Two labeled point clouds are projected onto a rotating line, showing how their one-dimensional distributions shift with angle. The animation demonstrates that large mean separation alone is not enough if class spread is also large, then builds the Fisher criterion J(w) from between-class scatter S_B and within-class scatter S_W. It shows scale invariance of J(w) and contrasts the supervised Fisher direction with the unsupervised PCA direction. Useful for undergraduate AI students learning the motivation behind linear discriminant analysis before its derivation.
Narrated · 16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
# Fisher projection separates means while controlling spread Two labeled point clouds are projected onto a rotating line. As the line turns, their one-dimensional projected distributions change. Show that a large distance between projected means is insufficient if each class is also widely spread. Introduce class means, within-class scatter S_W, and the two-class between-class scatter S_B = (m1-m2)(m1-m2)^T. Build the Fisher criterion J(w) = (w^T S_B w)/(w^T S_W w), linking the numerator to mean separation and the denominator to within-class spread. Double the length of w to show that both terms scale equally, leaving the criterion unchanged. Contrast this supervised direction with a PCA direction that maximizes total variance without labels. Keep the projection direction distinct from the later classification threshold. Use black backgrounds, cyan and orange samples, white formulas, and smooth orthogonal projection lines. Useful for understanding Fisher's objective before deriving its solution. Audience: 二年级人工智能专业大学生