Functions and Graphs
- (Harvard University - Joyce Yang)
- Overview
A function is a mathematical relationship between inputs and outputs, while a graph is a visual representation of that function.
A function is a rule that transforms one real number into another real number. A graph is a geometric representation of that rule.
A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function.
A graph can give us a good idea of what function may be applied to the situation to solve the problem.
A graph is a diagram showing the relationship of quantities. A curve or line showing a mathematical function or equation, typically drawn in a Cartesian coordinate system.
A function has only one output value for each input value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function.
Please refer to the following for more details:
- Wikipedia: Function
- Wikipedia: Graph of a Function
- Wikipedia: Linear Function
- Graph Machine Learning
Graph Machine Learning (GML) is a broad field that uses mathematical structures called graphs to model relationships between objects. GML involves many different supervised and unsupervised machine learning (ML) tasks.
A graph is a set of objects, called vertices or nodes, connected by links, known as edges or arcs. A connected graph is a graph where every pair of nodes has a path between them.
GML's primary purpose is to compress large sparse graph structures while maintaining important signals for prediction and inference.
Common use cases for GML include:
- Social networks
- Drug discovery
- Particle physics
- Traffic and route optimization
- Knowledge graphs
- Financial transaction networks
- Chemistry (atoms & molecules)
Graph representation learning is a fundamental task that aims to encode high-dimensional sparse graph-structured data into low-dimensional dense vectors.
- Function in Python ML Models
Functions are a fundamental part of Python and are used to encapsulate code that performs a specific task. In the context of machine learning, functions can be used to define models, pre-process data, and evaluate model performance.
For example, the following function defines a simple linear regression model:
def linear_regression(x, y):
"""
Linear regression model.
Args:
x: Input data.
y: Target data.
Returns:
Predicted target values.
"""
# Calculate the slope and intercept of the line.
slope = (y.mean() - x.mean() * y.mean()) / (x.var() + 1e-8)
intercept = y.mean() - slope * x.mean()
# Predict the target values.
y_pred = slope * x + intercept
return y_pred
- Graphs in Python ML Models
Graphs are a powerful tool for visualizing data and relationships between data points. In machine learning, graphs can be used to visualize model performance, identify patterns in data, and debug models.
For example, the following code creates a scatter plot of the input and target data, along with the predicted target values:
import matplotlib.pyplot as plt
# Create the scatter plot.
plt.scatter(x, y, c='blue')
plt.scatter(x, y_pred, c='red')
# Add labels and a title.
plt.xlabel('Input')
plt.ylabel('Target')
plt.title('Linear Regression Model')
# Show the plot.
plt.show()
This plot can be used to visually assess the performance of the model and identify any areas where the model is not performing well.
Overall, functions and graphs are essential tools for working with machine learning models in Python. By understanding how to use these tools, you can more effectively develop, train, and evaluate your models.
[More to come ...]
