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(M_w,μ,σ,D)= ½ p=1nt(t_p-M_w,μ,σ(d_p))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 n_t training instances; each training instance is composed of descriptive features, d, and a target feature t; M_w,_μ,_σ(d_p) 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)

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.
2. 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.
3. 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.
4. 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.