Update tech_docs/math_objects_python.md

tech_docs/math_objects_python.md

@@ -225,4 +225,105 @@ This document serves as a starting point for exploring these concepts. To dive d
 - [TensorFlow Tutorials](https://www.tensorflow.org/tutorials)
 - [PyTorch Tutorials](https://pytorch.org/tutorials/)
 
-By combining a strong understanding of mathematical objects with the power of Python libraries, you'll be well-equipped to tackle a wide range of data science and computational challenges.
+By combining a strong understanding of mathematical objects with the power of Python libraries, you'll be well-equipped to tackle a wide range of data science and computational challenges.
+
+---
+
+# Mathematical Objects and Their Applications in Python
+
+This document delves into different types of mathematical objects, their practical applications, and the specific datasets they are frequently associated with. It also highlights Python libraries that facilitate working with these objects.
+
+## Scalar
+
+A **scalar** is a single number used to represent a singular value or magnitude in various scientific and computational fields.
+
+### Example
+
+$$ a = 3 $$
+
+### Applications and Datasets
+- **Data Science**: Scalars are used as thresholds or coefficients in algorithms.
+- **Physics**: Scalars represent quantities like mass or temperature.
+- **Datasets**: In a healthcare dataset, a scalar could represent a patient's age or blood pressure level.
+
+### Python Libraries
+- **NumPy**: Manages numerical operations; scalars can be represented as `numpy.float64` or `numpy.int32`.
+
+## Vector
+
+A **vector** is a sequence of numbers arranged in an ordered array, often representing directional data in multi-dimensional space.
+
+### Example
+
+$$ \mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$
+
+### Applications and Datasets
+- **Machine Learning**: Feature vectors represent data points in models.
+- **Physics**: Vectors describe velocities and forces.
+- **Datasets**: In finance, a vector could represent the time series of stock prices.
+
+### Python Libraries
+- **NumPy**: Supports powerful array structures for vectors.
+- **SciPy**: Facilitates scientific computations involving vectors.
+
+## Matrix
+
+A **matrix** is a rectangular array of numbers arranged in rows and columns, used to represent complex data structures or linear transformations.
+
+### Example
+
+$$ \mathbf{M} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$
+
+### Applications and Datasets
+- **Computer Graphics**: Matrices transform the coordinates of shapes.
+- **Statistics**: Covariance matrices quantify the correlation between variables.
+- **Datasets**: In genomics, matrices can represent gene expression data across different conditions.
+
+### Python Libraries
+- **NumPy**: Essential for matrix creation and manipulation.
+- **Pandas**: Manages data in a matrix-like format for analysis.
+- **Matplotlib**: Visualizes data from matrices.
+
+## Tensor
+
+A **tensor** generalizes vectors and matrices to higher dimensions, crucial in fields like machine learning where they represent high-dimensional datasets.
+
+### Example
+
+$$
+\mathcal{T} = \begin{bmatrix}
+\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} & \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \\
+\begin{matrix} 9 & 10 \\ 11 & 12 \end{matrix} & \begin{matrix} 13 & 14 \\ 15 & 16 \end{matrix}
+\end{bmatrix}
+$$
+
+### Applications and Datasets
+- **Deep Learning**: Tensors represent multi-dimensional arrays of parameters in neural networks, such as weights.
+- **Computer Vision**: Used for image data with layers representing different color channels.
+- **Datasets**: In video processing, a tensor could represent frames of videos, where each frame is a matrix of pixel values.
+
+### Python Libraries
+- **TensorFlow**: Specialized for tensor operations in neural networks.
+- **PyTorch**: Provides a dynamic computational graph that allows manipulation of high-dimensional tensors.
+- **NumPy**: Handles lower-dimensional tensors effectively.
+
+## Machine Learning Models and Their Mathematical Foundations
+
+### Gradient Boosted Decision Trees (GBDT)
+- **Application**: Used for both regression and classification tasks, GBDT is particularly effective for handling non-linear relationships and interactions.
+- **Mathematical Objects**: Primarily utilizes vectors and matrices to manage datasets during the training and prediction phases.
+- **Python Libraries**: Includes libraries such as XGBoost, LightGBM, and CatBoost, which are optimized for performance and scalability.
+
+### Random Forest Regression
+- **Application**: Offers robust predictions for continuous variables and is valuable for understanding feature importance.
+- **Mathematical Objects**: Utilizes vectors and matrices to represent data points and features respectively.
+- **Python Libraries**: Scikit-learn’s `RandomForestRegressor` provides a user-friendly interface to implement Random Forest models in Python.
+
+### Linear and Logistic Regression
+- **Application**: Linear regression is used for predicting continuous outcomes, while logistic regression is ideal for binary classification tasks.
+- **Mathematical Objects**: Both models employ vectors and matrices to manage data and parameters effectively.
+- **Python Libraries**: Scikit-learn and Statsmodels support these regression models, offering tools for fitting, predicting, and analyzing the results.
+
+## Conclusion
+
+Understanding these fundamental mathematical objects and their interplay with specific datasets enhances efficiency in fields like data science, artificial intelligence, and engineering. Python’s comprehensive suite of libraries provides robust tools for manipulating these objects, positioning it as an essential language for scientific computing