In principal component analysis, the projected lower dimensional space corresponds to – A. subset of the original co-ordinate axis B. eigenvectors of the data covariance matrix C. eigenvectors of the data distance matrix D. orthogonal vectors to the original co-ordinate axis
Question
In principal component analysis, the projected lower dimensional space corresponds to – A. subset of the original co-ordinate axis B. eigenvectors of the data covariance matrix C. eigenvectors of the data distance matrix D. orthogonal vectors to the original co-ordinate axis
Solution
The answer is B. Eigenvectors of the data covariance matrix.
Here's a step-by-step explanation:
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Principal Component Analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.
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The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components.
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The principal components are orthogonal because they are the eigenvectors of the covariance matrix, which is symmetric. Eigenvectors of the covariance matrix correspond to the directions (i.e., the vectors) in the original data where there is the most variance. The corresponding eigenvalue indicates how much variance there is in the data along the eigenvector.
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Therefore, the projected lower dimensional space in PCA corresponds to the eigenvectors of the data covariance matrix.
Similar Questions
How does the term "principal component analysis" (PCA) contribute to dimensionality reduction in data analysis?By clustering similar data points togetherBy transforming features into a lower-dimensional spaceBy creating new features based on existing onesBy evaluating feature importance
What is Principal Component Analysis?
Which dimensionality reduction technique is affected by the curse of dimensionality?Review LaterPrincipal Component Analysis (PCA)UMAPt-SNENone of the above
Which of the given options is NOT associated with the properties of the Principal Component?Select an option Clear ResponseThe Principal Components are orthogonal.The variation in the Principal Components increases from the 1st first Principal Component to the last one.The linear combinations of the original variables from the Principal ComponentsThe Principal Components helps in dimensionality reduction without information loss.
1. Consider the case where Y is an n × p data matrix containing n observations on p-variables. The datamatrix has been demeaned. Use this information to answer the following questions. (4 marks).• S denotes the sample covariance matrix of Y. Using matrix notation, state the sample covariancematrix of Y. Be sure to clearly define any quantities needed to state S.• What is the dimension of the sample covariance matrix of Y.• What is the dimension of an arbitrary eigenvector associated with S?
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