Some tricks for estimating space-time point process models. Rick Paik Schoenberg, UCLA
|
|
- Gyles Griffith
- 5 years ago
- Views:
Transcription
1 Some tricks for estimating space-time point process models. Rick Paik Schoenberg, UCLA 1. Nonparametric estimation. 2. Parametric estimation of clustering. 3. Separable estimation for covariates. 1
2 1. Nonparametric estimation of clustering. Estimate the rate λ(t,x,y,m) = µ(x,y,m) + g(t-t i, x-x i, y-y i, m i ). i:ti<t from Marsan and Lengliné (2008). 2
3 1. Nonparametric estimation. Marsan, D. and Lengliné, O. (2008). Extending earthquakes' reach through cascading. Science 319,
4 λ(t,x,y,m) = µ(x,y,m) + g(t-t i, x-x i, y-y i, m i ). The idea in Marsan and Lengliné (2008) is to use a kind of E-M algorithm: 0. Start with some naive estimate of µ and g. 1. Using this, compute w ij = P(eq i triggered earthquake j), and i:ti<t w 0j = P(earthquake j was a mainshock). 2. Using w ij and w 0j, estimate µ and g by MLE. Repeat steps 1 and 2 until convergence. With µ a constant, the estimate of g in step 2 is simply g(δt, Δr, m) = A w ij / [N m S t], where A is the set of pairs of points whose distance ~ Δt, Δr, m i ~ m, i.e. t i t j in (Δt - t, Δt + t), etc. N m is the number of eqs with m i ~ m, and S is the total surface covered by the disk with radius Δr +/- r. 4
5 λ(t,x,y,m) = µ(x,y,m) + g(t-ti, x-xi, y-yi, mi). i:ti<t This nonparametric estimate can then be used as a guide to come up with a parametric form for the triggering function, g. Fox et al., 2015.
6 2. Parametric estimation of clustering. The Hawkes process (Hawkes 1971) is a useful form for modeling clusteted processes, where λ(t,x,y,m) = µ(x,y,m) + g(t-t i, x-x i, y-y i, m i ). i:ti<t An example is the Epidemic-Type Aftershock Sequence (ETAS) model of Ogata (1988, 1998), which has been used for earthquakes as well as invasive species (Balderama et al. 2012) and crime (Mohler et al. 2011). With ETAS, µ(x,y,m) = µ(x,y) f(m), and e.g. 6
7 parametric point process estimation is easy?! a. Point processes can generally be very well estimated by maximum likelihood estimation (MLE).!!MLEs are generally consistent, asymptotically normal, and efficient!!!fisher (1925), Cramer (1946), Ogata (1978), Le Cam (1990).! b. Maximizing a function is fast and easy.! 400 yrs of work on this since Newton (1669). Every computer package has a strong quasi-newton minimization routine, such as optim() in R.!!For instance, suppose the function you want to minimize is!!!!f(a,b) = (a-4.19) 4 + (b-7.23) 4.!!f = function(p) (p[1]-4.19)^4 + (p[2]-7.23)^4!!p = c(1,1) ## initial guess at (a,b)!!est = optim(p,f)$par! c. The likelihood function is fairly simple.! In practice, it s convenient to compute log(likelihood), and minimize that.!!log(likelihood) = log{λ(t i,x i,y i )} - λ(t,x,y) dt dx dy.! ETAS (Ogata 98), λ(t,x,y) = µ ρ(x,y) + g(t-t j, x-x j, y-y j; M j ),! where g(t,x,y,m) = K (t+c) -p e a(m-m0) (x 2 + y 2 + d) -q! or K (t+c) -p {(x 2 + y 2 )e -a(m-m0) + d} -q.! 7
8 Parametric point process estimation is easy?! ETAS (Ogata 98), λ(t,x,y) = µ ρ(x,y) + g(t-t j, x-x j, y-y j; M j )! where g(t,x,y,m) = K (t+c) -p e a(m-m0) (x 2 + y 2 + d) -q! or K (t+c) -p {(x 2 + y 2 )e -a(m-m0) + d} -q.! To make these spatial and temporal parts of g densities, I suggest writing g as!!g(t,x,y,m) = {K (p-1) c p-1 (q-1) d q-1 / π} (t+c) -p e a(m-m0) (x 2 + y 2 + d) -q! or!!g(t,x,y,m) = {K (p-1) c p-1 (q-1) d q-1 / π} (t+c) -p {(x 2 + y 2 )e -a(m-m0) + d} -q.! Two reasons for this:!!1) It is easy to see how to replace g by another density.!!2) For each earthquake (t j,x j,y j,m j ),!!! g(t-t j, x-x j, y-y j; M j ) dt dx dy = K e a(m-m0).! 8
9 Actually, parametric point process estimation is hard!!!!!!!!! Main obstacles:!!a. Small problems with optimization routines.!!!extreme flatness, local maxima, choosing a starting value.!!b. The integral term in the log likelihood.!!!log(likelihood) = log{λ(t i,x i,y i )} - λ(t,x,y) dt dx dy.!!!!!the sum is easy, but the integral term is extremely difficult to compute.!!by far the hardest part of estimation (Ogata 1998, Harte 2010).! Ogata 98: divide space around each pt into quadrants and integrate over them.! PtProcess, Harte 10: uses optim() or nlm(). User must calculate the somehow.! Numerical approximation is slow and is a poor approx. for some values of Θ.! 1,000 computations x 1,000 events x 1,000 function calls = 1 billion computations.!!problem B. contributes to reluctance to repeat estimation and check on A.! 9
10 Suggestions (Schoenberg 2013). 1. Write the triggering function as a density in time and space. 2. Approximate the integral term as K e a(mj- M0). 3. Given data from time 0 to time T, estimate ETAS progressively, using data til time T/100, then 2T/100,..., and assess convergence. 10
11 What is behind this integral approximation?! Recall that log(likelihood) = log{λ(t i,x i,y i )} - λ(t,x,y) dt dx dy,!!! where λ(t,x,y) = µ ρ(x,y) + g(t-t j, x-x j, y-y j; M j ).! After writing g as a density,!!! g(t-t j, x-x j, y-y j; M j ) dt dx dy = K e a(m-m0),! if the integral were over infinite time and space.! Technically, in evaluating λ(t,x,y) dt dx dy, the integral is just over time [0,T] and in some spatial region S.! I suggest ignoring this and approximating g(t-t j, x-x j, y-y j; M j ) dt dx dy ~ K e a(m-m0).! Thus, λ(t,x,y) dt dx dy = µt + g(t-t j, x-x j, y-y j; M j ) dt dx dy!!!!!! = µt + g(t-t j, x-x j, y-y j; M j ) dt dx dy!!!!! ~ µt + K e a(mj-m0),! which doesn t depend on c,p,q,d, and is extremely easy to compute.! 11
12 Simulation and example.! 100 ETAS processes, estimated my way.! 12
13 Simulation and example.!!% error in µ, from 100 ETAS simulations.! 13
14 Simulation and example.! CA earthquakes from 14 months after Hector Mine, M 3, from SCSN / SCEDC, as discussed in Ogata, Y., Jones, L. M. and Toda, S. (2003). 14
15 Simulation and example.! Hector Mine data. Convergence of ETAS parameters is evident after ~ 200 days.! Note that this repeated estimation over different time windows is easy if one uses the integral trick. This seems to lead to more stability in the estimates, since a little local maximum is less likely to appear repeatedly in all these estimations.! 15
16 Simulation and example.! Hector Mine data. ETAS parameter estimates as ratios of final estimates.! 16
17 Parametric point process estimation. Suggestions (Schoenberg 2013). 1. Write the triggering function as a density in time and space. 2. Approximate the integral term as K e a(mj- M0). 3. Given data from time 0 to time T, estimate ETAS progressively, using data til time T/100, then 2T/100,..., and assess convergence. It s fast, easy, and works great. Huge reduction in run time. Enormous reduction in programming. 17
18 3. Separable Point Process estimation for covariates. for example, when modeling wildfire incidence as a function of fuel moisture, precipitation, temperature, windspeed,...
19 Burning Index (BI) NFDRS: Spread Component (SC) and Energy Release Component (ERC), each based on dozens of equations. BI = [10.96 x SC x ERC] 0.46 Uses daily weather variables, drought index, and vegetation info. Human interactions excluded. Predicts: flame length, area/fire? # of fires? # of fires? Total burn area?
20 Some BI equations: (From Pyne et al., 1996:) Rate of spread: R = I R ξ (1 + φ w + φ s ) / (ρ b ε Q ig ). Oven-dry bulk density: ρ b = w 0 /δ. Reaction Intensity: I R = Γ w n h η M η s. Effective heating number: ε = exp(-138/σ). Optimum reaction velocity: Γ = Γ max (β / β op ) A exp[a(1- β / β op )]. Maximum reaction velocity: Γ max = σ 1.5 ( σ 1.5 ) -1. Optimum packing ratios: β op = σ A = 133 σ Moisture damping coef.: η M = M f /M x (M f /M x ) (M f /M x ) 3. Mineral damping coef.: η s = S e (max = 1.0). Propagating flux ratio: ξ = ( σ) -1 exp[( σ 0.5 )(β + 0.1)]. Wind factors: σ w = CU B (β/β op ) -E. C = 7.47 exp( σ 0.55 ). B = σ E = exp(-3.59 x 10-4 σ). Net fuel loading: w n = w 0 (1 - S T ). Heat of preignition: Q ig = M f. Slope factor: φ s = β -0.3 (tan φ) 2. Packing ratio: β = ρ b / ρ p.
21 Good news for BI: BI is positively associated with wildfire occurrence. Positive correlations with number of fires, daily area burned, and area per fire. Properly emphasizes windspeed, relative humidity.
22 Some problems with BI Correlations are low.! Corr(BI, area burned) = 0.09! Corr(BI, # of fires) = 0.13! Corr(BI, area per fire) = 0.076! Corr(date, area burned) = 0.06! Corr(windspeed, area burned) = Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)
23
24
25
26 Separable Estimation for Point Processes Consider λ(t, x 1,, x k ; θ). [For fires, x 1 =location, x 2 = area.]
27
28
29
30
31 Separable Estimation for Point Processes Consider λ(t, x 1,, x k ; θ). [For fires, x 1 =location, x 2 = area.] Say λ is multiplicative in mark x j if λ(t, x 1,, x k ; θ) = θ 0 λ j (t, x j ; θ j ) λ -j (t, x -j ; θ -j ), where x -j = (x 1,,x j-1, x j+1,,x k ), same for θ -j and λ -j If λ is multiplicative in x ~ j and if one of these holds, then ^ θ j, the partial MLE, is consistent (Schoenberg 2015): S λ -j (t, x -j ; θ -j ) dµ -j = γ, for all θ -j. S λ j (t, x j ; θ j ) dµ j = γ, for all θ j. ^ ~ S λ j (t, x; θ) dµ = S λ j (t, x j ; θ j ) dµ j = γ, for all θ.
32 Individual Covariates: Suppose λ is multiplicative, λ j (t,x j ; θ j ) = f 1 [X(t,x j ); β 1 ] f 2 [Y(t,x j ); β 2 ], X and Y are independent, and the log-likelihood is differentiable w.r.t. β 1, then the partial MLE of β 1 is consistent. Suppose λ is multiplicative and the jth component is additive: λ j (t,x j ; θ j ) = f 1 [X(t,x j ); β 1 ] + f 2 [Y(t,x j ); β 2 ]. If f 2 is small S f 2 (Y; β 2 ) 2 / f 1 (X; ~ β 1 ) dµ / Τ -> p 0], then the partial MLE β 1 is consistent.
33 Impact Model building. Model evaluation / dimension reduction. Excluded variables.
34 (sq m) r = 0.16
35
36 (sq m) (F)
37
38 Model Construction Wildfire incidence seems roughly multiplicative. (only marginally significant in separability test) Windspeed. RH, Temp, Precip. Tapered Pareto size distribution f, smooth spatial background µ. [*] λ(t,x,a) = f(a) µ(x) β 1 exp(β 2 RH + β 3 WS) (β 4 + β 5 Temp)(max{β 6 - β 7 Prec,β 8 }) Relative AICs (Poisson - Model, so higher is better): Poisson RH BI Model [*]
39 Comparison of Predictive Efficacy False alarms per year % of fires correctly alarmed BI 150: Model [*]: BI 200: Model [*]:
40 Earthquake weather example, Schoenberg model with temp model without temp
41 Conclusions: Start with nonparametric estimates of triggering and bg rate. For a parametric model with clustering, write the triggering function g as a density times K, so you can estimate by MLE, essentially approximating the integral of g as K. Build initial attempts at point process models by examining each covariate individually.
Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.
Stat 13, Intro. to Statistical Methods for the Life and Health Sciences. 1. Review exercises. 2. Statistical analysis of wildfires. 3. Forecasting earthquakes. 4. Global temperature data. 5. Disease epidemics.
More informationStatistical Properties of Marsan-Lengliné Estimates of Triggering Functions for Space-time Marked Point Processes
Statistical Properties of Marsan-Lengliné Estimates of Triggering Functions for Space-time Marked Point Processes Eric W. Fox, Ph.D. Department of Statistics UCLA June 15, 2015 Hawkes-type Point Process
More informationStatistics 222, Spatial Statistics. Outline for the day: 1. Problems and code from last lecture. 2. Likelihood. 3. MLE. 4. Simulation.
Statistics 222, Spatial Statistics. Outline for the day: 1. Problems and code from last lecture. 2. Likelihood. 3. MLE. 4. Simulation. 1. Questions and code from last time. The difference between ETAS
More informationUCLA UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Non Parametric Estimation of Inhibition for Point Process Data Permalink https://escholarship.org/uc/item/7p18q7d1 Author Beyor, Alexa Lake Publication
More informationA recursive point process model for infectious diseases. Harrigan, Ryan. Hoffmann, Marc. Schoenberg, Frederic P.
A recursive point process model for infectious diseases. Harrigan, Ryan. Hoffmann, Marc. Schoenberg, Frederic P. Department of Statistics, University of California, Los Angeles, CA 995 1554, USA. phone:
More informationResearch Article. J. Molyneux*, J. S. Gordon, F. P. Schoenberg
Assessing the predictive accuracy of earthquake strike angle estimates using non-parametric Hawkes processes Research Article J. Molyneux*, J. S. Gordon, F. P. Schoenberg Department of Statistics, University
More informationThreshold estimation in marginal modelling of spatially-dependent non-stationary extremes
Threshold estimation in marginal modelling of spatially-dependent non-stationary extremes Philip Jonathan Shell Technology Centre Thornton, Chester philip.jonathan@shell.com Paul Northrop University College
More informationSelf-exciting point process modeling of crime
Self-exciting point process modeling of crime G. O. Mohler M. B. Short P. J. Brantingham F. P. Schoenberg G. E. Tita Abstract Highly clustered event sequences are observed in certain types of crime data,
More informationLecture 2 APPLICATION OF EXREME VALUE THEORY TO CLIMATE CHANGE. Rick Katz
1 Lecture 2 APPLICATION OF EXREME VALUE THEORY TO CLIMATE CHANGE Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home
More informationMeasurement Error and Linear Regression of Astronomical Data. Brandon Kelly Penn State Summer School in Astrostatistics, June 2007
Measurement Error and Linear Regression of Astronomical Data Brandon Kelly Penn State Summer School in Astrostatistics, June 2007 Classical Regression Model Collect n data points, denote i th pair as (η
More informationClimate Change and Arizona s Rangelands: Management Challenges and Opportunities
Climate Change and Arizona s Rangelands: Management Challenges and Opportunities Mike Crimmins Climate Science Extension Specialist Dept. of Soil, Water, & Env. Science & Arizona Cooperative Extension
More informationSelf-exciting point process modeling of crime
Self-exciting point process modeling of crime G. O. Mohler M. B. Short P. J. Brantingham F. P. Schoenberg G. E. Tita Abstract Highly clustered event sequences are observed in certain types of crime data,
More informationGeneralized additive modelling of hydrological sample extremes
Generalized additive modelling of hydrological sample extremes Valérie Chavez-Demoulin 1 Joint work with A.C. Davison (EPFL) and Marius Hofert (ETHZ) 1 Faculty of Business and Economics, University of
More informationEVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH. Rick Katz
1 EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home page: www.isse.ucar.edu/staff/katz/
More informationTemporal point processes: the conditional intensity function
Temporal point processes: the conditional intensity function Jakob Gulddahl Rasmussen December 21, 2009 Contents 1 Introduction 2 2 Evolutionary point processes 2 2.1 Evolutionarity..............................
More informationTemporal and spatial variations in radiation and energy fluxes across Lake Taihu
Temporal and spatial variations in radiation and energy fluxes across Lake Taihu Wang Wei YNCenter Video Conference May 10, 2012 Outline 1. Motivation 2. Hypothesis 3. Methodology 4. Preliminary results
More informationSummary statistics for inhomogeneous spatio-temporal marked point patterns
Summary statistics for inhomogeneous spatio-temporal marked point patterns Marie-Colette van Lieshout CWI Amsterdam The Netherlands Joint work with Ottmar Cronie Summary statistics for inhomogeneous spatio-temporal
More informationGEOG 402. Soil Temperature and Soil Heat Conduction. Summit of Haleakalā. Surface Temperature. 20 Soil Temperature at 5.0 cm.
GEOG 40 Soil Temperature and Soil Heat Conduction 35 30 5 Summit of Haleakalā Surface Temperature Soil Temperature at.5 cm 0 Soil Temperature at 5.0 cm 5 0 Air Temp 5 0 0:00 3:00 6:00 9:00 :00 5:00 8:00
More informationA recursive point process model for infectious diseases. Schoenberg, Frederic P. 1. Hoffmann, Marc. 2. Harrigan, Ryan. 3. phone:
A recursive point process model for infectious diseases. Schoenberg, Frederic P. 1 Hoffmann, Marc. 2 Harrigan, Ryan. 3 1 Department of Statistics, University of California, Los Angeles, CA 995 1554, USA.
More informationAnalytic computation of nonparametric Marsan-Lengliné. estimates for Hawkes point processes.
Analytic computation of nonparametric Marsan-Lengliné estimates for Hawkes point processes. Frederic Paik Schoenberg 1, Joshua Seth Gordon 1, and Ryan J. Harrigan 2. Abstract. In 2008, Marsan and Lengliné
More informationAn Epidemic Type Aftershock Sequence (ETAS) model on conflict data in Africa
ERASMUS UNIVERSITY ROTTERDAM Erasmus School of Economics. Master Thesis Econometrics & Management Science: Econometrics An Epidemic Type Aftershock Sequence (ETAS) model on conflict data in Africa Author:
More informationSTATISTICAL METHODS FOR RELATING TEMPERATURE EXTREMES TO LARGE-SCALE METEOROLOGICAL PATTERNS. Rick Katz
1 STATISTICAL METHODS FOR RELATING TEMPERATURE EXTREMES TO LARGE-SCALE METEOROLOGICAL PATTERNS Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder,
More informationApplication of branching point process models to the study of invasive red banana plants in Costa Rica
Application of branching point process models to the study of invasive red banana plants in Costa Rica Earvin Balderama Department of Statistics University of California Los Angeles, CA 90095 Frederic
More informationAnalytic computation of nonparametric Marsan-Lengliné. estimates for Hawkes point processes. phone:
Analytic computation of nonparametric Marsan-Lengliné estimates for Hawkes point processes. Frederic Paik Schoenberg 1, Joshua Seth Gordon 1, and Ryan Harrigan 2. 1 Department of Statistics, University
More informationChapter 4 Water Vapor
Chapter 4 Water Vapor Chapter overview: Phases of water Vapor pressure at saturation Moisture variables o Mixing ratio, specific humidity, relative humidity, dew point temperature o Absolute vs. relative
More informationSPC Fire Weather Forecast Criteria
SPC Fire Weather Forecast Criteria Critical for temperature, wind, and relative humidity: - Sustained winds 20 mph or greater (15 mph Florida) - Minimum relative humidity at or below regional thresholds
More informationTheme V - Models and Techniques for Analyzing Seismicity
Theme V - Models and Techniques for Analyzing Seismicity Stochastic simulation of earthquake catalogs Jiancang Zhuang 1 Sarah Touati 2 1. Institute of Statistical Mathematics 2. School of GeoSciences,
More informationEstimating the Exponential Growth Rate and R 0
Junling Ma Department of Mathematics and Statistics, University of Victoria May 23, 2012 Introduction Daily pneumonia and influenza (P&I) deaths of 1918 pandemic influenza in Philadelphia. 900 800 700
More informationDisseminating Fire Weather/Fire Danger Forecasts through a Web GIS. Andrew Wilson Riverside Fire Lab USDA Forest Service
Disseminating Fire Weather/Fire Danger Forecasts through a Web GIS Andrew Wilson Riverside Fire Lab USDA Forest Service Hawaii Fire Danger System Supporters Hawaii Department of Forestry & Wildlife Pacific
More informationCox regression: Estimation
Cox regression: Estimation Patrick Breheny October 27 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/19 Introduction The Cox Partial Likelihood In our last lecture, we introduced the Cox partial
More informationSelf-Exciting Point Process Modeling of Crime
Self-Exciting Point Process Modeling of Crime G. O. MOHLER, M. B. SHORT, P.J.BRANTINGHAM, F.P.SCHOENBERG, andg.e.tita Highly clustered event sequences are observed in certain types of crime data, such
More informationLinear model A linear model assumes Y X N(µ(X),σ 2 I), And IE(Y X) = µ(x) = X β, 2/52
Statistics for Applications Chapter 10: Generalized Linear Models (GLMs) 1/52 Linear model A linear model assumes Y X N(µ(X),σ 2 I), And IE(Y X) = µ(x) = X β, 2/52 Components of a linear model The two
More informationPh.D. Qualifying Exam Friday Saturday, January 6 7, 2017
Ph.D. Qualifying Exam Friday Saturday, January 6 7, 2017 Put your solution to each problem on a separate sheet of paper. Problem 1. (5106) Let X 1, X 2,, X n be a sequence of i.i.d. observations from a
More informationAn Assessment of Crime Forecasting Models
An Assessment of Crime Forecasting Models FCSM Research and Policy Conference Washington DC, March 9, 2018 HAUTAHI KINGI, CHRIS ZHANG, BRUNO GASPERINI, AARON HEUSER, MINH HUYNH, JAMES MOORE Introduction
More informationPoint processes, spatial temporal
Point processes, spatial temporal A spatial temporal point process (also called space time or spatio-temporal point process) is a random collection of points, where each point represents the time and location
More informationHigh-frequency data modelling using Hawkes processes
High-frequency data modelling using Hawkes processes Valérie Chavez-Demoulin 1 joint work J.A McGill 1 Faculty of Business and Economics, University of Lausanne, Switzerland Boulder, April 2016 Boulder,
More informationOn the Goodness-of-Fit Tests for Some Continuous Time Processes
On the Goodness-of-Fit Tests for Some Continuous Time Processes Sergueï Dachian and Yury A. Kutoyants Laboratoire de Mathématiques, Université Blaise Pascal Laboratoire de Statistique et Processus, Université
More informationHigh-frequency data modelling using Hawkes processes
Valérie Chavez-Demoulin joint work with High-frequency A.C. Davison data modelling and using A.J. Hawkes McNeil processes(2005), J.A EVT2013 McGill 1 /(201 High-frequency data modelling using Hawkes processes
More informationStat 710: Mathematical Statistics Lecture 12
Stat 710: Mathematical Statistics Lecture 12 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 12 Feb 18, 2009 1 / 11 Lecture 12:
More information10. Composite Hypothesis Testing. ECE 830, Spring 2014
10. Composite Hypothesis Testing ECE 830, Spring 2014 1 / 25 In many real world problems, it is difficult to precisely specify probability distributions. Our models for data may involve unknown parameters
More informationScaling and trends of hourly precipitation extremes in two different climate zones: Hong Kong and the Netherlands
Scaling and trends of hourly precipitation extremes in two different climate zones: Hong Kong and the Netherlands Why hourly precipitation Long time series Extremes Trends in extremes Scaling Conclusions
More informationLocally stationary Hawkes processes
Locally stationary Hawkes processes François Roue LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay Talk based on Roue, von Sachs, and Sansonnet [2016] 1 / 40 Outline Introduction Non-stationary Hawkes
More informationQuasi-likelihood Scan Statistics for Detection of
for Quasi-likelihood for Division of Biostatistics and Bioinformatics, National Health Research Institutes & Department of Mathematics, National Chung Cheng University 17 December 2011 1 / 25 Outline for
More informationSTAT 730 Chapter 4: Estimation
STAT 730 Chapter 4: Estimation Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Analysis 1 / 23 The likelihood We have iid data, at least initially. Each datum
More informationA Conditional Approach to Modeling Multivariate Extremes
A Approach to ing Multivariate Extremes By Heffernan & Tawn Department of Statistics Purdue University s April 30, 2014 Outline s s Multivariate Extremes s A central aim of multivariate extremes is trying
More informationBayesian Inference for Clustered Extremes
Newcastle University, Newcastle-upon-Tyne, U.K. lee.fawcett@ncl.ac.uk 20th TIES Conference: Bologna, Italy, July 2009 Structure of this talk 1. Motivation and background 2. Review of existing methods Limitations/difficulties
More informationAME 513. " Lecture 8 Premixed flames I: Propagation rates
AME 53 Principles of Combustion " Lecture 8 Premixed flames I: Propagation rates Outline" Rankine-Hugoniot relations Hugoniot curves Rayleigh lines Families of solutions Detonations Chapman-Jouget Others
More informationProblem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv
V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More informationWildland Fire Modelling including Turbulence and Fire spotting
Wildland Fire Modelling including Turbulence and Fire spotting Inderpreet Kaur 1 in collaboration with Andrea Mentrelli 1,2, Frederic Bosseur 3, Jean Baptiste Filippi 3, Gianni Pagnini 1,4 1 BCAM, Bilbao,
More informationCounty-level analysis of the impact of temperature and population increases on California
County-level analysis of the impact of temperature and population increases on California wildfire data. M. Baltar 1, J. E. Keeley 2,3, and F. P. Schoenberg 4 Abstract The extent to which the apparent
More informationImpact of earthquake rupture extensions on parameter estimations of point-process models
1 INTRODUCTION 1 Impact of earthquake rupture extensions on parameter estimations of point-process models S. Hainzl 1, A. Christophersen 2, and B. Enescu 1 1 GeoForschungsZentrum Potsdam, Germany; 2 Swiss
More informationDiagnostics can identify two possible areas of failure of assumptions when fitting linear models.
1 Transformations 1.1 Introduction Diagnostics can identify two possible areas of failure of assumptions when fitting linear models. (i) lack of Normality (ii) heterogeneity of variances It is important
More informationFIRST PAGE PROOFS. Point processes, spatial-temporal. Characterizations. vap020
Q1 Q2 Point processes, spatial-temporal A spatial temporal point process (also called space time or spatio-temporal point process) is a random collection of points, where each point represents the time
More informationComparison of Short-Term and Time-Independent Earthquake Forecast Models for Southern California
Bulletin of the Seismological Society of America, Vol. 96, No. 1, pp. 90 106, February 2006, doi: 10.1785/0120050067 Comparison of Short-Term and Time-Independent Earthquake Forecast Models for Southern
More informationPYROGEOGRAPHY OF THE IBERIAN PENINSULA
PYROGEOGRAPHY OF THE IBERIAN PENINSULA Teresa J. Calado (1), Carlos C. DaCamara (1), Sílvia A. Nunes (1), Sofia L. Ermida (1) and Isabel F. Trigo (1,2) (1) Instituto Dom Luiz, Universidade de Lisboa, Lisboa,
More informationEstimation by direct maximization of the likelihood
CHAPTER 3 Estimation by direct maximization of the likelihood 3.1 Introduction We saw in Equation (2.12) that the likelihood of an HMM is given by ( L T =Pr X (T ) = x (T )) = δp(x 1 )ΓP(x 2 ) ΓP(x T )1,
More informationIntroduction to Reliability Theory (part 2)
Introduction to Reliability Theory (part 2) Frank Coolen UTOPIAE Training School II, Durham University 3 July 2018 (UTOPIAE) Introduction to Reliability Theory 1 / 21 Outline Statistical issues Software
More informationExponential families also behave nicely under conditioning. Specifically, suppose we write η = (η 1, η 2 ) R k R p k so that
1 More examples 1.1 Exponential families under conditioning Exponential families also behave nicely under conditioning. Specifically, suppose we write η = η 1, η 2 R k R p k so that dp η dm 0 = e ηt 1
More informationFireFamilyPlus Version 5.0
FireFamilyPlus Version 5.0 Working with the new 2016 NFDRS model Objectives During this presentation, we will discuss Changes to FireFamilyPlus Data requirements for NFDRS2016 Quality control for data
More informationSome Curiosities Arising in Objective Bayesian Analysis
. Some Curiosities Arising in Objective Bayesian Analysis Jim Berger Duke University Statistical and Applied Mathematical Institute Yale University May 15, 2009 1 Three vignettes related to John s work
More informationSTAT 512 sp 2018 Summary Sheet
STAT 5 sp 08 Summary Sheet Karl B. Gregory Spring 08. Transformations of a random variable Let X be a rv with support X and let g be a function mapping X to Y with inverse mapping g (A = {x X : g(x A}
More informationMachine Learning Practice Page 2 of 2 10/28/13
Machine Learning 10-701 Practice Page 2 of 2 10/28/13 1. True or False Please give an explanation for your answer, this is worth 1 pt/question. (a) (2 points) No classifier can do better than a naive Bayes
More informationj=1 π j = 1. Let X j be the number
THE χ 2 TEST OF SIMPLE AND COMPOSITE HYPOTHESES 1. Multinomial distributions Suppose we have a multinomial (n,π 1,...,π k ) distribution, where π j is the probability of the jth of k possible outcomes
More informationTheory of earthquake recurrence times
Theory of earthquake recurrence times Alex SAICHEV1,2 and Didier SORNETTE1,3 1ETH Zurich, Switzerland 2Mathematical Department, Nizhny Novgorod State University, Russia. 3Institute of Geophysics and Planetary
More information1 A Tutorial on Hawkes Processes
1 A Tutorial on Hawkes Processes for Events in Social Media arxiv:1708.06401v2 [stat.ml] 9 Oct 2017 Marian-Andrei Rizoiu, The Australian National University; Data61, CSIRO Young Lee, Data61, CSIRO; The
More informationExtreme Value Analysis and Spatial Extremes
Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models
More informationA Framework for Daily Spatio-Temporal Stochastic Weather Simulation
A Framework for Daily Spatio-Temporal Stochastic Weather Simulation, Rick Katz, Balaji Rajagopalan Geophysical Statistics Project Institute for Mathematics Applied to Geosciences National Center for Atmospheric
More informationSTOCHASTIC MODELING OF ENVIRONMENTAL TIME SERIES. Richard W. Katz LECTURE 5
STOCHASTIC MODELING OF ENVIRONMENTAL TIME SERIES Richard W Katz LECTURE 5 (1) Hidden Markov Models: Applications (2) Hidden Markov Models: Viterbi Algorithm (3) Non-Homogeneous Hidden Markov Model (1)
More informationCounty-level analysis of the impact of temperature and population increases on California wildfire data
Special Issue Paper Received: 05 February 2013, Revised: 19 October 2013, Accepted: 05 December 2013, Published online in Wiley Online Library: 23 January 2014 (wileyonlinelibrary.com) DOI: 10.1002/env.2257
More informationGaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008
Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:
More informationEXTREMAL MODELS AND ENVIRONMENTAL APPLICATIONS. Rick Katz
1 EXTREMAL MODELS AND ENVIRONMENTAL APPLICATIONS Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home page: www.isse.ucar.edu/hp_rick/
More informationEEL 851: Biometrics. An Overview of Statistical Pattern Recognition EEL 851 1
EEL 851: Biometrics An Overview of Statistical Pattern Recognition EEL 851 1 Outline Introduction Pattern Feature Noise Example Problem Analysis Segmentation Feature Extraction Classification Design Cycle
More informationFORECAST VERIFICATION OF EXTREMES: USE OF EXTREME VALUE THEORY
1 FORECAST VERIFICATION OF EXTREMES: USE OF EXTREME VALUE THEORY Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA Email: rwk@ucar.edu Web
More informationA Bivariate Point Process Model with Application to Social Media User Content Generation
1 / 33 A Bivariate Point Process Model with Application to Social Media User Content Generation Emma Jingfei Zhang ezhang@bus.miami.edu Yongtao Guan yguan@bus.miami.edu Department of Management Science
More informationTEXAS FIREFIGHTER POCKET CARDS
TEXAS FIREFIGHTER POCKET CARDS UPDATED: FEBRUARY 2014 Table of Contents Guide to Percentiles and Thresholds... 1 Fire Business... 2 Predictive Service Area Map... 4 Firefighter Pocket Cards Central Texas...
More informationOn Mainshock Focal Mechanisms and the Spatial Distribution of Aftershocks
On Mainshock Focal Mechanisms and the Spatial Distribution of Aftershocks Ka Wong 1 and Frederic Paik Schoenberg 1,2 1 UCLA Department of Statistics 8125 Math-Science Building, Los Angeles, CA 90095 1554,
More informationAlgorithms for Constrained Optimization
1 / 42 Algorithms for Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University April 19, 2015 2 / 42 Outline 1. Convergence 2. Sequential quadratic
More informationFire Weather Monitoring and Predictability in the Southeast
Fire Weather Monitoring and Predictability in the Southeast Corey Davis October 9, 2014 Photo: Pains Bay fire in 2011 (courtesy Donnie Harris, NCFWS) Outline Fire risk monitoring Fire risk climatology
More informationsimple if it completely specifies the density of x
3. Hypothesis Testing Pure significance tests Data x = (x 1,..., x n ) from f(x, θ) Hypothesis H 0 : restricts f(x, θ) Are the data consistent with H 0? H 0 is called the null hypothesis simple if it completely
More informationChapter 11: WinTDR Algorithms
Chapter 11: WinTDR Algorithms This chapter discusses the algorithms WinTDR uses to analyze waveforms including: Bulk Dielectric Constant; Soil Water Content; Electrical Conductivity; Calibrations for probe
More informationACTEX CAS EXAM 3 STUDY GUIDE FOR MATHEMATICAL STATISTICS
ACTEX CAS EXAM 3 STUDY GUIDE FOR MATHEMATICAL STATISTICS TABLE OF CONTENTS INTRODUCTORY NOTE NOTES AND PROBLEM SETS Section 1 - Point Estimation 1 Problem Set 1 15 Section 2 - Confidence Intervals and
More informationModelling geoadditive survival data
Modelling geoadditive survival data Thomas Kneib & Ludwig Fahrmeir Department of Statistics, Ludwig-Maximilians-University Munich 1. Leukemia survival data 2. Structured hazard regression 3. Mixed model
More informationFOREST FIRE HAZARD MODEL DEFINITION FOR LOCAL LAND USE (TUSCANY REGION)
FOREST FIRE HAZARD MODEL DEFINITION FOR LOCAL LAND USE (TUSCANY REGION) C. Conese 3, L. Bonora 1, M. Romani 1, E. Checcacci 1 and E. Tesi 2 1 National Research Council - Institute of Biometeorology (CNR-
More information12 - Nonparametric Density Estimation
ST 697 Fall 2017 1/49 12 - Nonparametric Density Estimation ST 697 Fall 2017 University of Alabama Density Review ST 697 Fall 2017 2/49 Continuous Random Variables ST 697 Fall 2017 3/49 1.0 0.8 F(x) 0.6
More informationK-means. Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University. November 19 th, Carlos Guestrin 1
EM Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University November 19 th, 2007 2005-2007 Carlos Guestrin 1 K-means 1. Ask user how many clusters they d like. e.g. k=5 2. Randomly guess
More informationPOLI 8501 Introduction to Maximum Likelihood Estimation
POLI 8501 Introduction to Maximum Likelihood Estimation Maximum Likelihood Intuition Consider a model that looks like this: Y i N(µ, σ 2 ) So: E(Y ) = µ V ar(y ) = σ 2 Suppose you have some data on Y,
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationModels for Multivariate Panel Count Data
Semiparametric Models for Multivariate Panel Count Data KyungMann Kim University of Wisconsin-Madison kmkim@biostat.wisc.edu 2 April 2015 Outline 1 Introduction 2 3 4 Panel Count Data Motivation Previous
More informationEEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as
L30-1 EEL 5544 Noise in Linear Systems Lecture 30 OTHER TRANSFORMS For a continuous, nonnegative RV X, the Laplace transform of X is X (s) = E [ e sx] = 0 f X (x)e sx dx. For a nonnegative RV, the Laplace
More informationNational Wildland Significant Fire Potential Outlook
National Wildland Significant Fire Potential Outlook National Interagency Fire Center Predictive Services Issued: September, 2007 Wildland Fire Outlook September through December 2007 Significant fire
More informationLimitations of Earthquake Triggering Models*
Limitations of Earthquake Triggering Models* Peter Shearer IGPP/SIO/U.C. San Diego September 16, 2009 Earthquake Research Institute * in Southern California Why do earthquakes cluster in time and space?
More informationIntroduction to gradient descent
6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our
More informationIntroduction to Maximum Likelihood Estimation
Introduction to Maximum Likelihood Estimation Eric Zivot July 26, 2012 The Likelihood Function Let 1 be an iid sample with pdf ( ; ) where is a ( 1) vector of parameters that characterize ( ; ) Example:
More informationMULTI-AGENCY COORDINATION SYSTEM PUBLICATION CALIFORNIA FIRE WEATHER PROGRAM RISK PREPAREDNESS GUIDE MACS 410-3
MULTI-AGENCY COORDINATION SYSTEM PUBLICATION CALIFORNIA FIRE WEATHER PROGRAM RISK PREPAREDNESS GUIDE MACS 410-3 August 26, 2013 California Fire Weather Program Risk Preparedness Guide Table of Contents
More informationWeather generators for studying climate change
Weather generators for studying climate change Assessing climate impacts Generating Weather (WGEN) Conditional models for precip Douglas Nychka, Sarah Streett Geophysical Statistics Project, National Center
More informationExtreme Precipitation: An Application Modeling N-Year Return Levels at the Station Level
Extreme Precipitation: An Application Modeling N-Year Return Levels at the Station Level Presented by: Elizabeth Shamseldin Joint work with: Richard Smith, Doug Nychka, Steve Sain, Dan Cooley Statistics
More informationCS Lecture 19. Exponential Families & Expectation Propagation
CS 6347 Lecture 19 Exponential Families & Expectation Propagation Discrete State Spaces We have been focusing on the case of MRFs over discrete state spaces Probability distributions over discrete spaces
More informationIs the superposition of many random spike trains a Poisson process?
Is the superposition of many random spike trains a Poisson process? Benjamin Lindner Max-Planck-Institut für Physik komplexer Systeme, Dresden Reference: Phys. Rev. E 73, 2291 (26) Outline Point processes
More informationA recursive point process model for infectious diseases. Frederic Paik Schoenberg 1. Marc Hoffmann 2. Ryan J. Harrigan 3. phone:
A recursive point process model for infectious diseases. Frederic Paik Schoenberg 1 Marc Hoffmann 2 Ryan J. Harrigan 3. 1 Corresponding author. Department of Statistics, University of California, Los Angeles,
More information