This example assesses the classification of size measurements for adult foraging penguins near Palmer Station, Antarctica. We used data on 11 variables obtained during penguin sampling in Antarctica.
This data is obtained from palmerpenguins package
The predicted variable can have three values corresponding to a penguin species: Adelie, Gentoo, and Chinstrap. Therefore, this is a multiple classification project.
To model the probability of each sample belonging to a penguin specie is the goal of this example.
The penguin_dataset.csv file contains the data for this example. Target variables have three values in our classification model: Adelie (0), Gentoo (1), and Chinstrap (2). The number of rows (instances) in the data set is 334, and the number of variables (columns) is 11.
To start, we use all instances. Each row contains the input and target variables of a different sample. The data set is subdivided into training, validation, and testing. Neural Designer automatically assigns 60% of the samples for training, 20% for selection, and 20% for testing. This values can be modified by the user to the desired ones.
Also, we can calculate the distributions of all variables. The following pie chart shows how many species we have.
The image shows the proportion of each penguin species: Adelie (44.18%), Gentoo (36,04%), and Chinstrap (19.76%).
The inputs-targets correlations might indicate to us which factors most differentiate between a penguins and therefore be more relevant to our analysis.
Here, the most correlated variables with penguin species are date_egg, culmen_depth_mm, delta_13_C and delta_15_N.
The next step is to set a neural network as the classification function. Usually, the neural network is composed of:
The scaling layer contains the inputs scaled from the data file and the method for doing so. Here the method selected is the minimum-maximum. As we use ten input variables, the scaling layer has ten inputs.
The perceptron layer has 3 neurons, and for that has 9 outputs for earch neurons.
The probabilistic layer contains the method for interpreting the outputs of the inner layers as probabilities. The output layer's activation function is the Softmax so that the output can be interpreted as a probability of class membership. The probabilistic layer has ten inputs. It has three outputs, representing the probability of a sample belonging to a class.
The following figure represents the neural network:
The network has ten inputs, obtaining three output values as mentioned above. These values are the probability of class membership for each patient.
The fourth step is to set the training strategy, which is composed of two terms:
The following image shows how the error decreases with the iterations during the training process. The final training and selection errors are training error = 0.004 and selection error = 0.005, respectively.
The curves have converged, as we can see in the previous image. However, the selection error is a bit higher than the training error.
Order selection algorithms train several network architectures with different number of neurons. Then it chooses the one with the smallest selection error.
However, we are going to use input selection to select features in the data set that provide the best generalization capabilities.
As we see in the following image, we reduce the selection error by increasing a bit the training error, thus improving our model.
At the end we obtain a training error = 0.01 and selection error = 0.003. Also, we have reduced the inputs to four. Our network is now like this:
Our final network, has 4 inputs corresponding to: culmen_length_mm, culmen_depth_mm, body_mass_g, sex.
The objective of the testing analysis is to validate the generalization properties of the trained neural network. The method to validate the performancer of our model, we compare the predicted values to the real values, using a confusion matrix. The next table contains the values of the confusion matrix. The rows represent the real classes in the confusion matrix, and the columns are the predicted classes for the testing data.
|Predicted Adelie||Predicted Gentoo||Predicted Chinstrap|
|Real Adelie||22 (32.353%)||0||1 (1.471%)|
|Real Gentoo||0||31 (45.588%)||1 (1.471%)|
|Real Chinstrap||0||0||13 (19.118%)|
As we can see, we can classify 66 (97.1%) of the samples, while we fail to do so for 2 (2.9%) samples
Once we have tested the neural network's performance, we can save it for the future using the model deployment mode.
The mathematical expression represented by the neural network is written below.
scaled_culmen_length_mm = (culmen_length_mm-43.9219017)/5.443640232; scaled_culmen_depth_mm = (culmen_depth_mm-17.15119934)/1.969030023; scaled_body_mass_g = (body_mass_g-4201.75)/799.6129761; scaled_sex = sex*(1+1)/(1-(0))-0*(1+1)/(1-0)-1; perceptron_layer_1_output_0 = tanh( -0.246009 + (scaled_culmen_length_mm*-2.33036) + (scaled_culmen_depth_mm*1.0486) + (scaled_body_mass_g*0.333745) + (scaled_sex*0.50902) ); perceptron_layer_1_output_1 = tanh( 0.158568 + (scaled_culmen_length_mm*0.0934119) + (scaled_culmen_depth_mm*0.819869) + (scaled_body_mass_g*-0.889764) + (scaled_sex*-0.0657631) ); perceptron_layer_1_output_2 = tanh( -0.160659 + (scaled_culmen_length_mm*-0.0955243) + (scaled_culmen_depth_mm*-0.822452) + (scaled_body_mass_g*0.88955) + (scaled_sex*0.0689181) ); probabilistic_layer_combinations_0 = 0.180645 +2.35908*perceptron_layer_1_output_0 +0.497254*perceptron_layer_1_output_1 -0.499956*perceptron_layer_1_output_2 probabilistic_layer_combinations_1 = 0.319912 -0.425221*perceptron_layer_1_output_0 -1.43431*perceptron_layer_1_output_1 +1.43744*perceptron_layer_1_output_2 probabilistic_layer_combinations_2 = -0.505925 -1.93948*perceptron_layer_1_output_0 +0.935236*perceptron_layer_1_output_1 -0.938556*perceptron_layer_1_output_2 sum = exp(probabilistic_layer_combinations_0) + exp(probabilistic_layer_combinations_1) + exp(probabilistic_layer_combinations_2); Adelie = exp(probabilistic_layer_combinations_0)/sum; Gentoo = exp(probabilistic_layer_combinations_1)/sum; Chinstrap = exp(probabilistic_layer_combinations_2)/sum;