Explore the fundamentals of Fisher’s Linear Discriminant Analysis (LDA) for high-dimensional data, focusing on concepts like class separation, projections, assumptions, and practical applications. This quiz is designed to strengthen understanding of how LDA works and its role in dimensionality reduction and classification tasks.
What is the primary goal of Fisher’s Linear Discriminant Analysis when applied to high-dimensional data?
Explanation: The main purpose of Fisher’s LDA is to find projections that maximize the separation between different classes. Unlike clustering, LDA requires class labels and does not work without them, so clustering is not the correct answer. LDA seeks optimal projections, not random ones, making the third option incorrect. Rather than increasing dimensionality, LDA reduces it, making the last option inappropriate.
Which key concept does Fisher’s LDA use to reduce the dimensionality of data while keeping classes separated as much as possible?
Explanation: LDA projects data onto a lower-dimensional space to maximize class separation. Orthogonal transformation can describe some dimensionality reduction methods, but LDA specifically seeks projections, not arbitrary orthogonal ones. Nonlinear kernel mapping is related to kernel methods, not classical LDA. Rotating the data matrix does not directly achieve dimensionality reduction or class separation.
In the context of Fisher’s LDA, what does the between-class scatter matrix represent?
Explanation: The between-class scatter matrix measures the variance between class means, reflecting how distinct the classes are from each other. The first option describes the within-class scatter, not the between-class scatter. The overall mean is a single value, not a scatter matrix. The total scatter is a broader concept that combines within and between-class scatter.
Which of the following is a key assumption required for Fisher’s LDA to perform optimally?
Explanation: LDA assumes that all classes share the same covariance matrix, which allows it to model class separation optimally. Feature independence is an assumption for other techniques like Naive Bayes, not LDA. LDA does not require a uniform distribution of data. Also, LDA is designed for class labels (categorical targets), not continuous targets.
Unlike Principal Component Analysis (PCA), what does Fisher’s LDA specifically use to decide on the directions for projection?
Explanation: LDA chooses directions that maximize the separation between classes, whereas PCA maximizes overall variance without considering class labels. The first and third options focus on total variance, a PCA concept rather than LDA. The centroid of the data is not used for projecting directions in either LDA or PCA.
If Fisher’s LDA is applied to a problem with five separate classes, what is the maximum number of discriminant axes LDA can provide?
Explanation: LDA can yield at most (number of classes minus one) discriminant axes, so for five classes it produces four. Five would imply one axis per class, but this is incorrect as there is always one less than the number of classes. One is too few for five classes. Ten is excessive and unrelated to the class count.
In supervised learning, Fisher’s LDA is typically used for which type of task?
Explanation: LDA is mainly used for classification, as it projects data to enhance separability between labeled classes. Clustering is an unsupervised method and does not fit LDA's use case. Regression relates to predicting continuous outcomes, not class labels, making it unsuitable. While LDA might be part of preprocessing for forecasting, its main use is for classification.
Which challenge may affect the performance of Fisher’s LDA when applied to very high-dimensional datasets?
Explanation: In high-dimensional settings, LDA can suffer from overfitting because there may be too few samples per feature, making reliable estimation of class statistics difficult. Non-linear class separation is generally an inherent limitation, not specific to high dimensions. Clustering does not require labels, but LDA does, so the third option is misleading. Higher dimensions do not usually increase interpretability.
After applying Fisher’s LDA, the original dataset is transformed into which kind of space?
Explanation: LDA projects data into a lower-dimensional discriminant space, increasing class separability along fewer axes. It does not increase dimensionality, making the second option incorrect. LDA does not produce binary codes or represent data in a time-frequency domain, so the third and fourth options are inapplicable here.
What will most likely happen if the assumption of equal covariance among classes in Fisher’s LDA is not met?
Explanation: If class covariances differ, the separation found by LDA may be suboptimal and less reliable, but the algorithm will still produce results. The algorithm does not stop with an explicit error, so option two is inaccurate. Performance usually degrades rather than improves (making option three incorrect). LDA does not switch to regression, so option four is also incorrect.