## 2 Theory  
### Ordinary and Partial Coherence  
The identification of a major noise source is typically difficult because it is challenging to evaluate the contribution between input and output when the inputs are partially coherent. In signal processing, the partial contribution analysis method calculates the contribution by using optimal linear frequency response functions in which the correlation between inputs is removed. This reveals the degree to which the input source directly contributes to the output. In the case of vehicle noise, the expected multiple-input/single-output model for the input/output and transfer function for the expected generation location of major BSR noise is shown in Figure 1.  
The equation for output with respect to a certain \(q\) number of inputs in this multiple-input/single-output spectrum model can be represented as shown in Equation (1) in the frequency domain. \[S_{yy}(f)=\sum_{j=1}^{q}\sum_{i=1}^{q}H_{iy}^{*}(f)H_{iy}(f)S_{ij}(f)+S_{nn}(f) \tag{1}\]  
where \(H_{iy}(f)\) represents a transfer function that relates the \(i\)-th input to output \(y\), and \(S_{ij}(f)\) denotes a cross-power spectrum between inputs \(X_{i}(f)\) and \(X_{j}(f)\). \(H_{iy}^{*}(f)\) is the conjugate complex number of the transfer function. \(H_{iy}(f)\) and represents a coherence relation between the transfer paths. \(S_{nn}(f)\) is expressed as a power spectrum of noise \(N(f)\). The coherence in a typical input/output model is represented using an ordinary coherence function (OCF) that represents correlations between inputs and between the input and output. This can be described as follows:  
\[Y_{ij}^{2}(f)=\frac{|S_{ij}(f)|^{2}}{S_{ii}(f)S_{ji}(f)}\quad(i,j=1,\;2,\ldots, \;q;i\
eq j) \tag{2}\]  
\[Y_{iy}^{2}(f)=\frac{|S_{iy}(f)|^{2}}{S_{ii}(f)S_{yy}(f)}\quad(i,j=1,\;2,\ldots, \;q) \tag{3}\]  
All of the inputs in the seat have transfer paths that are coupled to each other. In particular, it is difficult to distinguish the major noises in a complex structure, and the noises are coherent due to the close proximity of the noise measurement locations during measurement. For example, because two input sources are located close to each other, there are cases where similar signals are measured from the two microphones used in measuring the noises (Figure 2). Figure 1: Correlated multiple-input/single-output model. of transforming the projected values into independent ones by setting the axis in the direction with a high non-Gaussianity to remove the correlation between dimensions. Let a \(d\) number of independent signal inputs be \(S_{1}\), \(S_{2}\), \(S_{3}\),..., \(S_{d}\), and the new signals obtained through linear combinations of the inputs be \(x_{1}\), \(x_{2}\), \(x_{3}\),..., \(x_{n}\). Then, the relationship between the signals can be expressed as shown in Equation (10). \[x_{i}=\sum_{j=1}^{d}a_{ij}s_{j}\quad(i=1,\ 2,\ldots,\ n) \tag{10}\]  
where \(s_{j}\) and \(x_{i}\) are vectors which can be represented as \(s=(s_{1},s_{2},\cdots,s_{d})^{T}\) and \(x=(x_{1},x_{2},\cdots,x_{d})^{T}\), respectively, and \(a_{ij}\) is defined as the element of the mixing matrix. Then, Equation (10) can be represented as a vector matrix shown in Equation (11). \[x=As \tag{11}\]  
Here, in the state where \(s\) is not given and there is no information about mixing matrix \(A\), source \(s\) and mixing matrix \(A\) are estimated with only \(x\). This process is called blind source separation. Equation (12) can be written as follows to estimate the source:  
\[s=A^{-1}x=Wx \tag{12}\]  
where \(W\) is an inverse matrix of \(A\). In this paper, independent components were found using the FastICA algorithm proposed by [PERSON] and [PERSON] [23]. The FastICA method uses a fixed-point algorithm based on the theory that finding independent components with the central limit theorem is the same as finding the most non-Gaussian direction [24]. The central limit theorem is the theory that the sum of independent random variables becomes closer to a Gaussian distribution than each independent component.