This is perhaps the best known example in the field of machine learning.

The aim is to classify iris flowers among three species (setosa, versicolor or virginica) from measurements of length and width of sepals and petals.

The iris data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant.

The central goal here is to design a model which makes good classifications for new flowers or, in other words, one which exhibits good generalization.

This is a classification project, since the variable to be predicted is categorical (setosa, versicolor or virginica).

The goal here is to model the probabilities of class membership, conditioned on the flower features.

The first step is to prepare the data set, which is the source of information for the classification problem. For that, we need to configure the next concepts:

- Data source.
- Variables.
- Instances.

The data source is the file iris_flowers.csv. It contains the data for this example in comma separated values (CSV) format. The number of columns is 5 and the number of rows is 150.

The variables are:

**sepal_length**: Sepal length, in centimeters, used as input.**sepal_width**: Sepal width, in centimeters, used as input.**petal_length**: Petal length, in centimeters, used as input.**petal_width**: Petal width, in centimeters, used as input.**class**: Iris Setosa, Versicolor or Virginica, used as target.

Note that neural networks work with numbers. In this regard, the categorical variable "class" is transformed into three numerical variables as follows:

- iris_setosa: 1 0 0.
- iris_versicolor: 0 1 0.
- iris_virginica: 0 0 1.

The instances are divided into a training, a selection and a testing subsets. They represent 60% (90), 20% (30) and 20% (30) of the original instances, respectively, and are splitted at random.

We can calculate the distributions of all variables. The next figure is the pie chart for the iris flower class.

As we can see, the target is well-distributed, since there are the same number of virginica, setosa and versicolor samples.

The second step is to choose a neural network. For classification problems, it is usually composed by:

- A scaling layer.
- Two perceptron layers.
- A probabilistic layer.

The neural network must have four inputs, since the data set has four input variables (sepal length, sepal width, petal length and petal width).

The scaling layer normalizes the input values. As the sepal and petal lengths and widths have normal distributions, we use the mean and standard deviation scaling method.

Here we use 2 perceptron layers:

- A first layer with 4 inputs, 3 neurons and logistic activation function.
- A second layer with 3 inputs, 3 neurons and logistic activation function.

The probabilistic layer allows the outputs to be interpreted as probabilities, i.e., all outputs are between 0 and 1 and their sum is 1. The softmax probabilistic method is used here.

The neural network has three outputs, since the target variable contains 3 classes (setosa, versicolor and virginica).

The next figure is a graphical representation of this classification neural network:

The fourth step is to set the training strategy, which is composed by:

- Loss index.
- Optimization algorithm.

The loss index chosen for this application is the normalized squared error with L2 regularization.

The error term fits the neural network to the training instances of the data set. The regularization term makes the model more stable and improves generalization.

The optimization algorithm searches for the neural network parameters which minimize the loss index. The quasi-Newton method is chosen here.

The following chart shows how the training and selection errors decrease with the epochs during the training process.
The final values are **training error = 0.005 NSE** (blue), and **selection error = 0.195 NSE** (orange).

The objective of model selection is to find the network architecture with best generalization properties, that is, that which minimizes the error on the selection instances of the data set.

More specifically, we want to find a neural network with a selection error less than **0.195 NSE**,
which is the value that we have achieved so far.

Order selection algorithms train several network architectures with different number of neurons and select that with the smallest selection error.

The incremental order method starts with a small number of neurons and increases the complexity at each iteration. The following chart shows the training error (blue) and the selection error (orange) as a function of the number of neurons.

As we can see, the order that yields the minimum selection error is 2. Therefore, we select the neural network with 2 neurons in the first perceptron layer.

The purpose of testing analysis is to validate the generalization performance of the model. Here we compare the outputs provided by the neural network to the corresponding targets in the testing instances of the data set.

In the confusion matrix the rows represent the targets (or real values) and the columns the corresponding outputs (or predictive values). The diagonal cells show the cases that are correctly classified, and the off-diagonal cells show the misclassified cases.

Predicted setosa | Predicted versicolor | Predicted virginica | |
---|---|---|---|

Real setosa | 10 (33.3%) | 0 | 0 |

Real versicolor | 0 | 11 (36.7%) | 0 |

Real virginica | 0 | 1 (3.33%) | 8 (26.7%) |

Here we can see that all testing instances are well classified, but one. In particular, the neural network has classified a flowers as virginica being versicolor. Note that the confusion matrix depends on the particular testing instances that we have.

Next the classification tests are listed. These values are directly calculated from the confusion matrix.

**Classification accuracy: 3.33%**.**Error rate: 96.67%**.

The neural network is now ready to predict outputs for inputs that it has never seen. This process is called model deployment.

To classify a given iris flower, from the lengths and withs of its sepals and petals, we calculate the neural network outputs. For instance:

- Sepal length: 5.84 cm.
- Sepal width: 3.05 cm.
- Petal length: 3.76 cm.
- Petal width: 1.20 cm.
**Probability of Iris Setosa: 22 %.****Probability of Iris Versicolor: 57 %.****Probability of Iris Virginica: 21 %.**

For this particular case, the neural network would classify that flower as being of the virginica specie, since it has the highest probability.

The mathematical expression of the trained neural network is listed below.

scaled_sepal_length = 2*(sepal_length-4.3)/(7.9-4.3)-1; scaled_sepal_width = 2*(sepal_width-2)/(4.4-2)-1; scaled_petal_length = 2*(petal_length-1)/(6.9-1)-1; scaled_petal_width = 2*(petal_width-0.1)/(2.5-0.1)-1; y_1_1 = logistic(-8.9598 + (scaled_sepal_length*1.58675) + (scaled_sepal_width*-4.05111) + (scaled_petal_length*10.7863) + (scaled_petal_width*12.7033)); y_1_2 = logistic(1.8634 + (scaled_sepal_length*-0.789934) + (scaled_sepal_width*-2.53173) + (scaled_petal_length*3.47617) + (scaled_petal_width*5.04384)); non_probabilistic_iris_setosa = logistic(3.43875+ (y_1_1*0.950696)+ (y_1_2*-8.35938)); non_probabilistic_iris_versicolor = logistic(-4.41592+ (y_1_1*-11.319)+ (y_1_2*9.58436)); non_probabilistic_iris_virginica = logistic(-7.10419+ (y_1_1*11.5014)+ (y_1_2*1.88165)); (iris_setosa,iris_versicolor,iris_virginica) = softmax(non_probabilistic_iris_setosa, non_probabilistic_iris_versicolor, non_probabilistic_iris_virginica);

- UCI Machine Learning Repository. Iris Data Set.
- Fisher,R.A. "The use of multiple measurements in taxonomic problems" Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions to Mathematical Statistics" (John Wiley, NY, 1950).