Response and predictor folding to counter symmetric dependency in dimension reduction

Prendergast, Luke A.; Garnham, Alexandra L.

doi:10.1111/anzs.12170

Cited by 2 publications

(5 citation statements)

References 29 publications

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“…The FSIRR method aims to overcome the symmetric dependency by transforming the response variable, which folds up or down the link function f (·) around the axis of symmetry. In particular, in accordance with the idea of Prendergast & Garnham (2016), the transformation function is defined as follows:

t_{Y} false(Y, v false) = \{\begin{matrix} Y, & if ⟨ X, v ⟩ > ⟨ X, μ ⟩, \\ Y - 2 (Y - E [Y | ⟨ X, v ⟩ = ⟨ X, μ ⟩]), & if ⟨ X, v ⟩ \leq ⟨ X, μ ⟩, \end{matrix}

where μ is the mean function of the predictor X , and v is the axis of symmetry. To implement the proposed FSIRR method, we must first estimate μ , which is the mean function of X , and v is the initial estimation of the axis of symmetry and the regression function E[ Y |〈 X , v 〉=〈 X , μ 〉].…”

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

“…FSIRP folds over the predictor curve to break the symmetric dependency. In accordance with Prendergast & Garnham (2016), the specific transformation function is defined as follows:

t_{X} false(X - μ, v false) = Sign false(⟨ v, X - μ ⟩ false) false(X - μ false),

where μ and v are the same as those in the FSIRR method. We can use the same estimation method to obtain the estimations for μ and v .…”

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

“…direction significantly differ from those in Ferré & Yao (2003) because FFIR uses all samples to estimate E( X | Y ) in a single slice. Similar to theorem 1 in Prendergast & Garnham (2016), and FSIRR (FSIRP) will use the FSAVE estimation as an initial estimation. Lemma 2.1 still holds for FSIRR and FSIRP.…”

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

“…The FSIRR method aims to overcome the symmetric dependency by transforming the response variable, which folds up or down the link function f (•) around the axis of symmetry. In particular, in accordance with the idea of Prendergast & Garnham (2016), the transformation function is defined as follows:…”

Section: Fsirrmentioning

confidence: 99%

“…FSIRP folds over the predictor curve to break the symmetric dependency. In accordance with Prendergast & Garnham (2016), the specific transformation function is defined as follows:…”

Section: Fsirpmentioning

confidence: 99%

See 4 more Smart Citations

Functional dimension reduction based on fuzzy partition and transformation

Liang¹,

Gao

Bai

et al. 2022

Aus NZ J of Statistics

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Summary Functional sliced inverse regression (FSIR) is the among most popular methods for the functional dimension reduction. However, FSIR has two evident shortcomings. On the one hand, the number of samples in each slice must not be too small and selecting a suitable S is difficult, particularly for data with small sample size, where S indicates the number of slices. On the other hand, FSIR and its related methods are well‐known for their poor performance when the link function is an even (or symmetric) dependency. To solve these two problems, we propose three new types of estimation methods. First, we propose the functional fuzzy inverse regression (FFIR) method based on a fuzzy partition. Compared with FSIR that uses a hard partition, the fuzzy partition uses all samples with different weights to estimate the mean in each slice. Therefore, FFIR exhibits good performance even for data with small sample size. Second, we suggest two transformation approaches, namely, FSIRR and FSIRP, avoiding the symmetric dependency between the response and the predictor. FSIRR eliminates the symmetric dependency by transforming the response variable, while FSIRP overcomes the symmetric dependency by transforming the functional predictor. Third, we propose the FFIRR and FFIRP methods by combining the advantages of FFIR and two transformation methods. FFIRR and FFIRP replace the FSIR method on the transformation data via FFIR. Simulation and real data analysis show that three types of proposed methods exhibit better performance than FSIR in terms of the estimation accuracy and stability.

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t_{Y} false(Y, v false) = \{\begin{matrix} Y, & if ⟨ X, v ⟩ > ⟨ X, μ ⟩, \\ Y - 2 (Y - E [Y | ⟨ X, v ⟩ = ⟨ X, μ ⟩]), & if ⟨ X, v ⟩ \leq ⟨ X, μ ⟩, \end{matrix}

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

“…FSIRP folds over the predictor curve to break the symmetric dependency. In accordance with Prendergast & Garnham (2016), the specific transformation function is defined as follows:

t_{X} false(X - μ, v false) = Sign false(⟨ v, X - μ ⟩ false) false(X - μ false),

where μ and v are the same as those in the FSIRR method. We can use the same estimation method to obtain the estimations for μ and v .…”

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

Section: Ffir Fsirr Fsirp Ffirr and Ffirpmentioning

confidence: 99%

Section: Fsirrmentioning

confidence: 99%

“…FSIRP folds over the predictor curve to break the symmetric dependency. In accordance with Prendergast & Garnham (2016), the specific transformation function is defined as follows:…”

Section: Fsirpmentioning

confidence: 99%

See 3 more Smart Citations

Functional dimension reduction based on fuzzy partition and transformation

Liang¹,

Gao

Bai

et al. 2022

Aus NZ J of Statistics

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show abstract

Slice weighted average regression

Marina

Davies²,

Shaker

et al. 2023

Adv Data Anal Classif

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It has previously been shown that ordinary least squares can be used to estimate the coefficients of the single-index model under only mild conditions. However, the estimator is non-robust leading to poor estimates for some models. In this paper we propose a new sliced least-squares estimator that utilizes ideas from Sliced Inverse Regression. Slices with problematic observations that contribute to high variability in the estimator can easily be down-weighted to robustify the procedure. The estimator is simple to implement and can result in vast improvements for some models when compared to the usual least-squares approach. While the estimator was initially conceived with the single-index model in mind, we also show that multiple directions can be obtained, therefore providing another notable advantage of using slicing with least squares. Several simulation studies and a real data example are included, as well as some comparisons with some other recent methods.

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Response and predictor folding to counter symmetric dependency in dimension reduction

Cited by 2 publications

References 29 publications

Functional dimension reduction based on fuzzy partition and transformation

Functional dimension reduction based on fuzzy partition and transformation

Slice weighted average regression

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