Ciberseguridad Fusin de Sensores Dr. Ponciano Jorge Escamilla - - PowerPoint PPT Presentation

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Ciberseguridad Fusin de Sensores Dr. Ponciano Jorge Escamilla - - PowerPoint PPT Presentation

Nov 24, 2022 167 likes •792 views

INSTITUTO POLITCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION Laboratorio de Ciberseguridad Fusin de Sensores Dr. Ponciano Jorge Escamilla Ambrosio pescamilla@cic.ipn.mx http://www.cic.ipn.mx/~pescamilla/ CIC Sensor Fu Sensor

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INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION

Fusión de Sensores

  • Dr. Ponciano Jorge Escamilla Ambrosio

pescamilla@cic.ipn.mx http://www.cic.ipn.mx/~pescamilla/

Laboratorio de Ciberseguridad

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Sensor Fu Sensor Fusion sion Pr Probabil

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Based on Hugh Durrant-Whyte, Introduction to Sensor Data Fusion http://www.acfr.usyd.edu.au/teaching/graduate/Fusion/index.html

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 One of the simplest and most intuitive general

methods of sensor data fusion is to take a weighted average of redundant information provided by a group of sensors and use this as the fused value.

 Formally, the weighted average of N sensor

measurements xi with weights 0 ≤ wi ≤ 1 is,

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Sensor Fu Sensor Fusion sion Weighted Weighted average average

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 Formally, the weighted average of N sensor

measurements xi with weights 0 ≤ wi ≤ 1 is: where 𝑗 𝑥𝑗 = 1 and wi = 0 if xi is not within some specified thresholds.

 How to obtain the weights wi?

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 Example

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 If w1 = w2 = 1/2 then:

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 What if we have heterogeneous senosrs?

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 If the xi, i = 1, 2,…, n measurements are assumed

to be independent normally distributed random variables, with distribution 𝑂( 𝑦𝑗, 𝜏 𝑗

2), then a

linear weighted mean aggregation model combining these random variables into one random variable xf is given by: 𝑦𝑔 = 𝛾1𝑦1 + 𝛾1𝑦1 + ⋯ + 𝛾𝑜𝑦𝑜 with variance:

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Where 𝛾𝑗 is a positive weighting factor calculated by:

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w1 w2

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 Conditional probability definition:

  • If we multiply both sides of the definition of P(A|B)

by P(B) we obtain: P(A ∩ B) = P(A|B) P(B)

  • Similarly, if we multiply both sides of the definition
  • f P(B|A) by P(A) we obtain:

P(B ∩ A) = P(B|A) P(A)

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method method

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 Because P(A ∩ B) = P(B ∩ A), Bayes’ rule:

𝑄 𝐵 𝐶 = 𝑄 𝐶 𝐵 𝑄(𝐵) 𝑄(𝐶) 𝑄 𝐵 𝐶 = 𝑄 𝐶 𝐵 𝑄(𝐵) 𝑗 𝑄(𝐶|𝐵𝑗) 𝑄(𝐵𝑗)

where: P(A|B) = a posteriori probability. P(B|A) = likelihood function of A. P(A) = a priori probability of A.

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 Commonly, Bayes’ rule is thought of in terms of

updating the belief about a hypothesis A in the light of new evidence B. Thus, the posterior belief P(A|B) is calculated by multiplying the prior belief P(A) by the likelihood P(B|A) that B will occur if A is true.

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 Replacing A with x and B with z, we obtain the

following relation: 𝑄 𝐲 𝐴 = 𝑄 𝐴 𝐲 𝑄(𝐲) 𝑗 𝑄(𝐴|𝐲𝑗) 𝑄(𝐲𝑗)

 The items x and z are regarded as random

variables.

 x is a state or parameter of the system.  z is the sensor measurements.

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 The Bayes’ theorem is interpreted as the

computation of the posterior probability P(x|z), given the prior probability of the state or parameter (P(x)), and the observation probability (P(z|x)): the value of x that maximizes the term (x|data).

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 The term P(z|x) assumes the role of a sensor model in

the following way:

(1) First build a sensor model: fix x = x and then ask what probability density function (pdf) on z results. (2) Use the sensor model: observe z = z and then ask what the pdf on x is. (3) Practically P(z|x) is constructed as a function of both variables (or a matrix in discrete form). (4) For each fixed value of x, a distribution in z is defined. Therefore as x varies, a family of distributions in z is created.

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 1

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 1

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2a 2a

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2a 2a

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Example method: Example 2b 2b

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: method: Data fusion Data fusion

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Sensor Fu Sensor Fusion sion Bayesian Bayesian method: Data method: Data fusion fusion

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Sensor Sensor Fusion Fusion Recursive Recursive Bayes Bayes Updating Updating

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Sensor Fu Sensor Fusion sion Recursive Recursive Bayes Bayes Updating Updating

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Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Exampl ng: Example

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Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Example ng: Example

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Sensor Fu Sensor Fusion sion Recu Recursi rsive ve Bayes Bayes Updati Updating: Example ng: Example

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Generalised Generalised Bayesian Bayesian Filter Filtering: ing: Problem Statement Problem Statement

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Generalised Generalised Bayesian Bayesian Filter Filtering ing

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Generalised Generalised Bayesian Bayesian Filter Filtering ing

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Generalised Generalised Bayesian Bayesian Filter Filtering ing

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Generalised Generalised Bayesian Bayesian Filt Filteri ering ng

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