问题
I am trying to simulate data via bootstrapping to create confidence bands for my real data with a funnel plot. I am building on the strategy of the accepted answer to a previous question. Instead of using a single probability distribution for simulating my data I want to modify it to use different probability distributions depending on the part of the data being simulated.
I greatly appreciate anyone who can help answer the question or help me phrase the question more clearly.
My problem is writing the appropriate R code to do a more complicated form of data simulation.
The current code is:
n <- 1e4
set.seed(42)
sims <- sapply(1:80,
function(k)
rowSums(
replicate(k, sample((1:7)/10, n, TRUE, ps))) / k)
This code simulates data where each data point has a value which is the mean of between 1:80 observations.
For example, when the values of the data points are the mean of 10 observations (k=10) it randomly samples 10 values (which can be either 0.1,0.2,0.3, 0.4, 0.5,0.6 or 0.7) based on a probability distribution ps, which gives the probability of each value (based on the entire empirical distribution).
ps looks like this:
ps <- prop.table(table((DF$mean_score)[DF$total_number_snps == 1]))
# 0.1 0.2 0.3 0.4 0.5 0.6 0.7
#0.582089552 0.194029851 0.124378109 0.059701493 0.029850746 0.004975124 0.004975124
eg probability that the value of an observation is 0.1 is 0.582089552.
Now instead of using one frequency distribution for all simulations I would like to use different frequency distributions conditionally depending on the number of observations underlying each datapoint.
I made a table, cond_probs, that has a row for each of my real data points. There is a column with the total number of observations and a column giving the frequency of each of the values for each observation.
Example of the cond_probs table:
gene_name 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 total
A1 0.664 0.319 0.018 0.000 0.000 0.000 0.000 0.000 0.000 113.000
A2 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
So for the data point A2, there is only 1 observation, which has a value of 0.1. Therefore the frequency of the 0.1 observations is 1. For A1, there are 113 observations and the majority of those (0.664) have the value 0.1. The idea is that cond_probs is like ps, but cond_probs has a probability distribution for each data point rather than one for all the data.
I would like to modify the above code so that the sampling is modified to use cond_probs instead of ps for the frequency distribution. And to use the number of observations, k , as a criteria when choosing which row in cond_probs to sample from. So it would work like this:
For data points with k number of observations:
look in the cond_probs table and randomly select a row where the total number of observations is similar in size to k: 0.9k-1.1k. If no such rows exist, continue.
Once a datapoint is selected, use the probability distribution from that line in cond_probs just like ps is used in the original code, to randomly sample k number of observations and output the mean of these observations.
For each of the n iterations of replicate, randomly sample with replacement a new data point from cond_probs, out of all rows where the value of total is similar to the current value of k ( 0.9k-1.1k).
The idea is that for this dataset one should condition which probability distribution to use based on the number of observations underlying a data point. This is because in this dataset the probability of an observation is influenced by the number of observations (genes with more SNPs tend to have a lower score per observation due to genetic linkage and background selection).
UPDATE USING ANSWER BELOW:
I tried using the answer below and it works for the simulated cond_probs data in the example but not for my real cond_probs file. I imported and converted my cond_probs file to a matrix with
cond_probs <- read.table("cond_probs.txt", header = TRUE, check.names = FALSE)
cond_probs <- as.matrix(cond_probs)
and the first example ten rows (out of ~20,000 rows) looks like this:
>cond_probs
total 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
[1,] 109 0.404 0.174 0.064 0.183 0.165 0.009 0.000 0.000 0.000 0.000
[2,] 181 0.564 0.221 0.144 0.066 0.006 0.000 0.000 0.000 0.000 0.000
[3,] 289 0.388 0.166 0.118 0.114 0.090 0.093 0.028 0.003 0.000 0.000
[4,] 388 0.601 0.214 0.139 0.039 0.008 0.000 0.000 0.000 0.000 0.000
[5,] 133 0.541 0.331 0.113 0.000 0.008 0.008 0.000 0.000 0.000 0.000
[6,] 221 0.525 0.376 0.068 0.032 0.000 0.000 0.000 0.000 0.000 0.000
[7,] 147 0.517 0.190 0.150 0.054 0.034 0.048 0.007 0.000 0.000 0.000
[8,] 107 0.458 0.196 0.252 0.084 0.009 0.000 0.000 0.000 0.000 0.000
[9,] 13 0.846 0.154 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
If I run:
sampleSize <- 20
set.seed(42)
#replace 1:80 with 1: max number of SNPs in gene in dataset
sims_test <- sapply( 1:50, simulateData, sampleSize )
and look at the means from the sampling with x number of observations I only get a single result, when there should be 20.
for example:
> sims_test[[31]]
[1] 0.1
And sims_test is not ordered in the same way as sims:
>sims_test
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 0.1 0.1 0.1666667 0.200 0.14 0.2666667 0.2000000 0.2375 0.1888889
[2,] 0.1 0.1 0.1333333 0.200 0.14 0.2333333 0.1571429 0.2625 0.1222222
[3,] 0.1 0.1 0.3333333 0.225 0.14 0.1833333 0.2285714 0.2125 0.1555556
[4,] 0.1 0.1 0.2666667 0.250 0.10 0.1500000 0.2000000 0.2625 0.2777778
[5,] 0.1 0.1 0.3000000 0.200 0.16 0.2000000 0.2428571 0.1750 0.1000000
[6,] 0.1 0.1 0.3666667 0.250 0.16 0.1666667 0.2142857 0.2500 0.2000000
[7,] 0.1 0.1 0.4000000 0.300 0.12 0.2166667 0.1857143 0.2375 0.1666667
[8,] 0.1 0.1 0.4000000 0.250 0.10 0.2500000 0.2714286 0.2375 0.2888889
[9,] 0.1 0.1 0.1333333 0.300 0.14 0.1666667 0.1714286 0.2750 0.2888889
UPDATE 2
Using cond_probs <- head(cond_probs,n) I have determined that the code works until n = 517 then for all sizes greater than this it produces the same output as above. I am not sure if this is an issue with the file itself or a memory issue. I found that if I remove line 518 and duplicate the lines before several times to make a larger file, it works, suggesting that the line itself is causing the problem. Line 518 looks like this:
9.000 0.889 0.000 0.000 0.000 0.111 0.000 0.000 0.000 0.000 0.000
I found another 4 offending lines:
9.000 0.444 0.333 0.111 0.111 0.000 0.000 0.000 0.000 0.000 0.000
9.000 0.444 0.333 0.111 0.111 0.000 0.000 0.000 0.000 0.000 0.000
9.000 0.111 0.222 0.222 0.111 0.111 0.222 0.000 0.000 0.000 0.000
9.000 0.667 0.111 0.000 0.000 0.000 0.222 0.000 0.000 0.000 0.000
I don't notice anything unusual about them. They all have a 'total' of 9 sites. If I remove these lines and run the 'cond_probs' file containing only the lines BEFORE these then the code works. But there must be other problematic lines as the entire 'cond_probs' still doesn't work.
I tried putting these problematic lines back into a smaller 'cond_probs' file and this file then works, so I am very confused as it doesn't seem the lines are inherently problematic. On the other hand the fact they all have 9 total sites suggests some kind of causative pattern.
I would be happy to share the entire file privately if that helps as I don't know what to do next for troubleshooting.
One further issue that comes up is I'm not sure if the code is working as expected. I made a dummy cond_probs file where there are two data points with a 'total' of '1' observation:
total 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
1.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
So I would expect them to both be sampled for data points with '1' observation and therefore get roughly 50% of observations with a mean of '0.2' and 50% with a mean of '0.6'. However the mean is always 0.2:
sims_test[[1]]
[1] 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Even if I sample 10000 times all observations are 0.2 and never 0.6. My understanding of the code is that it should be randomly selecting a new row from cond_probs with similar size for each observation, but in this case is seems not to be doing so. Do I misunderstand the code or is it still a problem with my input not being correct?
The entire cond_probs file can be found at the following address:
cond_probs
UPDATE 3
Changing sapply to lapply when running the simulations fixed this issue.
Another reason I think leaving cond_probs as it is and choosing a distribution sampleSize number of times might be the best solution: The probability of choosing a distribution should be related to its frequency in cond_probs. If we combine distributions the odds of picking a distribution with total 9 or 10 will no longer depend on the number of observations with these totals. Example: If there are 90 distributions with total=10 and 10 with total=9 there should be a 90% chance to choose a distribution with total=10. If we combine distributions wouldn't the odds become 50/50 for choosing a distribution with 'total'= 9 or 10 (which would not be ideal)?
回答1:
I simply wrote a function ps that chooses an appropriate distribution from cond_probs:
N <- 10 # The sampled values are 0.1, 0.2, ... , N/10
M <- 8 # number of distributions in "cond_probs"
#-------------------------------------------------------------------
# Example data:
set.seed(1)
cond_probs <- matrix(0,M,N)
is.numeric(cond_probs)
for(i in 1:nrow(cond_probs)){ cond_probs[i,] <- dnorm((1:N)/M,i/M,0.01*N) }
is.numeric(cond_probs)
total <- sort( sample(1:80,nrow(cond_probs)) )
cond_probs <- cbind( total, cond_probs/rowSums(cond_probs) )
colnames(cond_probs) <- c( "total", paste("P",1:N,sep="") )
#---------------------------------------------------------------------
# A function that chooses an appropiate distribution from "cond_prob",
# depending on the number of observations "numObs":
ps <- function( numObs,
similarityLimit = 0.1 )
{
similar <- which( abs(cond_probs[,"total"] - numObs) / numObs < similarityLimit )
if ( length(similar) == 0 )
{
return(NA)
}
else
{
return( cond_probs[similar[sample(1:length(similar),1)],-1] )
}
}
#-----------------------------------------------------------------
# A function that simulates data using a distribution that is
# appropriate to the number of observations, if possible:
simulateData <- function( numObs, sampleSize )
{
if (any(is.na(ps(numObs))))
{
return (NA)
}
else
{
return( rowSums(
replicate(
numObs,
replicate( sampleSize, sample((1:N)/10, 1, prob = ps(numObs))))
) / numObs )
}
}
#-----------------------------------------------------------------
# Test:
sampleSize <- 30
set.seed(42)
sims <- lapply( 1:80, simulateData, sampleSize )
The distributions in cond_probs:
total P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
[1,] 16 6.654875e-01 3.046824e-01 2.923948e-02 5.881753e-04 2.480041e-06 2.191926e-09 4.060763e-13 1.576900e-17 1.283559e-22 2.189990e-28
[2,] 22 2.335299e-01 5.100762e-01 2.335299e-01 2.241119e-02 4.508188e-04 1.900877e-06 1.680045e-09 3.112453e-13 1.208647e-17 9.838095e-23
[3,] 30 2.191993e-02 2.284110e-01 4.988954e-01 2.284110e-01 2.191993e-02 4.409369e-04 1.859210e-06 1.643219e-09 3.044228e-13 1.182153e-17
[4,] 45 4.407425e-04 2.191027e-02 2.283103e-01 4.986755e-01 2.283103e-01 2.191027e-02 4.407425e-04 1.858391e-06 1.642495e-09 3.042886e-13
[5,] 49 1.858387e-06 4.407417e-04 2.191023e-02 2.283099e-01 4.986746e-01 2.283099e-01 2.191023e-02 4.407417e-04 1.858387e-06 1.642492e-09
[6,] 68 1.642492e-09 1.858387e-06 4.407417e-04 2.191023e-02 2.283099e-01 4.986746e-01 2.283099e-01 2.191023e-02 4.407417e-04 1.858387e-06
[7,] 70 3.042886e-13 1.642495e-09 1.858391e-06 4.407425e-04 2.191027e-02 2.283103e-01 4.986755e-01 2.283103e-01 2.191027e-02 4.407425e-04
[8,] 77 1.182153e-17 3.044228e-13 1.643219e-09 1.859210e-06 4.409369e-04 2.191993e-02 2.284110e-01 4.988954e-01 2.284110e-01 2.191993e-02
The means of the distributions:
> cond_probs[,-1] %*% (1:10)/10
[,1]
[1,] 0.1364936
[2,] 0.2046182
[3,] 0.3001330
[4,] 0.4000007
[5,] 0.5000000
[6,] 0.6000000
[7,] 0.6999993
[8,] 0.7998670
Means of the simulated data for 31 observations:
> sims[[31]]
[1] 0.2838710 0.3000000 0.2935484 0.3193548 0.3064516 0.2903226 0.3096774 0.2741935 0.3161290 0.3193548 0.3032258 0.2967742 0.2903226 0.3032258 0.2967742
[16] 0.3129032 0.2967742 0.2806452 0.3129032 0.3032258 0.2935484 0.2935484 0.2903226 0.3096774 0.3161290 0.2741935 0.3161290 0.3193548 0.2935484 0.3032258
The appopriate distribution is the third one:
> ps(31)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
2.191993e-02 2.284110e-01 4.988954e-01 2.284110e-01 2.191993e-02 4.409369e-04 1.859210e-06 1.643219e-09 3.044228e-13 1.182153e-17
来源：https://stackoverflow.com/questions/33498337/simulating-data-in-r-with-multiple-probability-distributions