I'm trying to write a function that takes in p1 probabilities for Mahalanobis distances and returns p2 probabilities. The formula for p2, along with a worked example is given at on the IBM website. I have written a function (below) that solves the problem, and allows me to reproduce the p2 values given in the worked example at the aforementioned webpage.
p1_to_p2 <- function(p1,N) {
p2 <- numeric(length(p1))
for (i in 1:length(p1))
{
k <- i;
p1_value <- p1[i];
start_value <- 1;
while (k >= 1)
{
start_value = start_value - choose(N,N-k+1) * (1-p1_value)^(N-k+1) * (p1_value)^(k-1)
k <- k-1;
}
p2[i] <- start_value;
}
return(p2)
}
p1 <- c(.0046132,.0085718,.0390278,.0437704,.0475222)
N <- 73
p1_to_p2(p1,N)
Although the function works, it's been suggested to me by a colleague that it's inefficient/poorly written as it's not vectorized. This is indeed potentially relevant since in general we will be converting a lot more than just 5 p1 values to p2 values.
I have some limited experience vectorizing code, but I am wondering if a vectorized solution is possible in this context since within the loop the variable start_value constantly needs to update itself. If vectorization is not possible, is there some other way I should improve the code so that it works better?
Here is one way to do it, Breaking the steps here can help(Please read the comments):
#Input:
N <- 73
p1 <- c(.0046132,.0085718,.0390278,.0437704,.0475222)
n <- N:(N-length(p1)+1)
# code:
mahalanobis_dist = function(x=x,n){
m = max(n)
max_min = Reduce(`*`,c(1, n[-length(n)]), accumulate = TRUE)
acc = c(1, Reduce(`*`, seq_along(n), accumulate = TRUE)[-length(n)])
comns = max_min/acc
exp <- comns*((1 - x)**n)*(x**(m - n))
return(1- sum(exp))
} ## the calculation of Mahalanobis distances
## This is just an iterator for each of the sequences we have to run the above function
ls <- lapply(n, function(x)(max(n):x))
## creating a list of iterators
## applying mapply, mapply or Map can iterate multiple inputs of the function,
## here the input p1 and ls , p1 is your input points, ls is the iterator created above
mapply(mahalanobis_dist,p1, ls)
## Applying the function on each iterators
#Output:
#> mapply(mahalanobis_dist,p1, ls)
#[1] 0.2864785 0.1299047 0.5461263 0.3973690
#[5] 0.2662369
Note:
Also, one can join the last two steps like below, with one function and correct iteration this can be achieved:
mapply(mahalanobis_dist,p1, lapply(n, function(x)(max(n):x)))
While trying to learn R, I want to implement the algorithm below in R. Consider the two lists below:
List 1: "crashed", "red", "car"
List 2: "crashed", "blue", "bus"
I want to find out how many actions it would take to transform 'list1' into 'list2'.
As you can see I need only two actions:
1. Replace "red" with "blue".
2. Replace "car" with "bus".
But, how we can find the number of actions like this automatically.
We can have several actions to transform the sentences: ADD, REMOVE, or REPLACE the words in the list.
Now, I will try my best to explain how the algorithm should work:
At the first step: I will create a table like this:
rows: i= 0,1,2,3,
columns: j = 0,1,2,3
(example: value[0,0] = 0 , value[0, 1] = 1 ...)
crashed red car
0 1 2 3
crashed 1
blue 2
bus 3
Now, I will try to fill the table. Please, note that each cell in the table shows the number of actions we need to do to reformat the sentence (ADD, remove, or replace).
Consider the interaction between "crashed" and "crashed" (value[1,1]), obviously we don't need to change it so the value will be '0'. Since they are the same words. Basically, we got the diagonal value = value[0,0]
crashed red car
0 1 2 3
crashed 1 0
blue 2
bus 3
Now, consider "crashed" and the second part of the sentence which is "red". Since they are not the same word we can use calculate the number of changes like this :
min{value[0,1] , value[0,2] and value[1,1]} + 1
min{ 1, 2, 0} + 1 = 1
Therefore, we need to just remove "red".
So, the table will look like this:
crashed red car
0 1 2 3
crashed 1 0 1
blue 2
bus 3
And we will continue like this :
"crashed" and "car" will be :
min{value[0,3], value[0,2] and value[1,2]} + 1
min{3, 2, 1} +1 = 2
and the table will be:
crashed red car
0 1 2 3
crashed 1 0 1 2
blue 2
bus 3
And we will continue to do so. the final result will be :
crashed red car
0 1 2 3
crashed 1 0 1 2
blue 2 1 1 2
bus 3 2 2 2
As you can see the last number in the table shows the distance between two sentences: value[3,3] = 2
Basically, the algorithm should look like this:
if (characters_in_header_of_matrix[i]==characters_in_column_of_matrix [j] &
value[i,j] == value[i+1][j-1] )
then {get the 'DIAGONAL VALUE' #diagonal value= value[i, j-1]}
else{
value[i,j] = min(value[i-1, j], value[i-1, j-1], value[i, j-1]) + 1
}
endif
for finding the difference between the elements of two lists that you can see in the header and the column of the matrix, I have used the strcmp() function which will give us a boolean value(TRUE or FALSE) while comparing the words. But, I fail at implementing this.
I'd appreciate your help on this one, thanks.
The question
After some clarification in a previous post, and after the update of the post, my understanding is that Zero is asking: 'how one can iteratively count the number of word differences in two strings'.
I am unaware of any implementation in R, although i would be surprised if i doesn't already exists. I took a bit of time out to create a simple implementation, altering the algorithm slightly for simplicity (For anyone not interested scroll down for 2 implementations, 1 in pure R, one using the smallest amount of Rcpp). The general idea of the implementation:
Initialize with string_1 and string_2 of length n_1 and n_2
Calculate the cumulative difference between the first min(n_1, n_2) elements,
Use this cumulative difference as the diagonal in the matrix
Set the first off-diagonal element to the very first element + 1
Calculate the remaining off diagonal elements as: diag(i) - diag(i-1) + full_matrix(i-1,j)
In the previous step i iterates over diagonals, j iterates over rows/columns (either one works), and we start in the third diagonal, as the first 2x2 matrix is filled in step 1 to 4
Calculate the remaining abs(n_1 - n_2) elements as full_matrix[,min(n_1 - n_2)] + 1:abs(n_1 - n_2), applying the latter over each value in the prior, and bind them appropriately to the full_matrix.
The output is a matrix with dimensions row and column names of the corresponding strings, which has been formatted for some easier reading.
Implementation in R
Dist_between_strings <- function(x, y,
split = " ",
split_x = split, split_y = split,
case_sensitive = TRUE){
#Safety checks
if(!is.character(x) || !is.character(y) ||
nchar(x) == 0 || nchar(y) == 0)
stop("x, y needs to be none empty character strings.")
if(length(x) != 1 || length(y) != 1)
stop("Currency the function is not vectorized, please provide the strings individually or use lapply.")
if(!is.logical(case_sensitive))
stop("case_sensitivity needs to be logical")
#Extract variable names of our variables
# used for the dimension names later on
x_name <- deparse(substitute(x))
y_name <- deparse(substitute(y))
#Expression which when evaluated will name our output
dimname_expression <-
parse(text = paste0("dimnames(output) <- list(",make.names(x_name, unique = TRUE)," = x_names,",
make.names(y_name, unique = TRUE)," = y_names)"))
#split the strings into words
x_names <- str_split(x, split_x, simplify = TRUE)
y_names <- str_split(y, split_y, simplify = TRUE)
#are we case_sensitive?
if(isTRUE(case_sensitive)){
x_split <- str_split(tolower(x), split_x, simplify = TRUE)
y_split <- str_split(tolower(y), split_y, simplify = TRUE)
}else{
x_split <- x_names
y_split <- y_names
}
#Create an index in case the two are of different length
idx <- seq(1, (n_min <- min((nx <- length(x_split)),
(ny <- length(y_split)))))
n_max <- max(nx, ny)
#If we have one string that has length 1, the output is simplified
if(n_min == 1){
distances <- seq(1, n_max) - (x_split[idx] == y_split[idx])
output <- matrix(distances, nrow = nx)
eval(dimname_expression)
return(output)
}
#If not we will have to do a bit of work
output <- diag(cumsum(ifelse(x_split[idx] == y_split[idx], 0, 1)))
#The loop will fill in the off_diagonal
output[2, 1] <- output[1, 2] <- output[1, 1] + 1
if(n_max > 2)
for(i in 3:n_min){
for(j in 1:(i - 1)){
output[i,j] <- output[j,i] <- output[i,i] - output[i - 1, i - 1] + #are the words different?
output[i - 1, j] #How many words were different before?
}
}
#comparison if the list is not of the same size
if(nx != ny){
#Add the remaining words to the side that does not contain this
additional_words <- seq(1, n_max - n_min)
additional_words <- sapply(additional_words, function(x) x + output[,n_min])
#merge the additional words
if(nx > ny)
output <- rbind(output, t(additional_words))
else
output <- cbind(output, additional_words)
}
#set the dimension names,
# I would like the original variable names to be displayed, as such i create an expression and evaluate it
eval(dimname_expression)
output
}
Note that the implementation is not vectorized, and as such can only take single string inputs!
Testing the implementation
To test the implementation, one could use the strings given. As they were said to be contained in lists, we will have to convert them to strings. Note that the function lets one split each string differently, however it assumes space separated strings. So first I'll show how one could achieve a conversion to the correct format:
list_1 <- list("crashed","red","car")
list_2 <- list("crashed","blue","bus")
string_1 <- paste(list_1,collapse = " ")
string_2 <- paste(list_2,collapse = " ")
Dist_between_strings(string_1, string_2)
output
#Strings in the given example
string_2
string_1 crashed blue bus
crashed 0 1 2
red 1 1 2
car 2 2 2
This is not exactly the output, but it yields the same information, as the words are ordered as they were given in the string.
More examples
Now i stated it worked for other strings as well and this is indeed the fact, so lets try some random user-made strings:
#More complicated strings
string_3 <- "I am not a blue whale"
string_4 <- "I am a cat"
string_5 <- "I am a beautiful flower power girl with monster wings"
string_6 <- "Hello"
Dist_between_strings(string_3, string_4, case_sensitive = TRUE)
Dist_between_strings(string_3, string_5, case_sensitive = TRUE)
Dist_between_strings(string_4, string_5, case_sensitive = TRUE)
Dist_between_strings(string_6, string_5)
Running these show that these do yield the correct answers. Note that if either string is of size 1, the comparison is a lot faster.
Benchmarking the implementation
Now as the implementation is accepted, as correct, we would like to know how well it performs (For the uninterested reader, one can scroll past this section, to where a faster implementation is given). For this purpose, i will use much larger strings. For a complete benchmark i should test various string sizes, but for the purposes i will only use 2 rather large strings of size 1000 and 2500. For this purpose i use the microbenchmark package in R, which contains a microbenchmark function, which claims to be accurate down to nanoseconds. The function itself executes the code 100 (or a user defined) number of times, returning the mean and quartiles of the run times. Due to other parts of R such as the Garbage Cleaner, the median is mostly considered a good estimate of the actual average run-time of the function.
The execution and results are shown below:
#Benchmarks for larger strings
set.seed(1)
string_7 <- paste(sample(LETTERS,1000,replace = TRUE), collapse = " ")
string_8 <- paste(sample(LETTERS,2500,replace = TRUE), collapse = " ")
microbenchmark::microbenchmark(String_Comparison = Dist_between_strings(string_7, string_8, case_sensitive = FALSE))
# Unit: milliseconds
# expr min lq mean median uq max neval
# String_Comparison 716.5703 729.4458 816.1161 763.5452 888.1231 1106.959 100
Profiling
Now i find the run-times very slow. One use case for the implementation could be an initial check of student hand-ins to check for plagiarism, in which case a low difference count very likely shows plagiarism. These can be very long and there may be hundreds of handins, an as such i would like the run to be very fast.
To figure out how to improve my implementation i used the profvis package with the corrosponding profvis function. To profile the function i exported it in another R script, that i sourced, running the code 1 once prior to profiling to compile the code and avoid profiling noise (important). The code to run the profiling can be seen below, and the most important part of the output is visualized in an image below it.
library(profvis)
profvis(Dist_between_strings(string_7, string_8, case_sensitive = FALSE))
Now, despite the colour, here i can see a clear problem. The loop filling the off-diagonal by far is responsible for most of the runtime. R (like python and other not compiled languages) loops are notoriously slow.
Using Rcpp to improve performance
To improve the implementation, we could implement the loop in c++ using the Rcpp package. This is rather simple. The code is not unlike the one we would use in R, if we avoid iterators. A c++ script can be made in file -> new file -> c++ File. The following c++ code would be pasted into the corrosponding file and sourced using the source button.
//Rcpp Code
#include <Rcpp.h>
using namespace Rcpp;
// [[Rcpp::export]]
NumericMatrix Cpp_String_difference_outer_diag(NumericMatrix output){
long nrow = output.nrow();
for(long i = 2; i < nrow; i++){ // note the
for(long j = 0; j < i; j++){
output(i, j) = output(i, i) - output(i - 1, i - 1) + //are the words different?
output(i - 1, j);
output(j, i) = output(i, j);
}
}
return output;
}
The corresponding R function needs to be altered to use this function instead of looping. The code is similar to the first function, only switching the loop for a call to the c++ function.
Dist_between_strings_cpp <- function(x, y,
split = " ",
split_x = split, split_y = split,
case_sensitive = TRUE){
#Safety checks
if(!is.character(x) || !is.character(y) ||
nchar(x) == 0 || nchar(y) == 0)
stop("x, y needs to be none empty character strings.")
if(length(x) != 1 || length(y) != 1)
stop("Currency the function is not vectorized, please provide the strings individually or use lapply.")
if(!is.logical(case_sensitive))
stop("case_sensitivity needs to be logical")
#Extract variable names of our variables
# used for the dimension names later on
x_name <- deparse(substitute(x))
y_name <- deparse(substitute(y))
#Expression which when evaluated will name our output
dimname_expression <-
parse(text = paste0("dimnames(output) <- list(", make.names(x_name, unique = TRUE)," = x_names,",
make.names(y_name, unique = TRUE)," = y_names)"))
#split the strings into words
x_names <- str_split(x, split_x, simplify = TRUE)
y_names <- str_split(y, split_y, simplify = TRUE)
#are we case_sensitive?
if(isTRUE(case_sensitive)){
x_split <- str_split(tolower(x), split_x, simplify = TRUE)
y_split <- str_split(tolower(y), split_y, simplify = TRUE)
}else{
x_split <- x_names
y_split <- y_names
}
#Create an index in case the two are of different length
idx <- seq(1, (n_min <- min((nx <- length(x_split)),
(ny <- length(y_split)))))
n_max <- max(nx, ny)
#If we have one string that has length 1, the output is simplified
if(n_min == 1){
distances <- seq(1, n_max) - (x_split[idx] == y_split[idx])
output <- matrix(distances, nrow = nx)
eval(dimname_expression)
return(output)
}
#If not we will have to do a bit of work
output <- diag(cumsum(ifelse(x_split[idx] == y_split[idx], 0, 1)))
#The loop will fill in the off_diagonal
output[2, 1] <- output[1, 2] <- output[1, 1] + 1
if(n_max > 2)
output <- Cpp_String_difference_outer_diag(output) #Execute the c++ code
#comparison if the list is not of the same size
if(nx != ny){
#Add the remaining words to the side that does not contain this
additional_words <- seq(1, n_max - n_min)
additional_words <- sapply(additional_words, function(x) x + output[,n_min])
#merge the additional words
if(nx > ny)
output <- rbind(output, t(additional_words))
else
output <- cbind(output, additional_words)
}
#set the dimension names,
# I would like the original variable names to be displayed, as such i create an expression and evaluate it
eval(dimname_expression)
output
}
Testing the c++ implementation
To be sure the implementation is correct we check if the same output is obtained with the c++ implementation.
#Test the cpp implementation
identical(Dist_between_strings(string_3, string_4, case_sensitive = TRUE),
Dist_between_strings_cpp(string_3, string_4, case_sensitive = TRUE))
#TRUE
Final benchmarks
Now is this actually faster? To see this we could run another benchmark using the microbenchmark package. The code and results are shown below:
#Final microbenchmarking
microbenchmark::microbenchmark(R = Dist_between_strings(string_7, string_8, case_sensitive = FALSE),
Rcpp = Dist_between_strings_cpp(string_7, string_8, case_sensitive = FALSE))
# Unit: milliseconds
# expr min lq mean median uq max neval
# R 721.71899 753.6992 850.21045 787.26555 907.06919 1756.7574 100
# Rcpp 23.90164 32.9145 54.37215 37.28216 47.88256 243.6572 100
From the microbenchmark median improvement factor of roughly 21 ( = 787 / 37), which is a massive improvement from just implementing a single loop!
There is already an edit-distance function in R we can take advantage of: adist().
As it works on the character level, we'll have to assign a character to each unique word in our sentences, and stitch them together to form pseudo-words we can calculate the distance between.
s1 <- c("crashed", "red", "car")
s2 <- c("crashed", "blue", "bus")
ll <- list(s1, s2)
alnum <- c(letters, LETTERS, 0:9)
ll2 <- relist(alnum[factor(unlist(ll))], ll)
ll2 <- sapply(ll2, paste, collapse="")
adist(ll2)
# [,1] [,2]
# [1,] 0 2
# [2,] 2 0
Main limitation here, as far as I can tell, is the number of unique characters available, which in this case is 62, but can be extended quite easily, depending on your locale. E.g: intToUtf8(c(32:126, 161:300), TRUE).
I have a non-square matrix and need to do some calculations on it's subdiagonals. I figure out that the best way is too turn subdiagonals to columns/rows and use functions like cumprod. Right now I use a for loop and exdiag defined as below:
exdiag <- function(mat, off=0) {mat[row(mat) == col(mat)+off]}
However it to be not really efficient. Do you know any other algorithm to achieve that kind of results.
A little example to show what I am doing:
exdiag <- function(mat, off=0) {mat[row(mat) == col(mat)+off]}
mat <- matrix(1:72, nrow = 12, ncol = 6)
newmat <- matrix(nrow=11, ncol=6)
for (i in 1:11){
newmat[i,] <- c(cumprod(exdiag(mat,i)),rep(0,max(6-12+i,0)))
}
Best regards,
Artur
The fastest but by far the most cryptic solution to get all possible diagonals from a non-square matrix, would be to treat your matrix as a vector and simply construct an id vector for selection. In the end you can transform it back to a matrix if you want.
The following function does that:
exdiag <- function(mat){
NR <- nrow(mat)
NC <- ncol(mat)
smalldim <- min(NC,NR)
if(NC > NR){
id <- seq_len(NR) +
seq.int(0,NR-1)*NR +
rep(seq.int(1,NC - 1), each = NR)*NR
} else if(NC < NR){
id <- seq_len(NC) +
seq.int(0,NC-1)*NR +
rep(seq.int(1,NR - 1), each = NC)
} else {
return(diag(mat))
}
out <- matrix(mat[id],nrow = smalldim)
id <- (ncol(out) + 1 - row(out)) - col(out) < 0
out[id] <- NA
return(out)
}
Keep in mind you have to take into account how your matrix is formed.
In both cases I follow the same logic:
first construct a sequence indicating positions along the smallest dimension
To this sequence, add 0, 1, 2, ... times the row length.
This creates the first diagonal. After doing this, you simply add a sequence that shifts the entire previous sequence by 1 (either down or to the right) until you reach the end of the matrix. To shift right, I need to multiply this sequence by the number of rows.
In the end you can use these indices to select the correct positions from mat, and return all that as a matrix. Due to the vectorized nature of this code, you have to check that the last subdiagonals are correct. These contain less elements than the first, so you have to replace the values not part of that subdiagonal by NA. Also here you can simply use an indexing trick.
You can use it as follows:
> diag1 <- exdiag(amatrix)
> diag2 <- exdiag(t(amatrix))
> identical(diag1, diag2)
[1] TRUE
In order to come to your result
amatrix <- matrix(1:72, ncol = 6)
diag1 <- exdiag(amatrix)
res <- apply(diag1,2,cumprod)
res[is.na(res)] <- 0
t(res)
You can modify the diag() function.
exdiag <- function(mat, off=0) {mat[row(mat) == col(mat)+off]}
exdiag2 <- function(matrix, off){diag(matrix[-1:-off,])}
Speed Test:
mat = diag(10, 10000,10000)
off = 4
> system.time(exdiag(mat,4))
user system elapsed
7.083 2.973 10.054
> system.time(exdiag2(mat,4))
user system elapsed
5.370 0.155 5.524
> system.time(diag(mat))
user system elapsed
0.002 0.000 0.002
It looks like that the subsetting from matrix take a lot of time, but it still performs better than your implementation. May be there are a lot of other subsetting approaches, which outperforms my solution. :)
I am trying to split my data set using two parameters, the fraction of missing values and "maf", and store the sub-data sets in a list. Here is what I have done (it's not working). Any help will be appreciated,
Thanks.
library(BLR)
library(missForest)
data(wheat)
X2<- prodNA(X, 0.4) ### creating missing values
dim(X2)
fd<-t(X2)
MAF<-function(geno){ ## markers are in the rows
geno[(geno!=0) & (geno!=1) & (geno!=-1)] <- NA
geno <- as.matrix(geno)
## calc_Freq for alleles
n0 <- apply(geno==0,1,sum,na.rm=T)
n1 <- apply(geno==1,1,sum,na.rm=T)
n2 <- apply(geno==-1,1,sum,na.rm=T)
n <- n0 + n1 + n2
## calculate allele frequencies
p <- ((2*n0)+n1)/(2*n)
q <- 1 - p
maf <- pmin(p, q)
maf}
frac.missing <- apply(fd,1,function(z){length(which(is.na(z)))/length(z)})
maf<-MAF(fd)
lst<-matrix()
for (i in seq(0.2,0.7,by =0.2)){
for (j in seq(0,0.2,by =0.005)){
lst=fd[(maf>j)|(frac.missing < i),]
}}
It sounds like you want the results that the split function provides.
If you have a vector, "frac.missing" and "maf" is defined on the basis of values in "fd" (and has the same length as the number of rows in fd"), then this would provide the split you are looking for:
spl.fd <- split(fd, list(maf, frac.missing) )
If you want to "group" the fd values basesd on of maf(fd) and frac.missing within the bands specified by your for-loop, then the same split-construct may do what your current code is failing to accomplish:
lst <- split( fd, list(cut(maf(fd), breaks = seq(0,0.2,by =0.005) ,
include.lowest=TRUE),
cut(frac.missing, breaks = seq(0.2,0.7,by =0.2),
right=TRUE,include.lowest=TRUE)
)
)
The right argument accomodates the desire to have the splits based on a "<" operator whereas the default operation of cut presumes a ">" comparison against the 'breaks'. The other function that provides similar facility is by.
the below codes give me exactly what i need:
Y<-t(GBS.binary)
nn<-colnames(Y)
fd<-Y
maf<-as.matrix(MAF(Y))
dff<-cbind(frac.missing,maf,Y)
colnames(dff)<-c("fm","maf",nn)
dff<-as.data.frame(dff)
for (i in seq(0.1,0.6,by=0.1)) {
for (j in seq(0,0.2,by=0.005)){
assign(paste("fm_",i,"maf_",j,sep=""),
(subset(dff, maf>j & fm <i))[,-c(1,2)])
} }