IAES International Journal of Robotics and Automation (IJRA) Vol.
No.
March 2025, pp.
ISSN: 2722-2586.
DOI: 10.
11591/ijra.
Optimizing robot anomaly detection through stochastic differential approximation and Brownian motion Branesh M.
Pillai1.
Arush Mishra2.
Rijo Jacob Thomas3.
Jackrit Suthakorn1 Center for Biomedical and Robotics Technology (BART LAB).
Faculty of Engineering.
Mahidol University.
Nakhon Pathom.
Thailand Bangkok International Preparatory and Secondary School.
Vadhana.
Bangkok.
Thailand Department of Mechanical Engineering.
TKM College of Engineering.
Kollam.
Kerala.
India
Article Info
ABSTRACT
Article history:
This paper presents an adaptive approximation method for detecting anomalous patterns in extensive data streams gathered by mobile robots operating in rough terrain.
Detecting anomalies in such dynamic environments poses a significant challenge, as it requires continuous monitoring and adjustment of robot movement, which can be resource To address this, a cost-effective solution is proposed that incorporates a threshold mechanism to track transitions between different regions of the data stream.
The approach utilizes stochastic differential approximation (SDA) and optimistic optimization of Brownian motion to determine optimal parameter values and thresholds, ensuring efficient anomaly detection.
This method focuses on minimizing the movement cost of the robots while maintaining accuracy in anomaly identification.
applying this technique, robots can dynamically adjust their movements in response to changes in the data stream, reducing operational expenses.
Moreover, the temporal performance of the data stream is prioritized, a key factor often overlooked by conventional search engines.
This paper demonstrates how the approach enhances the precision of anomaly detection in resource-constrained environments, making it particularly beneficial for real-time applications in rugged terrains.
Received Oct 25, 2024 Revised Jan 21, 2025 Accepted Jan 26, 2025 Keywords:
Brownian motion Data stream Differential approximation Mobile robot Optimistic optimization This is an open access article under the CC BY-SA license.
Corresponding Author:
Jackrit Suthakorn.
Center for Biomedical and Robotics Technology (BART LAB).
Department of Biomedical Engineering.
Faculty of Engineering.
Mahidol University 999.
Phuttamonthon Sai 4.
Salaya.
Nakorn Pathom, 73170.
Thailand Email: jackrit.
sut@mahidol.
INTRODUCTION
The primary signal and sensor processing issue is anomaly identification during the data stream .
This research's primary focus is identifying anomalies within data streams, particularly emphasizing the element of time.
The objective is to enhance the efficiency of representing a specific subset of temporal data streams through a sequential design of experiments, facilitating accurate and rapid anomaly detection .
, .
A significant challenge in time-based anomaly identification, especially in the context of mobile robots used in rough terrain rescue missions, is the associated cost of transitioning the data stream from one geographical region to another .
This cost primarily arises from the movement of robots.
Various approximation algorithms have been developed to address this issue, with the Brownian motion algorithm being a prominent choice due to its experience in managing time and cost-effective model construction .
However, the standard Brownian motion algorithm faces challenges, including handling vast datasets, memory limitations, and the inability to adapt to a behavior-based system with infinite variance .
, .
To address the challenges Journal homepage: http://ijra.
ISSN: 2722-2586
of real-time anomaly detection in dynamic, resource-constrained environments like rugged terrains, the authors propose a novel approach leveraging stochastic differential approximation (SDA) and optimistic optimization of Brownian motion .
This method approximates the shortest path characterized by Brownian motion within a defined time interval, achieving greater accuracy by minimizing path length.
The study aims to provide a cost-effective, adaptive solution that balances energy efficiency, operational cost, and real-time response while optimizing robot movement for precise anomaly detection.
Brownian motion is defined as the variation in the path length measurement, characterized by its continuous and natural nature and adaptability to showcase differentiation in erratic motions .
, .
presenting the minimum Brownian motion using stochastic differential approximation and optimistic optimization, the path is depicted in time intervals, each further divided into sub-intervals .
The critical objective is to determine how each sub-interval should be represented within the entire time interval.
This involves the selection of a mathematical model, in this case, stochastic differential equations (SDE), and an optimization technique, specifically an optimistic optimization algorithm .
In summary, this paper introduces an innovative approach to tackle the challenges associated with time-based anomaly identification, particularly in mobile robots, by presenting the stochastic differential approximation and optimistic optimization of Brownian motion as a more effective alternative to the traditional Brownian motion algorithm .
The proposed approach provides several key contributions to enhance cost efficiency and improve anomaly detection accuracy in robotic systems.
One of its primary advantages is the significant reduction in expenses related to the movement of mobile robots, particularly during the process of detecting anomalies in data streams .
By employing a mathematical model that approximates the minimal Brownian motion path, the method effectively minimizes unnecessary robot movements, thereby conserving both energy and resources.
Brownian motion, often associated with random movement patterns, is streamlined here to limit robot movement, focusing on pathways with minimal deviation.
This targeted movement strategy not only saves operational costs but also extends the operational lifespan of robotic components by reducing wear and tear on the machinery .
In addition to cost savings, the approach employs an advanced mathematical framework based on stochastic differential equations (SDE.
to refine the precision of the minimal path By leveraging SDEs, the system can dynamically adjust to unpredictable factors that impact robot movement and data collection .
This added layer of mathematical rigor enhances the accuracy of the Brownian motion model, ensuring that the robots follow a path close to the minimal distance required to detect anomalies effectively.
The use of SDEs ensures that the approach adapts well to fluctuating conditions, which are common in real-world applications where robotic systems encounter varied terrains and obstacles .
Ae.
This results in a robust system where the accuracy of anomaly detection remains high, even under challenging conditions.
To improve the approach, an optimistic optimization technique is incorporated to effectively tackle continuous optimization challenges associated with Brownian motion .
This framework plays a vital role in identifying the optimal paths for robots, striking a balance between minimizing path length and maximizing the probability of anomaly detection.
By focusing on optimizing the robot's path, the method significantly increases the chances of accurately identifying anomalies without necessitating extensive This optimization process does not solely focus on identifying anomalies but also emphasizes efficient resource allocation, minimizing computational power, and ultimately reducing the time and cost involved in anomaly detection.
An additional aspect of this methodology is its emphasis on accurately pinpointing the most likely minimum path of Brownian motion within a specific time frame, a feature that significantly enhances the effectiveness of anomaly detection .
The proposed approach effectively narrows down the set of probable paths, focusing on accurately identifying the minimum path that exhibits Brownian motion characteristics.
This ensures that anomalies in extensive data streams are detected promptly and with minimal resource consumption, reducing unnecessary robotic movements.
The time-bound nature of this identification process further enhances the system's responsiveness, enabling real-time detection of By maintaining high precision in anomaly detection while operating under resource constraints, the approach becomes highly suitable for applications where both accuracy and operational efficiency are critical, such as autonomous navigation, disaster response, and environmental monitoring in challenging METHOD Consider the data stream yce = { 1,2,3 A A yc.
where f creates the series of random variables are yce yce yce yce ya1 , ya2 , ya3 A A yaycu occurs with the data sample space A.
Each data stream is dependent on either one of the hypotheses such as ya0 ycuyc ya1 .
The observations made by two different probability distribution functions on IAES Int J Rob & Autom.
Vol.
No.
March 2025: 19-30
IAES Int J Rob & Autom
ISSN: 2722-2586
yce data sample space A of data stream are ya1 ycaycuycc ya2 .
Taken ya0 for ya1 where ya0 Ie yaycu is true for ya0 .
Similarly, yce ya1 for ya2 where ya1 Ie yaycu is true for ya1 where n denotes the set of integers.
ya0 is positive when the specific data stream defines the movement of data stream is normal and ya1 is positive when the movement of data stream is target.
Suppose for the taken data stream, ya1 is true and its probability is yuU and ya0 is true with its probability .
Oe yuU) where yuU OO .
When these criteria are not satisfied for yuU, it shows that target data stream happens to be too rare case.
During the expectation of hypothesis ycEycn .
The general structure of Brownian motion is shown in Figure 1.
Figure 1.
Structure of Brownian motion .
To reduce the movement cost of the mobile robot during the data stream from one point to another point, the problem observed and formulated during the data movements from stream ycO1 to the next stream ycOycu 1, so it exists with cost yuIycO within distribution ya OA ycO.
)yuIycO Ou 0 and C.
uIycO ] = yuIE is finite .
Assume that movement cost yuIycO and the remarks ycCycNycO are commonly independent OA W,ycO A &ycN.
The movement cost is expensive due to the movement of the robot while the data stream is monitored during the new data stream .
If the observations are not taken, then it is said to be zero time and if the observations are monitored and the time are recorded then it shows the observation for movement cost.
This can also be related to energy.
Problem 1: Consider the Brownian motion .
aAyc.
)ycOu0 with a Hurst value Hu > 1/2.
Stochastic linear equations of this type are investigated in the format in .
= ycI.
, y.
yccyc ycN.
, y.
yccyaAyc.
, .
c0 ) = ya0 .
Whereas yc0 OO .
, ycN), ya0 is an Unpredictable vector in Eoycu and the subsequent criteria hold true with likelihood 1 for the randomly generated functions S and T in .
, ycI yun ya(Eoycu O .
, ycN).
Eoycu ), ycNyun ya1 (Eoycu O .
, ycN).
Eoycu ) .
yuiycN(Oo,y.
yuiycN(Oo,y.
for every yc OO .
, ycN) the functions ycI(Oo, y.
, for all i in .
, 2.
, .
are localized Lipschitz.
yuiycu ycn yuiyc Consider the supplementary partial differential .
along the route on Eoycu O Eo O .
, ycN), yuiya yuiyc .
cu, yc, y.
= ycN.
cu, yc, y.
, y.
cU0 , ycU0 , yc0 ) = ya0 Wherein ycU0 is an array of random elements in some set Eoycu and ycU0 is an independent variable in some set Eo.
It derives from differentiated equation theory that in the neighbourhood N of .
cU0 , ycU0 , yc0 ) a local solution ya OO ya1 (Eoycu O Eo O .
, ycN).
Eoycu )with Lipschitz limited variations in the parameter x occurs with likelihood 1 in .
yuiycu yc cu, yc, y.
) O 0
For .
c, yc, y.
OO ycA yui2 ya yuiyc 2 .
c, yc, y.
= Ocycuyc=1 yuiycN yuiyc yc .
cu, yc, y.
, y.
ycNyc .
cu, yc, y.
, y.
Optimizing robot anomaly detection through stochastic differential approximation A (Branesh M.
Pilla.
A ISSN: 2722-2586 From .
, on .
T], additionally consider the path wise differential equation .
n matrix for.
as in .
= ( yuiya , yaAyc.
, y.
Oe1 , yaAyc.
, y.
, y.
, yaAyc.
, y.
yccyc, ya.
c0 ) = ya0 From .
, a maximum range, has a distinct local solution as .
c01 , yc02 ) OI .
, y.
with yc0 yun.
c01 , yc02 ).
Here, a use of stochastic application of the formula to the randomized function ycE.
c, y.
= ya.
, yc, y.
and the Brownian motion Br in .
, ya.
, yaAyc.
, y.
Oe ya.
c0 ), yaAyc.
c0 ), yc0 ) ycu = Oc O( yc=0 yc0 yc yc yc0 yc0 yuiya yuiya yuiya .
, yaAyc.
, y.
) yccycUyc .
O ( .
, yaAyc.
, y.
) yccyaAyc .
O ( .
, yaAyc.
, y.
) yccyc yuiycu yuiyc yuiycu yc yc yc = Oyc ycI.
, yaAyc.
, y.
, y.
yccyc Oyc ycI.
, yaAyc.
, y.
, y.
yccyaAyc.
Hence, ya.
Oi ya.
, yaAyc.
, y.
yc yc ya.
= ya0 Oyc ycI.
, y.
yccyc Oyc ycN.
, y.
yccyaAyc.
For each ycs OO E,a treat the path wise differential equation of .
epresented as a matri.
as an approximation yaAycycs .
of the original process yaAyc.
and the .
, yccyayc .
= ( Oe1 ycI.
, yaAycycs .
, y.
, y.
, .
, yuiya ya yaAycycs y.
[ ] yccyc yuiycu yc , yaAycycs .
, yc yuiyc yc c0 ) = ya0 A distinct localized equilibrium yaycA on the maximum period of presence .
c1, yc.
OI .
c01 , ycycu2 ).
Using the stochastic formula, ycE.
c, y.
= ya.
, yc, y.
for the random function ycE.
c, y.
and the procedure BN as in .
, ya.
, yaAycycs .
, y.
Oe ya.
c0 ), yaAycycs .
c0 ), yc0 ) ycu = Oc O( yc=0 yc0 , yaAycycs .
, y.
) yccyaycs yc .
O ( .
, yaAycycs .
, y.
) yccyaAycycs .
yuiycu yc ycs , yaAycycs .
, y.
) yccyc yuiyc yc0 yc yc = Oyc ycI.
, yaAycycs .
, y.
, y.
yccyc Oyc ycI.
, yaAycycs .
, y.
, y.
yccyaAycycs .
Hence, yaycs .
Oi ya.
, yaAycycs .
, y.
proves in .
, yc yc yaycs .
= ya0 Oyc ycI.
, y.
yccyc Oyc ycN.
, y.
yccyaAycycs .
The subsequent path wise condition arises from the proposed theorem lim supAnyaycs .
Oe ya.
An = 0 ycsIeO since L follows that lim ycycycyAnyaycs .
Oe ya.
An = 0 ycsIeO IAES Int J Rob & Autom.
Vol.
No.
March 2025: 19-30
IAES Int J Rob & Autom
ISSN: 2722-2586
This study effectively demonstrates the connection between the outcomes and approximations obtained from the provided stochastic differential equation and their association with the path wise differential equation, as referenced in sources .
Ae.
More specifically, it showcases how the solution C for the path wise differential .
can be inferred from the response A corresponding to the stochastic equation .
It's crucial to emphasize that this statement is made under the assumption that equation 6 possesses a unique global solution, thereby guaranteeing the uniqueness of the local solution for equation 1.
This paper establishes a clear link between the solutions and estimations of stochastic and path wise differential equations .
, .
Additionally, it underscores the significance of unique conditions, particularly in the context of .
and its implications for equation 1.
Let L represent the solution to .
It is well established that the solution is invertible in the neighborhood N of .
cU0 , ycU0 , yc0 ).
cu, yc, y.
OO ycA, ya(.
cu, yc, y.
, yc, y.
has the opposite.
K stands for the mapping that provides ya.
cu, yc, y.
, yc, y.
= ycu and ya.
cu, yc, y.
, yc, y.
= yc In area ycA around us, the matrix equivalence holds in .
c, yc, y.
= ( yuiyc c, yc, y.
, yc, y.
) Oe1 By using .
, the obtain .
c, yc, y.
= Oe Ocycuycn=0 yuiyc yuiya yuiyc .
c, yc, y.
= Oe Ocycuycn=0 yuiya yuiyc ycn yuiya yuiyc ycn .
c, yc, y.
ycN ycn .
c, y.
c, yc, y.
yuiyaycn yuiyc .
c, yc, y.
, yc, y.
By plugging the parameters of the yiOycu 1 -valued procedure .
, yaAyc.
) into the stochastic equation for an expression K .
, y, .
, get .
, ya.
, yaAyc.
, y.
Oe ya.
c0 ), yaAyc.
c0 ), yc0 ) .
yc yuiya yc yuiya = Ocycuyc=0 Oyc ( yc .
, yaAyc.
, y.
) yccycsyc .
Oyc ( .
, yaAyc.
, y.
) yccyaAyc .
0 yuiyc 0 yuiyc yc yuiya Oyc .
, yaAyc.
, y.
) yccyc ycu yc = Oc O( yc=0 yc0 yuiya .
, yaAyc.
, y.
) ya.
, y.
yccyc yuiyc yc ycu yc OeOc O yc=0 yc0 yuiya yuiyayc .
, yaAyc.
, y.
, yaAyc.
, y.
, yaAyc.
, y.
yccyc yc yuiyc yuiyc But ya.
= ya.
, yaAyc.
, y.
, yaAyc.
, y.
holds, hence ya.
Oi ya.
, yaAyc.
, y.
satisfies the path-wise .
on an individual scale.
Similarly, it may establish that yaycs .
Oi ya.
, yaAycycs .
, y.
satisfies the path-wise .
on a particular scale.
Let Br approximate a proportionate Brownian motion B z.
Let ycI, ycN: yiOycu O .
, y.
be predetermined in .
, yc yc ya.
= ya0 Oyc ycI.
, y.
yccyc Oyc ycN.
, y.
yccyaAyc.
yc yc yaycs .
= ya0 Oyc ycI.
, y.
yccyc Oyc ycN.
, y.
yccyaAycycs .
where ycs OO E, accepts, with high probability, a single local solution on the same interval .
1, t.
here t0 is outside of Z but still part of the interva.
In addition, the following approximate result has been obtained in .
Optimizing robot anomaly detection through stochastic differential approximation A (Branesh M.
Pilla.
A ISSN: 2722-2586 yaycyycyycycuycu.
( lim supAnyaycs .
Oe ya.
An = .
= 1
ycsIeO Problem 2 Find the nearer location from the source point where the data stream W begins and define optimal path for the data stream W moves from one point to another point.
Proof .
The anticipated quantity of near-optimal locations for any b is constrained in .
Eo.
uC )] O 6yuC 2 2yca It fixes the value of yuCyca Ou yuCyuA ( yc.
The speed of growth of yeyca .
uCyca ), the quantity of yuCyca -near-optimal locations in .
, .
of the form yco/2yca , is measured by the near-optimality dimension in dimension one with the pseudodistanceyco.
co, yc.
= yuCyuA (.
cu Oe yc.
It shows that, in general, this quantity grows at a constant rate regarding b.
This implies that there exists a metric where the Brownian is Lipschitz with likelihood at least 1 Oe yuA and has a near-optimality aspect = 0 with ya = ye.
/yuA)).
The ye.
/yuA)) term, originating from the standard DOO error for deterministic function optimisation, and a different ye.
/yuA)) term, originating from the need to adjust our pseudo-distance Ee ycycu yuA such that the Brownian is Ee -Lipschitz with likelihood 1- yuA, together represent the finalised study difficulty bound.
Combining these two bounds yields an upper limit on sample complexity of ye.
/yuA)).
Proof .
A Brownian motion whose optimum O is reached for the initial time at the location described as is denoted by U and the Brownian meander ycN0 can be defined as in .
, ycN0 Ou ycCOeycO.
c1Oeyc.
ycN1 Ou ycCOeycO.
c 1 yc.
Oeyc.
) .
Ooyc1 Oo1Oeyc1 Then the theorem 1 declares that ycN O ycN0 O ycN1 and t1 changes regardless of both ycN0 ycaycuycc ycN1 .
For each positive integer, it establishes a maximum constraint on the predicted amount of yuC-nearoptimal positions b>0 and any values of yuC > 0.
Eo.
uC )] = Eo [Ocyca=0 1 .
cO ( yca ) > ycC Oe yuC}] = Oc2yca=0 Eo .
cO ( yc.
> ycC Oe yuC}] yca = Oc2yca=0 Eo .
cO ( yc.
> ycC Oe yuC O yca O yc.
O .
cO ( yc.
> ycC Oe yuC O yca > yc.
}] yca yca Ae yc1 = Oc Eo .
cN0 .
] Oc Eo .
cN O yca > yc.
] yca yc12yca Oe yc1 Ooyc1 Oo1 Oe yc1 yca=0 yca=0 Since t1 changes regardless of ycN0 ycaycuycc ycN1 , utilizing the above equation with C= .
cN0 , ycN1 ).
D=t1 and function in .
, yco yca yceycycu: .
ca0 , yca1 ), ycc Ie Oc2yca=0 (.
Oe yc12 yc.
< yuC Ooycc yca yca yca Aeyc1 1Oeycc O yca O yc.
] 1 .
yuC Oo1Oeycc yca O yca > yc.
it has sufficient evidence to assert that.
Eo.
uC )] = Eo.
a, y.
] O sup Eo.
a, y.
] yca yca Aeyc O sup {Oc Eo .
cN0 .
Oe yca ) < O yca O y.
]} sup {Oc Eo .
cN1 .
O yca > y.
]} yc2 1Oeyc Ooyc 2 Oo1 Oe yc 2 yca=0 yca=0 yca UOyc 2 UU = sup {Ocyca=0 Ey .
cN0 .
Ooyc < yca UOyc 2 UU = 2 sup {Ocyca=0 Ey .
cN0 .
}} sup {Oc2 yca yca=UOyc yuC Ooyc yca Aeyc 2yca Ey .
cN1 ( UU 1Oeyc yuC Oo1Oeyc = 2 sup.
u1 yu2 yu3 yu4 } where .
is expanded as IAES Int J Rob & Autom.
Vol.
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March 2025: 19-30
IAES Int J Rob & Autom
ISSN: 2722-2586
UO2yca yuC 2 UU yco yuC yco yuC yu1 = Ocyca=0 Ey .
cN0 .
Oe yc.
< ], yu2 = Ocyca=UO2ycayuC2 UO Ey .
cN0 .
Oe yca ) < ], yc2 yc2 Ooyc Ooyc yu3 = Oc UOyc2yca UUOeUO2yca yuC 2 UO yc2yca yca=UO UO Ey .
cN0 .
Oe yco yc2yca UOyc2yca UU yuC Ooyc ], yu4 = Ocyca=UOyc2yca UUOeUO2ycayuC2UO Ey .
cN0 .
Oe yco yc2yca yuC Ooyc Given that 1 is the highest possible likelihood, it may simply place a limit on yu1 ycaycuycc yu4 as 2yca yuC 2 , to obtain that yu1 yu4 O 2.
yca yuC 2 ).
By accumulating across the Brownian meander distribution parameters, it can now place upper and lower bounds on the rest of the possibilities occurring in the aforementioned formula.
Ey.
cN0 .
< yc.
= 2Oo2yuU O ycc2 ycc exp (Oe ycc yca ycc2 2.
Oe y.
2yc O yccycc yccyca O O ycc2 exp (Oe ) yccycc 2yc ycOoyc O 2yuU ycOo.
Oe y.
2yuU) 0 0 ycOo.
Oe y.
O 2yuU yca ycc exp (Oe O 2yca 3 3ycOo.
Oe y.
2yuU) O 2yca 3 3yc.
Oe y.
Oo.
Oe y.
2yuU) yca 3Oo2yuU Ooyc.
Oe y.
The limit is then applied to the sum of yu2 ycaycuycc yu3 .
yu2 yu3 = Ocyca=UO2yca yuC2UO Ey .
cN0 .
3Oo2yuU Ooyc ] Oc UOyc2yca UUOeUO2yca yuC 2 UO yca=UO UO Ey .
cN0 .
Ooyc Ooyc UOyc2yca UUOeUO2yca yuC 2 UO yca=UO yc2yca 3Oo2yuU Oo.
yc2yca yc2yca ) yuC Ooyc 6OoyuU yca=UO2 yuC UO yca=UO UO UOyc2yca UUOeUO yc2yca ) o Ooyc 6OoyuU Ooyc UOyc2yca UUOeUO2yca yuC 2 UO o yca Oo.
Oe yca Oo yca ) yc2
yca=UO2yca yuC 2 UO
Ooyc
yca=UO2yca yuC 2 UO
6OoyuU
Oo1 Oe
yc2yca ) O
2yca yc
Ooyc
yca ycUU
UO2
yca=UO2yca yuC 2 UO ( yc2yca yc2yca ) Swapping out the indexing as yci = OeyciA UOyc2yca UU, discover the following.
yu2 yu3 O yca yuC2 )3/2 yc2yca 6OoyuU UOyc2yca UUOeUO yca=UO2 yuC UO yca=UO2yca yuC 2 UO yca yuC2 )2 3OoyuU yci3/2 yca=UO2yca yuC 2 UO yci 2 2yca yc yca yuC 2 )2 3OoyuU OoUO2yca yuC 2 UO OoyuU 2yca yuC 2 O 2yca yuC 2 , where it was utilized for anything in the previous row yci0 > 0.
Optimizing robot anomaly detection through stochastic differential approximation A (Branesh M.
Pilla.
A Oy
ISSN: 2722-2586
yci=yci0 yci 2 3 Oc O 3 yccyc = 3 O 3 yccyc = yci0 yc 2 yci0 2 Ooyci0 Ooyci0 yci=yci0 1 yciOe1 yc 2 At long last, the obtained .
OAyc, yu1 yu2 yu3 yu4 O 32yca yuC 2 Hence.
Eo.
uC )] O 6yuC 2 2yca
RESULTS AND DISCUSSION
In the MATLAB implementation, a series of numerical tests were conducted to assess the performance of a technique for evaluating the movement cost of data streams.
This process involved assigning specific cost values to the movement of data streams.
One key parameter, denoted as yuIC , was set to 0, simplifying the calculation by negating the contribution of to the cost function.
The system utilized a probability distribution where A = 0 and followed a standard normal distribution of n .
, which was essential for detecting data stream movements within a defined range.
The experiment aimed to evaluate the effectiveness of the proposed technique by simulating a series of data streams and measuring the associated movement costs.
However, a notable limitation arises when is set to 0, as it becomes impossible to calculate the optimal movement cost under such conditions.
Despite this, the experiment proceeded by generating target data stream values with a probability of 0.
1 and following a normal distribution of mean 0 and variance 1, n .
This probabilistic generation allowed for a controlled yet basic environment in which the technique could be evaluated.
For the simulation, the standard data stream was modeled using a distribution with a mean of 0 and a higher variance of 1.
7, allowing the system to compare the performance between the generated target data stream and the standard one.
This difference in distribution enabled the method to assess how well it could detect and evaluate movements across varied data patterns.
According to the results, the proposed method produced a 15% probability for detecting the values ycO0 and ycO1 , marking these as key points of interest within the data stream's movement.
Despite the insights gained, there were observable shortcomings.
Specifically, the movement cost for ycO0 reached its maximum, along with its associated error rate.
This indicates that while the method may offer some benefits, such as detecting movements in data streams, its efforts to provide optimal results under certain conditions, particularly when is set to 0.
To enhance the experiment, it would be beneficial to introduce a more complex scenario with varied parameter values and conditions to fully assess the technique's effectiveness and limitations.
Figure 2 presents a comparative visualization of three distinct data samples, labeled as sample 1, sample 2, and sample 3, each delineated by a unique colorAiblack, red, and blue, respectively.
Sample 1 exhibits a small KL divergence ya.
cO0 |.
cO1 ) and ya.
cO1 |.
cO0 ), indicating a predicted overshoot approaching zero.
Sample 2 has a small error proportion where ycE.
c ycu = yc0 ), and for a larger proportion, it shows a large value, denoted by .
-)/, which leads to data stream termination.
In the case of sample 3, which has a large value, the primary goal of the analysis is to reduce the number of algorithm changes before correctly identifying the target data stream.
Figure 2.
Samples with KL divergences IAES Int J Rob & Autom.
Vol.
No.
March 2025: 19-30
IAES Int J Rob & Autom
ISSN: 2722-2586
All three samples follow a similar trend with their own peculiar characteristics.
Sample 1 .
exhibits a steady, almost linear decline until it flattens out towards the end of the domain.
Sample 2 .
is characterized by a significant oscillation before sharply dropping, indicating a variable response or measurement before reaching a similar level as sample 1 towards the end.
Sample 3 .
mirrors the oscillation seen in sample 2, but with a less pronounced initial drop and a deeper final descent.
This comparison allows for an easy assessment of the similarities and differences in behavior, or responses captured by the three samples across the range of 'n' values.
Figure 3 presents a time-series linear approximation of Brownian motion, reflecting data stream flow dynamics within a unit interval.
The visualization begins at zero and rapidly reaches a maximum at T=0.
2, after which it moderately recedes to a plateau of 0.
8, sustained until T=0.
Subsequently, the series undergoes a pronounced drop to 0.
6, stabilizes momentarily, then descends precipitously to 0.
2, culminating in a final downturn back to zero as T This portrayal suggests a piecewise linear process with distinct, sustained levels before transitioning, indicative of a system exhibiting stepwise stability before entering new phases.
Figures 4 to 6 offer insights into the behavior of data stream movements and their relationship with the parameter .
Figure 4 demonstrates a minimum Brownian path with characteristic rise-and-fall patterns over time, displaying the inherent randomness of the motion through peaks and troughs between T=0 and T=1.
Figure 5 shows a decreasing trend in the number of movements starting from 40 at =0 and diminishing as increases, suggesting an inverse relationship.
Meanwhile.
Figure 6 contrasts this by depicting a direct, linear correlation between and the total number of observations, which increases proportionally from 110 to 145 as grows from 0 to 5.
This comparison highlights the significance in which values change data stream behavior, with the rise in observations possibly pointing to increased data collection or detection capabilities as grows, despite the decrease in movement frequency.
Together, these figures aid in identifying anomalies between different data stream paths by evaluating the movement cost over time.
Figure 3.
Stochastic differential equation with linear approximation for Brownian motion Figure 4.
Brownian path with time Optimizing robot anomaly detection through stochastic differential approximation A (Branesh M.
Pilla.
A ISSN: 2722-2586 Figure 5.
Data stream movements based on observations Figure 6.
Total of number of observation w.
t value CONCLUSION Anomaly detection is achieved by repositioning a data stream from one location to another, with movement costs evaluated through Brownian motion.
The process involves applying a mathematical model based on the stochastic differential equation of Brownian motion.
By optimizing the Brownian motion over time, the nearest position to the anomaly is identified.
The model developed and validated in this work effectively approximates the minimum path required to detect anomalies.
Its strength lies in its ability to accomplish the task within a relatively short crossing time, while accurately identifying anomalies through the observation of movement costs.
This approach proves to be a valuable tool for efficiently detecting anomalies, offering both speed and accuracy in the process.
ACKNOWLEDGEMENTS
This project is supported by the Reinventing University System through Mahidol University (IO 864102063.
, and This research is partially supported by the e-ASIA Joint Research Program (P-1950.
Grant through the National Science and Technology Development Agency (NSTDA) and Mahidol University.
Thailand.
REFERENCES