Consider the following spiral structure...

PCA simply rotates the data...

Nonlinear dimension reduction with t-SNE and UMAP

Both t-SNE and UMAP look at the local distances between points in the original p-dimensional space and try to reproduce them in a lower k-dimensional space

t-SNE: t-distributed stochastic neighbor embedding

• Construct conditional probability for similarity between observations in original space

  • i.e., probability xi will pick xj as its neighbor

pj∣i=exp(−‖xi−xj‖2/2σ2i)∑k≠iexp(−‖xi−xk‖2/2σ2i),pij=(pj∣i+pi∣j)2n

• σi is the variance of Gaussian centered at xi controlled by perplexity: log( perplexity )=−∑jpj∣ilog2pj∣i

  • loosely interpreted as the number of close neighbors to consider for each point

• Find points yi in lower dimensional space with symmetrized student t-distribution

qj∣i=(1+‖yi−yj‖2)−1∑k≠i(1+‖yi−yk‖2)−1,qij=qi∣j+qj∣i2n

• Match conditional probabilities by minimize sum of KL divergences C=∑ijpijlog(pijqij)

Starbucks t-SNE plot

set.seed(2013)
tsne_fit <- starbucks %>%
  dplyr::select(serv_size_m_l:caffeine_mg) %>%
  scale() %>%
  Rtsne(check_duplicates = FALSE)
starbucks %>%
  mutate(tsne1 = tsne_fit$Y[,1],
         tsne2 = tsne_fit$Y[,2]) %>%
  ggplot(aes(x = tsne1, y = tsne2, 
             color = size)) +
  geom_point(alpha = 0.5) + 
  labs(x = "t-SNE 1", y = "t-SNE 2")Copy Code

Starbucks t-SNE plot - involves randomness!

set.seed(2014)
tsne_fit <- starbucks %>%
  dplyr::select(serv_size_m_l:caffeine_mg) %>%
  scale() %>%
  Rtsne(check_duplicates = FALSE)
starbucks %>%
  mutate(tsne1 = tsne_fit$Y[,1],
         tsne2 = tsne_fit$Y[,2]) %>%
  ggplot(aes(x = tsne1, y = tsne2, 
             color = size)) +
  geom_point(alpha = 0.5) +
  labs(x = "t-SNE 1", y = "t-SNE 2")Copy Code

Starbucks t-SNE plot - watch the perplexity!

set.seed(2013)
tsne_fit <- starbucks %>%
  dplyr::select(serv_size_m_l:caffeine_mg) %>%
  scale() %>%
  Rtsne(perplexity = 100,
        check_duplicates = FALSE)
starbucks %>%
  mutate(tsne1 = tsne_fit$Y[,1],
         tsne2 = tsne_fit$Y[,2]) %>%
  ggplot(aes(x = tsne1, y = tsne2, 
             color = size)) +
  geom_point(alpha = 0.5) +
  labs(x = "t-SNE 1", y = "t-SNE 2")Copy Code

Back to the spirals: results depend on perplexity!

Criticisms of t-SNE plots

Interactive web apps with Shiny

Shiny is a framework to interactive web applications and dynamic dashboards in R

You do NOT need to be a web developer to create Shiny apps, you just need to learn some additional syntax to augment your R code

Every Shiny app consists of two scripts (could also be saved into one file app.R but that's annoying)

1. ui.R: controls user interface, sets up the display, widgets for user input

   • contains more code specific to Shiny

2. server.R: code to generate / display the results! Communicates with ui.R with reactive objects: processes user input to return output

   • will contain more traditional R code: load packages, data wrangling, create plots

Can be run locally or deployed on a Shiny app server for public viewing