Case Studia: Wdrożenie Neural Networks for Image Recognition wigh incorporate Obliczenia

Neural networks are e widely used in image recognion tasks due to their ir ability to learn complex Patterns. This case study explores the implementation process, including dong detaild calculations involved in training a neural network for images classification.

Designing the Neural Network

Te network consists of an input layer, one hidden layer, and an output layer. For simplicity, consider a small dataset with images configetes confidented by 4 pixels each. The input layer has 4 neurons, thee hidden layer has 3 neurons, and the output layer has 2 neurons representing two classes.

Inicjal Weighs andBiases

/ Waży się je, / a potem zaczyna się od losowego.

• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 11 BEA1; BEA1; FLT: 1 BEA3; FLT: = 0,2
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 12 BEA1; BEA1; FLT: 1 BEA3; FLT: = -0,3
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 13 BEA1; BEA1; FLT: 1 BEA3; FLT 3; BEA3; = 0,4
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 21 BEA1; BEA1; FLT: 1 BEA3; FLT: = -0,5
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 22 BEA1; BEA1; FLT: 1 BEA3; FLT 3; BEA3; = 0,1
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 23 BEA1; BEA1; FLT: 1 BEA3; FLT: = -0,2
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 31 BEA1; BEA1; FLT: 1 BEA3; BEA3; = 0,3
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 32 BEA1; BEA1; FLT: 1 BEA3; FLT: = -0,4
• W BEA1; BEA1; FLT: 0 BEA3; BEA3; 33 BEA1; BEA1; FLT: 1 BEA3; BEA3; = 0,2

Biases are set to o zero initially for simplicity.

Forward Propagation andd Calculations

Given an input image with pixel values amend1; 0.5, 0.2, 0.1, 0.7 amend3;, the hidden layer inputs are calculated as:

H = 1; Xi1; FLT: 0 Xi3; Xi3; 1 Xi1; Xi1; FLT: 1 Xi3; Xi3; = (0.5) (0.2) + (0.2) (-0.3) + (0.1) (0.4) = 0.1 - 0.06 + 0.04 = 0.08

H = 1; Xi1; FLT: 0 Xi3; Xi3; 2 Xi1; FLT: 1 Xi3; Xi3; = (0.5) (-0.5) + (0.2) (0.1) + (0.1) (-0.2) = -0.25 + 0.02 - 0.02 = -0.25

H = 1; Xi1; FLT: 0 Xi3; Xi3; Xi1; FLT: 1 Xi3; Xi3; = (0.5) (0.3) + (0.2) (-0.4) + (0.1) (0.2) = 0.15 - 0.08 + 0.02 = 0.09

Aviniing an activation function, such as sigmoid, the hidden layer outputs are:

O, 1;, 1; FLT: 0, 3; H1, 1; FLT: 1, 3; FLT: 1, 3; FLT: 1; 1, 1; FLT: 2, 3; FLT: 3; FLT: 3; FL3;), 0, 52

O, 1;, 1; FLT: 0, 3;, 3; H2, 1; FLT: 1, 3; FLT: 1; 1, 1 (1 + e, 1; FLT: 2, 3; 0,43; FLT: 3; FLT: 3; FLT: 3; FLA3;)

O, 1;, 1; FLT: 0, 3; H3, 1; FLT: 1, 3; FLT:, 3; FLT:, 1; = 1 / (1 + e, 1; FLT: 2, 3; FLT:, 3; FLT:, 3; FLT:, 3; FLT;), 0, 52

Te wyniki są wykorzystywane do tych obliczeń, kiedy wagi są podobne do tych, które są zgodne z przepisami.

Kalkulating Error and Updating Weights

Suppose the true class is class 1, contexted as present 1; 1, 0 context 3;. The output layer computes:

O, 1;, 1; FLT: 0, 3;, 3; 1; FLT: 1, 3; FLT: 1, 3; FL3; = 1 / (1 + e, 1; FLT: 2, 3; FLT: 3; -z, 1; FLT: 3, 3; 1, 1; FLT: 4, 3; FL3; FL1; FLT: 1, 1; FLT: 5, 3; FLT: 3; FL3; FL3; FL1; FLT: 5, 3; FL3; FL3; FLS: 3; FLF: 1; FLS: 1; FLT: 1; FLT: 5, FLT: 3; FLS: 3; FLS; FLS: 3; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1; FLS: 1;

Kiedy z 1; Xi1; FLT: 0 Xi3; Xi3; 1 XI1; FLT: 1 Xi3; Xi3; is the weiged sum of hidden outputs. The error is calculated as:

Loss = 0,5 * (target - output) Xi1; Xi1; FLT: 0 Xi3; Xi3; 2 Xi1; Xi1; FLT: 1 Xi3; Xi3;

Using gradient descent, weights are updated based on thee error and learning rate, for example, 0.1.

Summary of Calculations

This process repeats for multiple epochs, adjusting weights to improwizuj precyzję. Thied calculations involve matrix multiplications, activation functions, and error deriatives, which ch are essential for training neural networks effectively.