R Language and Classification Algorithm - Neural Network

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R Language and Classification Algorithm - Neural Network

Introduction

In this article, we explore the fundamental concepts of neural networks through the lens of the R programming language. We will delve into various aspects of neural networks, starting from simple perceptrons and progressing to more complex architectures like the Back-Propagation (BP) neural network. This journey will help us understand how these models can be used to classify data and simulate intelligent behavior.

First, Perceptron

A perceptron is essentially a single-layer neural network that can be used for binary classification tasks. It operates based on a linear combination of input features and weights, followed by a threshold function to produce a binary output.

Physical Structure: The perceptron consists of a set of input nodes, a single output node, and a threshold value. Each input node is connected to the output node with a weight.

Sensor Function: The output of the perceptron can be represented as:
[ y = \text{sign}(w \cdot x + b) ]
where ( w ) is the weight vector, ( x ) is the input vector, and ( b ) is the bias term.

Learning Process: The perceptron learning algorithm involves adjusting the weights to minimize misclassification errors. The steps are as follows:

  1. Define variables: ( x ) (input vector), ( w ) (weight vector), ( b ) (bias), ( y ) (actual output), ( d ) (desired output), and ( a ) (learning rate).
  2. Initialize: Set ( w = 0 ), ( b = 0 ).
  3. For each training sample, calculate the actual output ( y = \text{sign}(w \cdot x + b) ).
  4. Update the weights: ( w_{n+1} = w_n + a \cdot (d - y) \cdot x ).
  5. Repeat until convergence.

For the Iris dataset, we can apply this algorithm to classify “setosa” and “versicolor” species. Here’s the R code:

a <- 0.2
w <- rep(0, 3)
iris1 <- t(as.matrix(iris[, 3:4]))
d <- c(rep(0, 50), rep(1, 100))
e <- rep(0, 150)
p <- rbind(rep(1, 150), iris1)
max <- 100000
eps <- rep(0, 100000)
i <- 0
repeat {
  v <- w %*% p
  y <- ifelse(sign(v) >= 0, 1, 0)
  e <- d - y
  eps[i + 1] <- sum(abs(e)) / length(e)
  if (eps[i + 1] < 0.01) {
    print("finish:")
    print(w)
    break
  }
  w <- w + a * (d - y) %*% t(p)
  i <- i + 1
  if (i > max) {
    print("max time loop")
    print(eps[i])
    print(y)
    break
  }
}

Second, Neural Network Linear

While a perceptron can handle linearly separable data, it fails when the data is not linearly separable. To address this, we introduce the delta rule, which allows the network to converge to a best-fit weight vector even for non-linearly separable data.

Delta Rule: The delta rule uses gradient descent to adjust the weights in the direction that minimizes the error. The key steps are:

  1. Define variables: ( x ) (input vector), ( w ) (weight vector), ( b ) (bias), ( y ) (actual output), ( d ) (desired output), and ( a ) (learning rate).
  2. Initialize: Set ( w = 0 ).
  3. For each training sample, calculate the actual output ( e(n) = d - y ).
  4. Update the weights: ( w_{n+1} = w_n + a \cdot e(n) \cdot x ).
  5. Determine convergence and repeat.

Here’s an example using the Iris dataset:

p <- rbind(rep(1, 150), iris1)
d <- c(rep(0, 50), rep(1, 100))
w <- rep(0, 3)
a <- 1 / max(eigen(t(p) %*% p)$values)
max <- 1000
e <- rep(0, 150)
eps <- rep(0, 1000)
i <- 0
for (i in 1:max) {
  v <- w %*% p
  y <- v
  e <- d - y
  eps[i + 1] <- sum(e^2) / length(e)
  w <- w + a * (d - y) %*% t(p)
  if (i == max) print(w)
}

Three, BP Neural Network

The Back-Propagation (BP) neural network is a multi-layer network that can handle non-linearly separable data. It uses a sigmoid function as the activation function, which provides a smooth transition and better fitting capabilities.

Sigmoid Function: The sigmoid function is defined as:
[ y = \frac{1}{1 + e^{-x}} ]

BP Neural Network: The BP algorithm consists of two main phases: forward propagation and backward propagation. Forward propagation computes the output of each neuron, while backward propagation updates the weights to minimize the error.

Training Example: Using the Iris dataset, we can train a BP neural network to classify the species. Here’s the R code:

p <- rbind(rep(1, 150), iris1)
d <- c(rep(0, 50), rep(1, 100))
w <- rep(0, 3)
a <- 1 / max(eigen(t(p) %*% p)$values)
max <- 1000
e <- rep(0, 150)
eps <- rep(0, 1000)
i <- 0
for (i in 1:max) {
  v <- w %*% p
  y <- v
  e <- d - y
  eps[i + 1] <- sum(e^2) / length(e)
  w <- w + a * (d - y) %*% t(p)
  if (i == max) print(w)
}

Conclusion

In summary, neural networks provide a powerful framework for solving complex classification problems. From simple perceptrons to advanced BP networks, these models can simulate intelligent behavior and learn from data. By understanding the underlying principles and implementing them in R, we can effectively classify and predict outcomes in various domains.

This enhanced version maintains the technical accuracy while improving readability and flow, making it easier for readers to follow and understand the concepts.