Generate contaminated triangle data. Groups 1, 2, 3, and 4 are separable over the two dimensions of functional data. Groups 5 and 6 contain the contaminated curves of groups 1 and 3 respectively.
genTriangles()
List of functional data objects representing the two dimensions of triangle data.
Group classification for each curve
Cristina Anton and Iain Smith
Group 1:
\(X_1(t) = U + (0.6 - U)H_1(t) + \epsilon_1(t)\)
\( X_2(t) = U + (0.5 - U)H_1(t) + \epsilon_1(t)\)
Contaminated \(X_1(t) = \sin(t) + (0.6 - U)H_1(t) + \epsilon_2(t)\)
Contaminated \( X_2(t) = \sin(t) + (0.5 - U)H_1(t) + \epsilon_2(t)\)
Group 2:
\(X_1(t) = U + (0.6 - U)H_2(t) + \epsilon_1(t)\)
\(X_2(t) = U + (0.5 - U)H_2(t) + \epsilon_1(t)\)
Group 3:
\(X_1(t) = U + (0.5 - U)H_1(t) + \epsilon_1(t)\)
\( X_2(t) = U + (0.6 - U)H_2(t) + \epsilon_1(t)\)
Contaminated \(X_1(t) = \sin(t) + (0.5 - U)H_1(t) + \epsilon_3(t)\)
Contaminated \(X_2(t) = \sin(t) + (0.6 - U)H_2(t) + \epsilon_3(t)\)
Group 4:
\(X_1(t) = U + (0.5 - U)H_2(t) + \epsilon_1(t)\)
\(X_2(t) = U + (0.6 - U)H_1(t) + \epsilon_1(t).\) Here \(t\in [1,21]\), \(H_1(t) = (6-\vert t-7\vert)_+\), and \(H_2(t) = (6-\vert t-15\vert)_+\), with \((\cdot)_+\) representing the positive part. \(U \sim \mathcal{U}(0, 0.1)\), and \(\epsilon_1(t)\sim N(0, 0.5)\), \(\epsilon_2(t)\sim N(0, 2)\), \(\epsilon_3(t) \sim Cauchy(0, 4)\) are mutually independent white noises and independent of \(U\). We simulate 100 curves for each group, groups 1 and 3 consisting of 80 ordinary curves and 20 contaminated curves. Curves are smoothed using a 25 cubic B-spline basis.
- C.Bouveyron and J.Jacques (2011), Model-based Clustering of Time Series in Group-specific Functional Subspaces, Advances in Data Analysis and Classification, vol. 5 (4), pp. 281-300, <doi:10.1007/s11634-011-0095-6>
- Schmutz A, Jacques J, Bouveyron C, et al (2020) Clustering multivariate functional data in group-specific functional subspaces. Comput Stat 35:1101-1131
- Cristina Anton, Iain Smith Model-based clustering of functional data via mixtures of \(t\) distributions. Advances in Data Analysis and Classification (to appear).
plotTriangles
# Multivariate Contaminated Triangles
conTrig <- genTriangles()
cls = conTrig$groupd
plotTriangles(conTrig)
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