**3.2 Cumulative proportion of variance (CPV)**

When the PCA is generally utilized for some data, the Cumulative Proportion of Variance (CPV), *R*<sup>2</sup> ð Þ *K* , defined as Eq. (10), is used for the reference indicating how much variance is covered by using the first *K* PCs. The change of the CPV with PC(s) in each domain was plotted in **Figure 2**. It is found out from this figure that the CPV is monotonically increased and converges to 1.0 as the number of component(s) is increased in all domain. Among five domains, the domain CL has the largest CPV value for the first PC. The domain C has the fastest increase of the CPV against the number of PC(s), and its CPV value for the domain C is almost the same as that for the domain CL when the number of components is more than 7. In contrast, the domain L has the slowest increase, especially when the number of components is more than 15. This means that relatively large number of PCs is required to cover a certain proportion of variance in data.

Reference values in the CPV, more than which all the corresponding PCs are discarded, varied in the previous studies. Kistler *et al.* set this value to 0.90 [17], Chen *et al.* and Wu *et al.* set to 0.999 [18, 20], and Xie set this to about 0.98 [19]. Direct comparison of these values is impossible since the amount of data and analyzing purposes were different in those studies, but all of these values are more than 0.9. Therefore the least numbers of components to cover four values of the CPV, 0.90, 0.95, 0.99 and 0.999 for five domains are indicated in **Table 2**. Seeing **Table 2**, the domain CL has the smallest values among five domains in all of the CPV values, and the domain C has almost the same property. This means that the variance in the spatial variation of HRTFs can be covered by using relatively small number of PCs in these domains. The domains F and L also have smaller number of PCs when the set CPV value is small. In these domains the major PCs having large corresponding eigenvalues cover the major part of variance in data. The required number of PCs increases in the domain L when the set CPV value is large. Varying the CPV values from 0.90 to 0.999, the required number of PCs becomes five to six times in the domains I, C, F and CL, while more than ten times are required in the domain L.

**Figure 2.** *Change in cumulative proportion of variance (CPV) with number of components in SPCA for five domains.*

*Spatial Principal Component Analysis of Head-Related Transfer Functions and Its Domain… DOI: http://dx.doi.org/10.5772/intechopen.104449*


**Table 2.**

*The least number of PCs to cover the CPV in each case.*

#### **3.3 Reconstruction accuracy in time and frequency domains**

The CPV is known to be an effective criterion for the coverage of variance with a certain number of PCs. However, comparison of the CPVs among five domains is impossible since the covariance matrices as the target for the PCA are different from each other. Therefore, the reconstruction accuracy, defined as the accuracy between the original HRTFs/HRIRs and the ones reconstructed with a certain number of PCs in five domains. In this chapter, the following two measures were computed in order to evaluate the reconstruction accuracy for the SPCA in five domains in both time and frequency domains:

**Signal-to-Deviation Ratio (SDR):** Signal-to-Deviation Ratio (SDR) is defined as the level difference between the energy (Euclid norm) of the original impulse response and that of the deviation:

$$\text{SDR}\left[\mathbf{h}, \hat{\mathbf{h}}\right] = 10 \log\_{10} \frac{||\mathbf{h}||}{||\mathbf{h} - \hat{\mathbf{h}}||} \quad \text{[dB]}, \tag{19}$$

where **<sup>h</sup>** and **<sup>h</sup>**^ respectively indicate the original and the reconstructed HRIRs, j j j j� indicates the Euclid norm of the vector. The larger SDR corresponds to the closer **h** ^ to **h**.

**Spectral Distortion (SD):** Spectral Distortion (SD) is defined as standard deviation in log-amplitudes of two frequency spectra, as follows:

$$\text{SD}\left[\mathbf{A},\hat{\mathbf{A}}\right] = \sqrt{\frac{1}{N\_f} \sum\_{k=0}^{N\_f - 1} 20 \log\_{10} \left| \frac{\mathbf{A}(k)}{\hat{\mathbf{A}}(k)} \right|} \quad [\text{dB}], \tag{20}$$

where **A** and **A**^ are the frequency amplitude spectrum of the original and the reconstructed responses, respectively, and *A k*ð Þ and *A k* ^ð Þ are the *<sup>k</sup>*-th components of the vectors **A** and **A**^ , respectively. The value of *N <sup>f</sup>* is the number of frequency bin closest to 20 kHz. The smaller SD corresponds to the closer **A**^ to **A**.

Calculating the SDRs for the domains F and L, the corresponding original impulse responses are ones constructed with its minimum phase approximation, which are different from the ones in the domains I, C and CL. It is noted that the SDRs in each domain were computed as how much the reconstructed impulse response differs from the desired one. Such a treatment was not related to the calculation of SDs since the SD is defined by using only magnitude of the original and the reconstructed HRTFs.
