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If shuffle-playing playlist ×100 resulted in [10 13 10 3 2 2] different songs being repeated [1 2 3 4 5 6] times, what is the estimate for the total number of songs? (assuming shuffle play was completely random)

Update: (R code)

k <- 50 # k number of songs on the disk indexed 1:k
n <- 100 # n number of random song selections
m <- 20 # m number of repeat experiments
colnum <- 10
mat <- matrix(data=NA,nrow=m,ncol=colnum)
df <- as.data.frame(mat)
for(i in 1:m){
  played <- 1+floor(k*runif(n)) # actual song indices (1:k) selected 
  freq <- sapply(1:k,function(x){sum(played==x)})
  # = number of times song with index x is being played
  histo <- sapply(1:colnum,function(x){sum(freq==x)});
  for(j in 1:colnum){
   df[i,j] <- histo[j]
  }
}
df

Resulting in: e.g. 20 distributions (V1=number of single plays, V2=number of double plays, etc):

   V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
1  15 13 11  1  2  2  0  0  0   0
2  15 12 10  1  4  0  1  0  0   0
3  12 14  7  6  3  0  0  0  0   0
4  17 16  6  4  2  0  1  0  0   0
5  17 10 12  5  0  0  1  0  0   0
6  13 15 11  6  0  0  0  0  0   0
7  10 14  9  3  2  1  1  0  0   0
8  12 17  5  6  3  0  0  0  0   0
9   9 19  8  3  1  2  0  0  0   0
10 13  9 11  6  1  0  1  0  0   0
11 16  9 12  5  2  0  0  0  0   0
12 15  9 11  6  2  0  0  0  0   0
13 19  9  7  4  4  1  0  0  0   0
14 17 11  4  7  3  1  0  0  0   0
15 11 20  8  1  3  1  0  0  0   0
16 14 12 10  5  0  2  0  0  0   0
17  9 12  8  7  3  0  0  0  0   0
18 10 15  9  4  2  0  1  0  0   0
19 14 11 12  7  0  0  0  0  0   0
20 16 14 11  3  1  1  0  0  0   0

Now I need to get from here to the Poisson modelling--my R is a bit rusty (?lmer)...--Any help would be appreciated...

Attempted Poisson modelling: disappointing fit?!

plot(1:colnum,df[1,1:colnum],ylim=c(0,30),
   type="l",xlab="repeats",ylab="count")
for(i in 1:m){
  clr <- rainbow(m)[i]
  lines(1:colnum,df[i,1:colnum],type="l",col=clr)
  points(1:colnum,df[i,1:colnum],col=clr)
}

df.lambda=data.frame(lambda=seq(1,5,0.1),ssq=c(NA));df.lambda
for(ii in 1:dim(df.lambda)[1]){
  l <- df.lambda$lambda[ii]
      ssq <- 0
      for(i in 1:20){
        for(j in 1:10){
          ssq <- ssq + (df[i,j] - n*dpois(j,l))^2
        }
      }
      print(ssq)
      df.lambda$lambda[ii] <- l
  df.lambda$ssq[ii] <- ssq
    }
    df.lambda
    lambda.est <- df.lambda$lambda[which.min(df.lambda$ssq)]
lambda.est # 2.4
points(x <- 1:10, n*dpois(1:10,lambda.est),type="l",lwd=2)

100*dpois(1:10,3)
n/lambda.est

the estimated lambda stays around 2.3, with an n estimate of around 43; the fitted curve seems very discrepant, and seems to worsen with rising n !? Doesn't this have to do with the fact, that our repeats are different from the 'classical' Poisson distributions: it's not just ONE event that repeats itself x number of times, but the sum of repeats of different items (songs)?!

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1  
Related Wikipedia article: Mark and recapture. –  Henning Makholm Jun 16 '12 at 19:35
    
@henning-makholm: the reference to capture-recapture is interesting; although is it really applicable?--the sample size is actually (far) greater than the 'population'... –  ajo Jun 16 '12 at 19:54
    
This questions is (slightly) more appropiate for stats.stackexchange.com –  leonbloy Jun 16 '12 at 21:26
    
@leonbloy: you're right about stats.stackexchange.com -- I didn't see it when I browsed through the categories... –  ajo Jun 16 '12 at 21:36
    
@ajo: To couch this in capture-recapture terms, I think the appropriate phrasing would be that you're making a lot of samples of size 1 each. –  Henning Makholm Jun 16 '12 at 22:00
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3 Answers 3

Using the notation of leonbloy, also let $N_1=10$ be the number of songs that were played once. A simple Good-Turing estimator for $M$ is then given by $$ \hat M = {{P} \over {1-{N_1 \over N} }}= {{40} \over {1-{10 \over 100}}}=44.4,$$ which agrees nicely with the maximum likelihood approach.

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You can compute the likelihood: Let $M$ be the total (unknown) number of songs, $n_i$ the amount of songs that appeared $i$ times, so $ n_1 + 2 n_2 + 3 n_3 \cdots 6 n_6= N = 100$, and let $P=n_1 + n_2 +\cdots + n_6 = 40$

Then, the likelihood is

$${\mathbb l}(M) = \left(\frac{1}{M}\right)^N {M \choose n_1} {M -n_1 \choose n_2} \cdots {M - P + n_6 \choose n_6} K = \left(\frac{1}{M}\right)^N \frac{M!}{n_1! n_2! \cdots (M-P)!} K$$

where $K$ counts the permutations of $N$ elements with $n_2, n_3 \cdots$ repeated elements - which does not depend on $M$. Taking the loglikehood, ignoring elements that do not depend on $M$ and using the Stirling approximation, we get

$$L(M)= -N \log(M) + M ( \log(M) -1) -(M-p) (\log(M-P)-1)$$

The maximum of this log-likehood occurs between 44 and 45.

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You can approximate this by a Poisson distribution, which is more valid as the playlist gets longer. You just need to find the value of $\lambda$, the average number of times a song has been played that best fits your data. This is $\frac {100}{number of songs}$ Any numerical analysis book can help, or you can just use the goal seek in Excel on the sum squared error in each bin. As you played 40 different songs, the average song that got played was played $2.5$ times. If we use that for $\lambda$, the chance that a song is not played is about $8.2\%$ so there might be another $3$ songs or so. You even can make a start at checking whether the selection is really random by doing a chi-square test for the distribution.

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That's really helpful. I've added some code at the top, now I just need to get from the sample distributions to the poisson estimate... –  ajo Jun 16 '12 at 20:50
    
I'm trying along the lines of fitdistr(as.numeric(df[1,]),"Poisson") but with the 0 value missing that's difficult, and it only uses one of 20 experiments.... –  ajo Jun 16 '12 at 21:07
    
I don't know fitdistr. I would just make a line with nsongs in column 1 and the expected distribution of plays in columns 2 through 10. Then put your observations in the row below. Form the sum of the squares of the error between your observation and the prediction. Now use goal seek to minimize this squared error, changing the number of songs. –  Ross Millikan Jun 16 '12 at 21:13
    
the Poisson fitting attempt seems rather unsatisfactory (see code and comments above). It feels somehow that this is the wrong track. I think the 'repeats' are different from the usual Poisson repeats... –  ajo Jun 16 '12 at 23:40
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