 raise a little bit the volume or some maybe I should raise the volume somehow so can you hear me better now yeah sure sure much better yes yes can you share the screen I have tried to did I not succeed are you seeing my first slide no we don't we just see you Joel okay that's not good enough just a moment that's just getting let me just zoom me I see I picked maybe like that how do I get that there how about now are you just seeing me or are you are you seeing the screen we are seeing yes the screen with all participants and now yes yes you share the screen now we see the slide slide two perfect now let me see if I can make it so that I can see my slide just a moment yes we see slide okay now just get this out of the way just slide okay so shall I begin yes very good thank you very much yes please okay I'd like to thank the organizers for inviting me to join this meeting and I'd like to thank the other speakers for fascinating presentations from very different disciplinary backgrounds and perspectives I've learned a lot from the other speakers and from the questioners and I'm grateful to be a participant in this meeting I want to talk about the infinite variants of US COVID-19 cases and deaths and Taylor's law of heavy-tailed data the picture at the bottom of the screen shows where I am now in the Catskill Mountains northwest of New York City I'm sorry not to be with you in Trieste but I feel fortunate to be where I am there are four parts to this talk I want to introduce the concept of a variance function then tell you about Taylor's law then talk about heavy tails and regular variation and bring all of these concepts together in the fourth part to talk about the distribution of COVID-19 cases and deaths in the United States so we're on part one variance functions what is a variance function there's a mathematical definition and a practical definition the mathematical definition the population definition given a non-empty family of random variables that means quantities that you can measure if each random variable has a finite mean and a finite variance the population variance function is the mapping from the mean to the variance how does the variance change when the mean changes the sample variance function suppose you have a set of samples of observations how does the sample variance change as the sample mean changes so one has to do with the underlying probability distributions the sample variance function has to do with your observations that's part one part two is what is Taylor's law and this is a picture of Lionel Roy Taylor who was a British entomologist and he was the last person to discover Taylor's law it was discovered multiple times by other ecologists before him he collected many examples and it was then named after him Taylor's law says that if each random variable has a finite positive mean and variance then there exists a coefficient a and an exponent b such that the variance equals the coefficient a times the mean raised to the power b that's a power law so sometimes Taylor's law is called Taylor's power law because the variance is a power of the mean the sample Taylor's law holds if you have a samples with mean x bar and variance s squared then the power law holds and if you take the logarithm of the power law the log of the sample variance is roughly equal to a constant log of a plus a slope b times the logarithm of the sample mean or you can divide the sample variance by the sample mean raised to the b and that's roughly constant the difference between the population Taylor's law and the sample Taylor's law is crucial for this conversation the difference is for the sample Taylor's law the sample and the mean and the sample variance are always finite numbers even when the underlying distribution has no mean or variance or those are infinite so your observations are finite but the underlying law might be infinite here's the data structure for a Taylor's law you have a set of samples the columns s equals one s equals two s equals three and so on a set of samples in each sample you have a set of observations x11 x21 x31 for the first sample and similarly for the others for each sample you've got a bunch of numbers the number of numbers might be different from one sample to another and for each sample you can calculate a mean and you can calculate a variance so you get a set of pairs of mean and variance that is the data structure and you may have such data of your own let's illustrate that with tornadoes which is part of human ecology the united states has more tornadoes than any other country according to the Lloyds of London this is what a tornado looks like now what does tornadoes have to do with Taylor's law my colleagues Michael Tippett and I studied Taylor's law for tornadoes over the years 1954 to 2014 each year has a sequence of tornadoes we look at all the tornadoes and we group them into outbreaks and outbreaks outbreak is six or more consecutive tornadoes with not more than six hours between the start of one and the start of the next so we take all the tornadoes and f1 plus means Fuji to scale one or higher we only take credibly repeatedly observed tornadoes we don't take you know maybe I saw a tornado or not so this is to reduce variability due to population density and make the data more credible so for each year we can look at the number of tornadoes per outbreak for all the outbreaks in that year and we calculate the mean number of tornadoes per outbreak and the variance of the number of tornadoes per outbreak so it's an outbreak also geographically closer it has to be in united states can't lower 48 states they can be anywhere but in general when there is a seek an outbreak they are generally closely related because there's a storm system that is throwing off the tornadoes but we did not impose a geographic constraint except that they are in the lower 48 states and that's where I that's where there are more than any other elsewhere in the world although that's not the only place where they occur so an outbreak is defined as at least six tornadoes starting not more than six hours apart according to the insurance companies 79% of tornado fatalities and most economic losses occurred in outbreaks why if you have only one tornado okay it comes and goes if you have two or three the first one weakens the roof the second one takes it off there are no trends over the 60 years that we looked at in the number of reliably reported tornadoes and there are no trends in the number of outbreaks in the last half century but the tornadoes are increasingly concentrated in outbreaks the mean and the variance of the number of tornadoes per outbreak and the insured losses increased significantly in the last half century and here's our paper from nature communications the panel a number of tornado outbreaks per year no significant trend in fact a slight slight decrease but no trend however panel B the mean number of tornadoes per outbreak grew by 0.66% per year and the variance of the number of tornadoes per outbreak grew by 2.89% per year roughly four times faster on the right in panel D the number of tornadoes per outbreak the vertical axis is the variance the horizontal axis is the mean and we have a power law they are both axes are on logarithmic scales the mean raised to the power 4.33 is proportional to the number the variance of the number of tornadoes per outbreak so we have a power law relationship okay that's an example and the higher percentiles increased faster in this work with Chiara LePore and Michael Tippett in science in 2016 on the left panel in panel A the upper curve is the 80th percentile and you can see that that 80th percentile is growing faster over this half century than the bottom line the blue 20th percentile which hardly grew at all so the linear growth rate is plotted on the right and you can see that the higher the percentile the more extreme observations grew faster than the less extreme okay we are now on to part three heavy tails and regular variation Joel can you can you assign any meaning to the exponent you mean the exponent 4.33 yeah we do not have a theory for that that requires some physics and we're not there yet thank you good good question i wish i could answer it okay heavy tails and regular variation so let me tell you about a familiar distribution called the log normal distribution there's a plot of the probability density function on the top right corner a random variable with parameters mu and sigma squared is called log normal if the logarithm of that random variable is normal with mean mu and variance sigma squared okay that's the definition your log normal if your logarithm is normal here is the formula for the mean the expectation e and here's the formula for the variance and i hate to make people look at formulas but in this case just brace yourself and if sigma squared the variance parameter is constant and only mu changes look at the formulas for the expectation it's proportional to expo exponential of mu if the sigma squared is constant and for the variance it's proportional to exponential of two mu if the sigma squared is constant so in this case the variance changes as the square of the mean if the sigma squared is constant and that's what i say in the last line we'll need this fact later on when we're talking about covid if sigma squared is constant and only mu changes then variance is proportional to the mean squared that is taylor's law with x bone and two okay i promise not to do more mathematics here is the levy distribution and again the levy distribution is named after the last person who discovered it levy discovered it in france in 1924 but helmerth and helmerth and luroth discovered it about half a century earlier in germany if a random variable x is normal then one over x squared has the levy distribution it's called a stable distribution with index of half and there's the formula and there i plotted the probability density function and lo and behold it sort of resembles the log normal distribution now we compare the levy and the log normal the data in these two plots is our identical however the vertical axis on the left in the top is linear and the vertical axis of the probability density is logarithmic in the lower and what you see is the blue levy distribution tail in the lower picture is slightly higher than the red dotted log normal distribution tail in the lower picture what that means is that the tail of the levy falls slightly more slowly than the tail of the log normal and the consequence of that slight difference in dropping off as x gets large is that while the log normal has a finite mean and variance the levy has an infinite mean and variance now what does that mean for you and me it means something dramatic here are simulations which i did of samples of increasing size from one a two actually to 10 000 the bottom red line is the cumulative average of a sample of increasing size for a log normal distribution with mu equals zero and sigma squared equals one and as you can see it converges to one and stays there the blue lines jagged broken fragments are the sample cumulative sample averages from the levy distribution one over normal squared and you can see it goes along for a sample size and then jumps up because some observation is extreme and yanks the whole cumulative average up then you get regular you know nice observations and the cumulative average drops down then another extreme happens and another and another and things drift toward infinity in fact at a rate proportional to the sample size here's another one things going along very nicely for the blue levy curve all of a sudden after 7400 observations you get an extreme that yanks the sample average up to 14 times 10 to the fourth that's um 140 000 while the log normal red curve is going along very peacefully at one my colleagues let mark brown and victor de la penha at columbia proved in 2017 that even though the sample mean and the sample variance are infinite they obey a power law and here are some simulations in the lower left corner the horizontal axis is the logarithm of the sample mean the vertical axis is the logarithm of the sample average sorry a sample variance excuse me and one means a sample of size 10 and two along that diagonal line means a sample of size 100 three is a sample of size 1000 six is a sample of size a million nine is a sample of size billion and these points fall along a line of slope three which we calculated theoretically this is not fitted that that must be the slope the general formula if you have a stable law of index alpha it obeys tail line is alpha over one minus alpha and that's true even when the observations are not independent but are correlated 0.999 and this is work by Richard Davis and ganadi samarod nitski published in proceedings of the royal society 2020 and it's true for the higher central moments not just the mean and the variance but for the central third moment fourth moment and fifth moment our simulations show this our theorems prove it to be true many roads lead to Taylor's law many models yield exactly or asymptotically the power law form and parameters do not determine other test details we're now at the last part covid the new york times okay i'll go ahead publishes cumulative covid 19 hello yes can you hear me thank you new york times publishes covid cases and deaths at the end of each day when i downloaded the data in june 2021 there were 1.4 million counts and we arrange the data like this we have one column per state and for each state we have the cumulative number of cases in each county in that state and we have the cumulative number of deaths in each county in that state and we can calculate the mean number of cases per county and the variance of the number of cases per county now i used only the states that had at least one death or one case in at least seven counties and the data conformed to Taylor's law this is unpublished work but i believe it's a new finding there are 15 panels in this picture they range from the first of april 2020 then the first of may 2020 across to the right then the second line and the third line 15 months up to the first of june 2021 in the lower right corner the horizontal axis on a logarithmic scale is the mean number of cases per county and each x is one state the vertical axis is the variance of the number of cases per county and each x is one state and the x's fall along a line that's the yellow line and in the top left corner of each panel is my estimate of the slope of that line and below that is the confidence interval for the slope the red dotted line below the data is what would be predicted from a Poisson distribution that is a random purely random distribution Poisson in which the variance equals mean so that's the line of slope one the yellow line is this line of roughly slope two okay so all of these 15 panels have the same coordinates so you can see the data points moving from the lower left upward and to the right by june 2021 all the data are in the upper right corner but the slope is still two that's the accumulation of the pandemic before the introduction of vaccines this is the same plot for deaths and again taylor's law holds but the slope is three in april 2020 and then rapidly drops down to two and stays there so this is cases this is deaths these are the parameters the a coefficient that is the intercept of that line is steady around five and the b coefficient is never distinct from two for cases over the 15 months the same plot for deaths you see the three in the lower panel on the left and then it drops down to two so the slope converges to two to summarize this finding taylor's law describes countless cumulative cases and deaths we lost you the log variance increases linearly the slope can you hear me now yes yes i'm sorry i'm in the mountains and the internet connection is not stable the survival curve plots the probability that counts exceed x as a function of x that's the definition of a survival curve maybe if you switch off your video join things can be but or um can i continue now okay yes i will do that one second how do i do that just a second i gotta figure out how to do that turn off the camera turn off the camera i think i have to let's see what else can i do here should be a camera down at the lower left of your screen okay is that better yes okay now you don't have to look at me that's certainly an improvement all right um so here's what we did the black can you see this slide now please tell me if you can see it yes we can see perfect good okay so let's take the k up in the top left corner the solid black is the empirical survival curve of all roughly 3200 counties in the united states of the number the the probability that the number of cases exceeds the quantity on the x axis both axes are on logarithmic scales so you see that the black dots slowly come down that means the probability that you're more than 100 cases is maybe one percent of the probability that you're less greater than one case and it drops off then i fitted log normal distributions that's the blue curve and i fitted y-ball distributions that's the dotted red curve well the y-ball curve is no good it's just too low the log normal distribution fits pretty well for the first 99 percent so the vertical axis when it drops down to 10 to minus 2 10 to the minus 2 is 1 percent so the the log normal describes the data for the first 99 percent of counties but it doesn't describe the last 1 percent now if you look across all of these plots that's a consistent pattern go down to june 2021 the lower right corner the y-ball's no good the logarithmic log normal covers 99 percent or more but the top 1 percent of cases are dropping more slowly than the log normal predicts same story for the deaths okay the log normal describes 99 percent of the distributions of cases and deaths well if the count why do we get a log normal if the count has a log normal distribution if the variance is constant and only the mean only the mu sorry if sigma squared parameter is constant and only mu varies then taylor's law holds with slope two so we look state by state we fitted log normal distribution state by state and examined how does sigma which is the square root of sigma squared on the vertical axis change as mu changes and what you see is that effectively sigma squared or sigma is constant compared to mu so we that's for cases same story for deaths so we have an explanation of why taylor's law holds with slope two for the lower 99 percent of counts the log normal distribution is what predicts that but the very largest counts of cases and deaths are more extreme than the log normal distribution predicts we zoom into the counties with the highest one percent of counts and this is the upper tail of only the highest one percent of counties and we see again a straight we see a we see a straight survival curve curve and we ask this is for cases and this is for deaths and we fit a line to those data and we ask what's the slope and in fact here I'm using hill estimates of the tail index for stable laws and alpha is this quantity the hill tail index alpha shown in red here the number associated with each data point for each of the 15 months is the number of counties on which that's based the number of counties in the top one percent and the blue lines above and below are the confidence intervals from a thousand bootstrap resamples for each month and what you see is that alpha is between one and two that's for cases and this is for deaths and this is the same alpha is between one and two the empirical survival curve suggests that the variance is infinite why because the estimated upper tail index alpha falls between one and two a stable law with alpha between one and two has a finite mean but an infinite variance and when I came to this finding my feeling was expressed by this painting from bell in 1806 wonder fear and astonishment what's going on well I'm going to have to go fast here because of time but regularly varying upper tail with an index alpha between one and two explains why Taylor's law with b with exponent two holds even for the largest counts where the log normal distribution fails so we have two parts to the explanation log normal covers the lower 99 percent stable laws with alpha between one and two cover the top one percent and to demonstrate first I demonstrated this by doing some simulations and then my colleagues who are a lot smarter than I am did the mathematics and I'm going to show you the simulations and just you'll have to believe me or read the papers if you want to see the mathematics I'm not going to inflict that upon you I simulated four models of regularly varying upper tails based on the one over a normal random variable one over a uniform random variable one over a product of unit two uniform random variables and one over a product of three uniform random variables and I used indices of one half one and three half the upper lines are the survival curves analogous to the survival curves that I showed you for the data and the bottom three panels are the variance functions and they all conform to Taylor's power law with slope two or closer to slope two as your sample size increases there are four different functions here with 100 samples of size 100 for each and you can have correlations the data aren't independent and the models don't need to be independent and I assure you in the lower right corner yes we have theorems so what here is the bottom line from all of this analysis if the variance of variances of cases and deaths per county are infinite then facility and resource planning should prepare for unboundedly high counts no single county no single state no single nation can prepare for unboundedly high counts cooperative exchanges of support should be planned cooperatively here are my math collaborators and teachers mark brown victor de la penha richard davis gennady samarad nitsky and most recently chuan fa tang and cheung chi yam i've learned a lot from them and i'm grateful to them thank you for your attention i welcome your questions thank you well maybe i can ask a question we have part here is a question you're last your last slide you referred to uniform distribution and so then what was the range that you took zero to one ah is this the slide that you were asking about martha sorry sorry is this the slide that you were asking is this what you were asking about yes of course it is okay okay good you're looking at so let me let me answer that in a little more detail you take a uniform you take a random number between zero and one that's a uniform random barrier you take it's absolute it's positive okay then you take one over that and you raise that to the power one over alpha yeah yeah i got it that's how you generate these things and you can do this on your own home computer and see what it does for you i have better things to do joe good for you simon did you have a question yeah i did joe um thank you very much for this fascinating talk um it's always a good question when you have power laws is why you have power laws and there are lots of ways you can get power laws one of them is through near second order phase transitions you talked about earthquakes you talked about outbreaks is there any connection there to these being near critical points that's a good question and i don't have a quick answer i'm sorry i'm slow yeah well do you have any ideas to why power laws should be arising yes i do i have a definite idea about that so the upper one percent first of all the spread of a disease is a multiplicative process you know i get infected i infect a certain number of other people in the simplest possible world which doesn't really exist they each infect a certain number of other people and there's a cascade a multiplicative cascade think of a branching process but of course the factors a propagation differ from one person to another and the circumstances in which the infection is propagated differ it's not only the individuals it's not the only the infection the agent is different the different variants and the circumstances crowding or not those multiplicative processes can generate a pareto type distribution with a power law and that is a well-known mathematical fact i don't have to invent that so it's the details of the multiplicative process that determine whether you get the log normal or the pareto but we know processes that are capable of the pareto type survival function multiplicative processes that could generate it and those could be plausible models for transmission in states with very high number of cases and deaths per county that that i think that's a sensible answer to your question what do you think sigh does that i think i think i think that makes sense okay and i think it also may lead to some critical phenomena so there may be a secondary explanation there good i'm with you okay can i ask a question please yes so uh i just to connect this also to the rest of the of the workshop the workshop is about limits that exist in real physical systems and i'm not captured by models so here we have a an empirical distribution and we have an analytic distribution the empirical distribution fits so that fits very well the analytical distribution or the other around up to a certain scale up to a certain limit right but then there are physical cysts because of physical size there is a physical limit on the empirical distribution that the that is not there with analytic distribution so we know that up to the scale that you were able to observe in kovit or with the with with storm so the sample has not yet reached the physical limit and therefore we can i mean it looks like uh up to that that scale the analytic distribution is a very good fit but but uh at bigger scale so at bigger city if we if we if we observe for longer time or maybe we see that at some point the the the sample the empirical sample will be will be cut it will be limited will be truncated will be truncated and then at that point uh we would uh the math will be a bit different right so what i'm trying to say is that if you go back to the last slide you say in analytical distribution of levy for instance you have infinite mean infinite variance and therefore i'm bound to the process but in the real system you will always have some bound some physical bound the lesson that i take is that uh probably we have not seen the maximum value but this does not mean that the maximum value does not exist that that the the real distribution can be unbounded so in some sense this connect also to what we were discussing the previous days sometimes uh it's an artifact of the model that there is no bound no limit but but it's just a model right so perhaps the model will work only up to the scale where we have not experienced the limit but we should be careful in reasoning about the model uh right so maybe you can comment on that yes i can tell you right now what is the limit of the upper the upper limit of the number of deaths from covid it's approximately eight billion which is the population of the earth and that is a fact but it does not provide useful guidelines for how we respond to covid now and models have different purposes the purpose here is to shed light on what is a useful strategy in planning for response to this and similar viral disease outbreaks of which there will be more so i accept the fact that there are limits that the models don't represent they haven't appeared yet in the data but the way the the course of the event so far gives us some guidance that maybe we need to plan cooperative responses at the scale at which we experience the the covid pandemic does that make sense in other words i'm talking about what's now not you are absolutely right about a physical limit but that doesn't bear on how we should behave now absolutely i agree with the conclusion it's just the term unbounded on physical system that looked a little bit extreme so you raise your hand i think simon just forgot to lower his hand over the end thank you john for this talk if we develop the the title of law we'll find that the the b is the elasticity of the variance with respect to the mean that's correct as i if we have the same context of a phenomenon the same phenomenon if for example introduce more control on this phenomenon as more what more what i more what i couldn't get the word we have a market or a machine is it is functioning there is some relationship between the variance of the the length of a piece that is produced and the mean and we have we control more than matching the change the context of its functioning to control it do you think that this elasticity will decrease or will increase join i have to raise to explain more please i don't understand exactly what you're what you're proposing as the model i'm sorry i couldn't understand could you say it slowly please for the same phenomenon if we change the context and we find for example that the b the elasticity is increasing or decreasing what shape of conclusion can we draw okay now i understand thank you um that depends entirely on the details because you can get uh an exponent or slope of two from a variety of models and um you there's a descriptive consequence when the slope is less than two it means that the coefficient of variation decreases as the mean increases when the slope or exponent b is greater than two it means that the coefficient of variation increases as the mean increases when the slope b equals two itself exactly then the coefficient of variation is constant as the mean increases the coefficient of variation is defined as the standard deviation divided by the mean so it's a measure of varying a variation you know it's one over the signal to noise ratio but interpreting that mechanistically depends on the particular details of the content a context okay thanks for these questions and if anybody wants to communicate with me my email is cohen at rockerfeller.edu and i'm happy to continue the conversation thanks so much we have another question just a last question i was working i mean i know the thing because i got people who used to work in public health england the uk hsa working on this issue so what the what the government means i mean what the government cares about uh is the basic reproduction number it's basically the growth of your epidemic when the growth is above one uh something it's under it's out of control when the basic reproduction number is below one it's under control and you start relaxing uh measures so did you did you think looking at the same relationship before the basic reproduction number using cases to look at r no or r the t r t and look at the same thing because it's more or less the first day reality it's not exactly the same quantity looking at because that's what they care about that's what the government cares about and one might ask why does the government care about the basic reproduction number and the answer is because epidemiologists have told the government that is a parameter you need to care about that is a consequence of a theoretical perspective not a given in fact what the government cares about is or at least what the people involved care about is how many people are getting sick and how many people are dying the basic reproduction number scientists have told the government is a key to answering what will happen but there are assumptions in relying on the basic reproduction number and those assumptions derive from the models in which that is an important parameter I've tried to circumvent the conventional epidemiological theory by going directly to the counts which themselves are unreliable I admit and to discover sort of a positive model such as the models that invoke a basic reproduction number okay thank you thank you okay so I think we can take a break and uh what I can be in at uh uh Scotty is it okay if we take a half an hour break and uh recommend uh 10 plus four of course yes of course thank you okay thank you so so thanks bye bye see you all at uh 10 plus four