MATH 4910/5010 - R Lab 5
In this lab you will use topological data analysis tools to identify point cloud sample data in R.
Objectives:- Demonstrate understanding of TDA by identifying the shape of point clouds
You can find the R markdown template and the datasets used for this lab on Canvas in the course files under the "R Lab 5" folder. Follow the instructions below. Instructions in green indicate tasks that should be completed in the R markdown file for this lab.
Mystery Manifolds
In this lab you will use all the things you've learned to identify the shape of the data. You should have found four data sets on Canvas. Each dataset contains the coordinates for one of the six sets of manifolds below embedded in \(\mathbb{R}^10\).
| Set A | Set B |
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| Set C | Set D |
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| Set E | Set F |
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Possible Sets of Manifolds
Match the datasets with the manifolds from which they were sampled.
In code block 1, write code to load any packages needed to complete this lab.
In code block 2, write code to loads data from
mystery_manifold1.csv,mystery_manifold2.csv,mystery_manifold3.csv, andmystery_manifold4.csvinto R. The last three files, contains coordinates for points sampled from manifolds embedded in \(\mathbb{R}^8\).In code blocks 3 through 6, write code that draws the persistence diagrams and barcodes for each set of data. Then in space following each code block, identify the Betti numbers of the manifolds sampled. Then, indicate which set of manifolds each dataset is sampled from.
Useful Betti numbers
I've added the following Betti number facts to help with this task.
Contractable SpacesLet \(X\) be one of \(\mathbb{R}^n\), \(B^n\), \(B_O^n\), or \(H^n\) for \(n\geq 0\). \[ \beta_k(X)=\begin{cases} 1 & k=0\\ 0 & \mathrm{otherwise} \end{cases} \]
The \(n\)-Dimensional SphereConsider the \(n\)-dimensional sphere \(S^n\). \[ \beta_k(S^n)=\begin{cases} 1 & k=0\\ 1 & k=n\\ 0 & \mathrm{otherwise} \end{cases} \]
SurfacesLet \(S_g\) be the 2-sided genus \(g\) compact surface without boundary. \[ \beta_k(S_g)=\begin{cases} 1 & k=0\\ 2g & k=1\\ 1 & k=2\\ 0 & \mathrm{otherwise} \end{cases} \]
Congratulations! You've completed all the R Labs for this course. You're a master at doing TDA in R