Evolutionary Gaussian Decomposition

Abstract

We present a computational approach for performing the Gaussian decomposition (GD) of experimental spectral data, called evolutionary Gaussian decomposition (EGD). The key feature of EGD is its ability to estimate the optimal number of Gaussian components required to fit a target function, which can be any experimental functional dependence. The efficiency and robustness of EGD are achieved through the use of the differential evolution (DE) algorithm, which allows us to tune the performance of the method. Based on statistics from the independent trials of DE, EGD can determine the number of Gaussians above which further improvement in fit quality does not occur. EGD works by collecting statistics on local minima in the vicinity of the estimated optimal number of Gaussians, and, if necessary, repeats this process several times during optimization until the desired results are obtained. The method was tested using both synthetic spectral-like functions and measured spectra of photosynthetic pigments. In addition to the local minima statistics, the most significant factors that affect the results of the analysis were the median and minimum values of the cost function. These values were obtained for each different number of Gaussian functions used in the evaluation process.

1. Introduction

Experimental determination of many quantities in the natural sciences involves measuring the functional dependence of certain variables on a specific parameter. In physics, specifically in optics, this dependence is referred to as a spectrum and can be thought of as a set of intensity values for emitted or absorbed light particles. The shape of an optical spectrum is influenced by both the specific physical phenomena occurring within the studied object, as well as the spectroscopy measurement method and the characteristics of the recording device.

The presence of noise and distortions caused by instruments often makes it difficult to unambiguously interpret experimental spectra, especially when they include closely spaced bands. Technical analysis of spectra requires an accurate determination of peak numbers and shapes [1,2]. For example, in inorganic and organic chemistry, analytical formulas for simulating spectra are integral expressions defined by the optical response function of a sample. Semiclassical quantum theory can accurately predict the absorption line shape of any system of electronic transitions that interact with the vibrational modes of a molecule [3,4]. This line shape can be irregular, but, in most cases, it can be well approximated by a combination of Gaussian functions.

The Gaussian decomposition is widely used in various applications [5,6], including astrophysics [7,8,9,10,11], chromatography, spectroscopy [12,13,14,15,16,17,18,19], mass spectrometry [20], Raman spectroscopy [21], engineering [22], and optics design processing [23]. Fitting the spectra with Lorentzian functions is also used in X-ray diffraction analysis [24], overlapping signal decomposition [25], laser-induced breakdown spectroscopy [26,27,28], NMR [29], and Mössbauer spectroscopy [30].

The main challenge with Gaussian functions fitting is that the results of calculations are not unambiguous, since any given curve can theoretically be represented by an infinite number of different fitting functions. Therefore, the practical goal of using Gaussian decomposition is to estimate the minimum number of Gaussian components needed to accurately reproduce the modeled spectrum, while maintaining high fitting quality. If M is the number of independent Gaussian functions, the decomposition of a spectral curve can be seen as a nonlinear optimization problem with 3M fitting parameters. The development of evolutionary optimization algorithms [31,32,33,34,35] has made it possible to significantly advance the solution of applied problems, such as the decomposition of spectral curves. In particular, genetic algorithms [36] have been actively used to optimize fitting [25,29,37,38,39], as well as particle swarm optimization [40], machine learning [9,41], and deep learning [1].

A developed approach to solving the Gaussian decomposition problem involves using the differential evolution (DE) algorithm [42,43] to fit functions that represent the absorption spectrum profile of organic pigments. DE is a multiparametric metaheuristic technique designed to minimize a wide range of functions, particularly those that are nondifferentiable. The algorithm works on principle of natural selection. Initially, a population of different parameters sets of function parameters is generated, which are then modified during the algorithm’s evolution to ensure that the objective function reaches its global minimum. The most critical component of the algorithm is the strategy used to select the next generation of parameters. The main problem of all evolutionary optimization methods is the absence of a single effective and stable scheme that leads to a global optimum. Despite having some optimal settings available, additional convergence tests are still required for each specific problem.

Recently, we proposed a procedure that allows us to determine if DE has reached a local minimum and, after making partial modifications to the parameters of the population, guide the evolution of the algorithm towards the global optimum [44]. This procedure is not based on a specific strategy but rather serves as a supplement to any existing strategy. However, it turns out that a disadvantage of any optimization algorithm, such as falling into a local minimum, can be used to evaluate the quality of modeling. Specifically, based on the analysis of local minima statistics, this study has developed a criterion that allows us to determine the optimal number of Gaussians required to fit the target function.

The article is divided into the following parts: a statement of the problem is given in Section 2; a description of DE is given in Section 3; Section 4 is devoted to the strategy choice; Section 5 presents the results; Section 6 contains a discussion; and Section 7 concludes the paper.

2. Statement of the Problem

Let us consider an optical spectrum, where each frequency ${\omega }_i$ is associated with the intensity of the recorded signal $I\left({\omega }_i\right)$. The dependency of intensity on frequency, $I\left({\omega }_i\right)$, can be of any type. However, for a wide range of physical phenomena, the observed spectrum is typically considered to be a combination of Gaussian functions.

$$I\left({\omega }_i\right)~{\sum }_k{g}_k\left(A_k,{\sigma }_k,{\omega }_k^0,{\omega }_i\right),$$

(1)

$$g_k\left(A_k,{\sigma }_k,{\omega }_k^0,{\omega }_i\right)\equiv g_k\left(\omega \right)=A_k{e}^{-{\left(\omega -{\omega }_k^0\right)}^2/2{{\sigma }_k}^2},$$

(2)

Thus, the evaluation of Gaussian parameters $\left\{A_k,{\sigma }_k,{\omega }_k^0\right\}$ by fitting the spectra allows us to understand the phenomena under study. In fact, this approach is not only applicable to the study of physical phenomena; it has many applications in various fields of science where GDs occur.

Considering M as the number of Gaussian functions in the decomposition and x as the matrix of GD parameters, the simulated curve can be expressed as $G\left(\mathit{x}{,\omega }_i\right)={\sum }_{k=1}^M{g}_k\left(\omega \right)$. The comparison of the target and simulated data is carried out in accordance with the following expression:

$$f\left(\mathit{x}\right)={\displaystyle \frac{1}{K}}{\sum }_{i=1}^K{\left(G\left(\mathit{x}{,\omega }_i\right)-I\left({\omega }_i\right)\right)}^2,$$

(3)

where K is the number of frequencies ${\omega }_i$. Technically, any type of optimization algorithm can be used to minimize this expression (3), as long as it can find the correct solution. Thus, considering the results of benchmark testing of various optimization algorithms, DE was selected as the basis on which the EGD algorithm will be built.

Two main issues need to be addressed in order to achieve successful optimization. The first one is adjusting the performance of the DE algorithm when focusing on a specific problem. The second issue is developing a criterion to determine the optimal number of Gaussian functions for fitting. This latter point is particularly important, as it is obvious that any experimental curve with noise could be approximated to a given accuracy using an infinite number of Gaussians.

3. Differential Evolution

Let us consider the principles behind the operation of the classic DE algorithm, which has been widely used for both engineering and scientific applications [32,45]. It is designed to find the optimal solution for a function with a large number of variables. One of its advantages over other algorithms is that it does not require knowledge about the minima–maxima landscape of the function being minimized.

Several parameters can be adjusted in order to optimize the performance of the algorithm. These include the population size $Np=10N$ (where N is the number of free parameters), the number of generations g, the weighting factor $F\in \left[\mathrm{0,1}\right]$, and the crossover probability $Cr\in \left[\mathrm{0,1}\right]$. By adjusting these parameters, one can improve the efficiency of the algorithm for various type of problems. The number of simulation procedure calls $SPC=Np\ast g$.

The classic DE algorithm consists of a series of repetitive steps in which the best possible solution is found. At each step, a set of vectors in an N-dimensional space is generated. Among these vectors, the best one is selected and used to generate a new set of vectors. The best vector from the new set is then compared to the current best solution. There are five basic ways to create mutant vectors from the current ones, as follows:

$${\mathbf{v}}_j^g={\mathit{x}}_{r0}^g+F\left({\mathit{x}}_{r1}^g-{\mathit{x}}_{r2}^g\right)$$

(4)

$${\mathbf{v}}_j^g={\mathit{x}}_{best}^g+F\left({\mathit{x}}_{r1}^g-{\mathit{x}}_{r2}^g\right),$$

(5)

$${\mathbf{v}}_j^g={\mathit{x}}_{r0}^g+F\left({\mathit{x}}_{best}^g-{\mathit{x}}_{r0}^g\right)+F\left({\mathit{x}}_{r1}^g-{\mathit{x}}_{r2}^g\right),$$

(6)

$${\mathbf{v}}_j^g={\mathit{x}}_{best}^g+F\left({\mathit{x}}_{r1}^g-{\mathit{x}}_{r2}^g\right)+F\left({\mathit{x}}_{r3}^g-{\mathit{x}}_{r4}^g\right),$$

(7)

$${\mathbf{v}}_j^g={\mathit{x}}_{r0}^g+F\left({\mathit{x}}_{r1}^g-{\mathrm{x}}_{r2}^g\right)+F\left({\mathit{x}}_{r3}^g-{\mathit{x}}_{r4}^g\right).$$

(8)

where $j\in \left\{1,\dots ,Np\right\}$ of each member of the new population and $\left(r0\ne r1\ne r2\ne r3\ne r4\right)\in \left\{1,\dots ,Np\right\}$ of the current one.

After ${\mathbf{v}}_j^g$ is created, a crossover operation is performed. The trial vectors for a new generation are determined based on the crossover conditions, which are controlled by a single parameter $Cr$. There are two types of crossovers, namely binomial and exponential. Binomial crossover is applied to all elements of the vector, while exponential crossover works like a “while-do” loop; after the first replacement of the mutant vector, the remaining elements are taken from the previous vector.

At the selection, the best trial vector is compared with the best one of the current generation. The stopping point is determined by reaching a certain value of the cost function (usually a kind of “machine epsilon”) or the specified number of generations.

4. The Strategy Choice

As shown in [46], choosing the optimal strategy and tuning parameters is crucial for each task. There are many different variations of DE [32], but to achieve the best optimization performance, a proper combination of strategies and tuning parameters must be found.

Two types of crossover and five equations (Equations (4)–(8)) result in ten classic DE strategies. The efficiency of each strategy was evaluated by estimating the convergence rate during the fitting of simulated and experimental data. In these simulations, we preferentially use four strategies (DE/best/1/exp, DE/rand-to-best/1/exp, DE/best/1/bin, and DE/rand-to-best/1/bin) that have shown the best performance in previous studies [44,46].

In addition to the strategy, the selection of tuning parameters also has a significant impact on the results. The variability in the new generation is closely linked to F and Cr. F influences the relative diversity of each individual variable, while Cr determines the total number of variables that can be modified. If the level of variability is either too low or too high, the algorithm may take an excessively long time to find the optimal solution, or it may become stuck in a local minimum. Due to the narrow range of acceptable settings for many strategies, it is essential to conduct thorough testing, as it is impossible to predict the optimal settings in advance.

5. Results

To evaluate the effectiveness of EGD, two types of objective functions, namely synthetic and experimental, were used. Synthetic curves are generated by combining a set of Gaussian curves with different parameters. It is expected that when fitting these curves, the resulting parameters will match the parameters used in the generation of the synthetic data. This type of modeling is used to study the behavior of the algorithm and the fine-tuning of its performance.

The experimental curves are absorption spectra obtained directly from a spectrophotometer. These spectra do not necessarily follow a simple Gaussian distribution and may contain a significant percentage of noise, making them more challenging to interpret. By using both synthetic and experimental data, we can explore both idealized scenarios and real-world challenges that the EGD algorithm may encounter.

5.1. Synthetic Data Fitting

5.1.1. One and Two Gaussians

Analyzing data that are modeled using one or two Gaussian functions may seem like a simple task at first, but when processing spectrophotometry results, even this simple problem requires careful calculations [47]. Therefore, this example serves as a starting point for this research. The fitting results of the simulated spectra using one (A) and two (C) Gaussians are presented in Figure 1. The strategy DE/rand/1/bin is used with $F=0.7$ and $Cr=0.95$ for both cases. The number of generations is 200 for (A) and 500 for (C). The parameters of the Gaussian curve for (A) are $A=130, \sigma =150, \mu =1250$. The parameters of Gaussian curves for (C) are $A_1=100$, ${\sigma }_1=130$, ${\mu }_1=1100$, $A_2=120$, ${\sigma }_2=100$, ${\mu }_2=1400$. The corresponding dynamics of the cost function are shown on the right.

5.1.2. Gaussians with Almost Identical Frequencies

The next step of the research was to determine the extent to which the algorithm can distinguish between closely spaced Gaussian functions and whether there is a limit to its ability. Since the combination of two closely located Gaussians forms a function that resembles a single Gaussian, it is tempting for the optimization algorithm to try to take the easier route out and fit the combined objective function with a single Gaussian.

Based on strategy tests, the DE/rand-to-best/1/exp strategy was used with $F=0.7$ and $Cr=0.95$ in both cases. The number of generations was 5000, 15,000, and 30,000. Calculations were performed with six free parameters, i.e., two Gaussian functions with three parameters each (Figure 2). The intensities of the Gaussians were $A_1=A_2= 100$. Cases where the algorithm clearly identified two Gaussians were marked as “+”, and those where it failed were marked “−”. A case with two Gaussians with ${\sigma }_1={\sigma }_2= 100$, ${\mu }_1=1000,$ and ${\mu }_2=1050$ using 30,000 generations is presented in Figure 2B,C. A similar case with 150,000 generations is shown in Figure 2D.

Based on the panels of Figure 2A, it can be seen that the wider the distance between the peaks, and the narrower their width, the easier it is for the algorithm to find the specified Gaussians. As can be seen, in the vicinity of 75,000 generations (450,000 simulation procedure calls) the algorithm finds the correct solution, resulting in the target function moving towards machine epsilon (~10⁻³²). Therefore, 150,000 generations were sufficient for the algorithm to accurately determine two closely located Gaussians. At some point, the cost function starts to converge rapidly towards machine epsilon.

5.1.3. Test with a Set of Gaussian Curves

Let us now evaluate whether EGD can achieve an acceptable result for a combination of multiple Gaussians. In Figure 3, the fitting results of the simulated spectra using eight equidistant identical Gaussian functions (A) and the eight different Gaussians (C) are presented. The Gaussian parameters are given in

Table 1. Parameters of two datasets used to generate synthetic spectral-like curves to test the EGD method.

	Test Curve 1			Test Curve 2
K	${\mathit{A}}_{\mathit{k}}$	${\mathit{\sigma }}_{\mathit{k}}$	${\mathit{\omega }}_{\mathit{k}}^0$	${\mathit{A}}_{\mathit{k}}$	${\mathit{\sigma }}_{\mathit{k}}$	${\mathit{\omega }}_{\mathit{k}}^0$
1	100	100	400	150	120	400
2	100	100	650	85	150	650
3	100	100	900	90	70	900
4	100	100	1150	120	300	1150
5	100	100	1400	100	110	1400
6	100	100	1650	105	70	1650
7	100	100	1900	70	180	1900
8	100	100	2150	95	200	2150

. The DE/rand/1/bin strategy was used, with $F=0.9$ and $Cr=0.95$ for the equal frequencies set, and $F=0.7$ and $Cr=0.95$ for the unequal set. The number of generations was 10,000 for both cases. The corresponding dynamics of the cost function are shown on the right.

To determine the optimal number of Gaussian functions for the synthetic spectrum, calculations were performed where the fitting was performed using different combinations of Gaussians with the number of functions varying from 4 to 12. The number of independent runs was 25, and the results for medians and means are shown in Figure 4.

The results show that with a smaller number of Gaussians, there will be no complete overlap between the curves. However, with a larger number, the algorithm will find a better solution, although it will be significantly slower due to the increased number of parameters in the system. This is because it takes longer to solve the optimization problem due to overdetermination. Therefore, if each Gaussian function contributes significantly and can be distinguished from the others, the DE algorithm will still be able to find this solution using the same set of parameters.

5.2. Experimental Data Fitting

In order to extend the applicability of the EGD algorithm for fitting experimental data, simulations were made for absorption spectra of chlorophyll a, bacteriochlorophyll a, spheroidene, and spheroidenone [17]. The chemical structure of the studied pigments and their absorption spectra are presented in Figure 5. An additional challenge of such modeling is the unknown optimal number of Gaussians, which requires the development of a certain criterion for the fitting procedure. Taking into account the results of the preliminary simulations, the process for determining the optimal number of Gaussians for the experimental data can be algorithmized as follows.

Evolutionary Gaussian Decomposition Algorithm

This algorithm is implemented in the following way (Algorithm 1). Before simulation starts, a guess number for the optimal number of Gaussians, $M_0$, is chosen. It can take any value, but it is better to choose a reasonable number that roughly corresponds to the shape of the modeled curve. Then the search range of integer numbers $\left[M_0-\delta ,M_0+\delta \right]$ is set, where δ is the width of the range; here, $\delta =4$ appeared to be good enough for the test spectra. Then DE runs 25 times for each $M\in \left[M_0-\delta ,M_0+\delta \right]$ and the mean, median, and minimum values of the objective function are saved. When the cycle is finished, one of the three possible options occurs: (i) the optimum is within the given range; the optimal number of Gaussians in decomposition $M_{opt}$ is determined according to the developed criterium and optimization stops; (ii) the minimum value is on the left of $\left[M_0-\delta ,M_0+\delta \right]$ (i.e., at $M_0-\delta$), $M_0$ becomes $M_0-\delta$, and the optimization runs once again with the new search range; and (iii) the minimum value is on the right of $\left[M_0-\delta ,M_0+\delta \right]$, $M_0$ becomes $M_0+\delta$, and the optimization runs once again, as in (ii). The program runs until it reaches (i).

Algorithm 1. Pseudocode for the evolutionary Gaussian decomposition

1: Input: the objective function and the guess number of Gaussians $M_0$

2: Create the boundaries $\left[M_1, M_2\right]=\left[M_0-\delta ,M_0+\delta \right]$$, \mathrm{where} \delta \ge 1$$(\delta =4$)

3: if $M_0\le \delta$ then

4:  $M_1=1$,

5:   $M_2=M_0+\delta$,

6: else

7:   $M_1=M_0-\delta$

8:   $M_2=M_0+\delta$

9: end if

10: Fitting a curve with a different number of Gaussians from $M_1$$\mathrm{to} M_2$

11: do_next_statistics = true

12: while do_next_statistics do

13:  for  $i=M_1$$\mathbf{to} M_2$

14:  Creating statistic for a model with the current number of Gaussians

15:   for $\mathrm{j}=1$ to 25

16:      DE: initialization

17:      DE: main loop

18:      Save $f_{ij}\left(\mathit{x}\right)$

19:   end for

20:  Save the mean value ${Mean}_i$$\mathrm{for} f_{ij}\left(\mathit{x}\right)$

21:  Save the median ${Med}_i$$\mathrm{for} f_{ij}\left(\mathit{x}\right)$

22:  Save the minimum ${Min}_i$$\mathrm{for} f_{ij}\left(\mathit{x}\right)$

23:  Save statistic of local minima

24:  end for

25:  Analyze ${Mean}_i$$, {Med}_i$$, \mathrm{and} {Min}_i$

26:  if ${Mean}_{M_2}=$$\mathrm{minimum} ({Mean}_i$) then

27:     $M_1=M_1+\delta$,

28:     $M_2=M_2+\delta$,

29:  else

30:   if ${Mean}_{M_1}=$$\mathrm{minimum} ({Mean}_i$) then

31:       $M_1=M_1-\delta$,

32:       $M_2=M_2-\delta$,

33:     else

34:      if (Check statistic of the local minima $f_{ij}\left(\mathit{x}\right)$: if there is such $i=M_{opt}$ that

35:      for $\forall i<M_{opt}$ there are the same $f_{ij}\left(\mathit{x}\right)$ and $\forall i>M_{opt}$ all values of $f_j\left(\mathit{x}\right)$

36:      are different) then

37:      do_next_statistics = false

38:      else

39:      Vary slightly parameters of DE (F,Cr) and do another while code

40:      do_next_statistics = true

41:      end if

42:     end if

43:  end if

44: end while

45: Output: $M_1$$, M_2$$, f_{ij}\left(\mathit{x}\right)$$, {Mean}_i$$, {Med}_i$$, {Min}_i$

The results of the fitting are presented in Figure 6. The DE/rand-to-best/1/exp strategy was used with $F=0.6$ and $Cr=0.95$ for all calculations with 50,000 generations. Calculations were made for the number of Gaussians $N\in \left[2, 12\right]$, and the number of independent runs was 25. Means ${Mean}_i$, medians ${Med}_i$, and minima ${Min}_i$ are shown in Figure 6, and corresponding cost function values are presented in

Table 2. The statistics of $f\left(\mathit{x}\right)$ for different M values while fitting the chlorophyll a and bacteriochlorophyll a spectra.

	Chlorophyll a				Bacteriochlorophyll a
M	Mean	Median	Minimum	SD	Mean	Median	Minimum	SD
2	273.7382	68.9180	51.6628	430.8823	212.8532	91.2610	91.2608	206.3146
3	52.7892	49.5938	2.0770	73.4533	73.4819	21.5659	7.6429	119.1133
4	13.6238	2.0770	0.6624	20.8840	43.5470	5.3226	4.6018	113.1656
5	10.0222	0.6192	0.5385	19.2457	11.7928	5.0799	1.3315	22.7074
6	4.6872	0.4984	0.4416	13.7434	8.1310	1.3066	1.0411	21.2079
7	0.4670	0.4725	0.3827	0.0368	4.2458	1.2251	0.7876	15.5379
8	0.4496	0.4443	0.3580	0.0416	1.2482	1.1142	0.7465	0.8730
9	0.5369	0.4594	0.3571	0.4535	1.4332	1.0290	0.7245	1.2853
10	0.4458	0.4500	0.3617	0.0464	1.3725	1.1648	0.8140	0.8725
11	0.4795	0.4753	0.3772	0.0705	1.3162	1.1199	0.7983	0.5841
12	0.4589	0.4647	0.3768	0.0405	1.5879	1.0445	0.7645	1.2382

and

Table 3. The statistics of $f\left(\mathit{x}\right)$ for different M values while fitting the spheroidene and spheroidenone spectra.

	Spheroidene				Spheroidenone
M	Mean	Median	Minimum	SD	Mean	Median	Minimum	SD
2	2342.8376	2308.9890	1172.6170	776.6016	233.1206	164.2957	125.5443	165.0890
3	1124.2584	676.9159	230.3491	1062.3844	141.5495	122.3225	31.3535	176.3336
4	466.4389	104.4260	35.8062	759.1592	32.0442	17.4921	5.3045	34.2694
5	411.7361	18.4233	16.3121	765.5299	42.2465	16.6599	3.2486	51.1552
6	265.0876	10.4512	2.6606	736.5726	2.2583	0.9473	0.6401	3.1104
7	60.8308	2.6844	1.3607	235.9919	0.8035	0.7035	0.4939	0.5734
8	10.8681	2.1116	0.8876	41.2166	0.8102	0.6098	0.3550	0.7455
9	1.7094	1.6294	0.6613	0.7874	0.8764	0.5319	0.2399	0.8516
10	2.9402	2.0990	0.7176	2.9423	0.7540	0.5455	0.2532	0.8994
11	2.3697	1.9480	0.8178	2.0913	0.5967	0.5190	0.2181	0.3307
12	3.0945	1.7127	0.6203	3.1256	0.8200	0.6122	0.2082	0.7686

. The results indicate that the optimal number of Gaussian functions lies between 8 and 10. In contrast to the synthetic spectra, the minimum required number of Gaussians is not known beforehand and depends on the quality of the experimental data (signal-to-noise ratio).

Upon analyzing the results, an interesting pattern emerged: with a sufficient number of Gaussian functions, all solutions were unique, while with an insufficient number, groups of similar results were formed. This relationship is illustrated in more detail in Figure 7. The results suggest that the optimal number for chlorophyll is $M_{opt}=8$, while for the other three pigments, the optimal number is $M_{opt}=9$, which corresponds to the optimal range shown in Figure 6. An example of the best fit for each pigment can be seen in Figure 8, with the corresponding dynamics of the cost function shown on the right. Despite the significant differences in the initial generations, the algorithm achieves a good agreement between the experimental and simulated spectra by the end of the process for all pigments.

6. Discussion

6.1. Synthetic Data Fitting

In the synthetic data, a sharp decrease in the cost function can be observed when a sufficient number of Gaussian components is reached. This can be considered one of the indicators of an optimal value (Figure 4). With a number of Gaussian components less than in the synthetic spectrum (target function), the modeled spectrum cannot completely match the target one. For cases where the Gaussian functions are noticeably different from each other and their contributions to the spectrum are clearly visible, the optimal number equals the number of components in the target spectrum (Figure 3B).

When there are a large number of Gaussian components, some of them may be completely nullified or their intensities may be divided between several components, thereby fitting one Gaussian in the synthetic spectrum. If the number of Gaussian components significantly exceeds the number in the target spectrum, the situation becomes more complex. There may be a complete elimination of “extra” components or different variations in the intensity distribution, where each component will be significant.

6.2. Experimental Data Fitting

Similar calculations were made for the experimental spectra as target functions. One significant difference from previous fits is that the algorithm initially does not know how many Gaussian components are present in the spectrum, and it finds the optimal number during the modeling process.

For experimental spectra, the behavior of the cost function during the optimization differs from that of synthetic data. As the number of Gaussian components $M_i$ increases, the minimum ${Min}_i$ constantly decreases (Figure 6). This is because the experimental data always contain a noise component, and with each additional Gaussian function, the accuracy of the fit increases. It is also worth noting that when $M_i$ is clearly insufficient (Figure 6, range from 2 to 6), ${Mean}_i$, ${Med}_i$, and ${Min}_i$ decrease fast and then reach plateaus where they fluctuate slightly within the error bounds as the number of Gaussians is further increased. $M_{opt}$ is definitely located around the point when the plateau begins.

In analyzing the data, we tried to identify characteristics in the fitting process that could be used as a criterion for determining the minimum number of Gaussians needed. Parameters, such as the mean and minimum values of the object function, are used to assess the quality of the fitting. Interestingly, unlike DE, the genetic algorithm has been used more than once to optimize the decomposition of experimental optical spectra into Gaussian functions [29,37]. However, the main drawback of these studies is the use of the genetic algorithm itself and the lack of a clear criterion for determining how many Gaussian functions will be sufficient for the decomposition to be considered acceptable. In this study, an original criterion was developed based on estimating the frequency of local minima reached by the DE algorithm during optimization (see pseudocode). Figure 7 shows statistics on these local minima. When fitting experimental spectra with an insufficient number of Gaussian functions, many solutions form groups of local minima (bars in Figure 7). These groups disappear when the system is overdetermined, which can also be seen as an implicit indication of the optimal number of Gaussians. However, further study is needed to fully understand this feature. It is important to note that the disappearance of local minima often coincides with the alignment of the mean, median, and, in some cases, the minimum values of the objective function.

The decomposition of spectra into Gaussian components is widely used in research, especially in areas where the measured spectra are known to have a Gaussian line profile of a combination of Gaussians. For instance, there is a study that proposes a method for analyzing the spectra of neutral hydrogen in galaxy spectra [9]. This method combines derivative spectroscopy and a machine learning technique. Interestingly, the study compares the effectiveness of this method with traditional manual fitting of the spectra. Another method of identifying bands in spectra was proposed based on the Bayesian Monte Carlo algorithm [48], which demonstrated some success in detecting absorption lines with a low signal-to-noise ratio. Actually, Monte Carlo simulation is used in many applications to fit spectra [11,49]. It is important to note that evolutionary algorithms, such as DE, use Monte Carlo ideas at the initial stage of their evolution. However, they move on to a more sophisticated evolution of parameter modification. Evolutionary algorithms allow us to explore a large number of possibilities, especially if the computational procedures are not resource-consuming, such as Gaussian decomposition.

7. Conclusions

While investigating the convergence of the DE algorithm in solving problems related to modeling the optical response of organic pigments, we have encountered the classic issue of premature convergence, which is common for many types of optimization algorithms. Our recent research has led to the development of a specific procedure that can be applied in conjunction with the classic DE algorithm to significantly improve its performance in certain applications [44]. During this work, it became clear that even such problems as local minima can be utilized to analyze experimental data, as they provide valuable insights into the optimization process.

Thus, the idea of analyzing the frequency with which an optimization algorithm become stuck in local minima was incorporated into the proposed implementation of a widely used method known as Gaussian decomposition. By investigating the behavior of DE on both synthetic and experimental data, an evolutionary Gaussian decomposition (EGD) method was developed that allows us to estimate the minimum number of Gaussians required for the decomposition. This number represents the limit beyond which the quality of the fit for the studied spectrum or function does not significantly improve.

The main feature of EGD is its analysis of the statistics of reaching local minima (see Figure 7). EGD calculates the number of times optimization completes with the same objective function values. By comparing the mean, median, and minimum values of the objective function, we conclude that analyzing the statistic of local minima can be useful for determining the optimal number of Gaussians in a decomposition.