# Mapping Phonation Types by Clustering of Multiple Metrics
## 2 Methods
### The Choice of Observation Sets  
In this original data set, the \(i\)-th observation \(C\) is a list of several parameters, denoted as:  
\[C_{i}=(f_{\rm o},\text{SPL},\ m_{1},m_{2},\ldots,m_{in}) \tag{3}\]  
where \(n\) is the number of metrics (EGG and/or acoustic metrics, denoted as \(m\)). The grand total number of cycles is 3,732,124. The first two metrics, \(f_{\rm o}\) and SPL, are the indices for each observation on the voice map. \(f_{\rm o}\) is in semitones on the MIDI scale (60 = 261 Hz) and ranges from 30 to 90. SPL is in the range of 40-120 dB. We define a cell on the voice map as 1 semitone \(\times\) 1 dB.  
We then define two kinds of observations: (a) the cycle-by-cycle observations of metrics \(m_{1}\),..., \(m_{n}\) extracted directly from each phonated cycle are denoted as \(C_{i}^{cycle}\), defined as:  
\[C_{i}^{cycle}=[m_{i1},m_{i2},\ldots,m_{in}], \tag{4}\]  
and (b) the cell-by-cell averaged observations \(C_{j}^{cell}\) on the integer grid of \(f_{\rm o}\)\(\times\) SPL,  
\[C_{j}^{cell}=[\overline{m}_{j1},\overline{m}_{j2},\ldots,\ \overline{m}_{jn}], \tag{5}\]  
\[\overline{m}_{jn}=\frac{1}{p}\sum_{a=1}^{p}m_{an}, \tag{6}\]  
where \(p\) is the count of observations that lie inside each grid unit. The grand total number of cells is 43,235. Each cell typically contains a mean of some 100 cycles.  
In (a), each recording provides on the order of 100,000 observations of phonatory cycles per participant, while in (b), we have one observation per cell. On a map of the full voice range, there are on the order of 1000 cells, where each cell contains the metric averages for a number of cycles, which is on the order of 100. The latter method is much faster, but the per-cell averaging causes some loss of information. The tradeoff in quantityand computational time was assessed by comparing the classification results from these two types of observations.  
One set of K-means centroids was trained on the cycle-by-cycle observations, assigning a cluster number to each _cycle_. On the subsequent binning into cells, however, a given cell can receive points from more than one cluster, because the same \(f_{\rm o}\) and SPL can be produced with different kinds of phonation. In this case, with cycle-by-cycle clustering, the cell is displayed as belonging to the cluster that attracted the majority of cycles in that cell. For the faster method of clustering, each _cell_ is directly assigned a cluster number on the basis of the metric means that it contains.  
The agreement of the two methods for assigning cells to clusters was 94% (\(\pm 2\%\), \(p<0.05\)). This degree of agreement is acceptable because, in the default voice map, the \(f_{\rm o}\) ranges from 30 to 90 in semitones, and the SPL ranges from 40 to 120 in dB, i.e., a map of \(60\times 80\) units. An error rate of 6% on the sporadically distributed map is not easily discernible by the naked eye. Additionally, we are less concerned about to which cluster any particular SPL \(\times f_{\rm o}\) cell is assigned. Instead, the actual distribution and the changes in distribution and centroids should be focused on, to assess the stability of the K-means algorithm.  
Computations were much faster when using the voice map cell-by-cell representation rather than the data from the individual cycles. Even though K-means is known for its high-speed calculation, it is less efficient when the total number of observations reaches millions, which it does here when clustering groups of participants cycle-by-cycle. The chosen voice map representation instead holds one observation per cell, resulting in about a thousand observations per participant for a full voice range.