A Method of Local Influence Analysis in Sufficient Dimension Reduction

Chen, Fei; Shi, Lei; Zhu, Lin; Zhu, Lixing

doi:10.5705/ss.202019.0388

Cited by 2 publications

(14 citation statements)

References 35 publications

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“…Let a vector

ω = {(() ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

denote the perturbation introduced to the model, and

\overset{true^}{boldB} false(bold-italicω false) = ({\overset{b}{true}}_{1} false(bold-italicω false), {\overset{b}{true}}_{2} false(bold-italicω false), \dots, {\overset{b}{true}}_{\overset{K}{true}} false(bold-italicω false))

where

{\overset{b}{true}}_{1} false(bold-italicω false), {\overset{b}{true}}_{2} false(bold-italicω false), \dots, {\overset{b}{true}}_{\overset{K}{true}} false(bold-italicω false)

represent the estimates of the dimension reduction vectors under the perturbed model. Based on the trace correlation proposed by Hooper [25], Chen et al [14] constructed a space displacement function

D false(bold-italicω false)

for the local influence analysis of sufficient dimension reduction, which is defined by

D false(bold-italicω false) = 1 - \frac{1}{\overset{K}{true}} {false\sum}_{i = 1}^{\overset{K}{true}} r_{i}^{2} = 1 - \frac{1}{\overset{K}{true}} tr ({boldP}_{{\overset{X}{true}}^{T} \overset{true^}{boldB}} {boldP}_{{\overset{X}{true}}^{T} \overset{true^}{boldB} false(bold-italicω false)}),

where

r_{i}

denotes the

…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

“…Let

ω = {bold-italicω}_{0} + t h

where

{bold-italicω}_{0}

represents no perturbation and

h

denotes a standardized vector. Following Chen et al [14], we call the surface

{(() {bold-italicω}^{T}, D (ω))}^{T}

the influence graph and call the curve

L false(boldh false) = \{{(() {((), ω_{0} goodbreak+ t boldh)}^{T}, D ((), ω_{0} goodbreak+ t boldh))}^{T} : t \in {scriptR}^{1}\} .

the lifted line along the direction

h

on the influence graph, which passes through

{(() {bold-italicω}_{0}^{T}, D ((), {bold-italicω}_{0}))}^{T}

. To investigate the local behavior of

L false(boldh false)

t = 0

, the following assumptions are required.Assumption (1)

rk false(\overset{X}{true} false) = p

; (2) For any given

h

\overset{true^}{boldB} (ω_{0} + t h)

is continuous in a neighborhood of

t = 0

; (3) There is a matrix

{boldF}_{boldBh}

such that

…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

“…Under the above general framework proposed by Chen et al [14], we will deduce the statistic of the local influence assessment for SATME. In the remainder of this section, we will show the existence of

{boldF}_{boldBh}

and derive the expression of

{boldf}_{boldBh}^{(k)}

under SATME, which will be shown to be a vector‐valued linear function of

h

such that the quasi‐curvature can be expressed as a quadratic form of

h

with its maximization greatly simplified.…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

“…However, as SATME uses the third‐order conditional moment of the predictors given the response, it may not as robust as some other SDR methods that use lower‐order moments, say, SIR and SAVE. Actually, as Chen et al [14] pointed out, even the CS estimate obtained by SIR may be hurt by the outliers. Hence, for SATME, which depends on stricter assumptions and higher‐order conditional moment than SIR, it is necessary to assess the influence of the observations on the CS estimate and find out the harmful influential observations.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we will focus on the local influence analysis of SATME, which can avoid masking effects by introducing simultaneous perturbation. On the basis of the general and basic framework proposed by Chen et al [14], we construct a statistic for local influence assessment under SATME and a data‐trimming strategy based on the influence assessment of data points. In SATME, there is an irreplaceable standardization of the predictor vector.…”

Section: Introductionmentioning

confidence: 99%

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Local influence analysis for the sliced average third‐moment estimation

Rao

Liu

Chen

2022

Statistical Analysis

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Sliced average third‐moment estimation (SATME) is a typical method for sufficient dimension reduction (SDR) based on high‐order conditional moment. It is useful, particularly in the scenarios of regression mixtures. However, as SATME uses the third‐order conditional moment of the predictors given the response, it may not as robust as some other SDR methods that use lower order moments, say, sliced inverse regression (SIR) and slice average variance estimation (SAVE). Based on the space displacement function, a local influence analysis framework of SATME is constructed including a statistic of influence assessment for the observations. Furthermore, a data‐trimming strategy is suggested based on the above influence assessment. The proposed methodologies solve a typical issue that also exists in some other SDR methods. A real‐data analysis and simulations are presented.

show abstract

“…Let a vector

ω = {(() ω_{1}, ω_{2}, \dots, ω_{n})}^{T}

denote the perturbation introduced to the model, and

\overset{true^}{boldB} false(bold-italicω false) = ({\overset{b}{true}}_{1} false(bold-italicω false), {\overset{b}{true}}_{2} false(bold-italicω false), \dots, {\overset{b}{true}}_{\overset{K}{true}} false(bold-italicω false))

where

{\overset{b}{true}}_{1} false(bold-italicω false), {\overset{b}{true}}_{2} false(bold-italicω false), \dots, {\overset{b}{true}}_{\overset{K}{true}} false(bold-italicω false)

represent the estimates of the dimension reduction vectors under the perturbed model. Based on the trace correlation proposed by Hooper [25], Chen et al [14] constructed a space displacement function

D false(bold-italicω false)

for the local influence analysis of sufficient dimension reduction, which is defined by

D false(bold-italicω false) = 1 - \frac{1}{\overset{K}{true}} {false\sum}_{i = 1}^{\overset{K}{true}} r_{i}^{2} = 1 - \frac{1}{\overset{K}{true}} tr ({boldP}_{{\overset{X}{true}}^{T} \overset{true^}{boldB}} {boldP}_{{\overset{X}{true}}^{T} \overset{true^}{boldB} false(bold-italicω false)}),

where

r_{i}

denotes the

…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

“…Let

ω = {bold-italicω}_{0} + t h

where

{bold-italicω}_{0}

represents no perturbation and

h

denotes a standardized vector. Following Chen et al [14], we call the surface

{(() {bold-italicω}^{T}, D (ω))}^{T}

the influence graph and call the curve

L false(boldh false) = \{{(() {((), ω_{0} goodbreak+ t boldh)}^{T}, D ((), ω_{0} goodbreak+ t boldh))}^{T} : t \in {scriptR}^{1}\} .

the lifted line along the direction

h

on the influence graph, which passes through

{(() {bold-italicω}_{0}^{T}, D ((), {bold-italicω}_{0}))}^{T}

. To investigate the local behavior of

L false(boldh false)

t = 0

, the following assumptions are required.Assumption (1)

rk false(\overset{X}{true} false) = p

; (2) For any given

h

\overset{true^}{boldB} (ω_{0} + t h)

is continuous in a neighborhood of

t = 0

; (3) There is a matrix

{boldF}_{boldBh}

such that

…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

{boldF}_{boldBh}

and derive the expression of

{boldf}_{boldBh}^{(k)}

under SATME, which will be shown to be a vector‐valued linear function of

h

such that the quasi‐curvature can be expressed as a quadratic form of

h

with its maximization greatly simplified.…”

Section: The Framework Of Local Influence Analysis For Satmementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Local influence analysis for the sliced average third‐moment estimation

Rao

Liu

Chen

2022

Statistical Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

An example study of applications based on multiple machine learning algorithms

Hao,

Zhou,

Zhang

2023

2023 IEEE 6th International Conference on Automation, Electronics and Electrical Engineering (AUTEEE)

View full text Add to dashboard Cite

A Method of Local Influence Analysis in Sufficient Dimension Reduction

Cited by 2 publications

References 35 publications

Local influence analysis for the sliced average third‐moment estimation

Local influence analysis for the sliced average third‐moment estimation

An example study of applications based on multiple machine learning algorithms

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