Installation

if (!require("BiocManager", quietly = TRUE))
    install.packages("BiocManager")
BiocManager::install("TDbasedUFE")

Introduction

TDbasedUFE is a R package that intends to perform various analyses using Tensor decomposition based unsupervised feature extraction (TDbasedUFE) (Y-H. Taguchi 2020) as much as possible in an user friendly manner. Although in the future, I am planning to implement all the features possible by TDbasedUFE in this R package, at the moment, it includes only basic features.

What are tensor and tensor decomposition?

Although [tensor] (https://bioconductor.org/packages/release/bioc/vignettes/DelayedTensor/inst/doc/DelayedTensor_1.html#12_What_is_Tensor) and [tensor decomposition] (https://bioconductor.org/packages/release/bioc/vignettes/DelayedTensor/inst/doc/DelayedTensor_1.html#13_What_is_Tensor_Decomposition) were fully described in vignettes of DelayedTensor(Tsuyuzaki 2022), we also briefly describe these.

Tensor is a natural extension of matrix that has only rows and columns, whereas tensor has more number of “rows” (or “columns”). For example, three mode tensor, x_ijk ∈ ℝ^{N × M × K}, has three suffix, i ∈ ℕ, j ∈ ℕ, and k ∈ ℕ, each of which ranges [1, N], [1, K] and [1, M]. This representation fitted with that of genomic datasets, e. g., x_ijk can represent gene expression of ith gene of kth tissue of jth subject (person). If the measurements are performed over multiple conditions, e.g., different time points, we can add additional sufix that corresponds to time. Alternatively, we can also make use of tensor to store multiomics data if we can add additional suffix that represent which of multiomics data is stored in the elements of tensor. Since the experimental design has become more and more complicated, tensor is an ideal data structure to store modern genomic observations.

Although tensor can store very complicated structure well, it does not always mean that it has ability to help us to understand these complicated structure. In order to decompose tensor, tensor decomposition was developed. There are various way to compose tensor. Usually, tensor decomposition (TD) is to decompose tensor into product of some of vectors and tensors.

The most popular TD is CP decomposition, which can decompose tensor into product some of vectors. For example, x_ijk ∈ ℝ^{N × M × K} can be decomposed into the product sum of three vectors, ℝ^N, ℝ^M, and ℝ^K.

Tensor Decomposition

which can be mathematically expressed as x_ijk = ∑_ℓλ_ℓu_ℓiu_ℓju_ℓk Alternatively, more advanced way, tensor train decomposition,
can decomposes the tensor to product of a mixture of matrix and tensor as x_ijk = ∑_ℓ1∑_ℓ2R_ℓ1iR_ℓ1ℓ2jR_ℓ2k However, in this specific study, we employed more primitive way of decomposition, Tucker decomposition, as x_ijk = ∑_ℓ1∑_ℓ2∑_ℓ3G(ℓ₁ℓ₂ℓ₃)u_ℓ1iu_ℓ2ju_ℓ3k where G ∈ ℝ^{N × M × K} is core tensor that evaluate the weight of product $ u_{1 i} u{2 j} u{_3 k}$ and u_ℓ1i ∈ ℝ^N × N, u_ℓ2j ∈ ℝ^M × M, u_ℓ3k ∈ ℝ^K × K are singular value matrices and are orthogonal matrices.

Although there are no uniqueness of Tucker decomposition, we specifically employed higher order singular value decomposition (HOSVD) (Y-H. Taguchi 2020) to get Tucker decomposition. The reason why we employed Tucker decomposition was fully described in my book(Y-H. Taguchi 2020).

Tensor decomposition based unsupervised feature extraction.

Tensor decomposition based unsupervised feature extraction

Here we briefly describe how we can make use of HOSVD for the feature selection. Suppose x_ijk represents the expression of gene i of the kth tissue of the jth subject (person) and the task is to select genes whose expression is distinct between patients and healthy controls only at one specific tissue. In order that, we need to select u_ℓ2j attributed to jth subject which is distinct between healthy control and patients (see (1)) and u_ℓ3k attributed to kth tissue is kth tissue specific (see (2)). Then we next identify which G(ℓ₁ℓ₂ℓ₃) has the largest absolute value with fixed ℓ₂ and ℓ₃ that correspond to distinction between patient and healthy control and tissue specificity, respectively (see (3)). Finally, with assuming that u_ℓ1i obeys Gaussian distribution (null hypothesis), P-values are attributed to ith gene. P-values are corrected with considering multiple comparison corrections (specifically, we employed Benjamini Hochberg method(Y-H. Taguchi 2020)) and genes associated with adjusted P-values less than 0.01 are usually selected (see (4) and (5)).

Optimization of standard deviation

Although TD based unsupervised FE was proposed five years ago, it was successfully applied to wide range of genomic science, together with the proto type method, principal component analysis (PCA) based unsupervised FE proposed ten years ago, we recently recognize that the optimization of the standard deviation (SD) in Gaussian distribution employed as the null hypothesis drastically improved the performance(Y.-H. Taguchi and Turki 2022) (Y.-H. Taguchi and Turki 2023) (Roy and Taguchi 2022). How we can optimize SD and why the optimization of SD can improve the performance is as follows. Previously, we computed SD of u_ℓ1i, σ_ℓ1 as $$ \sigma_{\ell_1} = \sqrt{\frac{1}{N} \sum_i \left ( u_{\ell_1 i} - \langle u_{\ell_1i} \rangle\right)^2} $$ $$ \langle u_{\ell_1 i} \rangle = \frac{1}{N} \sum_i u_{\ell_1 i}. $$ However, σ_ℓ1 has tendency to be over estimated since some of ith does not follow Gaussian distribution and has too large absolute values. In order to exclude those i that does not follow Gaussian, we tried to minimize SD of h_n which is the frequency of P_i computed with assuming that u_ℓ1i obeys Gaussian and falling into nth bin of histogram P_i $$ \sigma_h = \sqrt{ \frac{1}{N_h} \sum_n \left ( h_n - \langle h_n \rangle \right)^2} $$ where N_h is the number of bins of the histogram of P_i and $$ \langle h_n \rangle = \frac{1}{N_h} \sum_n h_n $$ with excluding some of P_i which is supposed to be too small to assume that obeys Gaussian (Null hypothesis). This is because h_n should be constant if all of P_i follows Gaussian (Null hypothesis).

Left: σ_h as a function of σ_ℓ1. Right: Histogram, h_n, of 1 − P_i being computed with the optimized σ_ℓ1.

The above plot is output of examples of function selectFeature in TDbasedUFE package and represents the dependence of σ_h upon σ_ℓ1 as well as the resulting histogram h_n of 1 − P_i when u_ℓ1i is fully generated from Gaussian. The smallest σ_h correctly corresponds to the true σ_ℓ1( = 0.01) and h_n is almost independent of n as expected.

Thus it is expected that σ_h will take the smallest value when we correctly exclude is whose P_i does not obey Gaussian (Null hypothesis). The following is the step to optimized σ_ℓ1.

1. Prepare initial σ_ℓ1 and define threshold adjusted P-value, P₀.

2. Attribute P_is to is with assuming Gaussian distribution of u_ℓ1i using σ_ℓ1.

3. Attribute adjusted P_is to is with using Benjamini-Hochberg criterion.

4. Exclude is associated with adjusted P_i less than P₀.

5. Compute histogram of 1 − P_i and σ_h.

6. Modify σ_ℓ1 until σ_h is minimized.

7. Finally, we identify excluded is as the selected features because these is do not obey Gaussian (Null hypothesis)

Multiomics data analysis

When we have multiomics data set composed of K omics data measured on common M samples as x_ikjk ∈ ℝ^{N_k × M × K} where N_k is the number of features in kth omics data, we can make use of a tensor which is a bundle of linear kanel(Y. Taguchi and Turki 2022). In this case, we can generate liner kernel of kth omics data as x_jj′k = ∑_ikx_ikjkx_ikj′k ∈ ℝ^{M × M × K} to which HOSVD was applied and we get x_jj′k = ∑_ℓ1∑_ℓ2∑_ℓ3G(ℓ₁ℓ₂ℓ₃)u_ℓ1ju_ℓ2j′u_ℓ3k where ℓ₁ and ℓ₂ are exchangeable, since x_jj′k = x_j′jk. Singular values attributed to features can be retrieved as u_ℓik = ∑_ju_ℓjx_ikj Then the same procedure was applied to u_ℓi in order to select features.

Multiomics analysis

Conclusions

In this vignettes, we explain how TD based unsupervised FE can work with optimized SD. In the other vignettes, we introduce how it works well in the real examples.

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