Contents<\/h3>\n\n- Application type<\/a>.<\/li>\n
- Data set<\/a>.<\/li>\n
- Neural network<\/a>.<\/li>\n
- Training strategy<\/a>.<\/li>\n
- Model selection<\/a>.<\/li>\n
- Testing analysis<\/a>.<\/li>\n
- Model deployment<\/a>.<\/li>\n<\/ol>\n<\/section>\n
\n1. Application type<\/h2>\n
This is a classification<\/a> project since the variable to be predicted is binary: gamma (signal) or hadron (background).<\/p>\n<\/section>\n\n2. Data set<\/h2>\n
The first step is to prepare the\u00a0dataset<\/a>, which serves as<\/span>\u00a0the source of information for the classification problem.<\/p>\nFor that, we need to configure the following concepts:<\/p>\n
\n- Data source.<\/li>\n
- Variables.<\/li>\n
- Instances.<\/li>\n<\/ul>\n
Data source<\/h3>\n
The data source<\/a> is the file gamma.csv<\/a>. It contains the data for this example in comma-separated values (CSV) format. The number of columns is 11, and the number of rows is 19020.<\/p>\nThe CORSIKA simulation program generated the data set, utilizing a Monte Carlo code to simulate extensive air showers.<\/p>\n
Additionally, it has been utilized to simulate the registration of high-energy gamma particles from the Cherenkov gamma telescope using the imaging technique.<\/p>\n
Variables<\/h3>\n
The variables are:<\/a><\/p>\n<\/section>\nShape features<\/h4>\n\n- length<\/strong> \u2013 Major axis of the ellipse (mm).<\/li>\n
- width<\/strong> \u2013 Minor axis of the ellipse (mm).<\/li>\n
- M3Long<\/strong> \u2013 Cube root of the third moment along the major axis (mm).<\/li>\n
- M3Trans<\/strong> \u2013 Cube root of the third moment along the minor axis (mm).<\/li>\n<\/ul>\n
Intensity\/concentration features<\/h4>\n\n- size<\/strong> \u2013 Log of total pixel content (photons).<\/li>\n
- conc<\/strong> \u2013 Ratio of the two highest pixel sums to total size.<\/li>\n
- conc1<\/strong> \u2013 Ratio of the highest pixel to total size.<\/li>\n<\/ul>\n
Geometry and orientation features<\/h4>\n\n- asym<\/strong> \u2013 Distance from highest pixel to center, projected on major axis (mm).<\/li>\n
- alpha<\/strong> \u2013 Angle of the ellipse\u2019s major axis relative to the origin vector (\u00b0).<\/li>\n
- distance<\/strong> \u2013 Distance from the origin to the ellipse center (mm).<\/li>\n<\/ul>\n
Target variable<\/h4>\n\n- class<\/strong> \u2013 Event type: gamma<\/em> (signal) or hadron<\/em> (background).<\/li>\n<\/ul>\n
\nInstances<\/h3>\n
Neural Designer randomly splits the instances<\/a> into 60% training (11,412), 20% validation (3,804), and 20% testing (3,804).<\/p>\nVariables distributions<\/h3>\n
We can calculate the distributions<\/a> of all variables. The following figure is the pie chart for the gamma or hadron cases.<\/p>\n![](\"https:\/\/www.neuraldesigner.com\/images\/gamma-pie-chart.webp\")
<\/p>\n
As we can see, most of the samples are gamma signals. The hadron class (background) represents the majority of events in the actual data.<\/p>\n
Input-target correlations<\/h3>\n
Finally, the input-target correlations<\/a> might indicate to us what factors most influence.<\/p>\n![](\"https:\/\/www.neuraldesigner.com\/images\/gamma-correlation.webp\")
<\/p>\n
The most correlated variables with the classification are alpha and dist, which refer to ellipse variables.<\/p>\n
Additionally, there are not many correlated variables, such as fConc1 and fM3Trans.<\/p>\n<\/section>\n\n3. Neural network<\/h2>\n
The second step is to choose a neural network<\/a>. In classification problems, it is usually composed of:<\/p>\n\n- A scaling layer.<\/li>\n
- Two perceptron layers.<\/li>\n
- A probabilistic layer.<\/li>\n<\/ul>\n
Scaling layer<\/h3>\n
The scaling layer<\/a> contains the statistics on the inputs calculated from the data file and the method for scaling the input variables.<\/p>\nHidden dense layer<\/h3>\n
The number of perceptron layers<\/a> is 1. This perceptron layer has 10 inputs and 10 neurons.<\/p>\nOutput dense layer<\/h3>\n
Finally, we will set the binary probabilistic method<\/a> for the probabilistic layer<\/a>, as we want the predicted target variable to be binary.<\/p>\nNeural network graph<\/h3>\n
The following figure is a graphical representation of this classification neural network<\/a>.<\/p>\nIn this diagram, yellow circles are scaling neurons, blue circles are perceptron neurons, and red circles are probabilistic neurons.<\/p>\n
![](\"https:\/\/www.neuraldesigner.com\/images\/gamma-neural-network.webp\")
<\/p>\n
The number of inputs is 10, and the number of outputs is 1.<\/p>\n<\/section>\n\n4. Training strategy<\/h2>\n
The fourth step is to set the training strategy<\/a>, which is composed of:<\/p>\n\n- Loss index.<\/li>\n
- Optimization algorithm.<\/li>\n<\/ul>\n
Loss index<\/h3>\n
The loss index<\/a> chosen for this application is the\u00a0normalized squared error<\/a>\u00a0with L2 regularization<\/a>.<\/p>\nThe error term<\/a>\u00a0trains the neural network using the\u00a0training instances<\/a> of the dataset<\/span>.<\/p>\nThe regularization term<\/a> makes the model more stable and improves generalization.<\/p>\nOptimization algorithm<\/h3>\n
The optimization algorithm<\/a> searches for the neural network parameters that minimize the loss index.<\/p>\nThe quasi-Newton method<\/a>\u00a0is chosen here.<\/p>\nTraining<\/h3>\n
The following chart illustrates how the training and selection errors decrease over the course of training epochs.<\/p>\n
![](\"https:\/\/www.neuraldesigner.com\/images\/gamma-quasi-newton.webp\")
<\/p>\n
The final values are training error = 0.547 NSE<\/b> (blue) and selection error = 0.550 NSE<\/b> (orange).<\/p>\n<\/section>\n\n5. Model selection<\/h2>\n
The objective of model selection<\/a> is to find the network architecture with the best generalization properties, which minimizes the error on the selected instances<\/a> of the data set.<\/p>\nMore specifically, we\u00a0aim to develop a neural network with a selection error of less than\u00a00.550 WSE<\/strong>, the value we have previously <\/span>achieved.<\/p>\nOrder selection<\/a> algorithms train several network architectures with different numbers of neurons and select the one with the smallest selection error.<\/p>\nThe incremental order<\/a> method starts with a few neurons and increases the complexity at each iteration.<\/p>\n<\/section>\n\n6. Testing analysis<\/h2>\n
The last step is to test<\/a> the neural network\u2019s generalization by comparing its predicted values with the observed ones.<\/p>\nROC curve<\/h3>\n
1. Application type<\/h2>\n
This is a classification<\/a> project since the variable to be predicted is binary: gamma (signal) or hadron (background).<\/p>\n<\/section>\n The first step is to prepare the\u00a0dataset<\/a>, which serves as<\/span>\u00a0the source of information for the classification problem.<\/p>\n For that, we need to configure the following concepts:<\/p>\n The data source<\/a> is the file gamma.csv<\/a>. It contains the data for this example in comma-separated values (CSV) format. The number of columns is 11, and the number of rows is 19020.<\/p>\n The CORSIKA simulation program generated the data set, utilizing a Monte Carlo code to simulate extensive air showers.<\/p>\n Additionally, it has been utilized to simulate the registration of high-energy gamma particles from the Cherenkov gamma telescope using the imaging technique.<\/p>\n The variables are:<\/a><\/p>\n<\/section>\n Neural Designer randomly splits the instances<\/a> into 60% training (11,412), 20% validation (3,804), and 20% testing (3,804).<\/p>\n We can calculate the distributions<\/a> of all variables. The following figure is the pie chart for the gamma or hadron cases.<\/p>\n As we can see, most of the samples are gamma signals. The hadron class (background) represents the majority of events in the actual data.<\/p>\n Finally, the input-target correlations<\/a> might indicate to us what factors most influence.<\/p>\n The most correlated variables with the classification are alpha and dist, which refer to ellipse variables.<\/p>\n Additionally, there are not many correlated variables, such as fConc1 and fM3Trans.<\/p>\n<\/section>\n The second step is to choose a neural network<\/a>. In classification problems, it is usually composed of:<\/p>\n The scaling layer<\/a> contains the statistics on the inputs calculated from the data file and the method for scaling the input variables.<\/p>\n The number of perceptron layers<\/a> is 1. This perceptron layer has 10 inputs and 10 neurons.<\/p>\n Finally, we will set the binary probabilistic method<\/a> for the probabilistic layer<\/a>, as we want the predicted target variable to be binary.<\/p>\n The following figure is a graphical representation of this classification neural network<\/a>.<\/p>\n In this diagram, yellow circles are scaling neurons, blue circles are perceptron neurons, and red circles are probabilistic neurons.<\/p>\n The number of inputs is 10, and the number of outputs is 1.<\/p>\n<\/section>\n The fourth step is to set the training strategy<\/a>, which is composed of:<\/p>\n The loss index<\/a> chosen for this application is the\u00a0normalized squared error<\/a>\u00a0with L2 regularization<\/a>.<\/p>\n The error term<\/a>\u00a0trains the neural network using the\u00a0training instances<\/a> of the dataset<\/span>.<\/p>\n The regularization term<\/a> makes the model more stable and improves generalization.<\/p>\n The optimization algorithm<\/a> searches for the neural network parameters that minimize the loss index.<\/p>\n The quasi-Newton method<\/a>\u00a0is chosen here.<\/p>\n The following chart illustrates how the training and selection errors decrease over the course of training epochs.<\/p>\n The final values are training error = 0.547 NSE<\/b> (blue) and selection error = 0.550 NSE<\/b> (orange).<\/p>\n<\/section>\n The objective of model selection<\/a> is to find the network architecture with the best generalization properties, which minimizes the error on the selected instances<\/a> of the data set.<\/p>\n More specifically, we\u00a0aim to develop a neural network with a selection error of less than\u00a00.550 WSE<\/strong>, the value we have previously <\/span>achieved.<\/p>\n Order selection<\/a> algorithms train several network architectures with different numbers of neurons and select the one with the smallest selection error.<\/p>\n The incremental order<\/a> method starts with a few neurons and increases the complexity at each iteration.<\/p>\n<\/section>\n The last step is to test<\/a> the neural network\u2019s generalization by comparing its predicted values with the observed ones.<\/p>\n2. Data set<\/h2>\n
\n
Data source<\/h3>\n
Variables<\/h3>\n
Shape features<\/h4>\n
\n
Intensity\/concentration features<\/h4>\n
\n
Geometry and orientation features<\/h4>\n
\n
Target variable<\/h4>\n
\n
Instances<\/h3>\n
Variables distributions<\/h3>\n
<\/p>\nInput-target correlations<\/h3>\n
<\/p>\n3. Neural network<\/h2>\n
\n
Scaling layer<\/h3>\n
Hidden dense layer<\/h3>\n
Output dense layer<\/h3>\n
Neural network graph<\/h3>\n
<\/p>\n4. Training strategy<\/h2>\n
\n
Loss index<\/h3>\n
Optimization algorithm<\/h3>\n
Training<\/h3>\n
<\/p>\n5. Model selection<\/h2>\n
6. Testing analysis<\/h2>\n
ROC curve<\/h3>\n
![](\"https:\/\/www.neuraldesigner.com\/images\/gamma-roc.webp\")
<\/p>\n