Feature selection for predicting tool wear of machine tools

Abstract

In this study, the vibration transmitted solely from a spindle to the worktable is proposed to be a crucial feature of wear prediction models for machine tools. To validate the effectiveness of the proposed feature, a feature ranking and screening methodology was also used for developing a tool wear prediction model. First, the features extracted from vibration signals were ranked according to their contributions to tool wear prediction. The features were then filtered through a screening process based on singular value decomposition to eliminate redundant features, which exhibited collinearity with features of higher rankings. The aim of the aforementioned steps was to use a relatively small number of highly appropriate features to create an accurate real-time tool wear prediction model. The results indicated that the accuracy of the tool wear prediction model based on the proposed feature ranking and screening methodology is higher than that of models without feature ranking or screening. Moreover, the proposed feature was proven to be more important and effective than other features.

Appendix

1.1 Formulations of PCA and PCR

The original feature data set X is expressed in matrix form as follows:

$$ \mathbf{X}=\left[{\mathbf{X}}_1,\cdots, {\mathbf{X}}_n,\cdots, {\mathbf{X}}_N\right] $$

(9)

where X_n is the nth feature with M samples and N is the total number of features. Each feature must be normalized by its z-score to obtain a new matrix X_nor because each feature has various scale ranges. Subsequently, the eigenvalue λ_j and eigenvector p_j of the covariance matrix of X_nor are calculated, and the following formula is obtained:

$$ {\mathbf{S}}_{\mathrm{xx}}{\mathbf{p}}_j={\lambda}_j{\mathbf{p}}_j,j=1,2,\dots, n, $$

(10)

where ${\mathbf{S}}_{\mathrm{xx}}={\mathbf{X}}_{\mathrm{nor}}^T{\mathbf{X}}_{\mathrm{nor}}$ is the covariance matrix of X_nor, λ_j is the eigenvalue, and p_j is the corresponding eigenvector or principal component (PC) loading vector. The degree of importance or weight of the PC loading vector can be determined by calculating the contribution rate f_j of each eigenvalue as follows:

$$ {f}_j=\frac{\lambda_j}{\sum \limits_{k=1}^n{\lambda}_k},\kern0.5em j=1\dots n. $$

(11)

The dimensions of the original feature set can be reduced because of the partial mutual linear dependence among features when eigenvalues with small contributions are discarded and an r number of eigenvalues with large contributions (f_j) are retained.

Next, X_nor is projected onto each eigenvector (p_j) to obtain the corresponding weight t_j, which is known as the PC score vector. This vector represents the weights of X_nor on the PC loading vectors.

$$ {\mathbf{t}}_j={\mathbf{X}}_{\mathrm{nor}}{\mathbf{p}}_j,j=1,\dots, r,r\le n $$

(12a)

or

$$ \mathbf{T}={\mathbf{X}}_{\mathrm{nor}}\mathbf{P}, $$

(12b)

where T = {t₁, t₂, …, t_r} is an m × r matrix representing the projection of X_nor on the eigenspace and P = [p₁ p₂ … p_r] is the eigenmatrix formed by the PC loading vector p_j. If r < n, then the feature set X_nor is reduced to ${\hat{\mathbf{X}}}_{\mathrm{nor}}={\mathbf{TP}}^{\mathrm{T}}={\mathbf{X}}_{\mathrm{nor}}{\mathbf{PP}}^{\mathrm{T}}$ with low dimensionality. In this study, a cumulative contribution of at least 90% was selected as the threshold to determine the r value. Moreover, the summation of f₁–f_r was greater than 90% of the summation of f₁–f_n. The contribution of each feature in the original data set cannot be determined through PCA because P is a linear combination of the original features. Consequently, regression must be performed to determine the contribution of each feature according to the PCs. This process is known as PCR, which facilitates subsequent feature ranking, as described in the following text.

The relationship between the feature set and the corresponding true tool wear measured using the CCD camera can be expressed as a regression equation as follows:

$$ \mathbf{Y}={\mathbf{X}}_{\mathrm{nor}}\boldsymbol{\upbeta} +\boldsymbol{\upvarepsilon}, $$

(13)

where Y is the true tool wear, β is the n × 1 weight vector expressed as β = [β₁ β₂ … β_n]^T, and ε is the error vector of the regression equation. The features are not linearly independent; therefore, they must be transformed using PCA in an eigenspace in which the PC loading vectors are orthogonal to each other. Because PP^T = I, Eq. (13) can be rewritten as follows:

$$ \mathbf{Y}={\hat{\mathbf{X}}}_{\mathrm{nor}}{\mathbf{PP}}^{\mathrm{T}}\boldsymbol{\upbeta} +\boldsymbol{\upvarepsilon} =\mathbf{F}\boldsymbol{\uptheta } +\boldsymbol{\upvarepsilon} $$

(14)

where $\mathbf{F}={\hat{\mathbf{X}}}_{\mathrm{nor}}\mathbf{P}$ is the orthogonal transformation and ${\mathrm{T}}_{\mathrm{m}\times \mathrm{r}}={\mathbf{X}}_{{\mathrm{n}\mathrm{or}}_{\mathrm{m}\times \mathrm{n}}}{\mathbf{P}}_{\mathrm{n}\times \mathrm{r}}$θ = P^Tβ represents the regression coefficients in the eigenspace. The optimal regression coefficient $\hat{\boldsymbol{\uptheta}}$ can be obtained directly by using the least squares method as follows:

$$ \hat{\boldsymbol{\uptheta}}={\left({\mathbf{F}}^{\mathrm{T}}\mathbf{F}\right)}^{-1}\left({\mathbf{F}}^{\mathrm{T}}\mathbf{Y}\right). $$

(15)

After the optimal regression coefficient $\hat{\boldsymbol{\uptheta}}$ in the eigenspace is obtained, the contribution of each feature to tool wear, which is represented by the regression coefficient β, is determined by transforming $\hat{\boldsymbol{\uptheta}}$ back into the original feature space. Consequently, the following equation is obtained: β = P$\hat{\boldsymbol{\uptheta}}$. Moreover, the approximate tool wear $\hat{\mathbf{Y}}$ can be obtained as a linear combination of the feature set X_nor as follows:

$$ \hat{\mathbf{Y}}={\mathbf{X}}_{\mathrm{nor}}\left(\mathbf{P}\hat{\boldsymbol{\uptheta}}\right), $$

(16)

where the contribution of each feature to tool wear is ranked by $\mathbf{P}\hat{\boldsymbol{\uptheta}}$ in descending order.