In a previous post we have used mean squared displacement to understand diffusion of Brownian and Levy's walks. Alternatively we could use a more familiar statistical measure: variance (or standard deviation).

A well-known example of "normal" diffusion is the Brownian walk. Brownian walk, and "normal" diffusion by consequence, is one of the basic concepts in statistical physics and stochastic modeling. Needless to say, it is one of the most important concepts in our work. But in the empirical we can find examples of different kind of diffusion, which notably faster (known as super-diffusion) or notably slower (known as sub-diffusion) than the "normal" diffusion.

In this post we will compare normal and anomalous diffusion from the point-of-view of mean square displacement (abbr. MSD).

Another group of people using statistical methods have shown that Biden has stolen Trump's votes! To be more precise, they have misused statistical methods to prove their point. Well as you know there are lies, damned lies and statistics.

The "proof" relies on a fact that multiple elections were held at the same time (at least in some states). The people who have conducted the analysis have calculated the difference between the fraction of votes for Trump (in presidential elections), \( v_t \) and the fraction of votes for Republican candidates in other elections, \( v_r \):

\begin{equation} \Delta = v_t - v_r . \end{equation}

They have found that \( \Delta \) decreases with \( v_r \). Their claim is that with larger \( v_r \) we should observe larger \( v_t \), therefore \( \Delta \approx 0 \). Which appears logical from the first glance, but is actually false.

Let us imagine that there is some non-zero probability of defecting (voting for president representing another party), \( p_d \). Assuming that there were \( 1 - v_r \) votes cast for Democrats (equivalent to assumption that there are no third parties), fraction of votes cast for Trump will be given by:

\begin{equation} v_t = v_r (1-p) + (1-v_r) p . \end{equation}

Fraction of votes cast for Trump is a sum of votes from non-defecting Republican voters and votes from defecting Democrat voters. Let us now obtain the expression for difference:

\begin{equation} \Delta = \left[ v_r (1-p) + (1-v_r) p \right] - v_r
= p ( 1 - 2 v_r ) . \end{equation}

So, it appears qualitatively the same as this simplistic model predicts. And we have no election rigging built into the model.

We invite you to watch a video by Matt Parker from Stand-up maths, which explores this topic.

In the recent series of posts we have discussed estimation of the recovery time distribution from the empirical COVID-19 data (the confirmed cases time series). This was a quite easy task as we know both the confirmed cases time series and the recovered cases time series. In this post we will make an attempt to solve an inverse problem. Namely, I will try to recover the confirmed cases time series from the recovered cases time series.

There are many unsubstantiated claim about election fraud in the recent US presidential elections. Most of these claims provide no proofs or arguments, while there are few which are supposedly scientific. One of the example relies on the Benford's law, which specifies the expected frequency at which we should observe specific first digit of certain number. So if vote counts in polling stations do not follow Benford's law, it should be an indication of wide spread fraud!

Not so fast, as often in science, laws and models are applicable only when certain conditions (assumptions) are satisfied. In case of Benford's law, one main condition is that the original numbers (first digits of which we consider) should span multiple orders of magnitude. Also Benford's law is formulated as a rather general empirical observation. The law is supposedly observed in many naturally occurring data.

Vote counts is obviously an example of naturally occurring data. The issue is that vote counts do not span many order of magnitude. Also vote counts, especially in elections with two competitors, are not related to exponential growth (which is prevalent in many naturally occurring systems), which can actually be a driver for the Benford's law.

Watch the following video by Stand-up Maths for a video discussion on applicability of the Benford's law to the data from the recent presidential election in the US. The video below covers another simple method to check for the election fraud.