This is not actually a research question. It is more an exercise which I posed myself in mathematical/statistical modelling. I have some Whatsapp data of a chat with someone. I want to find a mathematical model to describe the data. I have manually cut the chat into meaningful conversation pieces. So far I have the following Ansatz: Let $t_{j,i}$ be the time at which something is said by Person A or Person B in the whatsapp-chat at conversation j. We have the following "waiting times": $0=t_{11}<t_{12}<\cdots<t_{1,a_1}<t_{2,1}<t_{2,2}<\cdots<t_{2,a_2}<\cdots<t_{n,1}<\cdots<t_{n,a_n}$ So we have $n$ "conversations" in this chat by two people. Now my modeling Ansatz is that we have between each conversation a pause $P_j$:
$t_{1,a_1}+P_1 = t_{2,1}$
$t_{2,a_2}+P_2 = t_{3,1}$
$\cdots$
$t_{n-1,a_{n-1}}+P_{n-1} = t_{n,1}$
I have verified with the Kolmogorov-Smirnov Test all my assumptions concerning distribution of variables. Now we have
$P_j \sim Exp(\lambda_P)$
$d_{j,i} = t_{j,i+1}-t_{j,i} \sim Exp(\lambda_d)$ "interarrival times"
$a_j \sim Pois(\lambda_a)$
Now one could think of this as a "nested Poisson process", by which I mean, that we have a Poisson Process which governs the distributions of the conversations, and in each conversation we have a homogeneous Poisson process. Two conversations might have different parameters.
Ok, so in reality we can not observe when one conversation ends and when it starts. So the question is, given the data $t_1 < \cdots < t_m$ is it possible to calibrate the above model to find out how many conversations there are in this chat and when a conversation ends / starts, or are there to many parameters in the model, which need to be estimated?
If it is of help: We also observe at each timestamp who is chatting (Person A / Person B).
We have
$t_{n,a_n} = \sum_{j=1}^n P_j + \sum_{j=1}^n\sum_{i=1}^{a_j-1}d_{j,i}$
From this I have computed the expected value and the variance of $t_{n,a_n}$:
$E(t_{n,a_n}) = n/\lambda_P + n(\lambda_a-1)/\lambda_d$
$Var(t_{n,a_n}) = n/\lambda_P^2 + n(\lambda_a-1)/\lambda_d^2$
Now the question is, given the data $t_1<\cdots<t_m$ how to estimate the parameters: $n, \lambda_P, \lambda_d, \lambda_a$?
EDIT: (by suggestion of Bjørn Kjos-Hanssen):
One idea, as suggested by Bjørn Kjos-Hanssen is to plot the differences (pauses) and then to cut them off at the mean of pauses:
The number of times the pauses are above the mean, could be estimated as $n$ the number of conversations. So to make it more precise let $d_i = t_{i+1}-t_i$ $i=1,\cdots,m-1$ Then $\widehat{d} = 1/(m-1) \sum_{i=1}^{m-1} d_i$. Now let $n = $ number of times we have $d_i > \widehat{d}$. What assumptions should I make to justify this procedure?
Suppose, that the above procedure can distinguish between a conversation and a pause, then we have $E(m) = \sum_{i=1}^nE(a_i) = n \lambda_a$ hence we can estimate $\lambda_a$ as $\widehat{\lambda_a} = m / n$. On the other hand we can estimate $\lambda_P$ as $\widehat{\lambda_P} = \frac{1}{1/n \sum_{d_j>\widehat{d}}d_j}$
And the Ansatz
$t_m = n/\widehat{\lambda_P}+n(\widehat{\lambda_a}-1)/\widehat{\lambda_d}$
gives an estimate of $\widehat{\lambda_d}$ as:
$\widehat{\lambda_d} = \frac{m/n-1}{t_m/n-1/n \sum_{d_j>\widehat{d}}d_j}$
So in order to make this argumentation more valid, my question is: What assumptions should I make to justify the procedure above?
The data is:
conversation time person
1 0 A
1 1 A
1 34 B
1 35 A
1 36 B
2 5585 B
2 5586 B
2 5911 A
3 8837 B
3 8838 A
3 8839 B
3 8840 B
3 8841 B
3 8850 A
3 8851 A
3 8870 A
3 8947 B
3 8948 B
3 9592 A
4 14406 B
4 14430 A
4 14435 B
4 14443 B
4 14446 A
4 14447 B
5 14857 B
5 15834 B
5 17125 A
5 17162 B
5 17163 A
5 17165 B
6 17251 A
6 17253 A
7 23330 B
7 23999 A
8 32968 A
8 32969 A
8 32970 B
8 32971 B
8 32972 B
8 32973 B
8 32988 B
9 39365 A
9 39742 B
9 46310 A
9 46330 B
9 46331 A
9 50791 A
9 50866 B
9 51368 A
9 51429 B
9 51441 A
9 51459 B
9 51461 A
9 51462 B
9 51467 A
9 51468 A
10 52890 A
10 52891 B
11 54825 B
11 54830 A
11 54831 A
11 54842 A
11 54843 B
11 54844 A
11 54859 B
11 54860 A
11 54861 A
11 54863 B
11 54865 A
12 70562 A
12 70566 B
12 70568 A
12 70570 A
12 70571 A
12 70572 B
12 70586 A
12 70587 B
13 71609 B
13 71611 A
13 71613 B
13 71617 A
13 71618 B
13 71619 A
14 96595 A
14 96625 A
14 96626 A
14 96627 A
14 96632 B
14 96633 B
14 96634 A
14 96635 A
15 96755 B
15 96782 A
15 96787 A
15 96792 B
15 96794 A
15 96867 A
15 96869 B
15 96870 B
15 96871 A
15 96873 B
15 96905 A
15 96911 A
15 96921 B
16 102817 A
16 102940 B
