Orthogonality and Least Squares
Orthogonality and Least Squares is the concept of orthogonality plays a pivotal role in understanding and solving various problems in mathematics and data analysis. It forms the backbone of methods like the least squares approach, which is essential for applications such as data fitting and regression. This blog explores the core ideas of orthogonality, the Gram-Schmidt process, orthogonal bases, and their connection to least squares solutions.
Orthogonal Vectors and Orthogonal Projections
a- Orthogonal Vectors –
Two vectors are said to be orthogonal if their dot product is zero. In Euclidean space, this means the vectors are perpendicular. For example, in a 2D plane, the vectors v=(1,0)\mathbf{v} = (1, 0) and u=(0,1)\mathbf{u} = (0, 1) are orthogonal since:
v⋅u=1⋅0+0⋅1=0.\mathbf{v} \cdot \mathbf{u} = 1 \cdot 0 + 0 \cdot 1 = 0.
Orthogonality is foundational in linear algebra because it simplifies computations and provides geometric insights into the structure of vector spaces.
b- Orthogonal Projections –
Orthogonal projections are a way to represent one vector onto another. If we project a vector b\mathbf{b} onto a vector a\mathbf{a}, the projection is given by:
projab=a⋅ba⋅aa.\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}.
This projection minimizes the distance between b\mathbf{b} and all vectors lying along a\mathbf{a}, making it crucial in optimization and data fitting problems.
Gram-Schmidt Process
The Gram-Schmidt process is an algorithm for transforming a set of linearly independent vectors into an orthogonal (or orthonormal) set. Given a basis {v1,v2,…,vn}\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}, the Gram-Schmidt process generates orthogonal vectors {u1,u2,…,un}\{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\} as follows:
- Set u1=v1\mathbf{u}_1 = \mathbf{v}_1.
- For i=2,…,ni = 2, \dots, n: ui=vi−∑j=1i−1projujvi,\mathbf{u}_i = \mathbf{v}_i – \sum_{j=1}^{i-1} \text{proj}_{\mathbf{u}_j} \mathbf{v}_i, where projujvi\text{proj}_{\mathbf{u}_j} \mathbf{v}_i is the projection of vi\mathbf{v}_i onto uj\mathbf{u}_j.
If normalization is added (dividing each ui\mathbf{u}_i by its norm), the result is an orthonormal basis. This method is a cornerstone for tasks like QR factorization in numerical linear algebra.
Orthogonal and Orthonormal Bases
An orthogonal basis for a vector space consists of mutually orthogonal vectors. If these vectors are also unit vectors (norm = 1), the basis is orthonormal.
For a vector space VV, any vector v∈V\mathbf{v} \in V can be uniquely expressed as a linear combination of the orthogonal basis vectors. The simplicity of computation with orthonormal bases makes them highly desirable, as dot products and projections are straightforward.
Least Squares Solutions to Linear Systems
In many practical scenarios, a linear system Ax=bA\mathbf{x} = \mathbf{b} does not have an exact solution because b\mathbf{b} may not lie in the column space of AA. The least squares approach seeks to minimize the residual ∥Ax−b∥2\|A\mathbf{x} – \mathbf{b}\|^2, resulting in the best approximation to b\mathbf{b}.
The solution is given by solving the normal equations:
ATAx=ATb.A^T A \mathbf{x} = A^T \mathbf{b}.
This approach projects b\mathbf{b} onto the column space of AA, ensuring the error vector b−Ax\mathbf{b} – A\mathbf{x} is orthogonal to the column space.
Application to Data Fitting and Regression
Data Fitting In data fitting, the least squares method is used to find the best-fit line or curve for a given set of data points. Given data (xi,yi)(x_i, y_i), the goal is to find parameters β0\beta_0 and β1\beta_1 such that the line y=β0+β1xy = \beta_0 + \beta_1 x minimizes the sum of squared errors:
∑i=1n(yi−(β0+β1xi))2.\sum_{i=1}^n (y_i – (\beta_0 + \beta_1 x_i))^2.
This leads to solving a normal equation derived from the least squares criterion.
Regression Analysis In regression, least squares is fundamental for estimating relationships between variables. In multiple linear regression, where y=Xβ+ϵy = X\beta + \epsilon, the coefficient vector β\beta is estimated by:
β=(XTX)−1XTy.\beta = (X^T X)^{-1} X^T y.
The orthogonality of residuals and the design matrix XX ensures the validity of the least squares estimates.
Conclusion
Orthogonality provides the mathematical framework for simplifying computations and solving complex problems. From constructing orthonormal bases using the Gram-Schmidt process to applying least squares for data fitting, these principles are integral to modern data analysis and linear algebra. Understanding and leveraging these concepts is essential for success in fields like machine learning, statistics, and applied mathematics.