Virtual Labs

Regression and curve fitting using procedures such as Least-Square and Weighted Least-Square curve fitting.

The Least Squares method determines parameters by minimizing:

a: Sum of absolute errors Explanation

Explanation

b: Sum of squared residuals Explanation

Explanation

c: Maximum error Explanation

Explanation

d: Determinant of matrix Explanation

Explanation

A residual in regression represents:

a: Observed value minus predicted value Explanation

Explanation

b: Predicted value minus observed value Explanation

Explanation

c: Slope of the line Explanation

Explanation

d: Matrix determinant Explanation

Explanation

In matrix form, the normal equation for least squares is:

a: Xβ = y Explanation

Explanation

b: XᵀXβ = Xᵀy Explanation

Explanation

c: XXᵀβ = y Explanation

Explanation

d: βX = y Explanation

Explanation

Least squares is typically used when the system of equations is:

a: Exactly determined Explanation

Explanation

b: Underdetermined Explanation

Explanation

c: Overdetermined Explanation

Explanation

d: Nonlinear only Explanation

Explanation

The coefficient of determination (R²) measures:

a: Slope of regression line Explanation

Explanation

b: Proportion of variance explained by the model Explanation

Explanation

c: Total number of data points Explanation

Explanation

d: Inverse of residual Explanation

Explanation

Weighted Least Squares differs from ordinary Least Squares because it:

a: Uses logarithmic transformation Explanation

Explanation

b: Assigns different importance (weights) to data points Explanation

Explanation

c: Ignores residuals Explanation

Explanation

d: Uses integration Explanation

Explanation

Weighted Least Squares is particularly useful when:

a: Errors have constant variance Explanation

Explanation

b: Errors have unequal variance (heteroscedasticity) Explanation

Explanation

c: Data is perfectly linear Explanation

Explanation

d: Matrix is square Explanation

Explanation

In linear regression y = β0 + β1x, β1 represents:

a: Intercept Explanation

Explanation

b: Slope Explanation

Explanation

c: Residual Explanation

Explanation

d: Variance Explanation

Explanation

The solution to least squares problem can be interpreted geometrically as:

a: Projection of vector y onto column space of X Explanation

Explanation

b: Intersection of two lines Explanation

Explanation

c: Matrix inversion only Explanation

Explanation

d: Derivative of function Explanation

Explanation

If XᵀX is singular in least squares computation, then:

a: Unique solution always exists Explanation

Explanation

b: Matrix is full rank Explanation

Explanation

c: Columns of X are linearly dependent Explanation

Explanation

d: Residual becomes zero Explanation

Explanation