# 统计代写|主成分分析代写Principal Component Analysis代考|”Understanding Vector Mathematics: Mean, Centering, Standard Deviation, Variance, Covariance, and Correlation”

Mean and Centered Vectors

Given a vector xxx, its mean xˉ\bar{x} x ˉ is the average of its elements. The centered version of xxx, denoted xcx_cx c ​ , is obtained by subtracting the mean from each element: xc=x−xˉx_c = x – \bar{x}x c ​ =x− x ˉ . Adding or subtracting centered vectors preserves centering: (x+y)c=xc+yc(x+y)_c = x_c + y_c(x+y) c ​ =x c ​ +y c ​ . The Pythagorean theorem applies to xxx, xˉ\bar{x} x ˉ , and xcx_cx c ​ , i.e., ∣∣x∣∣2=∣∣xˉ∣∣2+∣∣xc∣∣2||x||^2 = ||\bar{x}||^2 + ||x_c||^2∣∣x∣∣ 2 =∣∣ x ˉ ∣∣ 2 +∣∣x c ​ ∣∣ 2 , indicating that the length of the centered vector is always less than or equal to the original vector. Standard Deviation, Variance, and Scaled Vectors (2.3.2)

The standard deviation sxs_xs x ​ of vector xxx represents the amount of variation among its elements and is calculated after centering: sx=∣∣xc∣∣/n−1sx = ||x_c|| / \sqrt{n-1}sx=∣∣x c ​ ∣∣/ n−1 ​ . The variance V(x)V(x)V(x) is the square of the standard deviation: V(x)=∣∣xc∣∣2/(n−1)V(x) = ||x_c||^2 / (n-1)V(x)=∣∣x c ​ ∣∣ 2 /(n−1). Scaling a vector by its length and dividing by n−1\sqrt{n-1} n−1 ​ creates a standardized vector xcsx_{cs}x cs ​ . Covariance and Correlation between Vectors (2.3.3)

The covariance between two vectors xxx and yyy, denoted Cov(x,y)Cov(x, y)Cov(x,y), measures how much they vary together and is defined as Cov(x,y)=(x−xˉ,y−yˉ)/(n−1)Cov(x, y) = (x – \bar{x}, y – \bar{y}) / (n-1)Cov(x,y)=(x− x ˉ ,y− y ˉ ​ )/(n−1). The correlation Corr(x,y)Corr(x, y)Corr(x,y) is a normalized version of the covariance that compares the directional similarity of the centered and scaled vectors: Corr(x,y)=Cov(xcs,ycs)Corr(x, y) = Cov(x_{cs}, y_{cs})Corr(x,y)=Cov(x cs ​ ,y cs ​ ). The correlation is bounded by -1 and 1, indicating perfect negative, zero, or positive linear association, respectively. Absolute correlation values are invariant to scalar multiplication, while the sign of the correlation depends on the sign of the scalar applied to either vector. Overall, these vector-based functions provide tools for understanding and comparing datasets, particularly in data science contexts, where they are essential for assessing relationships between variables and preparing data for further analysis. The section also emphasizes that certain properties, such as the angle between vectors, might change when centering but remain invariant under scaling, while others, like variance and covariance, transform predictably under linear operations.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。