This document presents an assignment involving a single neuron artificial neural network (ANN) with a sigmoid activation function. Here's an outline of the tasks:

1. Output Calculation: Given two input vectors $\mathbf{x}^{(1)} = (0.6, 2.1)$ and $\mathbf{x}^{(2)} = (2, 1)$, and weights $(w_0, w_1, w_2) = (-1.5, 0.5, 1)$, calculate the neuron outputs using the sigmoid activation function $\varphi(s)$.

2. MSE Loss Function: Compute the Mean Squared Error (MSE) loss function for the given inputs and weights.

3. Error with Ground Truth: Calculate the error given the real outputs (ground truth) $y^{(1)} = 2$ and $y^{(2)} = -1$, with MSE.

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Step 1: Neuron Output Calculation

The neuron calculates the weighted sum of the inputs: $s = w_0 + w_1 x_1 + w_2 x_2$ Then, apply the sigmoid function: $\varphi(s) = \frac{1}{1 + e^{-s}}$

For both input vectors $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$, we will plug in the weights and calculate the corresponding output.

Step 2: MSE Loss

The Mean Squared Error (MSE) is given by: $MSE = \frac{1}{2} \sum_{i=1}^{2} \left( \hat{y}^{(i)} - y^{(i)} \right)^2$ where $\hat{y}^{(i)}$ is the predicted output and $y^{(i)}$ is the real output.

Would you like me to proceed with the detailed calculations for these steps or assist you in writing the Python program mentioned in the assignment? Let me know if you need help understanding the steps in more depth.

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Here are 5 follow-up questions related to this topic:

1. What is the general form of the sigmoid activation function, and why is it widely used in neural networks?
2. How does the choice of weights affect the output of a neuron?
3. What are alternative activation functions to the sigmoid function, and when are they preferred?
4. How is the Mean Squared Error (MSE) useful in training neural networks?
5. What other loss functions can be used for neural networks aside from MSE?

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Tip: When computing the weighted sum $s$ for a neuron, remember that $w_0$ often acts as a bias term and does not multiply any input.