---
title: "Optimizing the calculation of Hopkins statistic"
author: "Kevin Wright"
date: "2021-04-19"
bibliography: hopkins.bib
output: 
  rmarkdown::html_vignette:
    md_extensions: [ 
      "-autolink_bare_uris" 
    ]
vignette: >
  %\VignetteIndexEntry{Optimizing the calculation of Hopkins statistic}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

The `clustertend::hopkins` function used two nested `for()` loops:

```{r, eval=FALSE}
for (i in 1:n) {
  distp[1] <- dist(rbind(p[i, ], data[1, ]))
  minqi <- dist(rbind(q[i, ], data[1, ]))
  for (j in 2:nrow(data)) {
    distp[j] <- dist(rbind(p[i, ], data[j, ]))
    error <- q[i, ] - data[j, ]
    if (sum(abs(error)) != 0) {
      distq <- dist(rbind(q[i, ], data[j, ]))
      if (distq < minqi) 
        minqi <- distq
    }
  }
```

Whenever nested `for()` loops are used, you should immediately consider if it possible to vectorize one or both loops. In this case, we can define a function that calculates the euclidean distance from a vector to the nearest row of a matrix and then use this function to eliminate the inner loop:

```{r, eval=FALSE}
DistVecToMat <- function(x,Y){
  min(apply(Y, 1, function(yi) sqrt(sum((x-yi)^2))))
}
# DistVecToMat(c(1,2), matrix(c(4,5,6,7), nrow=2, byrow=TRUE))
# 4.242641 7.071068 # sqrt(c(18,50)) 
    
# For each ith row of sample, calculate the distance of:
#   U[i,] to X
#   W[i,] to X[-k[i] , ], i.e. omitting W[i,] from X
dux <- rep(NA, n)
dwx <- rep(NA, n)
for(i in 1:n) {
  dux[i] <- DistVecToMat(U[i,], X)
  dwx[i] <- DistVecToMat(W[i,], X[-k[i],])
}
```

When thinking about manipulating two vectors or two matrices, you should always keep in mind that there are R functions like `crossprod()`, `outer()`, and `apply()` that might come in handy. I played around with these functions but was having trouble getting the results I wanted. I then used Google to search for ideas and discovered the `pdist` package, which can efficiently compute the distance between all pairs of rows of two matrices. This is exactly what we need.

```{r, eval=FALSE}
# pdist(W,X) computes distance between rows of W and rows of X
tmp <- as.matrix(pdist(W,X))
dwx <- apply(tmp, 1, min)

# pdist(U,X) computes distance between rows of U and rows of X
tmp <- as.matrix(pdist(U,X))
dux <- apply(tmp, 1, min)
```

Finally, there's two ways two different ways to change some elements of the distance matrix to be Inf:

```{r, eval=FALSE}
library(microbenchmark)

# Method 1. Loop over vector 1:n
# for(i in 1:m) dwx[i,k[i]] <- Inf
microbenchmark(hopkins(X12, 500))
## Unit: milliseconds
##                expr     min      lq     mean  median       uq      max neval
##  hopkins(X12, 500) 38.9493 42.8045 50.73876 45.0668 47.69945 120.9366   100

# Method 2. Build a matrix of indexes to the cells that need changing
# dwx[ matrix( c(1:n, k), nrow=n) ] <- Inf
microbenchmark(hopkins(X12, 500))
## Unit: milliseconds
##                expr     min       lq     mean   median      uq      max neval
##  hopkins(X12, 500) 39.2668 42.41565 50.21628 43.74215 46.9462 126.3522   100
```

The median times across 100 calls to the function are virtually identical for this test case. Results could be different for smaller/larger datasets. In our (purely subjective) taste, the loop method is a bit easier to understand.

How good is the optimization? In one simulated-data example with 1000 rows and 2 columns, sampling 500 rows, the non-optimized function used about 17 seconds, while the optimized function used approximately 0.05 seconds.

```{r, eval=FALSE, echo=FALSE}
set.seed(1)
X1 <- matrix(rnorm(1000, mean=0, sd=1), ncol=2)
X2 <- matrix(rnorm(1000, mean=5, sd=1), ncol=2)
X12 <- rbind(X1, X2)
plot(X12)

# Old version
set.seed(42)
system.time( hop1 <- clustertend::hopkins(X12, 500) ) # 17 sec
1-hop1$H
# .8994528

# New version
set.seed(42)
system.time( hop2 <- hopkins::hopkins(X12, 500, d=1) ) # .05 sec
hop2
# .9054555
```
