minor improvements to counting events

_config.yml

@@ -226,3 +226,4 @@ jekyll-archives:
     #category: /categories/:name/
 
 future: true
+excerpt_separator: "<!--more-->"

_posts/2024-10-16-waiting-times.md

@@ -2,7 +2,7 @@
 title: 'Waiting times'
 layout: post
 date: 2024-10-16
-tags: [probability theory]
+tags: [probability theory,time series]
 pin: true
 math: true
 ---
@@ -139,3 +139,5 @@ The notable solutions when this is possible to do analytically are summarized in
 
 ![](/assets/images/erlang_events.png "Sample waiting times from Erlang(6,2) distribution")
 ![](/assets/images/nb_events.png "Sample waiting times from NegativeBinomial(4,2/3) distribution")
+
+Next read: [Counting events](/posts/counting-events/)

_posts/2024-10-17-counting-events.md

@@ -2,7 +2,7 @@
 title: 'Counting events'
 layout: post
 date: 2024-10-17
-tags: [probability theory]
+tags: [probability theory,time series]
 pin: true
 math: true
 ---
@@ -88,7 +88,7 @@ and here I show where the name comes from.
 
 As a reminder, the density of the exponential distribution has a single parameter $\lambda$ and is given by:
 
-$$p(t) = \lambda e^{-\lambda t}$$ for $t > 0$, otherwise $0$.
+$$p(t) = \lambda e^{-\lambda t}, \; t > 0$$ 
 
 Convolving this density with itself produces:
 
@@ -191,6 +191,7 @@ We derived the Poisson distribution as the count of events of a random process f
 Obtaining the distribution of counts of events for other waiting time distributions is possible in principle, at least numerically,
 but usually won't have a nice closed-form solution.
 
+Next read: [Inhomogeneous Poisson process](/posts/inhomogeneous-poisson-process/)
 
 ## References
 1. Renewal Theory, D.R.Cox, 1962, p. 37

_posts/2024-12-04-inhomogeneous-poisson-process.md

@@ -2,7 +2,7 @@
 title: 'Inhomogeneous Poisson process & case study'
 layout: post
 date: 2024-12-04
-tags: [probability theory,statistics,case study]
+tags: [probability theory,statistics,time series,case study]
 pin: true
 math: true
 ---
@@ -55,7 +55,7 @@ Fortunately, even one realization may be enough to reasonably learn,
 if there are enough event records compared to the number of parameters.
 The log-likelihood can be shown<sup>[1],[2]</sup> to be:
 
-$$\mathcal{l}(\mathbf{t}|\mathbf{\theta}) = \sum_{i=1}^n \log \lambda(t_i|\mathbf{\theta}) - \int_0^T \lambda(s|\mathbf{\theta}) \mathrm{d}s$$
+$$\mathscr{l}(\mathbf{t}|\mathbf{\theta}) = \sum_{i=1}^n \log \lambda(t_i|\mathbf{\theta}) - \int_0^T \lambda(s|\mathbf{\theta}) \mathrm{d}s$$
 
 Intuitively, the sum accounts for the events that have occurred and the integral accounts for the events that did not occur, but could have. 
 For more complicated functions, the integral will likely not have an analytic solution 
@@ -107,6 +107,7 @@ where we can use them to express arbitrary periodicities, which provides a natur
 ![](/assets/images/step_plot_sales_fit.png "Cumulative purchase counts with fit")
 
 
+Next read: [Generalized linear models](/posts/glms/)
 
 
 ## References