How do i construct a design matrix for a least square problem?

In summary, a design matrix for a least square problem is a matrix that represents the independent variables used in a linear regression model. To construct a design matrix, one needs to identify the independent variables and arrange them as columns, with a column of 1s added for the intercept term. The importance of a design matrix lies in its role in performing linear regression and easily calculating predicted values. There are specific rules to follow when constructing a design matrix, such as the number of columns being equal to the number of independent variables plus one, and the matrix being full rank. Additionally, a design matrix can also be used for other types of regression models, with some differences in the dependent variable's distribution.
  • #1
papasmurfff
3
0
Suppose we want to to fit an arbitrary function f(t) with a polynomial of
degree n - 1 using the values of the function in an arbitrary set of points z = [z1; z2; : : : ; zm].

how would do i construct the least squares problem Ax=b. in other words, how would i construct the matrix A and b in terms of f(t) and z ?
 
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  • #2
...using the values of the function in an arbitrary set of points z...

Not sure what you mean here. Do you know the relationship of t to z?
 

1. What is a design matrix for a least square problem?

A design matrix for a least square problem is a matrix that represents the independent variables used in a linear regression model. It is used to calculate the coefficients of the model that will minimize the sum of squared errors between the predicted values and the actual values of the dependent variable.

2. How do I construct a design matrix?

To construct a design matrix, you need to first identify the independent variables that will be used in your linear regression model. Then, arrange those variables as columns in a matrix, with each row representing a different data point or observation. Add a column of 1s to the left of the matrix to account for the intercept term. This resulting matrix is your design matrix.

3. What is the importance of a design matrix in a least square problem?

A design matrix is essential in a least square problem because it helps us to perform linear regression and obtain the coefficients that will give us the best-fit line for our data. It also allows us to easily calculate the predicted values of the dependent variable based on the values of the independent variables.

4. Are there any specific rules for constructing a design matrix?

Yes, there are some rules to follow when constructing a design matrix. The number of columns in the matrix should be equal to the number of independent variables plus one for the intercept term. The matrix should have more rows than columns to ensure that the model is not overfitting the data. The independent variables should not be linearly dependent, and the matrix should be full rank.

5. Can a design matrix be used for other types of regression models?

Yes, a design matrix can also be used for other types of regression models, such as logistic regression, ridge regression, and lasso regression. The only difference is that the dependent variable may be transformed or have a different distribution, but the construction of the design matrix remains the same.

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