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Power-law distribution of the number of virus victims is derived from heterogenity across spreaders. What makes worse is the convex cost with respect to the number of victims. Similarly to taking higher-order moments under power-law distribution, the expected cost can become surprisingly high even if the number is truncated at some population. Policy for cutting the tail by specifically focusing on the superspreaders should be encouraged.

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Intro to Quant Trading Strategies (Lecture 9 of 10)

This document provides an introduction and overview of algorithmic trading strategies and quantitative equity portfolio management. It discusses key concepts like alpha, factor models, and the fundamental law of active management. Specific strategies covered include exploiting the post-earnings announcement drift anomaly by going long after good news and short after bad news. Historical return examples are provided showing double-digit annual returns. Factor models are explained as decomposing returns into exposures to various risk factors. Least squares regression is described as a method to estimate factor premiums from historical return and exposure data.

Demystifying the Bias-Variance Tradeoff

Explaining the probabilistic essence in the derivation of the Bias-Variance Tradeoff, and articulating its importance in Modeling.

Intro to Quant Trading Strategies (Lecture 8 of 10)

This document provides an introduction to performance measures for algorithmic trading strategies, focusing on Sharpe ratio and Omega. It outlines some limitations of Sharpe ratio, such as ignoring likelihoods of winning and losing trades. Omega is introduced as a measure that considers all moments of a return distribution by taking the ratio of expected gains to expected losses. Sharpe-Omega is proposed as a combined measure that retains the intuitiveness of Sharpe ratio while using put option price to better measure risk, incorporating higher moments. The document concludes with a discussion of portfolio optimization using Omega.

Probablity normal

The document discusses several continuous probability distributions including the uniform, normal, and exponential distributions. It provides the probability density functions and key characteristics of each distribution. Examples are given to demonstrate calculating probabilities and parameter values for the uniform and normal distributions. Excel functions are also introduced for computing probabilities and values for normal distributions.

Ch8 slides

This document discusses testing for non-stationarity and unit roots in time series data. It introduces the Augmented Dickey-Fuller (ADF) test and Phillips-Perron test for determining if a time series is integrated of order zero (I(0)), one (I(1)), or two (I(2)). The ADF test regressions the change in a variable on its lag and lags of the change to test for a unit root. If the null of a unit root is not rejected, further tests are needed to determine higher orders of integration. While ADF and Phillips-Perron tests are commonly used, their power is low if the process is near but not at the non-station

Intro to Quant Trading Strategies (Lecture 6 of 10)

This document provides an outline and overview of using Kalman filter methods for pairs trading strategies based on modeling the spread between two assets as a mean-reverting process. It discusses modeling the spread as an Ornstein-Uhlenbeck process, computing the expected state from observations using the Kalman filter, and how to predict state estimates and minimize posterior variance in the Kalman filter updating process. References on stochastic spread methods and the application of Kalman filters to pairs trading are also provided.

Intro to Quant Trading Strategies (Lecture 2 of 10)

This document provides an introduction to hidden Markov models for algorithmic trading strategies. It discusses key concepts like Bayes' theorem, Markov chains, and the Markov property. It then covers the three main problems in hidden Markov models: likelihood, decoding, and learning. It presents solutions to these problems, including the forward-backward, Viterbi, and Baum-Welch algorithms. It also discusses extensions to non-discrete distributions and trading ideas using hidden Markov models.

Stat lesson 5.1 probability distributions

The document defines key terms related to probability distributions, including random variables, discrete and continuous distributions, and mean, variance and standard deviation. It provides examples of discrete and continuous random variables and describes the binomial, hypergeometric and Poisson distributions. Examples are given to show how to calculate the mean, variance and standard deviation of a discrete probability distribution.

Ch 5 numerical analysis of risks and their relationships and

This document discusses various numerical analysis techniques used in insurance underwriting. It covers measures of central tendency like mean, median, and mode to summarize claim data. It also discusses principles of probability, the law of large numbers, frequency and severity of claims, correlation and regression analysis to determine relationships between variables, and expected value calculations. Mathematical techniques like regression analysis are used to predict future claims. Homogeneous risk pools in personal and motor insurance allow for more precise underwriting based on past data trends. However, some claims like catastrophes are difficult to predict accurately.

Ch12 slides

This document discusses limited dependent variable models, where the dependent variable can only take on certain values, such as 0 or 1. It begins by providing examples of situations that would call for such models. It then examines the linear probability model and its flaws, such as producing probabilities outside the valid 0-1 range. Better approaches like the logit and probit models are discussed, which use functions to constrain probabilities to this range. The document also covers interpreting coefficients, goodness of fit measures, and estimating these models using maximum likelihood. As an application, it summarizes a study using a logit model to test theories of corporate financing decisions.

Data Analysison Regression

The document discusses linear regression analysis and its applications. It provides examples of using regression to predict house prices based on house characteristics, economic forecasts based on economic indicators, and determining optimal advertising levels based on past sales data. It also explains key concepts in regression including the least squares method, the regression line, R-squared, and the assumptions of the linear regression model.

Ch10 slides

1. The document discusses switching models, which allow for changes in the behavior of economic and financial variables over time. These switches can be one-time changes or occur frequently.
2. Markov switching models generalize the dummy variable approach to allow for multiple "states of the world" that a variable can occupy. The probability of switching between states is governed by a transition probability matrix.
3. An example application uses a Markov switching model with two states to analyze real exchange rates. This allows for multiple switches between regimes and provides evidence on purchasing power parity theory.

Ch5 slides

This document discusses the assumptions of the classical linear regression model (CLRM) and methods for testing violations of those assumptions. It covers the assumptions of no autocorrelation between error terms, homoscedasticity or constant error variance, and the errors being normally distributed. Tests for heteroscedasticity discussed include the Goldfeld-Quandt test and White's test. Tests for autocorrelation examined are the Durbin-Watson test and Breusch-Godfrey test. Consequences of violations include inefficient coefficient estimates and invalid inference. Potential remedies discussed are generalized least squares or using heteroscedasticity-robust standard errors.

Ch4 slides

1) The document introduces the multiple linear regression model, where the dependent variable depends on more than one independent variable. 2) It shows how to write the multiple regression model using a matrix formulation, with the dependent variable as a column vector, the independent variables as a matrix, and the coefficients and error term also as vectors/matrices. 3) It explains how to estimate the coefficients using ordinary least squares (OLS) and calculate the standard errors of the estimates.

Ch1 slides

This document introduces the key concepts in econometrics and financial econometrics. It defines econometrics as the application of statistical and mathematical techniques to economic and financial problems. Some examples of problems that can be solved using econometrics are testing market efficiency, modeling volatility, and forecasting correlations. The document discusses the different types of data used in econometrics, including time series, cross-sectional, and panel data. It also covers important financial concepts like returns, deflating nominal values for inflation, and the differences between classical and Bayesian statistical approaches.

Ch13 slides

The document discusses simulation methods in econometrics and finance. It covers topics such as the Monte Carlo method, conducting simulation experiments by generating data and repeating experiments, random number generation, variance reduction techniques like antithetic variates and control variates, and examples of simulations in econometrics and finance including deriving critical values for Dickey-Fuller tests and pricing financial options. Bootstrapping methods are also discussed as an alternative to simulation that samples from real data rather than creating new data.

Intro to Quant Trading Strategies (Lecture 3 of 10)

This document provides an introduction to trend following strategies in algorithmic trading. It discusses Brownian motion and stochastic calculus concepts needed to model asset price movements. Geometric Brownian motion is presented as a model for asset price changes over time. Optimal trend following strategies seek to identify the optimal times to buy and sell an asset to profit from trends while minimizing transaction costs. The strategy parameters and expected returns are defined.

Ch2 slides

This document provides an overview of mathematical and statistical foundations relevant to econometrics. It defines functions and their linear and nonlinear forms. It discusses straight lines, their slopes and intercepts. It also covers quadratic functions, their roots and shapes. Additionally, it introduces exponential functions, logarithms, and their properties. It describes summation and differentiation notation used in calculus. The overall summary is an introduction to functions, lines, and other mathematical concepts important for understanding econometrics.

Ch11 slides

The document discusses panel data analysis and its application to analyzing competition in the UK banking sector. It summarizes:
1) Panel data has both time series and cross-sectional dimensions, allowing examination of how variables change over time for the same objects. A fixed effects model accounts for heterogeneity across objects.
2) A study analyzed competition in UK banking from 1980-2004 using a fixed effects panel data model. It tested for market equilibrium and calculated a contestability parameter to indicate the degree of competition.
3) The results found evidence of equilibrium and showed the contestability parameter fell from 0.78 to 0.46, suggesting competition weakened over the period.

Intro to Quant Trading Strategies (Lecture 7 of 10)

This document provides an overview of constructing small mean reverting portfolios. It discusses using distance and cointegration methods to construct initial portfolios, but notes their shortcomings. It then formulates the problem as maximizing mean reversion to find sparse portfolios. Various algorithms are presented to solve this, including greedy search, least absolute shrinkage and selection operator (LASSO), and semidefinite programming (SDP) approaches. Key steps involve estimating relationships between assets, selecting subsets of assets, and optimizing portfolio weights to maximize mean reversion.

Intro to Quant Trading Strategies (Lecture 9 of 10)

Intro to Quant Trading Strategies (Lecture 9 of 10)

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Probablity normal

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Ch8 slides

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Intro to Quant Trading Strategies (Lecture 2 of 10)

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Stat lesson 5.1 probability distributions

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Ch12 slides

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Data Analysison Regression

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Ch10 slides

Ch10 slides

Ch5 slides

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Ch4 slides

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Ch1 slides

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Ch13 slides

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Intro to Quant Trading Strategies (Lecture 3 of 10)

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Ch2 slides

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Ch11 slides

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Intro to Quant Trading Strategies (Lecture 7 of 10)

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Skew Berlin2009

1) The document proposes a new measure called "probability of skill" or p-value that calculates the probability a hedge fund's returns were achieved through a simple negatively skewed strategy rather than skill.
2) An empirical analysis calculates the p-value for 8,000 hedge funds and uses it to predict which funds were more likely to "blow up".
3) A logistic regression found the prediction was statistically significant, though the p-values alone were not strong predictors of blow ups. Strategy type and fund inflows were better predictors than p-values.

Slides ensae-2016-9

This document discusses various modeling approaches for non-life insurance tariffication including frequency-severity models, Tweedie regression models, and high-dimensional modeling techniques like ridge regression and the LASSO. It compares individual risk and collective risk models, explores the impact of the Tweedie parameter, and applies regularization methods to insurance data.

Basic concepts and how to measure price volatility

Basic concepts and how to measure price volatility African Growth and Development Policy (AGRODEP) Modeling Consortium

Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshopSlides ensae 9

This document discusses various modeling techniques for non-life insurance ratemaking including individual and collective models, Tweedie regression, and the LASSO method. It explores using a Tweedie distribution for compound Poisson models and the relationship between individual and collective models. The document also examines issues with high-dimensional data in insurance, bias-variance tradeoffs, and regularization methods like ridge regression and the LASSO for variable selection.

OR2 Chapter1

The document provides an overview of key statistical concepts used to analyze financial data and return distributions, including measures of central tendency, dispersion, skewness, and kurtosis. It discusses frequency distributions and their use in summarizing return data. Various measures of central tendency, dispersion, and shape are defined, including the mean, median, standard deviation, skewness, and kurtosis. Performance metrics like the Sharpe ratio that measure risk-adjusted return are also introduced.

Introduction to Machine Learning Lectures

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Data Science Cheatsheet.pdf

This document provides a summary of key machine learning concepts and techniques:
- It outlines common probability distributions including binomial, normal, and Poisson distributions.
- It describes concepts like bias-variance tradeoff, cross-validation, and model evaluation metrics for regression and classification.
- It summarizes supervised learning algorithms like linear regression, logistic regression, decision trees, random forests, and support vector machines.
- It also covers unsupervised learning techniques including k-means clustering, hierarchical clustering, and evaluating cluster quality.

Cheatsheet recurrent-neural-networks

1. Recurrent neural networks (RNNs) allow information to persist from previous time steps through hidden states and can process input sequences of variable lengths. Common RNN architectures include LSTMs and GRUs which address the vanishing gradient problem of traditional RNNs.
2. RNNs are commonly used for natural language processing tasks like machine translation, sentiment classification, and named entity recognition. They learn distributed word representations through techniques like word2vec, GloVe, and negative sampling.
3. Machine translation models use an encoder-decoder architecture with an RNN encoder and decoder. Beam search is commonly used to find high-probability translation sequences. Performance is evaluated using metrics like BLEU score.

Shortfall Aversion

Shortfall aversion reflects the higher utility loss of a spending cut from a reference point than the utility gain from a similar spending increase, in the spirit of Prospect Theory's loss aversion. This paper posits a model of utility of spending scaled by a function of past peak spending, called target spending. The discontinuity of the marginal utility at the target spending corresponds to shortfall aversion. According to the closed-form solution of the associated spending-investment problem, (i) the spending rate is constant and equals the historical peak for relatively large values of wealth/target; and (ii) the spending rate increases (and the target with it) when that ratio reaches its model-determined upper bound. These features contrast with traditional Merton-style models which call for spending rates proportional to wealth. A simulation using the 1926-2012 realized returns suggests that spending of the very shortfall averse is typically increasing and very smooth.

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Estimating Market Risk Measures: An Introduction and Overview

This discusses the topics under Estimating Market Risk Measures: An introduction and Overview by Kevin Dowd
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Pareto Models, Slides EQUINEQ

This document discusses different Pareto models for fitting top incomes data: Pareto I, generalized Pareto distribution (GPD), and extended Pareto distribution (EPD). It finds that the choice of threshold can significantly impact estimates of the tail index parameter. EPD and GPD are generally more stable across thresholds than Pareto I. The document also warns of potential misspecification, estimation, and sampling biases when applying these models to real-world income and wealth data. It provides examples analyzing South African income data.

lecture8.ppt

This document provides summaries of key concepts related to statistical hypothesis testing and confidence intervals involving t-distributions. It discusses when to use a t-distribution versus a standard normal distribution, specifically when the population standard deviation is unknown and must be estimated from a sample. It provides examples of hypothesis tests and confidence intervals for a single population mean when the sample size is small as well as examples involving paired data. Key formulas are presented for t-tests, confidence intervals, and the t-distribution.

Lecture8

This document provides summaries of key concepts related to statistical hypothesis testing and confidence intervals involving t-distributions. It discusses when to use a t-distribution versus a standard normal distribution, specifically when the population standard deviation is unknown and must be estimated from a sample. It provides examples of hypothesis tests and confidence intervals for a single population mean when the sample size is small as well as examples involving paired data. Key formulas are presented for t-tests, confidence intervals, and the t-distribution.

Real time information reconstruction -The Prediction Market

This document summarizes an approach to analyzing prediction markets using convex optimization. It discusses both offline and online formulations of the problem. The offline formulation involves accepting bids to maximize profit, which can be modeled as a linear program. Uniqueness of the solution is analyzed. The online formulation updates the model sequentially as bids arrive in real time. Various choices for the objective function are discussed, balancing risk aversion with expected gain. Truthfulness of bids is also addressed.

Greeks And Volatility

The document discusses concepts related to volatility and options Greeks. It defines volatility, and describes methods like historical volatility and EWMA to estimate volatility over time. It then explains the Greeks - delta as the change in option price wrt the underlying's price, theta as the time decay, gamma as the change in delta, vega as the sensitivity to volatility, and rho as the sensitivity to interest rates. It provides examples of using options positions to hedge or make a portfolio delta, gamma and vega neutral.

Machine Learning - Probability Distribution.pdf

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CLIM Fall 2017 Course: Statistics for Climate Research, Statistics of Climate...

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1. This document summarizes statistics on climate extremes, including time series plots and extreme value analyses of temperature and precipitation data from Houston, Texas.
2. Fitted generalized extreme value (GEV) distributions to one-day, three-day, and seven-day maximum precipitation values show increasing return levels with longer durations.
3. Bayesian and frequentist methods are demonstrated for fitting GEV distributions and estimating return levels of extreme precipitation events.Market Risk Modelling

This document discusses various techniques for modeling market risk and estimating volatility, including calculating volatility, exponentially weighted moving average models, GARCH models, Greeks (delta, gamma, theta, vega, rho), value at risk using variance-covariance and Monte Carlo simulation methods, and historical simulation. Key concepts covered include estimating and updating volatility, incorporating mean reversion in models, hedging positions to achieve gamma and vega neutrality, and calculating value at risk over different time horizons using variance-covariance, Monte Carlo simulation, and historical simulation approaches.

Skew Berlin2009

Skew Berlin2009

Slides ensae-2016-9

Slides ensae-2016-9

Basic concepts and how to measure price volatility

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Slides ensae 9

Slides ensae 9

OR2 Chapter1

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Introduction to Machine Learning Lectures

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Data Science Cheatsheet.pdf

Data Science Cheatsheet.pdf

Cheatsheet recurrent-neural-networks

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Shortfall Aversion

Shortfall Aversion

Statistical tests /certified fixed orthodontic courses by Indian dental academy

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FParaschiv_Davos

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Estimating Market Risk Measures: An Introduction and Overview

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Pareto Models, Slides EQUINEQ

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lecture8.ppt

lecture8.ppt

Lecture8

Lecture8

Real time information reconstruction -The Prediction Market

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Greeks And Volatility

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Machine Learning - Probability Distribution.pdf

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CLIM Fall 2017 Course: Statistics for Climate Research, Statistics of Climate...

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Market Risk Modelling

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kneeling asana - Comprehensive understanding of kneeling asana

Comprehensive understanding of hip opening asana
Benefits of hip opening asana
Hip-opening asanas can offer a variety of benefits, including:
Improved flexibility: Hip-opening asanas can help increase range of motion and flexibility in the hips and surrounding areas, such as the lower back and thighs.
Reduced lower back pain: Tight hip flexors can contribute to lower back pain, so opening up the hips can help alleviate discomfort in the lower back.
Improved posture: Tight hips can lead to poor posture, so opening the hips can help improve posture and alignment.
Stress relief: Many hip-opening asanas involve deep stretching and relaxation, which can help reduce stress and tension in the body.
Improved circulation: Asanas that stretch and open the hips can improve circulation to the area, which can help reduce inflammation and promote healing.
Increased energy: Hip-opening asanas can stimulate the second chakra, which is believed to be associated with creativity and energy. This can help increase energy levels and feelings of vitality.
Emotional release: The hips are often referred to as the "emotional junk drawer" of the body, as they can hold onto tension and emotions. Hip-opening asanas can help release these emotions and promote a sense of emotional release and well-being.
Contraindications for hip opening asana
Hip opening asanas are generally safe for most people, but there are some contraindications that one should keep in mind. Some of the contraindications for hip opening asanas are:
Recent hip or knee injury: If you have a recent injury to the hip or knee, it is best to avoid hip opening asanas until the injury has healed.
Joint instability: People with joint instability or hypermobility should be cautious while performing hip opening asanas as they may put excessive strain on the hip joint.
Hip replacement: If you have had a hip replacement surgery, it is important to avoid hip opening asanas until your doctor clears you for these movements.
Sciatica: If you have sciatica, you should avoid any hip opening asanas that aggravate the pain. It is best to consult a healthcare professional before practicing any yoga asanas.
Pregnancy: Pregnant women should avoid deep hip opening asanas or modify them under the guidance of a qualified yoga instructor.
Osteoporosis: People with osteoporosis should be cautious while practicing hip opening asanas as they may put excessive stress on the hip joint.
It is always advisable to consult a healthcare professional before practicing any yoga asanas, especially if you have any health concerns or medical conditions.
Counterpose for hip opening asana
Counterposes for hip opening asanas can vary depending on the specific pose being practiced, but some common counterposes include:
Forward folds, such as Uttanasana (Standing Forward Bend) or Paschimottanasana (Seated Forward Bend), can help stretch and release the hamstrings and lower back muscles after hip opening asanas.

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Ruksha Mudra (Dry Mudra) -Mudra will reduce the bladder pressure

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- 1. Social Cost in Pandemic as Expected Convex Loss Rikiya Takahashi (Rikiya.Takahashi@gmail.com) *This material exhibits the author's personal opinion by mathematical thinking, and does not reflect the views by the author's employer.
- 2. Summary ● Low reproduction number R0 calibrated under homogeneous-agent assumption (e.g., SIR) cannot justify relaxing lockdown. ● The social cost is expectation of some convex function of #infections/deaths, integrated over a power-law (often heavy-tailed) distribution. – Unstably diverging cost, unless tail is truncated – Even with truncation, still unexpectedly high cost ● Public policy should focus on how to cut the tail, e.g., identify & contain super spreaders, instead of uniformly tightening/relaxing all people.
- 3. Outlines ● 1. Why #infections/deaths obey power-law – Heterogeneity across people ● 2. Why power-law is problematic even if the mean (average) is stable empirically and/or theoretically – Convex cost similar to diverging high-order moment ● Conclusion: risk mitigation policy should focus on cutting the tail rather than uniform control
- 4. 1. Heterogeneity to Power-Law Marginal is not the distribution by avg. parameter ● Let Y be the number of additionally infected persons by one spreader whose rate is θ. ● Marginal probability distribution of Y is not the distribution by “representative spreader”, i.e., ● What is worse: marginal implies more catastrophe! – For risk management, you must never use the distribution by a representative person. ∫ p(Y∣θ) p(θ)d θ≠ p(Y∣E[θ])
- 5. 1. Heterogeneity to Power-Law Parametric Example ● Each spreader has a thin-tailed exponential distribution – For simplicity here we forget that Y is an integer. ● Heterogeneity across spreaders is given by a gamma distribution ● Then marginal is a Pareto distribution that is heavy-tailed when α<=2 ● By contrast, distribution by representative is still thin-tailed exponential p(Θ=θ)=Gamma(θ;α ,β)≡ βα Γ(α) θ α−1 exp(−βθ) p(Y = y∣Θ)=exp( y;Θ)≡Θexp(−Θ y) p(Y = y)= α β (1+ y/β) −α−1 q(Y = y)=exp(y ; α β )
- 6. 1. Power-Law: Stupidity of Watching Representative in Crisis ● Interestingly, median of the marginal is lower than representative rate. But let's see >99%-tiles... – E.g., when α=1.5 and β=1.5 (average spread=1) ● Model forecast by a calibrated parameter (e.g., median=0.39) with homogeneous model leads wrong sense of safety and underestimation of crisis, due to wrong extrapolation in the tails. Median 99%-tile 99.9%-tile Pareto Marginal 0.39 13.9 67.7 Representative Exponential 0.69 4.6 6.91
- 7. 1. Heterogeneity to Power-Law Related Work ● Heterogeneity of spreaders as a stylised fact – E.g., negative-binomial + branching process (Lloyd-Smith+, 2005)) ● Pareto distribution of death toll, whose exponent leads even infinite mean unless truncated (Cirillo & Taleb, 2020) N. Becker and I. Marschner, The effect of heterogeneity on the spread of disease, Stochastic Processes in Epidemic Theory, 90-103, 1990. J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature, 438, 355–359, 2005. P. Cirillo and N. N. Taleb, Tail Risk of Contagious Diseases, to appear in Nature Physics, 2020.
- 8. 2. Not Linear but Convex Cost ● Some still say “Heavy-tail, so what? The average death rate is small in reported stats.” – In the Pareto example, mean is just ~2.0. ● Error #1: when heavy-tailed, empirical mean is highly erroneous as the estimate of true mean. ● Error #2 (more essential): what the society pays is not the mean but expected convex loss. – Not constant but increasing cost per infection ● Fatal: similarly to calibration of high-order moments, the cost can be unexpectedly large even if the tail is truncated at entire population.
- 9. 2. Not Linear but Convex Cost Considerable Example Functions ● Piecewise-linear – Higher death rate when exceeding the maximum capacity ● Power function – Huge patients spoil the work efficiency of essential workers
- 10. 2. Not Linear but Convex Cost Diverging High-Order Moments ● Example: Y = # of infections obeys a Zeta distribution and cost is quadratic – When α=2.5, mean exists as E[Y]=1.947 but E[Y2 ] is infinite – Finiteness requires (Power exponent) – (Moment order) >1. ● Monte Carlo example in action – Mean is well behaving and converging, but – Avg[Y2 ] diverges as sample size increases ! p(Y )∝1/Y α N=1,000 N=10,000 N=100,000 N=1,000,000 Empirical Avg of Y 1.82 2.17 1.98 1.95 Empirical Avg. of Y2 1.01x103 2.29x105 5.74x105 2.77x106
- 11. Conclusion ● If people behaviours are too heterogeneous, number of additionally infected persons obeys a power-law distribution, potentially with heavy tail. ● Due to the capacity constraint in hospital and essential workers, the society must pay a convex cost w.r.t. #infections and/or deaths. ● Convex cost over power-law distribution can be catastrophic. Cutting the tail, with a strong focus on heterogeneity across people, is essential. – Prohibition of large-population gathering, travel restriction on too-active or gregarious persons, etc.
- 12. Ref1. More Accurate Model of p(Y) ● If you want Y to be strictly an integer, then we can consider beta-binomial distribution. – Let n be a large fixed number, say n=10,000 – Each spreader has a binomial distribution – Heterogeneity across spreaders is given as – Then we get the beta-binomial marginal as p(Y = y∣Θ)=Bi( y ;Θ)≡ n! y!(n− y)! Θ y (1−Θ) n− y p(Θ=θ)=Beta(θ;α , p0)≡ Γ(α) Γ(α p0)Γ(α(1− p0)) θ α p0 (1−θ) α(1− p0) p(Y = y)= Γ(α)Γ(n+1) Γ(α+n) ⋅ Γ( y+α p0) Γ(y+1)Γ(α p0) Γ(n− y+α(1− p0)) Γ(n− y+1)Γ(α(1− p0))
- 13. Ref1. More Accurate Model of p(Y) ● Beta-binomial behaves likes a truncated power- law distribution – By using Gautschi's inequality, – Approximately – When α<1, it behaves like a truncated power-law distribution whose exponent is ● Corresponding to the case that majority of people do not spread virus but some people are super spreaders. C (α , p0)≡Γ(α p0)Γ(α(1− p0)) Γ(α+n) Γ(α) ( y+1) α p0−1 (n− y+1) α(1− p0)−1 <C (α , p0) p(Y = y)< y α p0−1 (n− y) α(1− p0)−1 p(Y = y)∝ { y −(1−α p0) ,if y≪n (n−y) −(1−α(1− p0)) ,if y≃n 1−α p0>0
- 14. Ref1. More Accurate Model of p(Y) ● Less value of α leads more catastrophe Rare but very-high- prob. People when α=1 P.d.f. & c.d.f. When avg. infection prob. p0 = 10%
- 15. Ref2. Equivalence with Forced Selling of Options ● Decomposition of convex power function as an interpolation of piecewise linear functions ● So the cost estimation is regarded as option pricing under power-law with many strike prices – Unfortunately we cannot buy this option portfolio. We are forced to be only in the seller side. = + + +...