Formulation of the . . . Need for Fuzzy . . . Pre-Processing of Gait . . . Assessment of Fuzzy Gait Assessment . . . Explanation of the . . . Functional Impairment Conclusion and Future . . . Acknowledgment in Human Locomotion: Proof of the Main Result Fuzzy-Motivated Approach Home Page Title Page Murad Alaqtash 1 , Thompson Sarkodie-Gyan 1 , ◭◭ ◮◮ and Vladik Kreinovich 2 ◭ ◮ 1 Department of Electrical and Computer Engineering Page 1 of 100 2 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968, USA msalaqtash@miners.utep.edu, tsarkodi@utep.edu, Full Screen vladik@utep.edu Close Quit
Formulation of the . . . 1. Formulation of the Problem Need for Fuzzy . . . Pre-Processing of Gait . . . • Neurological disorders – e.g., the effects of a stroke – Fuzzy Gait Assessment . . . affect human locomotion (such as walking). Explanation of the . . . • In most cases, the effect of a neurological disorder can Conclusion and Future . . . be mitigated by applying an appropriate rehabilitation. Acknowledgment Proof of the Main Result • For the rehabilitation to be effective, it is necessary to Home Page be able: Title Page – to correctly diagnose the problem, ◭◭ ◮◮ – to assess its severity, and ◭ ◮ – to monitor the effect of rehabilitation. Page 2 of 100 • At present, this is mainly done subjectively, by experts who observe the patient. Go Back • This is OK for the diagnosis, but for rehabilitation, a Full Screen specialist can see a patient only so often. Close Quit
Formulation of the . . . 2. Formulation of the Problem (cont-d) Need for Fuzzy . . . Pre-Processing of Gait . . . • It is desirable to automatically gauge how well the pa- Fuzzy Gait Assessment . . . tient progresses. Explanation of the . . . • To measure the gait x ( t ), we can use: Conclusion and Future . . . Acknowledgment – inertial sensors that measure the absolute and rel- Proof of the Main Result ative location of different parts of the body, and Home Page – electromyograph (EMG) sensors that measure the Title Page electric muscle activity during the motion. ◭◭ ◮◮ • By comparing x ( t ) with gait of healthy people and with previous patient’s gait, we can: ◭ ◮ – gauge how severe is the gait disorder, and Page 3 of 100 – gauge whether the rehabilitation is helping. Go Back • Problem: signals x ( t ) corresponding to patients and to Full Screen healthy people are similar. Close Quit
Formulation of the . . . 3. Need for Fuzzy Techniques Need for Fuzzy . . . Pre-Processing of Gait . . . • Specialists can distinguish between signals corr. to pa- Fuzzy Gait Assessment . . . tients and healthy people. Explanation of the . . . • We want to automate this specialists’ skill. Conclusion and Future . . . Acknowledgment • Specialists describe their decisions by using imprecise Proof of the Main Result (“ fuzzy ”) words from natural language. Home Page • Formalizing such words is one of the main tasks for Title Page which fuzzy techniques have been invented. ◭◭ ◮◮ • Fuzzy techniques have been used to design efficient ◭ ◮ semi-heuristic assessment systems. Page 4 of 100 • The objective of this paper is to provide a theoretical justification for the existing fuzzy systems. Go Back • The existence of such a justification makes the results Full Screen of the system more reliable . Close Quit
Formulation of the . . . 4. Pre-Processing of Gait Signal Need for Fuzzy . . . Pre-Processing of Gait . . . • Motions differ by speed: the same person can walk Fuzzy Gait Assessment . . . slower or faster. Explanation of the . . . • To reduce the effect of different speeds, we re-scale time Conclusion and Future . . . x ′ ( T ) = x ( t 0 + T · T 0 ), where Acknowledgment – t 0 is the beginning of the gait cycle, Proof of the Main Result Home Page – T 0 is the gain cycle, and – the new variable T describe the position of the sen- Title Page sor reading on the gait cycle. ◭◭ ◮◮ • For example: ◭ ◮ – the value x ′ (0) describes the sensor’s reading at the Page 5 of 100 beginning of the gait cycle, Go Back – the value x ′ (0 . 5) describes the sensor’s reading in the middle of the gait cycle, Full Screen – the value x ′ (0 . 25) describes the sensor’s reading at Close the quarter of the gait cycle. Quit
Formulation of the . . . 5. Pre-Processing of Gait Signal (cont-d) Need for Fuzzy . . . Pre-Processing of Gait . . . • Motions also differ by intensity. Fuzzy Gait Assessment . . . • To reduce the effect of different intensities, we re-scale Explanation of the . . . the signal x ( t ) so that: Conclusion and Future . . . – the smallest value on each cycle is 0, and Acknowledgment Proof of the Main Result – the largest value on each cycle is 1. Home Page • Such a scaling has the form X ( T ) = x ′ ( T ) − x , where: Title Page x − x – x is the smallest possible value of the signal x ′ ( T ) ◭◭ ◮◮ during the cycle, and ◭ ◮ – x is the largest possible value during the cycle. Page 6 of 100 • After re-scaling, all we have to do is compare: Go Back – the (re-scaled) observed signal X ( T ) with Full Screen – a similarly re-scaled signal X 0 ( T ) corresponding to Close the average of normal behaviors. Quit
Formulation of the . . . 6. Fuzzy Gait Assessment System Need for Fuzzy . . . Pre-Processing of Gait . . . • An expert describes the gait by specifying how the mo- Fuzzy Gait Assessment . . . tion looked like at different ( p ) parts of the gait cycle. Explanation of the . . . • For each part, we form a triangular membership func- Conclusion and Future . . . tion µ ( x ) that best describes the corr. values X ( T ). Acknowledgment • We want the support ( a, b ) of µ ( x ) to be narrow and Proof of the Main Result Home Page to contain many observed values x i . Title Page • Pedrycz’s approach: find parameters a, b, m for which n ◭◭ ◮◮ � µ ( x i ) i =1 → max . ◭ ◮ b − a Page 7 of 100 • The gait on each part is described by three parameters ( a, b, m ), so overall we need N = v 3 p parameters. Go Back • A patient’s gait is described by g 1 , . . . , g N ∈ [0 , 1]. Full Screen Close • The normal gait is described by n 1 , . . . , n N ∈ [0 , 1]. Quit
Formulation of the . . . 7. Fuzzy Gait Assessment System (cont-d) and Need for Fuzzy . . . Our Result Pre-Processing of Gait . . . Fuzzy Gait Assessment . . . • A sequence g 1 , . . . , g N can be viewed as a fuzzy set g . Explanation of the . . . • A sequence n 1 , . . . , n N can be viewed as a fuzzy set n . Conclusion and Future . . . • So, we can define degree of similarity between patient’s Acknowledgment gait and normal gait as Proof of the Main Result Home Page N � min( g i , n i ) s = | g ∩ n | Title Page i =1 | g ∪ n | = . N ◭◭ ◮◮ � min( g i , n i ) i =1 ◭ ◮ • Our result: when the number of parts p is large enough, Page 8 of 100 we have Go Back s ≈ 1 − 1 � C · | x ( t ) − x 0 ( t ) | dt. Full Screen • Thus, the larger the integral, the more severe the dis- Close order. Quit
Formulation of the . . . 8. Explanation of the Reformulated Formula Need for Fuzzy . . . Pre-Processing of Gait . . . def � • Let’s explain why | ∆ x ( t ) | dt , where ∆ x ( t ) = x ( t ) − Fuzzy Gait Assessment . . . x 0 ( t ), is a good measure of the disorder’s severity. Explanation of the . . . • The effect is different for different behaviors. Conclusion and Future . . . Acknowledgment • It is reasonable to gauge the severity of a disorder by Proof of the Main Result the worst-case effect of this difference. Home Page • For each objective, the effectiveness E of this activity Title Page depends on the differences ∆ x ( t i ). ◭◭ ◮◮ • The differences ∆ x ( t i ) are small, so we can linearize the dependence: ∆ E = � c i · ∆ x ( t i ). ◭ ◮ • There is a bound M on possible values of | c i | . Page 9 of 100 • The largest value of � c i · ∆ x ( t i ) under the constraint Go Back | c i | ≤ M is equal to M · � | ∆ x ( t i ) | . Full Screen • Thus, the worst-case effect is indeed proportional to Close � | ∆ x ( t i ) | , i.e., to � | ∆ x ( t ) | dt . Quit
Formulation of the . . . 9. Conclusion and Future Work Need for Fuzzy . . . Pre-Processing of Gait . . . • Many traumas and illnesses result in motion disorders. Fuzzy Gait Assessment . . . • In many cases, the effects of these disorders can be Explanation of the . . . decreased by an appropriate rehabilitation. Conclusion and Future . . . • Different patients react differently to the current reha- Acknowledgment bilitation techniques. Proof of the Main Result Home Page • To select an appropriate technique, it is therefore ex- Title Page tremely important to be able to gauge: ◭◭ ◮◮ – how severe is the current disorder and – how much progress has been made in the process ◭ ◮ of rehabilitation. Page 10 of 100 • At present, this is mostly done subjectively, by a medi- Go Back cal doctor periodically observing the patient’s motion. Full Screen • When a certain therapy does not help, the doctor can Close change the rehabilitation procedure. Quit
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