Math Problem Statement
Solution
The questions in the image refer to Principal Component Analysis (PCA) and involve identifying the direction of projections based on ellipses representing data distributions in two-dimensional space.
Question Analysis:
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Figure 1 Analysis:
- The ellipse is elongated along the -axis. This suggests that the direction of maximum variance (first principal component) is along the -axis.
Answer:
- Option a) -axis is correct.
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Figure 2 Analysis:
- The ellipse is elongated at an angle between the and axes, indicating that the direction of maximum variance (first principal component) is along a diagonal direction.
Answer:
- Option c) Direction making an angle of 45° with -axis in anticlockwise direction is correct.
Would you like a more detailed explanation or have any other questions?
Relative Questions:
- What is Principal Component Analysis (PCA) and how is it used in data analysis?
- How does the orientation of an ellipse indicate the direction of maximum variance in PCA?
- What are the implications of projecting data onto the first principal component?
- How does PCA reduce the dimensionality of data?
- What is the significance of the eigenvectors in determining the directions of the principal components?
Tip:
In PCA, the first principal component represents the direction of maximum variance, and each subsequent component represents the next highest variance orthogonal to the previous ones.
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Math Problem Analysis
Mathematical Concepts
Principal Component Analysis
Linear Algebra
Multivariate Statistics
Formulas
Eigenvector decomposition
Covariance matrix calculation
Theorems
Principal Component Theorem
Suitable Grade Level
Undergraduate (Statistics/Mathematics)
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