Least Squares Estimation - Problem with Symbols

In summary, the conversation revolves around understanding a question about finding the minimizer of a function, specifically a norm function. The question involves a matrix A and vector b, and the answer is that the minimizer can be found when Ax=b. It is suggested to refer to the textbook for more information on finding the value of x.
  • #1
Schnurmann
3
0
Hi folks,
1. Homework Statement
I don't fully understand the question statement, how is it supposed to be read?
Question:
Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank n. Justify your answer.

Homework Equations


How do i read the part where set symbols are mentioned?
 
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  • #2
Okay i have another question:
How do we find the minimizer of a norm function?
 
  • #3
Schnurmann said:
Hi folks,
1. Homework Statement
I don't fully understand the question statement, how is it supposed to be read?
Question:
Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank n. Justify your answer.

Homework Equations


How do i read the part where set symbols are mentioned?
A is an m x n matrix, and b is an m-dimensional vector.
Schnurmann said:
Okay i have another question:
How do we find the minimizer of a norm function?
f(x) is always nonnegative, with a minimum value of 0, which occurs when Ax = b. Your textbook ought to have a discussion of how to find x in this case.
 

1. What is the concept of Least Squares Estimation?

Least Squares Estimation is a statistical method used to estimate the parameters of a mathematical model that minimizes the sum of the squares of the differences between the observed data and the predicted values.

2. What is the purpose of using Least Squares Estimation?

The purpose of using Least Squares Estimation is to find the best-fit line or curve that represents the relationship between two or more variables. It is commonly used in regression analysis to determine the relationship between a dependent variable and one or more independent variables.

3. What is the main problem with symbols in Least Squares Estimation?

The main problem with symbols in Least Squares Estimation is that they can be difficult to interpret and understand, especially for those who are not familiar with the notation used in statistics. This can make it challenging for individuals to fully grasp the concept and apply it correctly.

4. How is the Least Squares Estimation calculated?

The Least Squares Estimation is calculated by finding the values of the model parameters that minimize the sum of the squared differences between the observed data and the predicted values. This is typically done using the method of least squares, which involves finding the values that minimize the sum of the squared residuals.

5. What are some limitations of Least Squares Estimation?

Some limitations of Least Squares Estimation include the assumptions that the data is normally distributed and that the relationship between variables is linear. It can also be sensitive to outliers and may not perform well with small sample sizes. Additionally, it may not be the best method to use if the true relationship between variables is not well represented by the chosen model.

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