3.1. Basis as an Approximation of Functions

In mathematics and data science, the concept of a basis plays a fundamental role in approximating complex functions. A basis is a set of functions (or vectors) that can be combined to represent other functions within a given space. This idea allows us to break down a complicated function into simpler building blocks, making it easier to analyze and compute. 

Basis: From \(n\)-Dimensional Vectors to Functions. 

In \(n\)-dimensional space, a vector can be represented as a combination of basis vectors. For example, a vector \(\mathbf{v}\) in \(n\)-dimensional space can be written as: \[ \mathbf{v} = c_1 \mathbf{e}_1 + c_2 \mathbf{e}_2 + \cdots + c_n \mathbf{e}_n, \] where \(\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\) are the basis vectors, and \(c_1, c_2, \ldots, c_n\) are the coordinates of the vector with respect to this basis. The basis vectors provide a "coordinate system" for the space, allowing us to describe any vector in terms of these building blocks. 

    • For example, in 3D space, the standard basis vectors are: \[ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \] A vector \(\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}\) can then be expressed as: \[ \mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3. \] 

Extending the Idea to Functions Just like vectors in \(n\)-dimensional space can be expressed using basis vectors, functions can be expressed as combinations of **basis functions**. For example, consider a function \(f(x)\). It can be approximated as: \[ f(x) \approx c_1 \phi_1(x) + c_2 \phi_2(x) + \cdots + c_n \phi_n(x), \] where \(\phi_1(x), \phi_2(x), \ldots, \phi_n(x)\) are the basis functions, and \(c_1, c_2, \ldots, c_n\) are the coefficients (analogous to the coordinates of a vector). 

 Why Is This Useful? Using basis functions to approximate a function is extremely useful in data science and applications like biology and medicine. For example: 

    • Polynomial Approximation: Using basis functions like \(1, x, x^2, \ldots, x^n\) to fit data. 
    •  Fourier Series: Using sine and cosine functions as a basis to analyze periodic data. 
    •  Wavelets: Using localized basis functions to analyze signals or images. 
    • Principal component analysis (PCA): Finding a basis that captures the maximum variance in data for dimensionality reduction. 
    • Machine learning models: Using feature transformations or kernel functions that act as a basis for learning complex patterns. 
By choosing an appropriate basis, we can efficiently approximate and work with functions in a way that is computationally and conceptually practical for biomedical applications.

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2.1 Vectors

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Under Construction Vectors are mathematical entities characterized by both magnitude and direction. They are essential for describing physical quantities such as blood flow velocity, forces acting on joints, and features in image analysis for deep learning. A vector in three-dimensional space is written as: \[ \mathbf{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}, \] where \(\hat{i} = (1,0,0)\), \(\hat{j} = (0,1,0)\), and \(\hat{k} = (0,0,1)\) are the unit vectors along the \(x\)-, \(y\)-, and \(z\)-axes, respectively. The components \(v_x\), \(v_y\), and \(v_z\) represent the magnitude of the vector in the \(x\)-, \(y\)-, and \(z\)-directions.  Vectors can describe the speed and direction of blood flow in vessels, facilitating the analysis of hemodynamics. They are also used to model forces acting on prosthetic joints or tissues, aiding in the study of biomechanics and prosthetic design. A vector in \(n\)-dimensional space is written as: \[ \mathbf{x} = (x_1, x_2, x_3, \l...
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2.4. Taylor Expansion and Approximation

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Taylor expansion is a fundamental tool for approximating functions using their derivatives.  1D Taylor Expansion. Let \( f(x) \) be a smooth and differentiable scalar function of one variable \( x \) near a point \( x_0 \). The Taylor expansion of \(f(x)\) around \( x_0 \), up to the second order, is given by: \[ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x - x_0)^2 + \frac{1}{3!}f'''(x_0)(x - x_0)^3+O(|x - x_0|^4). \] The term \( O(|x - x_0|^4) \) (pronounced "Big-O of \(|x - x_0|^4\)") describes the error in the approximation. Specifically: \[ \frac{O(|x - x_0|^4)}{|x - x_0|^4} \text{ is bounded. }  \] This notation simplifies the representation of higher-order terms, which become egligible compared to the lower-order terms when \( x \) is close to \( x_0 \).   In machine learning, Taylor expansion is used to approximate non-linear activation functions, such as the sigmoid function: \[ \sigma(x) = \frac{1}{1 + e^{-x}}. \] For small \( x ...
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2.5. Vector Fields and Line integral

  Vector Fields A  vector field  assigns a vector to every point in space. Examples include hemodynamics, where vector fields are used to understand blood flow patterns and detect abnormalities such as stenosis or aneurysms, and respiratory dynamics, where vector fields model airflow in the lungs to study ventilation efficiency.  Let \(\mathbf{r} = (x, y, z)\). A vector field is defined as: $$F(\mathbf{r}) = F_x(\mathbf{r}) \hat{i} + F_y(\mathbf{r}) \hat{j} + F_z(\mathbf{r}) \hat{k},$$ where:  \( F_x(\mathbf{r}) \): The \(x\)-component of the vector field.  \( F_y(\mathbf{r}) \): The \(y\)-component of the vector field.  \( F_z(\mathbf{r}) \): The \(z\)-component of the vector field.  \( \hat{i}, \hat{j}, \hat{k} \): Unit vectors in the \(x\)-, \(y\)-, and \(z\)-directions, respectively.  Example 1: Blood Velocity Field   In a cylindrical blood vessel, the velocity of blood flow can be described as a vector field. For laminar flow, ...
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