Automatic Analysis of Composite Physical Signals Using Non-Negative Factorization and Information Criterion

In time-resolved spectroscopy, composite signal sequences representing energy transfer in fluorescence materials are measured, and the physical characteristics of the materials are analyzed. Each signal sequence is represented by a sum of non-negative signal components, which are expressed by model functions. For analyzing the physical characteristics of a measured signal sequence, the parameters of the model functions are estimated. Furthermore, in order to quantitatively analyze real measurement data and to reduce the risk of improper decisions, it is necessary to obtain the statistical characteristics from several sequences rather than just a single sequence. In the present paper, we propose an automatic method by which to analyze composite signals using non-negative factorization and an information criterion. The proposed method decomposes the composite signal sequences using non-negative factorization subjected to parametric base functions. The number of components (i.e., rank) is also estimated using Akaike's information criterion. Experiments using simulated and real data reveal that the proposed method automatically estimates the acceptable ranks and parameters

Estimated parameters.

<p>The estimated time constant (<i>τ_r</i>) and existence ratio (<i>h_r</i>) are shown as the mean of the results shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032352#pone-0032352-g003" target="_blank">Fig. 3</a> (<i>R_s</i>, <i>dim</i>) = (3, 145). The results obtained by BzNMF + AIC and BzNMF + AICc are the same when the optimization criterion is the same.</p

Rank estimation results for different ranks and sample dimensions.

<p>The simulation rank <i>R_s</i> and the dimension of the signal sequence (<i>dim</i>) are set to <i>R_s</i> = {2, 3, 4, 5} and <i>dim</i> = {75, 145, 715, 1,430, 7,150}, respectively. The input matrix is constructed from 50 signal sequences in a set. The ranks are estimated by three sets of input matrices. The blue, yellow, green, and purple bars show the mean of estimated ranks in <i>R_s</i> = 2, <i>R_s</i> = 3, <i>R_s</i> = 4, and <i>R_s</i> = 5, respectively. The red error bars show the maximum and minimum estimated ranks.</p

Comparison of computation times and estimated ranks.

<p>The computation times (CPU times) and the estimated ranks are evaluated using three sets of input matrices, similar to the case for <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032352#pone-0032352-g003" target="_blank">Fig. 3</a> (<i>dim</i> = 145). Parameter <i>k</i> in CV is set to 3.</p

Decomposition results for the Rh6G signal sequence.

<p><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032352#pone-0032352-g005" target="_blank">Figure 5</a> shows an example of the decomposition results for one signal sequence. The input matrix is the Rh6G measurement data in aqueous solution and consists of 54 signal sequences. The signal sequence is represented by a 92-dimensional vector. The rank was estimated to be 3 using the AIC. The open circles, the solid line, and the broken lines show the input signal sequence, the approximated signal sequence, and the decomposed components, respectively.</p

Rank estimation results by AIC and AICc.

<p>The input matrix setting is the same as <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032352#pone-0032352-g001" target="_blank">Figure 1</a>. The experimental results are obtained from one set of input matrices (50 signal sequences). The AIC and AICc are optimized in the LSE. The solid line and the broken line show the results obtained by the AIC and the AICc, respectively.</p

Error rates of parameters estimated by BzNMF + AIC optimized in the LSE.

<p>The simulation rank <i>R_s</i> and the dimension of signal sequence (<i>dim</i>) are set to <i>R_s</i> = {2, 3, 4, 5} and <i>dim</i> = {145, 715, 1,430, 7,150}, respectively. The input matrix is constructed from 50 signal sequences in a set. The parameters (existence rate and time constant) are estimated from the three sets of input matrices. The error rates of the parameters are calculated from 50×3 signal sequences (error rate of existence rate) and three sets of input matrices (error rate of the time constant). The blue, yellow, green, and purple bars show the averaged error rates for <i>dim</i> = 145, <i>dim</i> = 715, <i>dim</i> = 1,430, and <i>dim</i> = 7,150, respectively.</p

Decomposition results for the simulated signal sequence.

<p><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032352#pone-0032352-g001" target="_blank">Figure 1</a> shows examples of the decomposition results for one signal sequence. The input matrix is rank 2 and consists of 50 vectors (signal sequences). The signal sequence is represented by a 75-dimensional vector. The open circles, the solid line, and the broken lines show the input signal sequence, the approximated signal sequence, and the decomposed components, respectively. (a) shows the decomposition results obtained using the NMF optimized in the LSE. The rank of (a) is assumed to be 2. (b) shows the decomposition result obtained using BzNMF + AIC optimized in the LSE. The rank of (b) is estimated to be 2 using the AIC.</p