1. Extend one kind of non-homogeneous Poisson process (NHPP) and apply to single sequences of event times:
   • original tweets of a hashtag
   • retweets of one original
2. Model retweet count vs follower count by censored regression
3. Model retweets of all originals by a hierarchical NHPP model
   • follower count as a covariate
4. Application

If a sequence of events is assumed to arise from an NHPP with intensity function \(h(t)\geq 0\)

• The random variable of the number of events within \([t_1,t_2]\) follows the Poisson distribution with mean \(\displaystyle\int_{t_1}^{t_2}h(t)dt\)
• Independent of the random number of events in any other disjoint interval
• Also define the cumulative intensity \(H(t)=\displaystyle\int_0^t h(t)dt\)

• \(h(t)=\gamma t^{-\lambda}\), where \(\gamma>0,\lambda<1\)
• Some call it Weibull process in reliability
• N.B. \(\neq\) a renewal process with power-law distributed interarrival times
• Common special case: homogeneous Poisson process
  • Power law process with \(\lambda=0\)
  • Renewal process with exponential distributed interarrival times

How to check if the power law process is appropriate for the data?

• Let \(\{t_i,i=1,2,\ldots,n\}\) be a sequence of event times, \(0\leq t_1\leq t_2\leq\cdots\leq t_n\)
• Plot \(t_i/i\) vs \(t_i\) on log-log scale
• If the data is indeed generated from the power law process, the plot will give approximately a straight line with slope \(\lambda\)

Why \(t_i/i\) vs \(t_i\) on log-log scale?

• Denote \(N(t)\) as the number of events in the interval \([0,t]\)
• Expectation \(\displaystyle E[N(t)]=H(t)=\int_0^t h(t)dt=\frac{\gamma t^{1-\lambda}}{1-\lambda}\)
• So \(t_i\) should be such that \[ \frac{\gamma t_i^{1-\lambda}}{1-\lambda}\approx i \quad\Leftrightarrow\quad \log\left(\frac{t_i}{i}\right)\approx\log\left(\frac{1-\lambda}{\gamma}\right)+\lambda\log t_i\]

How to check if some (non-temporal) data are Weibull distributed?

• Consider a sequence \(\{x_i,i=1,2,\ldots,n\}\) assumed to be iid
• Plot \(-\log\hat{S}(x_i)\) vs \(x_i\) on log-log scale, where \(\hat{S}(x_i)\) is the empirical survival function
• If \(\{x_i\}\) indeed comes from the Weibull distribution, the plot will give approximately a straight line with slope equal to the shape parameter

• Latter events seem to occur at a rate slower than expected by the power law process
• We call an NHPP with \(h(t)=\gamma t^{-\lambda}e^{-\theta t}\) the hybrid process, where \(\gamma>0,\lambda<1,\theta\geq0\)
• Alternatively called "power law with exponential cutoff" (Mathews et al. 2017)
• When \(\theta=0\), the hybrid process becomes the power law process

• Data: retweets of top original only
• Inference: maximum likelihood
• Estimates: \(\left(\hat{\lambda},\hat{\theta},\hat{\gamma}\right)=(0.45,0,0.261)\)
• Including \(\theta\) doesn't improve the fit here

• Data: retweets of all originals
• Hierarchical approach: a distribution to govern the parameter values for retweets of each original
• Question: can we actually explain the distribution of the parameters (and that of the retweet count) by some covariates?

For \(i\)-th original, where \(i=1,2,\ldots,n\)

• Censored regression (Tobin 1958) \[ m_i^{*}=\max(0,\alpha+\beta x_i^{*}+\epsilon_i) \] \[ \epsilon_i~\overset{\text{iid}}{\sim}~\text{N}(0,\tau^{-1}) \]
• Note the difference with \[ m_i^{*}=\max(0,\alpha+\beta x_i^{*})+\epsilon_i \]
• \(x_i^{*}\) orthogonalises \(\alpha\) and \(\beta\) in inference

• Terms with higher powers of \(x_i^{*}\) can be added to potentially improve fit
• Linear and quadratic fits give maximum log-likelihood of 828.4 and 841.47, respectively
• Therefore, the following censored regression will be used in the hierarchical model: \[ m_i^{*}=\max\left(0,\alpha+\beta x_i^{*}+\kappa\left(x_i^{*}\right)^2+\epsilon_i\right) \]

• \(h(t)=\gamma t^{-\lambda}e^{-\theta t}, H(t)=\gamma\Gamma(1-\lambda,\theta t)\theta^{\lambda-1}\)
  • \(\Gamma(.,.)\) is the incomplete Gamma function
• \(h(t)\) and \(H(t)\) increase linearly with \(\gamma\)
• \(\gamma\) can be seen as a proxy for the retweet count
• Making \(\gamma\) depend on (mean-centred) follower count removes the need to model (transformed) retweet count separately

• The \(i\)-th original occurs at time \(t_i\)
• Its \(m_i\) retweets follow a hybrid process with intensity \[ h_i(t)=\gamma_i(t-t_i)^{-\lambda}e^{-\theta(t-t_i)}, t\geq t_i \]
• Location shift from \(t\) to \(t-t_i\): retweets occur relative to the creation of the \(i\)-th original
• Relationship of \(\gamma_i\) and \(x_i^{*}\) (mean-centred follower count): \[ \log(1+\gamma_i) = \max\left(0,\alpha+\beta x_i^{*}+\kappa\left(x_i^{*}\right)^2+\epsilon_i\right) \]
• As before, \(\epsilon_i~\overset{\text{iid}}{\sim}~\text{N}(0,\tau^{-1})\)

• Scalars
  • Observation period: \(T~\left(\geq t_n\geq t_{n-1}\geq\cdots\geq t_1\geq0\right)\)
  • Parameters: \(\alpha,\beta,\kappa,\lambda,\tau,\theta\)
• For each original
  • \(i\)-th original's & its retweets' times: \(\boldsymbol{t}_i=(t_i,t_{i1},t_{i2},\ldots,t_{im_i})\)
  • \(\gamma_i\) not a parameter but a function of \(\alpha,\beta,\kappa,\epsilon_i\) and \(x_i^{*}\)
• Vectors of length \(n\)
  • Retweet counts: \(\boldsymbol{m}=(m_1,m_2,\ldots,m_n)\)
  • All event times: \(\boldsymbol{t}=\{\boldsymbol{t}_1,\boldsymbol{t}_2,\ldots,\boldsymbol{t}_n\}\)
  • (Mean-centred) follower counts: \(\boldsymbol{x}^{*}=(x_1^{*},x_2^{*},\ldots,x_n^{*})\)
  • Random errors / latent variables: \(\boldsymbol{\epsilon}=(\epsilon_1,\epsilon_2,\ldots,\epsilon_n)\)

• We are actually interested which one is adequate
  • Hierarchical model of power law processes \((\theta=0)\)
  • Hierarchical model of hybrid processes \((\theta>0)\)
• Treat them as two different models
  • \(f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\alpha,\beta,\kappa,\lambda,\tau,\boldsymbol{\epsilon},\theta=0) \rightarrow f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_0,M=0)\), where \(\boldsymbol{\eta}_0=(\alpha,\beta,\kappa,\lambda,\tau,\boldsymbol{\epsilon})\)
  • \(f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\alpha,\beta,\kappa,\lambda,\tau,\boldsymbol{\epsilon},\theta>0) \rightarrow f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_1,M=1)\), where \(\boldsymbol{\eta}_1=(\alpha,\beta,\kappa,\lambda,\tau,\boldsymbol{\epsilon},\theta)\)
• Infer \(M\) via model selection in inference

Bayesian approach because of:

• The presence of latent variables \(\boldsymbol{\epsilon}\)
• Model selection required between \(M=0\) and \(M=1\)

Metropolis-within-Gibbs (MWG) algorithm

• Update each scalar parameter individually
• Metropolis steps:
  • \(\alpha,\beta,\kappa,\lambda\)
  • \(\epsilon_i, i = 1,2,\ldots,n\)
  • (For \(M=1\)) \(\theta\)
• Gibbs step: \(\tau\)

Gibbs variable selection (Dellaportas, Forster, and Ntzoufras 2002)

Required quantities

1. Likelihoods: \(f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_0,M=0)\) & \(f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_1,M=1)\)
2. Model priors: \(\pi(M=0)\) & \(\pi(M=1)\)
3. Prior of \(\theta\) under \(M=1\): \(\pi(\theta|M=1)\)
4. Pseudoprior of \(\theta\) under \(M=0\): \(\pi(\theta|M=0)\)

If current value of \(M=1\):

1. Draw \(\boldsymbol{\eta}_1\) using the individual MWG algorithm (previous slide)
2. Split \(\boldsymbol{\eta}_1\) into \(\boldsymbol{\eta}_0\) and \(\theta\) \(\quad\Leftrightarrow\quad\boldsymbol{\eta}_1=(\boldsymbol{\eta}_0,\theta)\)
3. Calculate \(A_0\) and \(A_1\):
   • \(A_0 = f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_0,M=0)\pi(\theta|M=0)\pi(M=0)\)
   • \(A_1 = f(\boldsymbol{m},\boldsymbol{t}|\boldsymbol{x}^{*},\boldsymbol{\eta}_1,M=1)\pi(\theta|M=1)\pi(M=1)\)
4. Set \(M\) to \(0\) with prob. \(\displaystyle\frac{A_0}{A_0+A_1}\) or \(1\) with prob. \(\displaystyle\frac{A_1}{A_0+A_1}\)

If current value of \(M=0\):

1. Draw \(\boldsymbol{\eta}_0\) using the individual MWG algorithm
2. Draw \(\theta\) from pseudoprior \(\pi(\theta|M=0)\)
3. Combine \(\boldsymbol{\eta}_0\) and \(\theta\) to form \(\boldsymbol{\eta}_1\) \(\quad\Leftrightarrow\quad\boldsymbol{\eta}_1=(\boldsymbol{\eta}_0,\theta)\)
4. Calculate \(A_0\) and \(A_1\) as before
5. Set \(M\) to \(0\) or \(1\) with probabilities as before

• The empirical proportion of \(M=1\) in the converged chain is the estimate for the posterior probability \(\pi(M=1|\boldsymbol{m},\boldsymbol{t},\boldsymbol{x}^{*})\)
• Similarly for \(\pi(M=0|\boldsymbol{m},\boldsymbol{t},\boldsymbol{x}^{*})\)
• Bayes factor for model \(1\) over model \(0\) is \[ B_{10}=\frac {\hat{\pi}(M=1|\boldsymbol{m},\boldsymbol{t},\boldsymbol{x}^{*})} {\hat{\pi}(M=0|\boldsymbol{m},\boldsymbol{t},\boldsymbol{x}^{*})} \left/\frac {\pi(M=1)} {\pi(M=0)} \right. \]

• Observation period: \(T\approx21\) hours
• Originals: \(n=2043\), of which \(265\) were ever retweeted
• Retweets: \(\displaystyle\sum_{i=1}^nm_i=971,~\underset{1\leq i\leq n}{\max}m_i=204\)

• Observation period: \(T\approx4.4\) hours
• Originals: \(n=25420\), of which \(3145\) were ever retweeted
• Retweets: \(\displaystyle\sum_{i=1}^nm_i=29751,~\underset{1\leq i\leq n}{\max}m_i=3204\)

#thehandmaidstale

• \(B_{10}=6.572\)
  • Hierarchical model of hybrid processes selected
• Mathews et al. (2017): "exponential cutoff" phenomenon for tweets collected over a similar duration of 24 hours

#gots7

• \(B_{10}=1.219\times 10^{-9}\)
  • Hierarchical model of power law processes selected
• Mathews et al. (2017): power law phenomenon for tweets collected over a similar duration of 3 hours

• Use Duane plot as a diagnostic for the power law process, which is extended to the hybrid process
• Model relationship between follower count and retweet count using censored regression
• Construct a hierarchical model for retweets by combining the hybrid process and censored regression
• Infer the model choice between power law and hybrid processes in MCMC algorithm
• Selected models for different data consistent with previous findings