is P(X|D), which is used to evaluate student’s mastery over concept (hypothesis) X given evidence D. Eq. (1) specifies CPT of D when D is binary (0 and 1)
\nEq (1) is our first relationship conversion. It implies
\nEvidence D can be used to diagnose hypothesis X if the so-called sufficient diagnostic proposition is satisfied, as seen in Table 1.
\nD is equivalent to X in diagnostic relationship if P(X|D) = kP(D|X) given uniform distribution of X and the transformation coefficient k is independent from D. In other words, k is constant with regards to D and so D is called sufficient evidence. | \n
Sufficient diagnostic proposition.
The concept of sufficient evidence is borrowed from the concept of sufficient statistics and it is inspired from equivalence of variables T and T’ in the research ([4], pp. 292-295). The proposition can be restated that evidence D is only used to assess hypotheses if it is sufficient evidence. As a convention, the proposition is called diagnostic condition and hypotheses have uniform distribution. The assumption of hypothetic uniform distribution (P(X = 1) = P(X = 0)) implies that we cannot assert whether or not given hypothesis is true before we observe its evidence.
\nIn learning context, D can be totally used to assess student’s mastery of X if diagnostic condition is satisfied. Derived from such condition, Eq. (2) specifies transformation coefficient k given uniform distribution of X.
\nWe need to prove that Eq. (1) satisfies diagnostic condition. Suppose the prior probability of X is uniform.
\nwe have
\nIt is easy to infer that the transformation coefficient k is 1, if D is binary. In practice, evidence D is often a test whose grade ranges within an interval {0, 1, 2,…, η}. Eq. (3) specifies CPT of D in this case
\nWhere
\nAs a convention, \n
We need to prove that Eq. (3) satisfies diagnostic condition. Suppose the prior probability of X is uniform.
\nThe assumption of prior uniform distribution of X implies that we do not determine if student has mastered X yet. Similarly, we have
\nSo, the transformation coefficient k is \n
In the most general case, discrete evidence D ranges within an arbitrary integer interval \n
Where
\nNote, \n
Similarly, we have
\nIf evidence D is continuous in the real interval [a, b] with note that a and b are real numbers, Eq. (5) specifies probability density function (PDF) of continuous evidence \n
where
\nAs a convention, [a, b] is called domain of continuous evidence, which can be replaced by open or half-open intervals such as (a, b), (a, b], and [a, b). Of course we have \n
Functions p(D|X = 1) and p(D|X = 0) are valid PDFs due to
\nAccording to the diagnostic condition, we need to prove the equality
\nwhere,
\nWhen D is continuous, its probability is calculated in ε-vicinity where ε is very small number. As usual, ε is bias if D is measure values produced from equipment. The probability of D given X, where D + ε\n
In fact, we have
\nIn general, Eq. (6) summarizes CPT of evidence of single diagnostic relationship.
\nWhere,
\nIn general, if the conditional probability P(D|X) is specified by Eq. (6), the diagnostic condition will be satisfied. Note that the CPT P(D|X) is the PDF p(D|X) in case of continuous evidence. The diagnostic relationship will be extended with more than one hypothesis. The next section will mention how to determine CPTs of a simple graph with one child node and many parent nodes based on X-gate inferences.
\nGiven a simple graph consisting of one child variable Y and n parent variables Xi, as shown in Figure 2, each relationship from Xi to Y is quantified by normalized weight wi where 0 ≤ wi ≤ 1. A large graph is an integration of many simple graphs. Figure 2 shows the DAG of a simple BN. As aforementioned, the essence of constructing simple BN is to convert graphic relationships of simple graph into CPTs of simple BN.
\nSimple graph or simple network.
Child variable Y is called target and parent variables Xis are called sources. Especially, these relationships are adhered to X-gates such as AND-gate, OR-gate, and SIGMA-gate. These gates are originated from logic gate [1]. For instance, AND-gate and OR-gate represent prerequisite relationship. SIGMA-gate represents aggregation relationship. Therefore, relationship conversion is to determined X-gate inference. The simple graph shown in Figure 2 is also called X-gate graph or X-gate network. Please distinguish the letter “X” in the term “X-gate inference” which implies logic operators (AND, OR, XOR, etc.) from the “variable X”.
\nAll variables are binary and they represent events. The probability P(X) indicates event X occurs. Thus, P(X) implicates P(X = 1) and P(not(X)) implicates P(X = 0). Eq. (7) specifies the simple NOT-gate inference.
\nX-gate inference is based on three assumptions mentioned in Ref. ([2], p. 157), which are as follows
X-gate inhibition: Given a relationship from source Xi to target Y, there is a factor Ii that inhibits Xi from being integrated into Y. Factor Ii is called inhibition of Xi. That the inhibition Ii is turned off is prerequisite of Xi integrated into Y.
Inhibition independence: Inhibitions are mutually independent. For example, inhibition I1 of X1 is independent from inhibition I2 of X2.
Accountability: X-gate network is established by accountable variables Ai for Xi and Ii. Each X-gate inference owns particular combination of Ais.
Figure 3 shows the extended X-gate network with accountable variables Ais ([2], p. 158).
\nExtended X-gate network with accountable variables Ais.
The strength of each relationship from source Xi to target Y is quantified by a weight 0 ≤ wi ≤ 1. According to the assumption of inhibition, probability of Ii = OFF is pi, which is set to be the weight wi.
\nIf notation wi is used, we focus on the strength of relationship. If notation pi is used, we focus on probability of OFF inhibition. In probabilistic inference, pi is also prior probability of Xi = 1. However, we will assume each Xi has uniform distribution later on. Eq. (8) specifies probabilities of inhibitions Iis and accountable variables Ais.
\nAccording to Eq. (8), given probability P(Ai=ON | Xi=1, Ii=OFF), it is assured 100% confident that accountable variables Ai is turned on if source Xi is 1 and inhibition Ii is turned off. Eq. (9) specifies conditional probability of accountable variables Ai (s) given Xi (s), which is corollary of Eq. (8).
\nAppendix A1 is the proof of Eq. (9). As a definition, the set of all Xis is complete if and only if
\nThe set of all Xis is mutually exclusive if and only if
\nFor each Xi, there is only one Ai and vice versa, which establishes a bijection between Xis and Ais. Obviously, the fact that the set of all Xis is complete is equivalent to the fact that the set of all Ai (s) is complete. We will prove by contradiction that “the fact that the set of all Xi (s) is mutually exclusive is equivalent to the fact that the set of all Ai (s) is mutually exclusive.” Suppose \n
By similar proof, we have
\nThe extended X-gate network shown in Figure 3 is interpretation of simple network shown in Figure 2. Specifying CPT of the simple network is to determine the conditional probability P(Y = 1 | X1, X2,…, Xn) based on extended X-gate network. The X-gate inference is represented by such probability P(Y = 1 | X1, X2,…, Xn) specified by Eq. (10) ([2], p. 159).
\nAppendix A2 is the proof of Eq. (10). It is necessary to make some mathematical notations because Eq. (10) is complicated, which is relevant to arrangements of Xi (s). Given the set Ω = {X1, X2,…, Xn} where all variables are binary, Table 2 specifies binary arrangements of Ω.
\nGiven Ω = {X1, X2,…, Xn} where |Ω| = n is cardinality of Ω. Let a(Ω) be an arrangement of Ω which is a set of n instances {X1=x1, X2=x2,…, Xn=xn} where xi is 1 or 0. The number of all a(Ω) is 2|Ω|. For instance, given Ω = {X1, X2}, there are 22=4 arrangements as follows: \n \nLet a(Ω:{Xi}) be the arrangement of Ω with fixed Xi. The number of all a(Ω:{Xi}) is 2|Ω|−1. Similarly, for instance, a(Ω:{X1, X2, X3}) is an arrangement of Ω with fixed X1, X2, X3. The number of all a(Ω:{X1, X2, X3}) is 2|Ω|−3. Let c(Ω) and c(Ω:{Xi}) be the number of arrangements a(Ω) and a(Ω:{Xi}), respectively. Such c(Ω) and c(Ω:{Xi}) are called arrangement counters. As usual, counters c(Ω) and c(Ω:{Xi}) are equal to 2|Ω| and 2|Ω|−1, respectively but they will vary according to specific cases. Let \n Let x denote the X-gate operator, for instance, x = ⊙ for AND-gate, x = ⊕ for OR-gate, x = not ⊙ for NAND-gate, x = not ⊕ for NOR-gate, x = ⊗ for XOR-gate, x = not ⊗ for XNOR-gate, x = ⊎ for U-gate, \n \n \nFor example, s(Ω) and s(Ω:{Xi}) for OR-gate are: \n \nSuch s(Ω) and s(Ω:{Xi}) are called arrangement sum. They are acting function F. Note that Ω can be any set of binary variables. | \n
Binary arrangements.
It is not easy to produce all binary arrangements of Ω. Table 3 shows a code snippet written by Java programming language for producing such all arrangements.
\n public class ArrangementGenerator { private ArrayList<int[]> arrangements; private int n; private int r; \n private ArrangementGenerator(int n, int r) { this.n = n; this.r = r; this.arrangements = new ArrayList(); } private void create(int[] a, int i) { for(int j = 0; j < n; j++) { a[i] = j; if(i < r - 1) create(a, i + 1); else if(i == r -1) { int[] b = new int[a.length]; for(int k = 0; k < a.length; k++) b[k] = a[k]; arrangements.add(b); } } } public int[] get(int i) { return arrangements.get(i); } public long size() { return arrangements.size(); } public static ArrangementGenerator parse(int n, int r) { ArrangementGenerator arr = new ArrangementGenerator(n, r); int[] a = new int[r]; for(int i=0; i<r; i++) a[i] = -1; arr.create(a, 0); return arr; } } | \n
Code snippet generating all binary arrangements.
Each element of the list “arrangements” is a binary arrangement a(Ω) presented by an array of bits (0 and 1). The method “create(int[] a, int i)” which is recursive method, is the main one that generates arrangements. The method call “ArrangementGenerator.parse(2, n)” will list all possible binary arrangements.
\nEq. (11) specifies the connection between s(Ω:{Xi = 1}) and s(Ω:{Xi = 0}), between c(Ω:{Xi = 1}) and c(Ω:{Xi = 0}).
\nIt is easy to draw Eq. (11) when the set of all arrangements a(Ω:{Xi = 1) is complement of the set of all arrangements a(Ω:{Xi = 0).
\nLet K be a set of Xis whose values are 1 and let L be a set of Xis whose values are 0. K and L are mutually complementary. Eq. (12) determines sets K and L.
\nThe AND-gate inference represents prerequisite relationship satisfying AND-gate condition specified by Eq. (13).
\nFrom Eq. (10), we have
\n(Due to Eq. (9))
\nIn general, Eq. (14) specifies AND-gate inference.
\nThe AND-gate inference was also described in ([3], p. 33). Eq. (14) varies according to two cases whose arrangement counters are listed as follows
\nThe OR-gate inference represents prerequisite relationship satisfying OR-gate condition specified by Eq. (15) ([2], p. 157).
\nThe OR-gate condition implies
\nFrom Eq. (10), we have ([2], p. 159)
\n(Due to Eq. (9))
\nIn general, Eq. (16) specifies OR-gate inference.
\nwhere K is the set of Xis whose values are 1. The OR-gate inference was mentioned in Refs. ([2], p. 158) and ([3], p. 20). Eq. (16) varies according to two cases whose arrangement counters are listed as follows
\nAccording to De Morgan’s rule with regard to AND-gate and OR-gate, we have
\n(Due to Eq. (16))
\nAccording to Eq. (14), we also have
\nIn general, Eq. (17) specifies NAND-gate inference and NOR-gate inference derived from AND-gate and OR-gate
\nwhere K and L are the sets of Xis whose values are 1 and 0, respectively.
\nSuppose the number of sources Xis is even. Let O be the set of Xis whose indices are odd. Let O1 and O2 be subsets of O, in which all Xis are 1 and 0, respectively. Let E be the set of Xis whose indices are even. Let E1 and E2 be the subsets of E, in which all Xis are 1 and 0, respectively.
\nThus, O1 and E1 are the subsets of K. Sources Xis and target Y follow XOR-gate if one of two XOR-gate conditions specified by Eq. (18) is satisfied.
\nFrom Eq. (10), we have
\nIf both XOR-gate conditions are not satisfied then,
\nIf the first XOR-gate condition is satisfied, we have
\nWe have
\n(Due to Eq. (9))
\nWe also have
\n(Due to Eq. (9))
\nGiven the first XOR-gate condition, it implies
\nSimilarly, given the second XOR-gate condition, we have
\nIf one of XOR-gate conditions is satisfied then,
\nThis implies Eq. (19) to specify XOR-gate inference.
\nWhere,
\nGiven n ≥ 2 and n is even, Eq. (19) varies according to six cases whose arrangement counters are listed as follows
\nSuppose the number of sources Xis is even. According to XNOR-gate inference [1], the output is on if all inputs get the same value 1 (or 0). Sources Xi (s) and target Y follow XNOR-gate if one of two XNOR-gate conditions specified by Eq. (20) is satisfied.
\nFrom Eq. (10), we have
\nIf both XNOR-gate conditions are not satisfied then,
\nIf Ai = ON for all i, we have
\n(Please see similar proof in AND-gate inference)
\nIf Ai = OFF for all i, we have
\n(Please see similar proof in OR-gate inference)
\nIf one of XNOR-gate conditions is satisfied then,
\nThis implies Eq. (21) to specify XNOR-gate inference.
\nwhere K and L are the sets of Xis whose values are 1 and 0, respectively. Eq. (21) varies according to three cases whose arrangement counters are listed as follows
\nLet U be a set of indices such that Ai = ON and let α ≥ 0 and β ≥ 0 be predefined numbers. The U-gate inference is defined based on α, β and cardinality of U. Table 4 specifies four common U-gate conditions.
\n|U|=α \n | \n
|U|≥α \n | \n
|U|≤β \n | \n
α≤|U|≤β \n | \n
U-gate conditions.
Note that U-gate condition on |U| can be arbitrary and it is only relevant to Ais (ON or OFF) and the way to combine Ais. For example, AND-gate and OR-gate are specific cases of U-gate with |U| = n and |U| ≥ 1, respectively. XOR-gate and XNOR-gate are also specific cases of U-gate with specific conditions on Ai (s). However, it must be assured that there is at least one combination of Ais satisfying the predefined U-gate condition, which causes that U-gate probability is not always equal to 0. In this research, U-gate is the most general nonlinear gate where U-gate probability contains products of weights (see Table 5). Later on, we will research a so-called SIGMA-gate that contains only linear combination of weights (sum of weights, see Eq. (23)). Shortly, each X-gate is a pattern owning a particular X-gate inference that is X-gate probability P(X1 × X2 ×…× Xn). Each X-gate inference is based on particular X-gate condition(s) relevant to only variables Ais.
\nLet, | \n\n\n | \n
As a convention, | \n\n\n | \n
|U|=0 | \n\n\n \n\n | \n
|U|≥0 | \n\n\n The case |U|≥0 is the same to the case |U|≤n | \n
|U|=n | \n\n\n \n\n | \n
|U|=α 0<α<n | \n\n\n | \n
|U|≥α 0<α<n | \n\n\n | \n
|U|≤β 0<β<n | \n\n\n \n\n | \n
α≤|U|≤β 0<α<n 0<β<n | \n\n\n | \n
U-gate inference.
From Eq. (10), we have
\nLet \n
If \n
This implies all sets U (s) must be subsets of K. The U-gate probability is rewritten as follows
\n(Due to Eq. (9))
\nLet PU be the U-gate probability; Table 5 specifies U-gate inference and cardinality of \n
Note that the notation \n
Arrangement counters relevant to U-gate inference and the set K are listed as follows
\nThe SIGMA-gate inference [9] represents aggregation relationship satisfying SIGMA-gate condition specified by Eq. (22).
\nwhere the set of Ai is complete and mutually exclusive
\nThe sigma sum \n
This implies
\nThe sigma sum \n
SIGMA-gate inference requires the set of Ais is complete and mutually exclusive, which means that the set of Xis is complete and mutually exclusive too. The SIGMA-gate probability is [9]
\nIt implies
\n(Due to Eq. (9))
\nIn general, Eq. (23) specifies the theorem of SIGMA-gate inference [9]. The base of this theorem was mentioned by Millán and Pérez-de-la-Cruz ([4], pp. 292-295).
\nwhere the set of Xis is complete and mutually exclusive.
\nThe arrangement counters of SIGMA-gate inference are c(Ω:{Xi = 1}) = c(Ω:{Xi = 0}) = 2n−1, c(Ω) = 2n.
\nEq. (9) specifies the “clockwise” strength of relationship between Xi and Y. Event Xi = 1 causes event Ai = ON with “clockwise” weight wi. There is a question “given Xi = 0, how likely the event Ai = OFF is”. In order to solve this problem, I define a so-called “counterclockwise” strength of relationship between Xi and Y denoted ωi. Event Xi = 0 causes event Ai = OFF with “counterclockwise” weight ωi. In other words, each arc in simple graph is associated with a clockwise weight wi and a counterclockwise weight ωi. Such graph is called bi-weight simple graph shown in Figure 4.
\nBi-weight simple graph.
With bi-weight simple graph, all X-gate inferences are extended as so-called X-gate bi-inferences. Derived from Eq. (9), Eq. (24) specifies conditional probability of accountable variables with regard to bi-weight graph.
\nThe probabilities P(Ai = ON | Xi = 0) and P(Ai = OFF | Xi = 1) are called clockwise adder di and counterclockwise adder δi. As usual, di and δi are smaller than wi and ωi. When di = 0, bi-weight graph becomes normal simple graph.
\nThe total clockwise weight or total counterclockwise weight is defined as sum of clockwise weight and clockwise adder or sum of counterclockwise weight and counterclockwise adder. Eq. (25) specifies such total weights Wi and \n
where
\nGiven Eq. (25), the set of all Ais is complete if and only if \n
By extending aforementioned X-gate inferences, we get bi-inferences for AND-gate, OR-gate, NAND-gate, NOR-gate, XOR-gate, XNOR-gate, and U-gate as shown in Table 6.
\nThe largest cardinalities of K (L) are 2n−1 and 2n with and without fixed Xi. Thus, it is possible to calculate arrangement counters. As a convention, the product of probabilities is 1 if indices set is empty.
\nWith regard to SIGMA-gate bi-inference, the sum of all total clockwise weights must be 1 as follows
\nDerived from Eq. (23), the SIGMA-gate probability for bi-weight graph is
\nShortly, Eq. (26) specifies SIGMA-gate bi-inference.
\nwhere the set of Xi(s) is complete and mutually exclusive.
\nThe next section will research diagnostic relationship which adheres to X-gate inference.
\nGiven a simple graph shown in Figure 2, if we replace the target source Y by an evidence D, we get a so-called multihypothesis diagnostic relationship whose property adheres to X-gate inference. Maybe there are other diagnostic relationships in which X-gate inference is not concerned. However, this research focuses on X-gate inference and so multi-hypothesis diagnostic relationship is called X-gate diagnostic relationship. Sources X1, X2,…, Xn become hypotheses. As a convention, these hypotheses have prior uniform distribution.
\nAccording to aforementioned X-gate network shown in Figures 2 and 3, the target variable must be binary whereas evidence D can be numeric. It is impossible to establish the evidence D as direct target variable. Thus, the solution of this problem is to add an augmented target binary variable Y and then, the evidence D is connected directly to Y. In other words, the X-gate diagnostic network have n sources {X1, X2,…, Xn}, one augmented hypothesis Y, and one evidence D. As a convention, X-gate diagnostic network is called X-D network. The CPTs of the entire network are determined based on combination of diagnostic relationship and X-gate inference mentioned in previous sections. Figure 5 depicts the augmented X-D network. Note that variables X1, X2,…, Xn, and Y are always binary.
\nAugmented X-D network.
Appendix A3 is the proof that the augmented X-D network is equivalent to X-D network with regard to variables X1, X2,…, Xn and D. As a convention, augmented X-D network is considered as same as X-D network.
\nThe simplest case of X-D network is NOT-D network having one hypothesis X1 and one evidence D, equipped with NOT-gate inference. NOT-D network satisfies diagnostic condition because it essentially represents the single diagnostic relationship. Inferred from Eqs. (1) and (7), the conditional probability P(D|X1) and posterior probability P(X1|D) of NOT-D network are
\n(Due to Bayes’ rule and uniform distribution of X1)
\nIt implies NOT-D network satisfies diagnostic condition. Let
\nWe will validate whether the CPT of diagnostic relationship, P(D|X) specified by Eq. (6), still satisfies diagnostic condition within general case, X-D network. In other words, X-D network is general case of single diagnostic relationship.
\nRecall from dependencies shown in Figure 5, Eq. (27) specifies the joint probability of X-D network.
\nEq. (28) specifies the conditional probability of D given Xi (likelihood function) and the posterior probability of Xi given D.
\nwhere Ω = {X1, X2,…, Xn} and the sign “\\” denotes the subtraction (excluding) operator in set theory [10]. Eq. (29) specifies the joint probability P(Xi, D) and the marginal probability P(D) given uniform distribution of all sources. Appendix A4 is the proof of Eq. (29).
\nwhere s(Ω) and s(Ω:{Xi}) are specified in Table 2. From Eqs. (28–30) specifies conditional probability P(D|Xi), posterior probability P(Xi|D), and transformation coefficient for X-gate inference.
\nThe transformation coefficient is rewritten as follows
\nNote that S, D, and M are abstract symbols and there is no proportional connection between 2n−1S and D for all D, specified by Eq. (6). Assuming that such proportional connection 2n−1S = aDj exists for all D where a is arbitrary constant. Given binary case when D = 0 and S = 1, we have
\nThere is a contradiction, which implies that it is impossible to reduce k into the following form
\nTherefore, if k is constant with regard to D then,
\nwhere C is constant. We have
\nIt is implied that
\nThis holds
\nAssuming ND = S we have
\nThere is a contradiction because M is maximum value of D. Therefore, if k is constant with regard to D then s(Ω) = 2n−1. Inversely, if s(Ω) = 2n−1 then k is
\nIn general, the event that k is constant with regard to D is equivalent to the event s(Ω) = 2n−1. This implies diagnostic theorem stated in Table 7.
\n\n \n \n\n \n\n \n\n \n\n \n\n \n\n \n | \n
There are four common conditions of U: |U|=α, |U|≥α, |U|≤β, and α≤|U|≤β. Note that \n \n \nThe largest cardinality of \n \n \n | \n
Bi-inferences for AND-gate, OR-gate, NAND-gate, NOR-gate, XOR-gate, XNOR-gate, and U-gate.
Given X-D network is combination of diagnostic relationship and X-gate inference: \n \n\n \nThe diagnostic condition of X-D network is satisfied if and only if \n \nAt that time, the transformation coefficient becomes: \n \nNote that weights pi = wi and ρi = ωi, which are inputs of s(Ω), are abstract variables. Thus, the equality s(Ω) = 2|Ω|−1 implies all abstract variables are removed and so s(Ω) does not depend on weights. | \n
Diagnostic theorem.
The diagnostic theorem is the optimal way to validate the diagnostic condition.
\nThe Eq. (30) becomes simple with AND-gate inference. Recall that Eq. (14) specified AND-gate inference as follows
\nDue to only one case X1 = X2 =…= Xn = 1, we have
\nDue to Xi = 0, we have
\nDerived from Eq. (30), Eq. (31) specifies conditional probability P(D|Xi), posterior probability P(Xi|D), and transformation coefficient according to X-D network with AND-gate reference called AND-D network.
\nFor convenience, we validate diagnostic condition with a case of two sources Ω = {X1, X2}, p1 = p2 = w1 = w2 = 0.5, \n
Given AND-gate inference, by applying Eq. (14), we have
\nGiven OR-gate inference, by applying Eq. (16), we have
\nGiven XOR-gate inference, by applying Eq. (19), we have
\nGiven XNOR-gate inference, by applying Eq. (21), we have
\nGiven SIGMA-gate inference, by applying Eq. (23), we have
\nIt is asserted that AND-gate, OR-gate, XOR-gate, and XNOR-gate do not satisfy diagnostic condition and so they should not be used to assess hypotheses. However, it is not asserted if U-gate and SIGMA-gate satisfy such diagnostic condition. It is necessary to expend equation for SIGMA-gate diagnostic network (called SIGMA-D network) in order to validate it.
\nIn case of SIGMA-gate inference, by applying Eq. (23), we have
\nIt is necessary to validate SIGMA-D network with SIGMA-gate bi-inference. By applying Eq. (26), we recalculate these quantities as follows
\nObviously, quantities s(Ω), s(Ω:{Xi=1}), and s(Ω:{Xi = 0}) are kept intact. According to diagnostic theorem, we conclude that SIGMA-D network does satisfy diagnostic condition due to s(Ω)=2n−1. Thus, SIGMA-D network can be used to assess hypotheses.
\nEq. (32), an immediate consequence of Eq. (30), specifies conditional probability P(D|Xi), posterior probability P(Xi|D), and transformation coefficient for SIGMA-D network.
\nIn case of SIGMA-gate, the augmented variable Y can be removed from X-D network. The evidence D is now established as direct target variable. Figure 6 shows a so-called direct SIGMA-gate diagnostic network (direct SIGMA-D network).
\nDirect SIGMA-gate diagnostic network (direct SIGMA-D network).
Derived from Eq. (23), the CPT of direct SIGMA-D network is determined by Eq. (33).
\nwhere the set of Xi (s) is complete and mutually exclusive.
\nEq. (33) specifies valid CPT due to
\nFrom dependencies shown in Figure 6, Eq. (34) specifies the joint probability of direct SIGMA-D network.
\nInferred from Eq. (29), Eq. (35) specifies the joint probability P(Xi, D) and the marginal probability P(D) of direct SIGMA-D network, given uniform distribution of all sources.
\nwhere s(Ω) and s(Ω:{Xi}) are specified in Table 2.
\nBy browsing all variables of direct SIGMA-D network, we have
\nSimilarly, we have
\nBy applying Eq. (35), s(Ω:{Xi = 0}), s(Ω:{Xi = 1}), and s(Ω), we get the same result with Eq. (32).
\nTherefore, it is possible to use direct SIGMA-D network to assess hypotheses. It is asserted that SIGMA-D network satisfy diagnostic condition when single relationship, NOT-D network, direct SIGMA-D network are specific cases of SIGMA-D network. There is a question: does an X-D network that is different from SIGMA-D network and not aforementioned exist such that it satisfies diagnostic condition?
\nRecall that each X-D network is a pattern owning a particular X-gate inference which in turn is based on particular X-gate condition (s) relevant to only variables Ais. The most general nonlinear X-D network is U-D network whereas SIGMA-D network is linear one. The U-gate inference given arbitrary condition on U is
\nLet f be the arrangement sum of U-gate inference.
\nThe function f is sum of many large expressions and each expression is product of four possible sub-products (Π) as follows
\nIn any case of degradation, there always exist expression Expr (s) having at least 2 sub-products (Π), for example,
\nConsequently, there always exist Expr (s) having at least 5 terms relevant to pi and ρi if n ≥ 5, for example,
\nThus, degree of f will be larger than or equal to 5 given n ≥ 5. According to diagnostic theorem, U-gate network satisfies diagnostic condition if and only if f(pi, ρi) = 2n−1 for all n ≥ 1 and for all abstract variables pi and ρi. Without loss of generality, each pi or ρi is sum of variable x and a variable ai or bi, respectively. Note that all pi, ρi, ai are bi are abstract variables.
\nThe equation f−2n−1 = 0 becomes equation g(x) = 0 whose degree is m ≥ 5 if n ≥ 5.
\nwhere coefficients Ci s are functions of ai and bis. According to Abel-Ruffini theorem [11], equation g(x) = 0 has no algebraic solution when m ≥ 5. Thus, abstract variables pi and ρi cannot be eliminated entirely from g(x) = 0, which causes that there is no specification of U-gate inference P(X1xX2x…xXn) so that diagnostic condition is satisfied.
\nIt is concluded that there is no nonlinear X-D network satisfying diagnostic condition, but a new question is raised: does there exist the general linear X-D network satisfying diagnostic condition? Such linear network is called GL-D network and SIGMA-D network is specific case of GL-D network. The GL-gate probability must be linear combination of weights.
\nwhere C is arbitrary constant.
\nThe GL-gate inference is singular if αi and βi are functions of only Xi as follows
\nThe functions hi and gi are not relevant to Ai because the final equation of GL-gate inference is only relevant to Xi (s) and weights (s). Because GL-D network is a pattern, we only survey singular GL-gate. Mentioned GL-gate is singular by default and it is dependent on how to define functions hi and gi. The arrangement sum with regard to GL-gate is
\nSuppose hi and gi are probability mass functions with regard to Xi. For all i, we have
\nThe arrangement sum becomes
\nGL-D network satisfies diagnostic condition if
\nSuppose the set of Xis is complete.
\nThis implies C = 0. Shortly, Eq. (36) specifies the singular GL-gate inference so that GL-D network satisfies diagnostic condition.
\nFunctions hi(Xi) and gi(Xi) are always linear due to Xim = Xi for all m ≥ 1 when Xi is binary. It is easy to infer that SIGMA-D network is GL-D network with following definition of functions hi and gi.
\nAccording to Millán and Pérez-de-la-Cruz [4], a hypothesis can have multiple evidences as seen in Figure 7. This is multi-evidence diagnostic relationship opposite to aforementioned multihypothesis diagnostic relationship.
\nDiagnostic relationship with multiple evidences (M-E-D network).
M-HE-D network.
Figure 7 depicts the multi-evidence diagnostic network called M-E-D network in which there are m evidences D1, D2,…, Dm and one hypothesis Y. Note that Y has uniform distribution.
\nIn simplest case where all evidences are binary, the joint probability of M-E-D network is
\nThe product \n
The posterior probability P(Y | D1, D2,…, Dm) given uniform distribution of Y is
\nThe possible transformation coefficient is
\nM-E-D network will satisfy diagnostic condition if k = 1 because all hypotheses and evidence are binary, which leads that following equation specified by Eq. (37) has 2m real roots P(Dj|Y) for all m ≥ 2.
\nEq. (37) has no real root given m = 2 according to following proof. Suppose Eq. (37) has 4 real roots as follows
\nFrom Eq. (37), it holds
\nThe final equation leads a contradiction (b1 = 2 or b1 = 1.5) and so it is impossible to apply the sufficient diagnostic proposition into M-E-D network. Such proposition is only used for one-evidence network. Moreover, X-gate inference absorbs many sources and then produces out of one targeted result whereas the M-E-D network essentially splits one source into many results. It is impossible to model M-E-D network by X-gates. The potential solution for this problem is to group many evidences D1, D2,…, Dm into one representative evidence D which in turn is dependent on hypothesis Y but this solution will be inaccurate in specifying conditional probabilities because directions of dependencies become inconsistent (relationships from Dj to D and from Y to D) except that all Djs are removed and D becomes a vector. However, evidence vector does not simplify the hazardous problem and it changes the current problem into a new problem.
\nAnother solution is to reverse the direction of relationship, in which the hypothesis is dependent on evidences so as to take advantages of X-gate inference as usual. However, the reversion method violates the viewpoint in this research where diagnostic relationship must be from hypothesis to evidence. In other words, we should change the viewpoint.
\nAnother solution is based on a so-called partial diagnostic condition that is a loose case of diagnostic condition for M-E-D network, which is defined as follows
\nwhere k is constant with regard to Dj. The joint probability is
\nM-E-D network satisfies partial diagnostic condition. In fact, given all variables are binary, we have
\n(Let Ψ = {D1, D2,…, Dm})
\n(Due to uniform distribution of Y)
\nPartial diagnostic condition expresses a different viewpoint. It is not an optimal solution because we cannot test a disease based on only one symptom while ignoring other obvious symptoms, for example. The equality P(Y|Dj) = 0.5P(Dj|Y) indicates the accuracy is decreased two times. However, Bayesian network provides inference mechanism based on personal belief. It is subjective. You can use partial diagnostic condition if you think that such condition is appropriate to your application.
\nIf we are successful in specifying conditional probabilities of M-E-D network, it is possible to define an extended network which is constituted of n hypotheses X1, X2,…, Xn and m evidences D1, D2,…, Dm. Such extended network represents multi-hypothesis multi-evidence diagnostic relationship, called M-HE-D network. Figure 8 depicts M-HE-D network.
\nThe M-HE-D network is the most general case of diagnostic network, which was mentioned in Ref. ([4], p. 297). We can construct any large diagnostic BN from M-HE-D networks and so the research is still open.
\nIn short, relationship conversion is to determine conditional probabilities based on logic gates that are adhered to semantics of relationships. The weak point of logic gates is to require that all variables must be binary. For example, in learning context, it is inconvenient for expert to create an assessment BN with studying exercises (evidences) whose marks are only 0 and 1. In order to lessen the impact of such weak point, the numeric evidence is used for extending capacity of simple Bayesian network. However, combination of binary hypothesis and numeric evidence leads to errors or biases in inference. For example, given a student gets maximum grade for an exercise but the built-in inference results out that she/he has not mastered fully the associated learning concept (hypothesis). Therefore, I propose the sufficient diagnostic proposition so as to confirm that numeric evidence is adequate to make complicated inference tasks in BN. The probabilistic reasoning based on evidence is always accurate. Application of the research can go beyond learning context whenever probabilistic deduction relevant to constraints of semantic relationships is required. A large BN can be constituted of many simple BN (s). Inference in large BN is hazardous problem and there are many optimal algorithms for solving such problem. In future, I will research effective inference methods for the special BN that is constituted of X-gate BN (s) mentioned in this research because X-gate BN (s) have precise and useful features of which we should take advantages. For instance, their CPT (s) are simple in some cases and the meanings of their relationships are mandatory in many applications. Moreover, I try my best to research deeply M-E-D network and M-HE-D network whose problems I cannot solve absolutely now.
\nTwo main documents that I referred to do this research are the book “Learning Bayesian Networks” [2] by the author Richard E. Neapolitan and the article “A Bayesian Diagnostic Algorithm for Student Modeling and its Evaluation” [4] by authors Eva Millán and José Luis Pérez-de-la-Cruz. Especially, the SIGMA-gate inference is based on and derived from the work of the Eva Millán and José Luis Pérez-de-la-Cruz. This research is originated from my PhD research “A User Modeling System for Adaptive Learning” [12]. Other references relevant to user modeling, overlay model, and Bayesian network are [13–16]. Please concern these references.
\nA1. Following is the proof of Eq. (9)
\nIt implies
\nA2. Following is the proof of Eq. (10)
\n(Because Y is conditionally independent from Xis given Ais)
\n(Because Ais are mutually independent)
\n(Because each Ai is only dependent on Xi) ■
\nA3. Following is the proof that the augmented X-D network (shown in Figure 5) is equivalent to X-D network (shown in shown in Figures 2 and 3) with regard to variables X1, X2,…, Xn, and D.
\nThe joint probability of augmented X-D network shown in Figure 5 is
\nThe joint probability of X-D network is
\nBy applying total probability rule into X-D network, we have
\n(Because D is conditionally independent from all Xi (s) given Y)
\nA4. Following is the proof of Eq. (29)
\nGiven uniform distribution of Xi (s), we have
\nThe joint probability becomes
\nThe joint probability of Xi and D is
\n(Due to Eq. (6))
\nThe marginal probability of D is
\nBy applying Table 2, the joint probability P(Xi, D) is determined as follows
\nSimilarly, the marginal probability P(D) is
\nSpinocerebellar ataxia (SCA), which is included in spinocerebellar degeneration (SCD), is a genetically heterogeneous group of autosomal dominantly inherited progressive disorders [1]. Cerebellar atrophy is the most prominent clinical feature of this condition and is accompanied by spinal cord and sequential brain stem and basal ganglion damage. Therefore, coordinated movement of the eyes, head, trunk, and extremities is impaired. Therefore, the activities of daily living (ADL) and participation in social activities are limited, and the quality of life (QOL) is undisputedly impaired in these patients [2].
The effects of medication and surgery in this clinical setting depend on the cause of ataxia and the extent of neuronal damage [3, 4]; however, there is no rational effective treatment for SCA and it is difficult to slow the progression of the disease. Rehabilitation [5, 6], including physical therapy [7, 8], aimed at improving/maintaining motor function, ADL, and QOL [5] is an important intervention for patients with SCA. Here we provide a narrative review of physical rehabilitation for SCA.
For the clinical diagnosis of cerebellar ataxia, specific blood studies and magnetic resonance imaging (MRI) have been performed [9]. Furthermore, genetic techniques improve the diagnosis of degenerative cerebellar ataxia [10]. Although the details of the findings of these genetic and blood studies are beyond the scope of this review of rehabilitation, cerebellar atrophy and cerebellar motor deficits are traditionally common observations in patients with degenerative cerebellar ataxia [9]. Furthermore, recently, the absence of motor cerebellar symptoms has also been recognized as being important for rehabilitation [11].
The cerebellum is the motor-control system in humans [12]. Clinically, the oculomotor deficit, speech deficits, ataxia in the trunk and extremities, balance disorder, and gait disturbance are the targets of rehabilitation in SCA [9, 13]. The possible underlying pathogenetic mechanisms include distorted timing, abnormal sensory acquisition, impaired sensory motor synchronization, impaired triggering of corticomotor excitability, and abnormal visuokinesthetic cerebro-cerebellar interactions [13].
Oculomotor deficits cause deoptimized vision. The vestibulo-ocular reflex and smooth pursuit [14] partially depend on motor prediction in static and dynamic movement and contribute to dynamic gazing [15]; moreover, the cerebellum contributes to the trainability of eye-head coordinated movements [16].
Abnormal excitability and modulation in the motor cortex and corticospinal tract causes a voluntary contraction deficit in [17, 18]. Cerebellar stimulation modulates the motor-evoked potential induced by transcranial magnetic stimulation (TMS) of the primary motor cortex [19, 20, 21]; however, this modulation is absent in patients with SCA [22, 23]. Furthermore, the cortical silent period, which reflects the excitability of the inhibitory GABAergic neural circuit in the primary motor cortex, is abnormal in these patients [24, 25, 26, 27, 28, 29], and this cerebellar effect on the cortical silent period is characteristic of the healthy population [30]. Before muscle contraction for movement, the corticospinal excitability increases in healthy individuals; in contrast, this facilitation is insufficient in SCA [31]. In addition, in patients with SCA, muscle tones are decreased [11] and the spinal reflex excitability is facilitated by cerebellar stimulation [32, 33, 34]. The long latency spinal reflex, which is correlated with the cortical circuit, is disturbed in SCA [35]. Although this functional cerebellum-spine connection may contribute to the preparation for muscle contraction, there is insufficient evidence that these connections contribute to motor control in healthy and cerebellar ataxia populations.
In simple movements, such as extension of the elbow, coordinated activity of the biceps and triceps is needed. For ballistic elbow-extension movement practice, the triphasic muscle agonist and antagonist contraction patterns contribute to the smooth movement, but under/overshooting appears during the uncoordinated contraction pattern of patients with SCA [36, 37]. Furthermore, this contraction pattern may be obtained by temporal electrical stimulation in these individuals [37].
The cerebellar internal model contributes to predictable/online/offline motor control and motor learning/adaptation [38]. The symptoms associated with motor learning do not appear at the onset of the cerebral atrophy [39], because several brain areas, i.e., the prefrontal cortex, primary motor cortex, and basal ganglia, compensate for cerebellar function in early-stage SCA [5, 6, 39]. Recently, the motor learning deficit at the early stage of the disorder was reportedly detected using an adaptation task [40]. Therefore, the assessment of the capacity for motor learning may be important to strategize the interventions that are concretely described in the following sections.
Representative nonataxia symptoms include hyperreflexia, areflexia, extensor plantar, spasticity, paresis, muscle atrophy, fasciculations, myoclonus, rigidity, chorea/dyskinesia, dystonia, resting tremor, sensory symptoms, urinary dysfunction, cognitive impairment, and brain stem oculomotor signs [41]. The Inventory of NonAtaxia Symptoms (INAS) [41] is used to estimate these nonataxia symptoms. The appearance of these symptoms depends on the type of SCA [41].
We should conduct assessment to detect the degree of motor dysfunction and consider more effective intervention of physical rehabilitation. The first, the imaging technology such as MRI provides us with structural information about the atrophic areas of the brain associated with the disease. We described about neuroimaging technique in Section 3.1. The next, we can use some outcome measurement to estimate the motor dysfunction and verification in the physical rehabilitation. Then, we introduce the representable outcome measures for physical rehabilitation in SCA in Section 3.2. However, we had not established method to estimate the remaining of motor learning ability, which is one of the most important factors to predict the effect of physical rehabilitation. Therefore, we propose the possible assessment of motor learning ability in Section 3.3.
Neuroimaging is a technique that is used to visualize the structural and functional activities of the brain. MRI measurements, such as diffusion tensor imaging and surface-based morphometry, visualize the brain structures. Functional activity imaging is achieved using fMRI and NIRS, which are indicators of cerebral blood flow, and electroencephalogram (EEG) and magnetoencephalography, which are indicators of electrical activity. Positron emission tomography and single-photon emission computed tomography with nuclear tracers are also used in this setting. The application of neuroimaging in the rehabilitation of cerebellar disorders includes voxel-based lesion symptom mapping in patients with stroke, to investigate the recovery of upper arm reach [42] and walking ability [43] depending on the lesion site.
Although conventional MRI [44] is widely used for the neuroimaging of spinocerebellar degeneration, to obtain diagnostic findings, few studies have used neuroimaging as a guideline or outcome of rehabilitation. The lack of reports in this context hampers the quantification of cerebellar degeneration in SCA and its correlation with motor dysfunctions. In terms of measurement techniques, the cerebellum exhibits a much tighter folding compared with the cerebral cortex, with individual cortical sheets with a thickness of 1–2 mm and a sheet area of 1500–2000 cm2, compared with a sheet area of 2200 cm2 with a thickness of 1.5–4 mm in the cerebral cortex. Therefore, the typical 2–4 mm3 spatial resolution of neuroimaging techniques is insufficient to capture local cerebellar changes. Patient factors include the difficulty in limiting the brain regions involved in movement disorders to the cerebellum, because the degenerative regions in SCD extend beyond this structure to multiple brain regions [45].
Among the neuroimaging modalities, the role of voxel-based morphometry (VBM) is notable in SCA rehabilitation. VBM is a statistical analysis of the entire brain in voxel units (1 mm3) that is used to identify the behavioral patterns and related brain morphological characteristics of patients [46]. Burciu et al. assessed the degree of cerebellar atrophy concerning motor and learning functions using VBM to evaluate brain structure changes after 2 weeks of balance training in patients with SCD; these authors reported the association between an increased volume of the dorsal premotor cortex and increased balance ability [47]. Matsgi et al. reported an association between VBM and neurophysiological markers in cerebellar brain inhibition (CBI), with atrophy of the dentate nucleus at VBM observed in cases of pure cerebellar ataxia that did not show CBI [48]. Bando et al. reported a correlation between adaptive learning ability and gray matter volume of the cerebellar IV-VII lobules and the supramarginal gyrus in a prismatic adaptation task in SCA [49]. Thus, VBM may be a biomarker to explain motor dysfunction in patients with SCA.
Conversely, VBM is not an ideal tool to show a causal relationship between brain structural changes and behavioral differences. As a solution to this problem, we can propose a combination of VBM and neurostimulation [50], as neurostimulation of the brain regions associated with the behavioral patterns obtained by VBM and the observation of behavioral changes before and after stimulation allow us to examine brain degeneration sites and behavior.
Gait disturbance is a major symptom of the cerebellar pathology in SCA [51]. The functional ambulation categories (FAC) is useful for the comprehensive assessment of walking ability; the FAC assesses gait for about 15 m and climbing stairs and classifies gait levels into 6 levels [52]. The FAC is also used in the exercise program created by Research Committee for Ataxia Disease (Research team under the jurisdiction of the Ministry of Health, Labour and Welfare in Japan,
The quantitative assessment of cerebellar ataxia is very important in clinical practice. The International Cooperative Ataxia Rating Scale (ICARS) has been used as a quantitative assessment of ataxia symptoms. However, it has been noted that the test reliability of the eye movement items is low [53]. The Scale for Assessment and Rating of Ataxia (SARA) is an 8-item performance-based scale that yields a total score of 0–40 (most severe ataxia). The minimal detectable change (MDC) for individual score difference from the baseline to the 1-year follow-up in SARA was <3.5 (n = 171; SCA1, n = 43; SCA2, n = 61; SCA3, n = 37; and SCA6, n = 30; mean age, 50.9 ± 13.5 years; mean disease duration, 11.8 ± 5.6 years) [54]. SARA does not include an eye movement section. Schmahmann et al. noted the importance of assessing oculomotor abnormalities and developed the Brief Ataxia Rating Scale, a modification of ICARS [55]. Each SCA genotype exhibits specific symptoms [56]. Therefore, these assessments should be used differently for different symptoms. However, one feature that is consistent among these assessments is that the scoring range is large and does not allow the assessment of minute symptom changes. Honda et al. developed a system to measure the evaluation of SARA using a depth sensor [57]. Using this system, the degree of ataxia can be measured numerically. In addition, because the system is inexpensive, it can be installed at the patient’s home, making it a useful tool for telemedicine.
The balance dysfunction in SCA has a significant impact on QOL [58]. The Berg Balance Scale and the Timed Up and Go test are widely used to assess balance dysfunction in SCA [59]. However, despite their widespread use, these assessments have not been examined for reliability and validity in SCA. Kondo et al. examined the test reliability of the Balance Evaluation Systems Test (BESTest) [60]. The BESTest is a multitask balance assessment tool that was developed to identify specific postural control problems (i.e., biomechanical constraints, stability limits, anticipatory postural adjustments, postural responses, sensory orientation, dynamic balance during gait, and cognitive effects) [61]. The MDC for an individual score difference from the baseline to the 4-week follow-up in BESTest was <8.7 (n = 20; SCA3, n = 4; SCA6, n = 9; SCA31, n = 7; mean age, 63.7 ± 10.1 years; age at onset, 53.9 ± 10.5 years; baseline SARA, 9.9 ± 3.5) [61]. Many types of balance function measures have been reported. However, BESTest is the only scale that is considered to have absolute reliability in SCA.
Gait speed is often used as an outcome of intervention studies in SCA [62, 63]. However, some changes in the gait pattern (e.g., base of support and gait speed) most likely reflect cerebellar-unspecific, compensatory strategies, and a high spatiotemporal gait variability appears to be a distinctive feature of ataxic gait [58, 64]. The Gait Variability Index (GVI) is a measure of gait variability that has been examined regarding reliability and validity [65]. The MDC for an individual score difference from day 1 to day 2 in GVI was <8.6 (Friedreich’s ataxia, n = 81; baseline ICARS, 70.4 ± 7.9) [65]. It has been suggested that gait instability in SCA are characterized by a stronger effect of balance-related impairments of cerebellar control during slow walking and a stronger effect of impaired intra-limb coordination during fast walking [58]. Therefore, in clinical practice, it is necessary to evaluate not only the optimal gait speed, but also slow walking and fast walking, to extract the characteristics of gait instability.
The cerebellum has the ability to compensate for tissue damage and loss of function. This is called the cerebellar reserve [6]. Mitoma et al. suggested that this is important for motor rehabilitation at a time when the cerebellar reserve is functioning [6]. Motor rehabilitation in the early stages may maintain and improve the cerebellar reserve [66, 67]. Therefore, it is important to assess this parameter.
Cerebellar ataxia is the main symptom of SCA. Ataxia symptoms may represent a compensation for predictive control using feedback control [6]. Predictive control requires a mechanism called internal model [38]. The internal model is constantly updated by motor learning [68]. In turn, motor learning is one of the most important functions of the cerebellum. Thus, a measure of motor learning ability may be useful as an assessment of the cerebellar reserve.
Prism adaptation (PA) is widely used as an assessment of motor learning ability in patients with SCA [40, 69]. The basic procedure of PA is shown in Figure 1. First, at the “baseline,” the task is performed without a prism lens. Subsequently, the prism lens is introduced and the task is performed. In the initial phase, the lens is set off to either the left or right side of the target, but the error is corrected as the number of repetitions increases. This period is called the “initial error correction phase.” Thereafter, a spatial realignment phase is performed under the prism lens. The purpose of this phase is to gather visuospatial information including the errors. Next, the prism is removed and an “after-effect phase” is performed. If the spatial information is being re-learned, errors are generated in the opposite direction to the initial error correction phase. Recently, Hashimoto et al. developed the Adaptability Index (AI), which is a composite index computed from several parameters measured PA (Figure 2). The clinical efficacy of the AI in discriminating patients with SCA from healthy individuals has been demonstrated [70]. Furthermore, Bando et al. found that a reduced AI was correlated with gray matter atrophy in the cerebellum in the SCA group [49]. In particular, the right lobule VI and the left Crus I showed the most robust correlation. These cerebellar regions are consistent with the correlates of PA detected in previous human and nonhuman primate studies [71, 72]. AI is considered as a motor learning index that reflects the cerebellar reserve (in this case, the degree of cerebellar atrophy).
Overview of prism adaptation. The ordinate shows the finger-touch error represented from the target to the touch point. Three phases are generally used: (1) absence of a prism lens (prism off), (2) presence of a prism lens (prism on), and (3) absence of a prism lens (prism off).
Calculation of the adaptability index (AI). The AI is calculated as follows: AI = a × b × c, where “a” is the adaptation index defined as the probability of correct touches in the last 10 trials of the spatial realignment phase 1, “b” is the retention index defined as the probability of incorrect touches in the initial 5 trials of the after-effect phase, and “c” is the extinction index designated as the probability of correct touches in the last 10 trials of the spatial realignment phase 2.
PA can be implemented using a simple system. In addition, it takes only 20 min to complete a PA. Reaching tasks can be performed even in the period during which the patient is unable to walk, and the fact that the PA can be assessed continuously over a long period is an advantage. However, only cross-sectional studies have been conducted in previous reports [40, 49, 69, 70, 73, 74]. Future studies need to be designed to examine long-term changes and intervention effects.
The targets of rehabilitation in cerebellar ataxia are mainly disability in ADL, gait, and motor dysfunction. Therefore, GAS, FIM, 10-m walking test, TCA, SARA, ICARS, and BESTest are used as important outcomes in rehabilitation. The most important strategies of rehabilitation for cerebellar ataxia including SCA consists in balance training (see Section 4.3), gait training (see Section 4.2), and muscle strengthening training using a high-intensity program (see Section 4.1). Further, optional possible interventions are using assistive technology (see Section 4.4) and neuromodulation technique (see Section 4.5).
Rehabilitation methods for cerebellar ataxia have been reported [75]. The most important strategy is the increase in the intensity of physical training, such as balancing, gait, and strength [76]. Several systematic reviews [77, 78, 79] and narrative reviews [3, 75, 80, 81] introduced and recommended intensive physical therapy for cerebellar ataxia in patients with SCA. Miyai et al. [62] reported that physical and occupational therapies of 2 h × 5 days +1 h × 2 days per week for 4 weeks were applied to inpatients and improved the SARA score and gait speed; however, the effect was carried over only up to 12 weeks after the training, and had disappeared at 24 weeks [62]. Conversely, Ilg et al. reported that intensive coordinative physiotherapy delivered over 4 weeks improved motor performance in degenerative cerebellar ataxia in a study with an intraindividual control design [63].
An outpatient rehabilitation program for 6 weeks applied to 19 participants with Friedreich’s ataxia improved the motor domain item in the FIM score and Friedreich’s Ataxia Impact Scale, but the posthome program could not maintain the effect [82]. Therefore, this finding indicates that continuous outpatient rehabilitation programs are important for maintaining the ADL in patients with Friedreich’s ataxia. Additional large-scale studies are needed to investigate the long-term effect of outpatient rehabilitation programs and identify the characteristics of patients who respond to treatment. Therefore, the development of optimal individual programs is important to obtain the effect of training, regardless of the inpatient, outpatient, or home-self-training setting [83]. The semi-order program of the Research Committee for Ataxia Disease (Research team under the jurisdiction of the Ministry of Health, Labour and Welfare in Japan,
Subsequently, the continuity of the intensive training is an important factor, because degradation in physical function was reported. Therefore, approaches aimed at upkeeping these programs in a way that suits the patients are needed. For example, exergames contribute to the practice of exercise at home. In the future, tele-rehabilitation systems [84] should be tested for the improvement (or maintenance) of the function and continuity of exercise.
Gait training has been reported to improve spatiotemporal gait parameters (cadence, step length/width, gait speed, etc.) [85, 86, 87], complex gait (Timed Up and Go test, Dynamic Gait Index) [85], independence (FAC) [86], ataxia (SARA) [88], and adaptive locomotor adjustments (ALA) [88]. Patients with SCA exhibit problems other than the gait disturbance itself, i.e., stiffening of the body in an attempt to avoid the occurrence of gait disturbances. Therefore, it is important to focus on gait disturbances and increasing the number of walking patterns when considering gait training in a person with SCA.
Disturbances of gait are the core features of SCA [89, 90, 91, 92], thus leading to a risk of falling down [93]. Patients with cerebellar ataxia walk with a reduced walking speed and cadence, as well as reduced step length, stride length, and swing phase; increased walking base width, stride time, step time, stance phase, and double limb support phase; and increased variability of step length, stride length, and stride time [94]. These items are affected by both balance-related impairments and deficits related to limb control and intra-limb coordination [95]. We believe that balance training and coordination training are key to the improvement of gait disturbances. Regarding the details of balance training, please refer to the Section 4.3.
In addition, stiffening of the body leads to a decrease in the number of walking patterns; as a result, ALA deteriorates [96, 97]. ALA implies that obstacle avoidance is achieved by modifying basic walking patterns in response to obstacle properties, e.g., a sloping road, stepping over an obstacle, or dynamically changing the spaces created by pedestrians in a hallway. In persons with SCA, feelings of anxiety as a result of the frequent experience of falls, as well as deficits related to limb control by ataxia, could negatively affect their ALA because of increased muscular co-contractions and reduced joint movements [98]. We will describe the approaches to improve ALA in the next paragraph.
The proposals for gait training are as follows: gait training without or with a treadmill. First, in gait training without a treadmill, we refer the reader to Section VI of the BESTest as gait adaptability training [61]. Section VI of the BESTest consists of a 7-item scale: (1) Gait Natural, (2) Change Speed, (3) Head Turns, (4) Pivot Turn, (5) Obstacles, (6) “Get Up & Go” Test, and (7) Cognitive Task “Get Up & Go” Test, aimed at evaluating the stability of the gait. These elements are important to improve ALA. As an example of gait training, persons with SCA are asked to walk while making an effort to change their walking speed according to therapist’s instructions to engage is “fast (or slow)” walking as fast (or slow) as possible. If patients need assistance when walking, you might want to change the walking speed with the support of a therapist.
Second, gait training using a treadmill has advantages in that patients can practice a relatively large amount of gait training over a short period and the therapists can control the speed and incline easily. Gait training using a treadmill has been reported as a potentially promising tool for improving ALA in a person with SCA [88], as well as gait disturbances in a person with Parkinson’s disease [99, 100]. It has been reported that variability was increased during slow and fast walking, but was normal during the preferred walking speed in a person with cerebellar ataxia [101]. Another study reported that, in ataxia, walking at the preferred speed minimizes the gait abnormalities, and the analysis of gait at a wide range of speeds is recommended [94]. For this reason, when using a treadmill in gait training, we suggest that walking be practiced at the speed at which the gait disturbance increases (i.e., slow or fast walking speed) for specific patients. When the fear of falling increases, the use of a harness is recommended, to provide a safe environment for gait without the fear of falling.
It is important to improve the balance ability and ALA during gait training in a person with SCA. Gait training is a relatively easy method; however, it is left to the therapist’s discretion and experience. By changing the task itself or adjusting the difficulty level of the task, gait training may be able to overcome the limited walking patterns of these patients.
All patients with SCA will develop balance difficulties during the course of the disease. Balance is essential for mobility, and is very important for QOL. Although there is no effective pharmacological treatment for decreasing the ataxia or slowing disease progression, physical therapy plays an important role in controlling ataxia and improving or maintaining function through training [76]. In general, the physical therapy programs for degenerative cerebellar ataxia are based on intensive static and dynamic balance and coordination training. There is some evidence that such therapeutic training programs alleviate the ataxic symptoms and improve functional activities in a person with cerebellar ataxia [63, 78, 102, 103]. In these patients, the disease progressively damages the cerebellar structure that plays a crucial role in motor learning [104]; however, these studies have indicated that it is necessary for highly repetitive balance training for balance impairment in SCA. For this reason, highly repetitive balance training in patients with SCA should be the focus of future studies.
More concretely, balance training exercises in early stages of the disease, i.e., ambulation, include the following categories: (1) static balance training, (2) dynamic balance training, and (3) coordination training (Figure 3). In addition, combining a dual task with balance training improves balance and reduces the number of falls in individuals with cerebellar ataxia [105].
National Center of Neurology and Psychiatry (NCNP) balance training program. This balance training program was devised through consultations with patients with SCA, medical doctors, and therapists at the NCNP in Japan. In the advanced stage of SCA, it is recommended to perform the programs indicated by an asterisk.
Moreover, it is important to provide support for these approaches and make them a habit of exercising. For instance, if the patients with SCA have no habit of exercising, they should start with a small number of exercises (i.e., the minimum necessary) to get used to exercising, followed by the gradual increase in the number of exercises. If the patients with SCA have a habit of exercising, the therapist should teach them to adjust the exercise load (e.g., exercise more slowly and/or provide a small base of support). It is also important to adopt balance training that can be enjoyed, e.g., video games [106] and Tai Chi [107], as a means of continuing balance training.
In advanced stages of the disease (i.e., no ambulation), it is necessary to perform balance training under safe conditions (e.g., prone, supine, crawl, and sitting positions), to prevent the decrease in physical activity. Even in advanced stages, it has been reported that a person with degenerative ataxia may benefit from balance training [108]. In addition, it is necessary to focus on ADL and living infrastructure at this stage. If a patient with SCA requires assistance during transfer, engaging in repetitive transfer training with assistance and/or modification of the living infrastructure (e.g., installation of handrails) are necessary.
Focusing on highly repetitive balance training in patients with SCA might preserve the balance function. There is no scientific basis for the number of balance training exercises that are necessary to achieve this goal; however, we would like to recommend engaging in 30 repetitions at least per balance training session. Furthermore, the balance training must be designed to provide a significant challenge to the person’s balance. If a person with SCA wants to preserve the balance function, they have to continue engaging in repetitive balance training, “use it or lose it.” However, few studies have reported the effect of gait and balance training in persons with SCA. Therefore, further studies are needed to clarify the clinical effectiveness of gait and/or balance training.
In recent years, various technologies have been used in the assessment of and treatment based on rehabilitation, as well as to support daily life in patients with SCD. Curara, a wearable robotic system, assists both hip and knee movements and supports the wearer’s rhythmic gait using a synchronization control based on a central pattern generator [109]. Gait support using the curara system has been reported to improve gait smoothness in patients with SCD [110]. In addition to these findings, a recent study addressed the effects of robotic gait training combined with noninvasive brain stimulation. This report showed that robot gait training using Lokomat-Pro in combination with cerebellar tDCS improved the functional scores on SARA, especially the scores on the subitems of gait, stance, sitting, and heel-shin slide compared with robot gait training alone [111]. Thus, hybrid training using robots and noninvasive brain stimulation will be applied to the rehabilitation treatment of patients with SCD in the future.
Accordingly, the use of walking aids is a complementary method for balance and gait impairment. In general, walking aids such as canes and walkers improve postural stability, but their improper use increases the risk of falling [112]. Because the manipulation of a cane requires coordinated upper limb movements [113], patients with SCD who have upper limb ataxia are likely to experience difficulty in using a cane. Conversely, because a walker does not require much coordinated movement of the upper limbs, technology-based walkers are being developed. Recently, a smart walker for mobility assistance and monitoring system aid, ASBGo, was developed and reported to improve gait parameters and postural stability in patients with SCA [114, 115]. In addition to technology, some studies on walking assistance using dogs and handkerchiefs have also been reported. Walking with a rehabilitation dog that has been specifically trained for goal-directed interventions or with an assistance dog that helps people with physical disability and mobility impairments has been reported to improve balance while walking in patients with SCD [116]. Furthermore, the handkerchief-guided gait, in which the patient with SCD walks along with the caregiver while maintaining light tension on a handkerchief by pulling lightly, has been shown to decrease body swaying and increase stride length and gait velocity during walking [117].
Moreover, technology is also being used as a tool to assess ataxia in patients with SCD living at home. Most of them represent attempts to evaluate SARA, which is a typical measure of ataxia, at home. In recent years, a technology aimed at objectively evaluating the speech, upper and lower limb, balance, and gait functions using wearable inertial sensors and a Kinect camera was developed, which makes it possible to discriminate between normal and abnormal functions and to detect ataxia at an early stage [118]. In addition, SaraHome has been developed to allow the remote evaluation of SARA items using Kinect and Leap Motion Controller [119]. Moreover, a spoon equipped with an inertial sensor, called Ataxia Instrumented Measurement-Spoon, has been developed, which allows the evaluation of upper limb function in ataxia while eating with a spoon [120, 121, 122]. Because SCD is an intractable neurological disease, it is difficult for many patients to leave their houses. Therefore, the contribution of technology to home-based rehabilitation is expected to increase in the future if a low-cost and easy method of assessing ataxia at home is established using the technologies and products of daily living described above.
Regarding the support of ADL, BMI studies have been reported. Patients with severe SCA often have difficulty in communicating because of language impairment. The application of BMI using event-related potentials and frequency bands of EEG is being investigated as a solution to this problem. The operational accuracy of BMI using P300 for event-related potentials was 82.9% in patients with SCA, which was similar to the accuracy observed in healthy subjects (83.2%) [123]. There are also reports of BMI manipulation in patients with SCD using the EEG frequency band associated with motor imagery [124]. BMI has a wide range of applications in diseases of the central nervous system, such as communication tools, transportation, and life support, and is expected to contribute to the QOL of patients with SCD.
Neuromodulation via noninvasive brain stimulation (NIBS) is a potential method for the treatment of cerebellar ataxia [19, 125]. A previous systematic review [126] reported the effectiveness of cerebellar neuromodulation using the TMS technique of transcranial direct current stimulation (tDCS). The SARA and ICARS scores in patients with SCA3, multiple system atrophy, and postlesion ataxia, as assessed using real cerebellar rTMS (1 Hz), were significantly lower than those detected in the sham stimulation group [126]. Furthermore, no harmful side effects were noted [126]. Cerebellar rTMS can modulate the plasticity of the vestibular reflex [16, 127]; therefore, cerebellar rTMS has potential for application in balance training to enhance vestibular contributions.
A single session of anodal cerebellar tDCS (2 mA, 20 min) significantly improved SARA, ICARS, 9-hole-peg test, and 8-m walking test scores [128]. Furthermore, combined anodal cerebellar tDCS and cathodal spinal DCS (5 days/week, 2 weeks) improved SARA score, ICARS score, 9-peg test, and 8-m walking time in patients with degenerative cerebellar ataxia [129]. There is insufficient evidence regarding whether simultaneous stimulation is more effective than single stimulation [130]; however, it is possible that this intervention method will produce improvements. Based on these findings, which were gleaned from small-sample studies, we suggest that a neuromodulation montage will improve the ataxia, balance, and gait ability. Therefore, we should perform further studies using a larger population.
Individualized physical rehabilitation programs for patients with SCA may improve/maintain their motor function, balance, gait ability, and ADL. In particular, the intensity and continuity of gait and balance training need to be considered to achieve effectiveness. Furthermore, several technologies, such as depth sensors, robotics, and NIBS, have contributed to the development of methods for the assessment and treatment of motor dysfunction in individuals with SCA. We should continue to study populations suffering from dysfunction caused by SCA.
This work was supported by Shijonawate Gakuen University and JSPS KAKENHI (Grant Number 20 K11298).
The authors declare no conflict of interest.
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