This study examines the available experiment data for a copper bromide vapor laser (CuBr laser), emitting in the visible spectrum at 2 wavelengths—510.6 and 578.2 nm. Laser output power is estimated based on 10 independent input parameters. The CART method is used to build a binary regression tree of solutions with respect to output power. In the case of a linear model, an approximation of 98% has been achieved and 99% for the model of interactions between predictors up to the the second order with an relative error under 5%. The resulting CART tree takes into account which input quantities influence the formation of classification groups and in what manner. This makes it possible to estimate which ones are significant from an engineering point of view for the development and operation of the considered type of lasers, thus assisting in the design and improvement of laser technology.
Metal vapor lasers, including copper and copper halide lasers, have long been recognized to possess unique properties and capabilities with wide area of applications [
This paper examines a copper bromide vapor laser which is from the class of copper halide vapor lasers. There is continuing interest in further improving the output characteristics of this laser and its applications [
As a set off to that, during the last few years, statistical models were developed and applied on the basis of accumulated experiment data. The models are in the form of explicit statistical relationships, dependencies, and classifications of the basis laser parameters. These give the opportunity to estimate the strength and the form of the relationship between the laser parameters. All this make it possible to direct the experiment towards increased output laser parameters and to make a preliminary estimate of experiment results using the models. Traditional parametric models of metal vapor lasers have been developed and analyzed in [
In this paper, another powerful nonparametric modeling method—CART (classification and regression trees)—is applied to available data for a copper bromide vapor laser. This method allows for the separation of all observations from the considered independent variables (predictors) in noninteracting groups in the form of a binary tree according to the degree of influence on the dependent variable, in this case, laser output power.
The objective of this study is to determine the influence 10 input laser characteristics (supplied electric power, geometric design of the tube, neon pressure, reservoir temperature, etc.) on the average output power based on available experiment data. For the first time, the powerful nonparametric technique CART, described in [
The results have been obtained using the CART software package [
The copper bromide vapor laser is an improved version of a pure copper vapor laser. It is the most powerful and effective laser in the visible spectrum demonstrating high coherence and convergence of the laser beam. We are investigating variations of this laser invented and developed at the Laboratory of Metal Vapor Lasers at the Georgi Nadjakov Institute of Solid State Physics of the Bulgarian Academy of Sciences, Sofia. The first patents related to this type of laser are [
Copper bromide vapor lasers are sources of pulse radiation in the visible spectrum (400–720 nm) emitting at two wavelengths: green, 510.6 nm, and yellow, 578.2 nm. They are considered to be high-pulse lasers. Neon is used as a buffer gas. In order to improve efficiency, small quantities of hydrogen are added. Unlike the high-temperature pure copper vapor laser, the copper bromide vapor laser is a low-temperature one, with an active zone temperature of about 500°C. The laser tube is made out of quartz glass without high-temperature ceramics as a result of which it is significantly cheaper and easier to manufacture. The discharge is heated by electric current (self-heating laser). It produces light impulses tens of nanoseconds long. Its main advantages are short initial heating period, stable laser generation, relatively long service life, high values of output power, and laser efficiency. A simple scheme of the laser is given in Figure
Structural diagram of the laser tube of a copper bromide vapor laser; 1—copper bromide reservoirs, 2—heat insulation of the active volume, 3—copper electrodes, 4—inner rings, and 5—mirrors.
The specific technical parameters of the investigated copper bromide vapor lasers are given in Table
Technical parameters of a copper bromide vapor laser [
Characteristic | Description |
---|---|
Emission wavelength | 510.6 and 578.2 nm |
Operating mode | Pulse-periodic, self-heating |
Pulse frequency | 10–125 kHz |
Average volume power density | 1.4–2 W/cm3 |
Measured temperature of the wall | 500°C |
Pulse length | 20–50 ns |
Average output power | 1–125 W |
Coefficient of efficiency (laser efficiency) | >1% |
Total service life | >1000 hours |
Pulse energy | 6.9 mJ |
Temperature of the active medium | 500–550°C |
Start time | 10–15 min |
Structural elements | Quartz tube, outer electrodes, |
This paper takes into account the following 10 independent input variables (predictors) and one dependent variable (response)—laser output power
The study uses the values of these variables taken from
The statistical summary for the whole set is given in Table
Descriptive statistics of CuBr laser experimental data.
Minimum | Maximum | Mean | Std. deviation | Skewness | Kurtosis | |||
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Statistic | Statistic | Statistic | Statistic | Statistic | Std. error | Statistic | Std. error | |
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Pout (W) |
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Valid |
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It should be noted that the variables are not normally distributed, which is observed from the values of asymmetry and excess. The same is valid for the multivariate distribution of the data. For this reason, nonparametric methods which have no requirements towards the type of data distribution, both as a whole and for subsets, are more suitable.
The CART method algorithm, as indicated by the name, solves the classification and regression problem. It was developed between 1974–1984 by Breiman et al. [
CART is a nonparametric solution tree technique which builds classification or regression trees depending on whether the dependent variable is categorical or numerical. In our case, this is a regression tree.
The algorithm is intended for the building of a binary solutions tree. The initial set of observations is divided into groups at the terminal nodes (leaves) of the tree. The goal is to find a tree which allows for a good distribution of the data with the lowest possible relative error of prediction. Each branch of the tree ends with one or two terminal nodes and each observation falls into exactly one terminal node, defined by a unique set of rules.
More specifically, the objective of the regression tree approach is to distribute the data in relatively homogeneous (with minimum least squares or minimum standard deviation) terminal nodes and to obtain a mean observed value at each node in the form of a predicted value. The building of a tree starts from a root node, containing all observations. At each step (at each running node) a rule is applied to divide the set of observations within the node into two subsets (two children) according to some condition for an independent variable (predictor)
Validation is usually applied when building regression trees, since they may be sensitive to random errors in the data. This helps diminish by “pruning” the initial tree, maintaining its regression characteristics and accuracy. In the case of fewer observations and variables, the use of the statistical method of cross-validation with V-fold is recommended. This validation technique in CART allows for the construction of very reliable models superior to standard regression models. In general case, CART applies the least squares splitting rule to build the maximal tree and a cross-validation procedure to select the optimal tree.
In this study, we have used the standard 10-fold cross-validation, recommended for small samples. The data have been randomly divided into 10 equal nonintersecting subgroups, each containing approximately 10% of the dataset. The tree has been built using 9/10 of the data (learn sample) and the remaining 1/10 (test sample) have been used for prediction and to determine the level of the error. The tree construction process is repeated 10 times and the average error of the 10 series is taken as a general estimate. This procedure ensures accurate estimation of the dependent variable and allows for the tree to be used for the classification or regression of another dataset.
The estimate
First we will build and analyze a linear model, that is, where the predictors are the independent variables participating only with their first degree, as described in Section
A CART model has been built in order to determine the relationship between laser output power and the 10 basis input laser variables. The minimum number of observations has been set at 10 for parent nodes and 5 for terminal nodes. It was established using a special feature Battery ATOM of the software CART [
Diagram of the relative errors in linear models for different minimum numbers of cases (ATM) in terminal nodes.
One more specific objective of our investigation is to build a tree which classifies and predicts well experiments with high values of output power. For this reason, further on we will concentrate on the node which contains the highest values of output power
In order to specify the tree and its reverse prune so as to find a tree with an optimal small relative error for the data, we apply the 10-fold cross-validation procedure described in Section
By setting the minimum number of the cases in the terminal nodes equal to 5 and 10 for the parent node, and setting the classification/regression criterion to least squares, an optimal regression tree is found. In practice, there exists a subset of trees that exhibit an accuracy performance statistically indistinguishable from the optimal tree. All of these models are candidates for optimal models too. This is called a “1 standard error” or 1 SE rule to identify these trees [
The curve of relative errors of generated models, including the optimal model with the smallest error is shown in Figure
Curve of the relative errors of linear CART models with 10 predictors.
The selected regression CART model with 27 terminal nodes accounts for
A detailed specific information about the hot spot node
Specific characteristics of the nodes with maximum values of output power
Figure
Statistics for a linear CART model of Pout with 27 terminal nodes and 10 predictors.
Terminal node | Minimum |
Maximum |
Mean | Observations | Splitting classification rules |
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1 | 0.4 | 2.8 | 159 | 17 |
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2 | 0.5 | 6.2 | 4.29 | 82 |
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3 | 12.8 | 19 | 15.43 | 14 |
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4 | 5.8 | 11.5 | 9.08 | 6 |
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5 | 1.6 | 10.8 | 6.92 | 5 |
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6 | 5 | 10.9 | 8.33 | 62 |
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7 | 0.25 | 8.27 | 3.95 | 22 |
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8 | 16.8 | 32 | 22.94 | 35 |
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9 | 36 | 51.8 | 46 | 11 |
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10 | 53 | 73 | 63.91 | 20 |
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11 | 70 | 90 | 80.33 | 6 |
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12 | 88 | 92 | 89.2 | 5 |
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13 | 60 | 90 | 74.2 | 5 |
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14 | 55 | 70 | 63.88 | 8 |
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15 | 35 | 50 | 44.29 | 7 |
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16 | 64 | 96 | 80.56 | 9 |
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17 | 90 | 102 | 97.83 | 6 |
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18 | 80 | 94 | 88.6 | 5 |
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19 | 90 | 104 | 100 | 6 |
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20 | 76 | 96 | 87.33 | 6 |
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21 | 98 | 112 | 107.5 | 6 |
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23 | 85 | 102 | 97 | 9 |
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24 | 58 | 82 | 68.5 | 8 |
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25 | 45 | 76 | 63.38 | 8 |
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26 | 57 | 82 | 73.83 | 6 |
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27 | 63 | 90 | 80.43 | 7 |
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Distribution of splitters for each node in the regression CART model with 10 predictors and 27 terminal nodes.
Experiment values of
In order to build a CART tree including up to second-degree polynomials, from the 10 independent variables we form 65 predictors of the following type:
Relative error curve for all generated CART trees using 65 predictors from (
To bring into comparison with the linear model we chose again a tree with 27 terminal nodes. It satisfies the selection criteria as in the linear case. More exactly, this model has 4.1% relative error (see Figure
Statistics for a CART model of Pout with 27 terminal nodes and 65 predictors.
Terminal node | Minimum Pout | Maximum Pout | Mean | Observations | Splitting classification rules |
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1 | 0.4 | 7 | 3.87 | 116 |
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2 | 12.8 | 19 | 15.43 | 14 |
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3 | 5.8 | 11.5 | 9.08 | 6 |
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4 | 0.25 | 10.9 | 7.89 | 72 |
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5 | 16.8 | 30.9 | 23.68 | 32 |
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6 | 23 | 41 | 31.4 | 5 |
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7 | 40 | 51 | 47.15 | 8 |
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8 | 51.8 | 58 | 55.4 | 5 |
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9 | 55 | 72 | 62.6 | 5 |
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10 | 63 | 81 | 71.67 | 12 |
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11 | 88 | 92 | 89.14 | 7 |
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12 | 64 | 90 | 76.4 | 5 |
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13 | 60 | 70 | 67.6 | 5 |
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14 | 53 | 62 | 57.67 | 6 |
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15 | 35 | 50 | 44.29 | 7 |
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16 | 64 | 91 | 78.29 | 8 |
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17 | 90 | 104 | 98.23 | 13 |
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18 | 96 | 112 | 105.67 | 6 |
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19 | 80 | 100 | 88.8 | 5 |
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21 | 68 | 98 | 86.67 | 6 |
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22 | 45 | 80 | 58.5 | 6 |
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23 | 96 | 102 | 99.33 | 6 |
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24 | 60 | 76 | 70.67 | 6 |
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25 | 76 | 90 | 84.13 | 8 |
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26 | 70 | 85 | 77.6 | 5 |
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27 | 57 | 77 | 66.33 | 6 |
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A detailed view of the hot spot nodes with the maximum values of
Specific characteristics of the hot spot nodes with maximum values of output power
The node with the highest values is number 20 (see Figure
The splitting rules for node 20 are as follows:
The general distribution diagram of the tree splitters according to variables is shown in Figure
Splitters for the CART tree with 27 terminal nodes for 65 predictors.
Figure
Experiment values for
In the obtained linear model of the 10 independent physical parameters, only 6 participate in the constructed regression tree. These defining parameters are
As shown in Figure
The analysis of the second-degree tree model (Figure
After reviewing the predictive capabilities of both models, we can conclude that the linear and second-degree models are almost equivalent. They describe quite well the various groups of classified cases and predict the values for the nodes with maximum output power within a relative error less than 5%. Since the second-degree model is the same in structure and better at predicting the group of higher output power values, it is recommended for engineering applications which aim at increasing output power. However, the results of both models can be combined for experiment planning. Another important comparison can be made with the models obtained using another powerful nonparametric technique—MARS. For the same data, second-degree MARS models also concur with 98-99% of the data, but are more precise in prediction of the output laser power than the CART models (see [
We will also discuss the influence within the models of the main parameters which define high
Influence of
Influence of
Influence of
Ensuring the combined action of these basic processes under the set conditions (
Regression models based on a CART tree, which classifies groups of similar experiments, have been built for a copper bromide vapor laser. The variables which play the main role in increasing laser output power have been identified for classified groups, as well as the intervals these should be within when conducting future studies and developing laser sources of the same type for improving laser technology.
This paper is published in cooperation with project of the Bulgarian Ministry of Education, Youth and Science, BG051PO001/3.3-05-0001 “Science and business” and financed under Operational program “Human Resources Development” by the European Social Fund.