Virtual Labs

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

In Least Squares regression, the optimal solution satisfies:

a: Residuals are maximized Explanation

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b: Xᵀ(y - Xβ) = 0 Explanation

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c: Determinant of X is zero Explanation

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d: β = y Explanation

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The geometric interpretation of Least Squares solution is:

a: Projection of y onto column space of X Explanation

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b: Intersection of curves Explanation

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c: Matrix inversion only Explanation

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d: Gradient of polynomial Explanation

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If the design matrix X has full column rank, then the Least Squares solution is:

a: Non-unique Explanation

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b: Unique Explanation

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c: Always zero Explanation

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d: Complex only Explanation

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Weighted Least Squares modifies the normal equation to:

a: XᵀXβ = y Explanation

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b: XᵀWXβ = XᵀWy Explanation

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c: WXβ = y Explanation

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d: XWβ = y Explanation

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Weighted Least Squares is most appropriate when:

a: All errors have equal variance Explanation

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b: Errors exhibit heteroscedasticity Explanation

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c: Data is perfectly linear Explanation

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d: Sample size is small only Explanation

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If two predictor variables are highly correlated, the regression model suffers from:

a: Homoscedasticity Explanation

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b: Multicollinearity Explanation

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c: Orthogonality Explanation

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d: Singularity of y Explanation

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The residual vector in Least Squares is orthogonal to:

a: Row space of X Explanation

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b: Column space of X Explanation

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c: Null space of X Explanation

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d: Determinant Explanation

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Overfitting in curve fitting typically occurs when:

a: Model is too simple Explanation

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b: Model has too many parameters relative to data Explanation

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c: Residuals are large Explanation

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d: Matrix is square Explanation

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If R² is close to 1, it indicates:

a: Poor model fit Explanation

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b: High proportion of variance explained Explanation

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c: High bias Explanation

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d: Model instability Explanation

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In matrix-based regression, solving via QR decomposition instead of normal equations improves:

a: Graph color Explanation

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b: Numerical stability Explanation

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c: Number of variables Explanation

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d: Data randomness Explanation

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