1.2.2.1Locally linear embedding (LLE)

LLE is a well-known nonlinear dimension reduction method proposed by Saul and Roweis in 2000 [12]. Its basic idea is to transform the global nonlinearity into local linearity, with the overlapping local neighbourhoods providing the global structure of the information. In accordance with certain rules, each part of the linear dimension reduction is then combined with the results to give a low-dimensional global coordinate representation.

In the dynamic fuzzy high-dimensional space $\left({\overleftarrow{R}}^D,{\overleftarrow{R}}^D\right),$, there is a dynamic fuzzy dataset $\left(\overleftarrow{X},\overrightarrow{X}\right)=\left\{\left({\overleftarrow{X}}_1,{\overleftarrow{X}}_1\right),\left({\overleftarrow{X}}_2,{\overleftarrow{X}}_2\right),\mathrm{...},\left({\overleftarrow{X}}_n,{\overleftarrow{X}}_n\right)\right\}.$ LLE reduces the dimension $\left(\overleftarrow{F},\overrightarrow{F}\right)$ to a relatively low dimension dynamic fuzzy space $\left({\overleftarrow{R}}^d,{\overleftarrow{R}}^d\right)$ and obtains the corresponding low-dimensional dynamic fuzzy vector (, ) = $\left\{\left({\overleftarrow{y}}_1,{\overleftarrow{y}}_1\right),\left({\overleftarrow{y}}_2,{\overleftarrow{y}}_2\right),\mathrm{...},\left({\overleftarrow{y}}_n,{\overleftarrow{y}}_n\right)\right\}$ while retaining as much information as possible from the original dynamic fuzzy data (DFD). That is, the topology of the dynamic fuzzy dataset (as determined by the neighbourhood relation of each point) is maintained. An effective approach is to replace the linear approximation with a linear structure in the local approximation, that is, to reduce the nonlinear dimensionality ($\overleftarrow{F}$, $\overrightarrow{F}$) by approximating the dimensional reduction locally, so that the problem is transformed into one of local linear dimension reduction.

Definition 1.7 The model for the dynamic fuzzy dimensionality reduction problem is $\left(\left(\overleftarrow{X},\overrightarrow{X}\right),\left(\overleftarrow{F},\overrightarrow{F}\right)\right),$, where the mapping $\left(\overleftarrow{F},\overrightarrow{F}\right):\left(\overleftarrow{X},\overrightarrow{X}\right)\to \left(\overrightarrow{Y},\overrightarrow{Y}\right).\text{Let}\left(\overleftarrow{F},\overrightarrow{F}\right)$ be the dimension reduction from $\left(\overleftarrow{X},\overrightarrow{X}\right)\text{to}\left(\overleftarrow{Y},\overrightarrow{Y}\right)$ for the dynamic fuzzy datasets.

Definition 1.8 The mapping $\left(\overleftarrow{f},\overrightarrow{f}\right):\left(\overleftarrow{Y},\overrightarrow{Y}\right)\to \left(\overleftarrow{T},\overrightarrow{T}\right)\subset \left({\overleftarrow{R}}^D,{\overleftarrow{R}}^D\right)$ is called an embedded mapping.

Consider the arbitrary point $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\in \left(\overleftarrow{X},\overrightarrow{X}\right),\text{Let}\left\{\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right),l=1,2,\mathrm{...},k\right\}$ represent the k-nearest neighbourhood of $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)$ [the DFS of k points nearest to $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\text{in}\left(\overleftarrow{X},\overrightarrow{X}\right)]$] and let the linear approximation of the local $\left(\overleftarrow{F},\overrightarrow{F}\right)\text{be}\left({\overleftarrow{F}}^i,{\overrightarrow{F}}^i\right).$ Let $\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)=\left({\overrightarrow{F}}^i,{\overrightarrow{F}}^i\right)\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\left({\overleftarrow{y}}_i^1,{\overleftarrow{y}}_i^1\right)=\left({\overrightarrow{F}}^i,{\overrightarrow{F}}^i\right)\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right).$ Then, we have the following theorem:

Theorem 1.1 If $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)={\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)}\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right).$, then [13]

where $\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\left(i=1,2,\mathrm{...},n;l=1,2,\mathrm{...},k\right)$ is a linear combination of dynamic fuzzy coefficients satisfying ${\displaystyle \sum _{l-1}^k\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)}=\left(\overleftarrow{1},\overrightarrow{1}\right).$

Proof: Because $\left({\overrightarrow{F}}^i,{\overrightarrow{F}}^i\right)$ is a linear dimensional reduction, there exists a d × D matrix ($\overleftarrow{A}$, $\overrightarrow{A}$ and a d-dimensional vector ($\overleftarrow{b}$, $\overrightarrow{b}$) such that

We have

This suggests that the local linear dimension reduction is a constant. $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)$ is approximated by a linear combination of $\left\{\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right)|l=1,2,\mathrm{...},k\right\}:\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)~$${\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right),}$ so the dynamic fuzzy coefficient $\left\{\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\right\}$ depicts the invariant features of the data structure under the reduced dimension. Then, according to $\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)=\left({\overleftarrow{F}}^i,{\overleftarrow{F}}^i\right)\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)~\left({\overleftarrow{F}}^i,{\overleftarrow{F}}^i\right){\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right)}={\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\left({\overrightarrow{y}}_i^l,{\overleftarrow{y}}_i^l\right)},$ allowing us to identify $\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)$ through the information provided by $\left\{\left({\overleftarrow{w}}_i^l,{\overleftarrow{w}}_i^l\right)\right\}.$

The core of the LLE method is to find the k-dimensional dynamic fuzzy vector $\left({\overleftarrow{W}}_i,{\overleftarrow{W}}_i\right)=\left(\left({\overleftarrow{W}}_i^1,{\overleftarrow{W}}_i^1\right),\left({\overleftarrow{W}}_i^1,{\overleftarrow{W}}_i^2\right),\mathrm{...},\left({\overleftarrow{W}}_i^k,{\overleftarrow{W}}_i^k\right)\right)$ and to minimize

where ${\displaystyle \sum _{l-1}^k\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)=\left(\overleftarrow{1},\overleftarrow{1}\right),}\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)=\left(\overleftarrow{0},\overrightarrow{0}\right),$ and, when $\left({\overleftarrow{x}}_j,{\overleftarrow{x}}_j\right)\cancel{\in }\left\{\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right)\right\}$$l=1,2,\mathrm{...},k\},\mathrm{and}\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)=\left(\overleftarrow{0},\overrightarrow{0}\right),\left({\overleftarrow{W}}_i,{\overleftarrow{W}}_i\right)$ records the neighbourhood informa tion of point $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right),$ that is, the local topological structure of each point, and the dynamic fuzzy matrix of all the components is denoted by ($\stackrel{\leftharpoonup }{W}$, $\stackrel{\rightharpoonup }{W}$). At this point, choose the appropriate dimension d and perform Rl d-dimensional embeddings, that is, seek $\left(\overleftarrow{Y},\overrightarrow{Y}\right)=\left(\left({\overleftarrow{y}}_1,{\overrightarrow{y}}_1\right),\left({\overleftarrow{y}}_2,{\overrightarrow{y}}_2\right),\mathrm{...},\left({\overleftarrow{y}}_n,{\overrightarrow{y}}_n\right)\right)\in \left({\overleftarrow{R}}^{d\times n},{\overleftarrow{R}}^{d\times n}\right)$ to satisfy

because (, ) is solved by the dynamic fuzzy dataset (, , and the computation is very sensitive to noise, especially when the eigenvalues of ${\left(\overleftarrow{X},\overrightarrow{X}\right)}^T\left(\overleftarrow{X},\overrightarrow{X}\right)$ are small. In this case, the desired result may not be obtained.

1.2.2.2Analysis of noise interference

Let $\left({\overleftarrow{x}}_l^1,{\overleftarrow{x}}_l^1\right)\text{'}=\left({\overleftarrow{x}}_l^1,{\overleftarrow{x}}_l^1\right)+\left({\overleftarrow{\epsilon }}_i^1,{\overleftarrow{\epsilon }}_i^1\right),\left(i=1,2,\mathrm{...},k\right)$ denote the corresponding noise-affected point and $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\text{'}={\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)\text{'}\left({\overleftarrow{x}}_i^l,{\overleftarrow{x}}_i^l\right)\text{'},}{\displaystyle \sum _{l=1}^k\left({\overleftarrow{w}}_i^1,{\overleftarrow{w}}_i^1\right)\text{'}=\left(\overleftarrow{1},\overrightarrow{1}\right);\left(\overleftarrow{U}\left({\overleftarrow{x}}_i\right)\right),}$$\overrightarrow{U}\left({\overrightarrow{x}}_i\right))=\left(\left({\overleftarrow{x}}_i^1,{\overleftarrow{x}}_i^1\right),\left({\overleftarrow{x}}_i^2,{\overleftarrow{x}}_i^2\right),\mathrm{...},\left({\overleftarrow{x}}_i^k,{\overleftarrow{x}}_i^k\right)\right)$ is the k-neighbourhood DFS of $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right),$$\left(\overleftarrow{U}\left({\overleftarrow{x}}_i\right),\overleftarrow{U}\left({\overleftarrow{x}}_i\right)\right)\text{'}=\left(\left({\overleftarrow{x}}_i^1,{\overleftarrow{x}}_i^1\right)\text{'},\left({\overleftarrow{x}}_i^2,{\overleftarrow{x}}_i^2\right)\text{'},\mathrm{...},\left({\overleftarrow{x}}_i^k,{\overleftarrow{x}}_i^k\right)\text{'}\right)$ is the k-neighbourhood DFS of ${\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)}^{\text{'}},$, and