# A Weighted-LSM Method to Improve Classification and Concentration Evaluation from Laser-Induced Fluorescence Spectra

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## Abstract

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## 1. Introduction

## 2. Theory of the Method

#### 2.1. Laser-Induced Fluorescence

_{0}), jump to an excited state (S

_{1}) due to the energy received from an electromagnetic wave [1,10]. The following return to the ground state of electrons takes place with the emission of energy in form of the electromagnetic wave as luminescence [1]. In fluorescence phenomena, the processes that take place during the absorption and the following emission of light can be represented by the Jablonsky diagram [1].

#### 2.2. Classical Least Square Minimisation Method (C-LSM)

#### 2.3. Weighted LSM Method Based on Feature Differences (W_{DIF}-LSM)

## 3. Numerical Analyses

- Database spectra generation by analytical equations;
- Generation of the “measured” spectrum;
- Noise addition to simulate a real spectrum;
- Spectrum analysis with C-LSM and W
_{DIF}-LSM.

#### 3.1. Method

_{DIF}-LSM algorithms were applied to the spectrum.

_{DIF}-LSM algorithm, the gain of efficiency function (G) was calculated for the W

_{DIF}-LSM algorithm, as follows:

_{DIF}-LSM algorithm has better efficiency versus the C-LSM algorithm.

#### 3.2. Results of the Numerical Tests

_{DIF}-LSM. It is worth mentioning that the average value does not considerably differ from the expected.

_{DIF}-LSM respect to C-LSM as a function of the variations in parameter A simulating two spectra with a ${f}_{sim}$ = 77.9% with the same concentrations (5 μM). The figure shows that:

- For very small A parameters, W
_{DIF}-LSM places importance on a small number of features (only the ones where the database spectra strongly differ). This makes the inversion excessively sensitive to noise. - For values of A ranging from 0.3 to 3, the gain is greater than 1, showing that the efficiency of W
_{DIF}-LSM is better. - When parameter A tends to higher values (>3), the weight matrix becomes ineffective, and the results converge to the classical LSM result (G = 1).

_{DIF}-LSM to increase the accuracy. For larger wavelengths, i.e., ${\lambda}_{s}$ > 500 nm, the spectra is so different that the inversion is almost perfect. The influence of A confirms what was observed in Figure 3.

_{DIF}-LSM allows the increase in the performances in the most challenging cases (similar spectra), while W

_{DIF}-LSM converges to C-LSM in the case of very dissimilar spectra.

_{DIF}-LSM algorithm results in all the cases and for each value of parameter A higher than the efficiency of the C-LSM algorithm.

_{DIF}-LSM algorithm. In particular, there is a range of parameter A values; for those values, a higher peak of efficiency in this algorithm is revealed if compared to the C-LSM one.

#### 3.3. Reconstruction Error: A Quality Identifications and Measurements Indicator

## 4. Preliminary Experimental Analyses

#### 4.1. Materials and Methods

- BC spore solution with a concentration of 0.8 × 10
^{9}spores per mL; - RF solution with a concentration of 2.6 × 10
^{−4}M.

- For the RF database spectrum, the fluorescence spectra were acquired at two different concentrations: 1.33 × 10
^{−7}M and 2.66 × 10^{−8}M. - For the BC database spectrum, the fluorescence spectra were acquired at two different concentrations: 0.04 × 10
^{9}spores per mL and 0.008 × 10^{9}spores per mL.

_{DIF}-LSM obtained through Equation 24 using the spectrum data reported in Figure 7a. It is important to see how, when the two signals intersect having the same intensity ($\lambda <500\mathrm{nm}$, Figure 7a), the same point in the weight matrix (Figure 7b) is equal to 0 because there are not differences between the two Bas spectra. At the same time, when the spectra signals are equal to 0 ($\lambda >750\mathrm{nm}$, Figure 7a), the values of weight matrix (Figure 7b) are equal to 0 because there are no differences.

#### 4.2. Results of the Preliminary Experimental Tests

_{DIF}-LSM results) in comparison with the expected concentration (black line in each plot) as a function of parameter A. For each concentration’s value, the uncertainty measurements calculated through the uncertainty propagation theory are reported [13]. In particular, the relative error on concentration measurements made by both algorithms (Equation (19)) is between 13% and 36%. These large errors are mostly due to two different limits. First, a small portion of elastic scattering is still present, and it makes the reconstruction not perfect (error ranging from 2% to 10%). Moreover, the spectrometer has a nonlinear response (a doubling of the intensity does not involve a doubling of the measured counts). Therefore, these two factors reduced the quality of the reconstruction and the extrapolation of the concentrations.

_{DIF}-LSM converges to C-LSM.

## 5. Conclusions

_{DIF}-LSM) and their behavior simulating the conditions in which the algorithms must operate. As expected, it was possible to observe the presence of an influence represented by the background noise on the tested algorithm.

_{DIF}-LSM, it is possible to see how the efficiency of this algorithm is higher than the efficiency of C-LSM. In particular, it is possible to see how the regularization of parameter A plays an important role in the algorithm’s performance. For very small values of A, W

_{DIF}-LSM does not place importance on many wavelengths. The information used to extract the concentration decreases and thus the uncertainties increase. Increasing the A values, the algorithm finds a balance between “weighting all intensities” and “weighting only the most relevant” and the inversion performances increases significantly. For very large A values, W-LSM converges to C-LSM.

_{DIF}-LSM, is able to return higher gains when the problem is more complex, i.e., when the two spectra are more similar. This is of course due to the fact that, when the spectra are strongly different, C-LSM obtains very high performances, which is probably impossible to improve.

_{DIF}-LSM algorithm did not have appreciable advantages (demonstrated also by the numerical analysis). Secondly, the spectrometer has a non-linear uncalibrated response that makes the spectra counts not directly proportional to the concentration of the agents. At last, a small amount of scattering radiation was still present, making the spectra harder to analyze. However, the last two points are typical problems in real applications that can not always be excluded. Therefore, it is important to note that, in such unfavorable conditions, even if the results are not perfect, the concentration estimation is correlated to the expected ones and that no absurd results were obtained (see Figure 9).

_{DIF}-LSM may truly increase the performance of the concentration extrapolation, allowing more reliable and accurate results.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Concentration measurements made by the C-LSM algorithm varying the percentage of noise value. In the plot are reported the increment in the uncertainty in concentration measurements as a function of the increase in noise level.

**Figure 2.**Efficiency gain (P) of the C-LSM algorithm versus the W

_{DIF}-LSM algorithm. In the plot are reported the trend of the efficiency gain through the two algorithms as a function of the increase in the fundamental parameter A.

**Figure 3.**Influence of the spectrum shapes and A on W

_{DIF}-LSM varying σ (

**a**–

**c**) and ${\lambda}_{s}$ (

**b**–

**d**) of spectrum 2.

**Figure 4.**Influence of the similitude factor on the W

_{DIF}-LSM algorithm. Figure (

**a**–

**c**) shows the efficiency gain results (G) between the two algorithms as a function of the similitude factor (${f}_{sim}$) obtained from the parametric study on σ. Figure (

**b**–

**d**) shows the efficiency gain results (G), for the parametric study on ${\lambda}_{s}$, which is reported as a function of the relative variation of the similitude factor (${f}_{sim}$). Every test was conducted varying parameter A.

**Figure 5.**Influence of concentration differences on the efficiency gain (P) between the two algorithms. The plot describes the parametric study on concentration variations and its effect on the accuracy of the W

_{DIF}-LSM algorithm as a function of the parameter A variation, using for every test the same couple of synthetic spectra.

**Figure 6.**Experimental apparatus for preliminary tests of LIF classification and measurements using the supposed algorithms.

**Figure 9.**Concentration measurement results for each biological agent (Riboflavin (

**a**) and Bacillus clausii (

**b**)) for each sample. In particular, in each plot, the expected concentrations of each biological agent in each sample (black star) are plotted with its bar of uncertainty value related to the dilution and mixture preparation. The concentrations of each agent calculated by C-LSM are reported in blue, while the concentrations of each agent calculated by W

_{DIF}-LSM are shown in red. Each calculated concentration value is indicated with its error bar, which is the uncertainty of the value calculated as the product between the reconstruction error and the concentration calculated by the algorithms.

**Table 1.**Sample composition used for experimental tests and exposure time used for each acquisition.

Samples | RF Concentration [μM] | BC Concentration [10 ^{6} Spores/mL] | Exposure Time [s] |
---|---|---|---|

Sample C01 | 0.132 | 0 | 5 |

Sample C02 | 0.026 | 0 | 5 |

Sample C03 | 0 | 40 | 5 |

Sample C04 | 0 | 8 | 5 |

Sample C05 | 0.066 | 20 | 5 |

Sample C06 | 0.066 | 40 | 5 |

Sample C07 | 0.132 | 40 | 5 |

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**MDPI and ACS Style**

Gabbarini, V.; Puleio, A.; Rossi, R.; Malizia, A.; Gaudio, P.
A Weighted-LSM Method to Improve Classification and Concentration Evaluation from Laser-Induced Fluorescence Spectra. *Sensors* **2022**, *22*, 7721.
https://doi.org/10.3390/s22207721

**AMA Style**

Gabbarini V, Puleio A, Rossi R, Malizia A, Gaudio P.
A Weighted-LSM Method to Improve Classification and Concentration Evaluation from Laser-Induced Fluorescence Spectra. *Sensors*. 2022; 22(20):7721.
https://doi.org/10.3390/s22207721

**Chicago/Turabian Style**

Gabbarini, Valentina, Alessandro Puleio, Riccardo Rossi, Andrea Malizia, and Pasqualino Gaudio.
2022. "A Weighted-LSM Method to Improve Classification and Concentration Evaluation from Laser-Induced Fluorescence Spectra" *Sensors* 22, no. 20: 7721.
https://doi.org/10.3390/s22207721