线性最小二乘回归项目：多类模型探讨与误差评估

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本项目主要关注线性回归问题的相关内容，涉及MATH728D机器学习课程中的实践任务。在该项目中，目标是使用一组数据对（ti, yi）来建立一个线性模型，该模型通过n个参数（x1, x2, ..., xn）与基础函数集合Φ相联系，表达式为y(t) = x1φ1(t) + x2φ2(t) + ... + xnφn(t)。这个模型的选择通常基于不同的基础函数系列，如多项式（1, t, t^2, ..., t^(n-1)）、三角函数（1, sin(t), cos(t), sin(2t),...）或者组合了多项式和三角函数。项目的核心是线性最小二乘法，即寻找使得残差向量r（由yi - y(ti)定义）的L2范数最小的参数值。残差向量ri被定义为每个数据点预测值yi与实际值y(ti)之间的差。评估模型性能的关键指标是均方根误差（Root Mean Square Error, RMSE），它是残差向量的平方和除以数据点数量的平方根。在MATLAB中，可以使用rms函数计算RMSE，而在Python中则通过np.sqrt求得。实施该项目时，你需要： 1. **选择合适的基函数**：根据问题特性选择基础函数集，如多项式、三角或混合模型。 2. **构建模型**：将数据映射到选定的基函数上，找到最优的参数x。 3. **计算残差**：对于每一个数据点，计算y(ti)和yi之间的差值，形成残差向量ri。 4. **计算RMSE**：利用上述公式计算残差的平方和，然后取平均并开平方根得到最终的RMSE值。 5. **模型验证与优化**：通过调整模型参数，不断减小RMSE，确保模型的准确性。在整个项目过程中，你需要深入理解线性最小二乘方法的原理，熟练运用编程工具（如MATLAB或Python）进行模型构建和评估，并能够解释模型性能的好坏以及可能的改进方向。这不仅涉及数学建模，还包含编程技能和数据分析实践，对理论知识和实践能力都有较高要求。

MATH 728D: Machine Learning

Project 1: Regression with Linear Least Squares

This project will consider aspects of the linear regression problem. Such problems start with a set of m data

pairs (t

, y

), and propose a model function y(t) which depends linearly on n parameters x:

y(t) = x

(t) + x

(t) + ... + x

(t)

where

Φ = {φ

(t), φ

(t), ..., φ

(t)}

is our set of basis functions. To deﬁne the model, we seek values for the unknown x. If we determine these

values as the minimizers of the `

norm of the residual, we are implementing a linear least squares algorithm,

as we have discussed in class.

A polynomial model chooses the n basis functions {1, t, t

, ..., t

n−1

}, A trigonometric model the ﬁrst n terms

in the sequence {1, sin(t), cos(t), sin(2t), ...}. Other options use exponentials, or a mix of polynomials and

trigonometric functions.

For a given model, the residual vector r is deﬁned by

= y(t

) − y

The error function that we will report is the root-mean-square (RMS) norm of the residual:

||r|| =

i=1

(y(t

) − y

)

which can be evaluated in MATLAB by

rnorm = rms ( r ) ;

or in Python by:

rnorm = np . s q r t ( ( r ∗∗2 ) . mean ( ) )

We use the RMS norm instead of the usual `

norm simply because it is scaled to have a more uniform

behavior as the vector size varies. When we refer to ||r||, we will mean the RMS of the vector r.

The following datasets will be used, and will be referred to in the text by their 1-letter code abbreviation:

Code ﬁle m

F ﬁlip.txt 82

M mexico.txt 13

R rising.txt 25

S sinus.m / sinus.py any

Each dataset is a text ﬁle, containing m lines, and line i contains the single pair of values t

, y

In Matlab, we can easily read dataset F by a command like:

data = lo ad ( ’ f i l i p . txt ’ ) ;

or in Python by:

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线性最小二乘回归项目：多类模型探讨与误差评估

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