Consider a radial basis function neural network (RBFNN) designed for a binary classification task. The RBFNN the
Question:
Consider a radial basis function neural network (RBFNN) designed for a binary classification task. The RBFNN the structure consists of:
- An input layer with three units (two input and one bias unit).
- A single hidden layer with thirteen units. Each unit implements the Gaussian function. The center and width of each Gaussian function have been predetermined and are provided in Table 1.
- An output layer with a single neuron. The single neuron implements the sigmoid activation function. The output of the network is converted to a classification label as follows:
-When the output of the network is equal to or greater than 0.5, predict the label 1.
-When the output of the network is less than 0.5, predict the label 0.
Note: The neuron in the output layer is connected to every unit in the hidden layer by the weight vector, w (no bias unit).
In this assignment, we will train the RBFNN using batch gradient descent so that the sum of squared error, L2, is minimized. Formally, L2 is defined as
L2(Mw,μ,σ,D)= ½ p=1nt(tp-Mw,μ,σ(dp))2………………(1)
Table 1: Pre-calculated RBFNN weights
j | μj1 | μj2 | σj |
1 | 0.3 | 0.75 | 0.08 |
2 | 0.65 | 0.75 | 0.08 |
3 | 0.7 | 0.45 | 0.09 |
4 | 0.4 | 0.6 | 0.07 |
5 | 0.9 | 0.65 | 0.09 |
6 | 0.4 | 0.5 | 0.05 |
7 | 0.6 | 0.55 | 0.06 |
8 | 0.95 | 0.5 | 0.06 |
9 | 0.18 | 0.53 | 0.07 |
10 | 0.15 | 0.85 | 0.07 |
11 | 0.5 | 0.85 | 0.07 |
12 | 0.7 | 0.95 | 0.06 |
13 | 0.8 | 0.85 | 0.07 |
where the training set is composed of nt training instances; each training instance is composed of descriptive features, d, and a target feature t; Mw,μ,σ(dp) is the prediction made by the RBFNN for a training instance
with descriptive features d; and the RBFNN is defined by the model parameters w, μ, and σ.
For the exercise assume that:
- The center, μ, and width, σ, of each Gaussian function is fixed. In other words, do not adjust these model parameters.
- The weight vector, w is initialized to the values:
(0.1,-0.1, 0.1,-0.1, 0.1,-0.1, 0.1,-0.1, 0.1,-0.1, 0.1,-0.1,0.1)
- Calculate the output of the fourth hidden neuron i.e. j = 4 for the input z = (0.5; 0.5). Provide your final answer rounded to three decimal places. Provide your supporting calculations.
- Calculate the output of the remaining hidden units for the same input. Complete the table below, and round off your answers to three decimal places.
- Calculate the output of the RBFNN for the input z = (0.5; 0.5). Provide your final answer rounded to three decimal places. Provide your supporting calculations.
- Calculate the sum of squared error (defined by Equation 1)) for the input z = (0.5; 0.5) given that the label (target) for the input is one. Provide your final answer rounded to three decimal places. Provide your supporting calculations.