added a rudimentary walktrough of some maths.

dsbook/_toc.yml

@@ -38,3 +38,7 @@ parts:
     chapters:
     - file: network/pathway.ipynb
     - file: network/gsea.ipynb
+  - caption: Appendix
+    numbered: True
+    chapters:
+    - file: appendix/math.ipynb

dsbook/appendix/math.ipynb

@@ -0,0 +1,174 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c3a1dbd7",
+   "metadata": {},
+   "source": [
+    "# Appendix: Some Maths\n",
+    "\n",
+    "## Row Vector\n",
+    "\n",
+    "A **row vector** is a 1-dimensional array consisting of a single row of elements. For example, a row vector with three elements can be written as:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{r} = [r_1, r_2, r_3]\n",
+    "$$\n",
+    "\n",
+    "## Column Vector\n",
+    "\n",
+    "A **column vector** is a 1-dimensional array consisting of a single column of elements. It can be thought of as a matrix with one column. For instance, a column vector with three elements appears as:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{c} = \\begin{bmatrix}\n",
+    "c_1 \\\\\n",
+    "c_2 \\\\\n",
+    "c_3\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "## Matrix\n",
+    "\n",
+    "A **matrix** is a rectangular array of numbers arranged in rows and columns. For example, a matrix with two rows and three columns is shown as:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{A} = \\begin{bmatrix}\n",
+    "a_{11} & a_{12} & a_{13} \\\\\n",
+    "a_{21} & a_{22} & a_{23}\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "## Transpose Operator\n",
+    "\n",
+    "The **transpose** of a matrix is obtained by swapping its rows with its columns. The transpose of matrix $\\mathbf{A}$ is denoted $\\mathbf{A}^T$. For the given matrix $\\mathbf{A}$, the transpose is:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{A}^T = \\begin{bmatrix}\n",
+    "a_{11} & a_{21} \\\\\n",
+    "a_{12} & a_{22} \\\\\n",
+    "a_{13} & a_{23}\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "## Multiplication between Vectors\n",
+    "\n",
+    "**Vector multiplication** can result in either a scalar or a matrix:\n",
+    "\n",
+    "- **Dot product**: Multiplication of the transpose of a column vector $\\mathbf{a}$ with another column vector $\\mathbf{b}$ results in a scalar. This is also known as the inner product:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{a}^T \\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\n",
+    "$$\n",
+    "\n",
+    "- **Outer product**: The multiplication of a column vector $\\mathbf{a}$ by the transpose of another column vector $\\mathbf{b}$ results in a matrix:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{a} \\mathbf{b}^T = \\begin{bmatrix}\n",
+    "a_1b_1 & a_1b_2 & a_1b_3 \\\\\n",
+    "a_2b_1 & a_2b_2 & a_2b_3 \\\\\n",
+    "a_3b_1 & a_3b_2 & a_3b_3\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "## Matrix Multiplication\n",
+    "\n",
+    "The product of two matrices $\\mathbf{A}$ and $\\mathbf{B}$ is a third matrix $\\mathbf{C}$. Each element $c_{ij}$ of $\\mathbf{C}$ is computed as the dot product of the $i$-th row of $\\mathbf{A}$ and the $j$-th column of $\\mathbf{B}$:\n",
+    "\n",
+    "$$\n",
+    "c_{ij} = \\sum_{k} a_{ik} b_{kj}\n",
+    "$$\n",
+    "\n",
+    "## Projection\n",
+    "\n",
+    "The **projection** of vector $\\mathbf{u}$ onto vector $\\mathbf{v}$ is given by:\n",
+    "\n",
+    "$$\n",
+    "\\text{proj}_{\\mathbf{v}} \\mathbf{u} = \\frac{\\mathbf{u}^T\\mathbf{v}}{\\mathbf{v}^T\\mathbf{v}} \\mathbf{v}\n",
+    "$$\n",
+    "\n",
+    "This represents the orthogonal projection of $\\mathbf{u}$ in the direction of $\\mathbf{v}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "18f60404",
+   "metadata": {
+    "tags": [
+     "hide-input"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "u = np.array([1, 3])\n",
+    "v = np.array([3, 1])\n",
+    "\n",
+    "# Recalculate the projection of u onto the new v\n",
+    "proj_u_on_v = np.dot(u, v) / np.dot(v, v) * v\n",
+    "orthogonal_component = u - proj_u_on_v\n",
+    "\n",
+    "# Update plot with new vector v and its projection\n",
+    "plt.figure(figsize=(6, 6))\n",
+    "plt.quiver(0, 0, u[0], u[1], angles='xy', scale_units='xy', scale=1, color='r', label=r'$\\mathbf{u}=(1,3)^T$')\n",
+    "plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='b', label=r'$\\mathbf{v}=(3,1)^T$')\n",
+    "plt.quiver(0, 0, proj_u_on_v[0], proj_u_on_v[1], angles='xy', scale_units='xy', scale=1, color='g', label='proj$_{\\mathbf{v}} \\mathbf{u}$')\n",
+    "\n",
+    "# Plot orthogonal line as a dotted line segment\n",
+    "end_point = proj_u_on_v + orthogonal_component\n",
+    "plt.plot([proj_u_on_v[0], end_point[0]], [proj_u_on_v[1], end_point[1]], 'purple', linestyle='dotted', label='Orthogonal Component')\n",
+    "\n",
+    "# Set plot limits and aspect\n",
+    "plt.xlim(0, 4)\n",
+    "plt.ylim(0, 4)\n",
+    "plt.gca().set_aspect('equal', adjustable='box')\n",
+    "\n",
+    "# Add a grid, legend, and labels\n",
+    "plt.grid(True)\n",
+    "plt.legend()\n",
+    "plt.title('Projection of Vector $\\mathbf{u}$ onto Vector $\\mathbf{v}$')\n",
+    "plt.xlabel('X axis')\n",
+    "plt.ylabel('Y axis')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cacd922c",
+   "metadata": {},
+   "source": [
+    "## Eigenvalue and Eigenvector\n",
+    "\n",
+    "An **eigenvalue** $\\lambda$ and its corresponding **eigenvector** $\\mathbf{v}$ of a matrix $\\mathbf{A}$ satisfy the equation:\n",
+    "\n",
+    "$$\n",
+    "\\mathbf{A} \\mathbf{v} = \\lambda \\mathbf{v}\n",
+    "$$\n",
+    "\n",
+    "## Gradient\n",
+    "\n",
+    "The **gradient** of a multivariable function $f(\\mathbf{x})$ is a vector of partial derivatives, which points in the direction of the steepest ascent of $f$:\n",
+    "\n",
+    "$$\n",
+    "\\nabla f(\\mathbf{x}) = \\begin{bmatrix}\n",
+    "\\frac{\\partial f}{\\partial x_1} \\\\\n",
+    "\\frac{\\partial f}{\\partial x_2} \\\\\n",
+    "\\vdots \\\\\n",
+    "\\frac{\\partial f}{\\partial x_n}\n",
+    "\\end{bmatrix}\n",
+    "$$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

dsbook/appendix/math.md

@@ -0,0 +1,153 @@
+---
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+jupytext:
+  formats: md:myst
+  text_representation:
+    extension: .md
+    format_name: myst
+---
+
+# Appendix: Some Maths
+
+## Row Vector
+
+A **row vector** is a 1-dimensional array consisting of a single row of elements. For example, a row vector with three elements can be written as:
+
+$$
+\mathbf{r} = [r_1, r_2, r_3]
+$$
+
+## Column Vector
+
+A **column vector** is a 1-dimensional array consisting of a single column of elements. It can be thought of as a matrix with one column. For instance, a column vector with three elements appears as:
+
+$$
+\mathbf{c} = \begin{bmatrix}
+c_1 \\
+c_2 \\
+c_3
+\end{bmatrix}
+$$
+
+## Matrix
+
+A **matrix** is a rectangular array of numbers arranged in rows and columns. For example, a matrix with two rows and three columns is shown as:
+
+$$
+\mathbf{A} = \begin{bmatrix}
+a_{11} & a_{12} & a_{13} \\
+a_{21} & a_{22} & a_{23}
+\end{bmatrix}
+$$
+
+## Transpose Operator
+
+The **transpose** of a matrix is obtained by swapping its rows with its columns. The transpose of matrix $\mathbf{A}$ is denoted $\mathbf{A}^T$. For the given matrix $\mathbf{A}$, the transpose is:
+
+$$
+\mathbf{A}^T = \begin{bmatrix}
+a_{11} & a_{21} \\
+a_{12} & a_{22} \\
+a_{13} & a_{23}
+\end{bmatrix}
+$$
+
+## Multiplication between Vectors
+
+**Vector multiplication** can result in either a scalar or a matrix:
+
+- **Dot product**: Multiplication of the transpose of a column vector $\mathbf{a}$ with another column vector $\mathbf{b}$ results in a scalar. This is also known as the inner product:
+
+$$
+\mathbf{a}^T \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3
+$$
+
+- **Outer product**: The multiplication of a column vector $\mathbf{a}$ by the transpose of another column vector $\mathbf{b}$ results in a matrix:
+
+$$
+\mathbf{a} \mathbf{b}^T = \begin{bmatrix}
+a_1b_1 & a_1b_2 & a_1b_3 \\
+a_2b_1 & a_2b_2 & a_2b_3 \\
+a_3b_1 & a_3b_2 & a_3b_3
+\end{bmatrix}
+$$
+
+## Matrix Multiplication
+
+The product of two matrices $\mathbf{A}$ and $\mathbf{B}$ is a third matrix $\mathbf{C}$. Each element $c_{ij}$ of $\mathbf{C}$ is computed as the dot product of the $i$-th row of $\mathbf{A}$ and the $j$-th column of $\mathbf{B}$:
+
+$$
+c_{ij} = \sum_{k} a_{ik} b_{kj}
+$$
+
+## Projection
+
+The **projection** of vector $\mathbf{u}$ onto vector $\mathbf{v}$ is given by:
+
+$$
+\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u}^T\mathbf{v}}{\mathbf{v}^T\mathbf{v}} \mathbf{v}
+$$
+
+This represents the orthogonal projection of $\mathbf{u}$ in the direction of $\mathbf{v}$.
+
+```{code-cell}ipython3
+:tags: [hide-input]
+import numpy as np
+import matplotlib.pyplot as plt
+
+u = np.array([1, 3])
+v = np.array([3, 1])
+
+# Recalculate the projection of u onto the new v
+proj_u_on_v = np.dot(u, v) / np.dot(v, v) * v
+orthogonal_component = u - proj_u_on_v
+
+# Update plot with new vector v and its projection
+plt.figure(figsize=(6, 6))
+plt.quiver(0, 0, u[0], u[1], angles='xy', scale_units='xy', scale=1, color='r', label=r'$\mathbf{u}=(1,3)^T$')
+plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='b', label=r'$\mathbf{v}=(3,1)^T$')
+plt.quiver(0, 0, proj_u_on_v[0], proj_u_on_v[1], angles='xy', scale_units='xy', scale=1, color='g', label='proj$_{\mathbf{v}} \mathbf{u}$')
+
+# Plot orthogonal line as a dotted line segment
+end_point = proj_u_on_v + orthogonal_component
+plt.plot([proj_u_on_v[0], end_point[0]], [proj_u_on_v[1], end_point[1]], 'purple', linestyle='dotted', label='Orthogonal Component')
+
+# Set plot limits and aspect
+plt.xlim(0, 4)
+plt.ylim(0, 4)
+plt.gca().set_aspect('equal', adjustable='box')
+
+# Add a grid, legend, and labels
+plt.grid(True)
+plt.legend()
+plt.title('Projection of Vector $\mathbf{u}$ onto Vector $\mathbf{v}$')
+plt.xlabel('X axis')
+plt.ylabel('Y axis')
+plt.show()
+
+```
+
+## Eigenvalue and Eigenvector
+
+An **eigenvalue** $\lambda$ and its corresponding **eigenvector** $\mathbf{v}$ of a matrix $\mathbf{A}$ satisfy the equation:
+
+$$
+\mathbf{A} \mathbf{v} = \lambda \mathbf{v}
+$$
+
+## Gradient
+
+The **gradient** of a multivariable function $f(\mathbf{x})$ is a vector of partial derivatives, which points in the direction of the steepest ascent of $f$:
+
+$$
+\nabla f(\mathbf{x}) = \begin{bmatrix}
+\frac{\partial f}{\partial x_1} \\
+\frac{\partial f}{\partial x_2} \\
+\vdots \\
+\frac{\partial f}{\partial x_n}
+\end{bmatrix}
+$$
+