Bayesian Nonparametric General Regression

Yuen, Ka‐Veng; Ortiz, Gilberto A.

doi:10.1615/int.j.uncertaintyquantification.2016016055

Cited by 17 publications

(13 citation statements)

References 30 publications

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“…In the following, a noninformative prior distribution is utilized for the uncertain parameters in b so it is absorbed into the normalizing constant. This can be done because the likelihood function is integrable . Furthermore, note that p ( b ) = p ( b | Λ ) because the modal data alone does not have any saying on the uncertain parameters in b .…”

Section: Bayesian Nonparametric General Regressionmentioning

confidence: 99%

“…The likelihood function can be expressed as

p (()|,, Θ, b, Λ) = \prod_{k = 1}^{N_{italicθ}} \prod_{i = 1}^{N} p ((), θ_{i}^{(k)} (|,,,,, {italicθ}_{1}^{(k)} {italicθ}_{2}^{(k)} \dots {italicθ}_{i - 1}^{(k)} b Λ)),

where

θ_{i}^{(k)}

represents the i th sample for the k th structural parameter. For i = 1,

p ((), θ_{i}^{(k)} (|,,,,, {italicθ}_{1}^{(k)} {italicθ}_{2}^{(k)} \dots {italicθ}_{i - 1}^{(k)} b Λ))

p ((), θ_{i}^{(k)} (|,, b, Λ))

.…”

Section: Bayesian Nonparametric General Regressionmentioning

confidence: 99%

“…In this case, the smoothing parameters

σ_{i}^{2}, i = 2, \dots, N

, and the predictionerror variances

ϑ_{i}^{2}, i = 2, \dots, N

, depend on the number of elements in the conditioning part of each PDF. The smoothing parameters

σ_{i}^{2}, i = 2, \dots, N

, representing the sparseness of the data points, are assumed proportional to the average squared distance between the point for prediction and the points in the corresponding conditioning part:

σ_{i}^{2} = \frac{b_{1}}{i - 1} \sum_{j = 1}^{i - 1} {‖{boldλ}_{i} - {boldλ}_{j}‖}^{2},

where b 1 is the smoothing scale parameter to be determined. According to Yuen and Ortiz, the prediction error variances

ϑ_{i}^{2}, i = 2, \dots, N

, take the following form:

ϑ_{i}^{2} = \frac{b_{2}}{\sum_{j = 1}^{i - 1} exp ([], - 2 {‖{boldλ}_{i} - {boldλ}_{j}‖}^{2})},

where b 2 is the prediction error scale parameter to be determined.…”

Section: Bayesian Nonparametric General Regressionmentioning

confidence: 99%

“…This can be done because the likelihood function is integrable. [28] Furthermore, note that p(b) = p(b| Λ) because the modal data alone does not have any saying on the uncertain parameters in b.…”

Section: Bayesian Inference For Grnnmentioning

confidence: 99%

“…It is a kernel‐based nonparametric regression technique that has been applied successfully to several research areas such as energy conservation, environmental engineering, soil dynamics, and oil drilling . Recently, the Bayesian nonparametric general regression (BNGR) was proposed. Because BNGR is a reformulation of GRNN under the Bayesian framework, the estimation uncertainty can be quantified.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Multiresolution Bayesian nonparametric general regression for structural model updating

Yuen

Ortiz

2017

Struct Control Health Monit

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Summary A novel Bayesian method, namely, multiresolution Bayesian nonparametric general regression (MR‐BNGR), is proposed for structural model updating using modal data, that is, identified natural frequencies and mode shapes. In this method, the model updating problem is posed as a non‐linear regression problem from the modal data to the structural parameters. The proposed method is nonparametric, so it does not require an explicit functional form of this mapping. Instead, it utilizes the input–output data to adaptively model its relationship. Its multiresolution nature allows to zoom into the significant region in stages to search the optimal point. Furthermore, the estimation uncertainty can be quantified. Training of the MR‐BNGR network is very straightforward and computationally economical. Examples of a 20‐storey shear building and a three‐dimensional truss are provided to demonstrate the capabilities of the proposed MR‐BNGR method, and the results confirm the effectiveness of this novel method.

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Section: Bayesian Nonparametric General Regressionmentioning

confidence: 99%

“…The likelihood function can be expressed as

p (()|,, Θ, b, Λ) = \prod_{k = 1}^{N_{italicθ}} \prod_{i = 1}^{N} p ((), θ_{i}^{(k)} (|,,,,, {italicθ}_{1}^{(k)} {italicθ}_{2}^{(k)} \dots {italicθ}_{i - 1}^{(k)} b Λ)),

where

θ_{i}^{(k)}

represents the i th sample for the k th structural parameter. For i = 1,

p ((), θ_{i}^{(k)} (|,,,,, {italicθ}_{1}^{(k)} {italicθ}_{2}^{(k)} \dots {italicθ}_{i - 1}^{(k)} b Λ))

p ((), θ_{i}^{(k)} (|,, b, Λ))

.…”

Section: Bayesian Nonparametric General Regressionmentioning

confidence: 99%

“…In this case, the smoothing parameters

σ_{i}^{2}, i = 2, \dots, N

, and the predictionerror variances

ϑ_{i}^{2}, i = 2, \dots, N

, depend on the number of elements in the conditioning part of each PDF. The smoothing parameters

σ_{i}^{2}, i = 2, \dots, N

, representing the sparseness of the data points, are assumed proportional to the average squared distance between the point for prediction and the points in the corresponding conditioning part: