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A. H. El-Shaarawi National Water Research Institute and McMaster University Southern Ontario Statistics, Graduate Student Seminar Days, 2006 McMaster University May 12, 2006

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Outline

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What is statistical science? A coherent system of knowledge that has its own methods and areas of applications. The success of the methods is measured by their universal acceptability and by the breadth of the scope of their applications. Statistics has broad applications (almost to all human activities including science and technology). Environmental problems are complex and subject to many sources of uncertainty and thus statistics will have greater role to play in furthering the understanding of environmental problems. The word “ENVIRONMETRICS” refers in part to Environmental Statistics

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What are the Sources of the foundations? Concepts and abstraction. Schematization == Models Models and reality (deficiency in theory leads to revision of models)

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What are the Tools? Philosophy “different schools of statistical inference”. Mathematics. Science and technology.

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How to become a successful statistician? Continue to upgrade your statistical knowledge. Improve your ability to perform statistical computation. Be knowledgeable in your area of application. Understand the objectives and scope of the problem in which you are involved. Read about the problem and discuss with experts in relevant fields. Learn the art of oral and written communication. The massage of communication is dependent on the interest of to whom the message is intended.

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Environmental Problem Tools for: Data Acquisition Analysis & Interpretation Modeling Model Assessment Trend Analysis Regulations Improving Sampling Network Estimation of Loading Spatial & Temporal Change E Canada H Canada DFO INAC Provincial EPA International Hazards Exposure Control

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Data Acquisition Data Analysis Empirical Models Process Models Information Prior Information

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Modeling Data Time Space Seasonal TrendInput-output Net-work Error +Covariates

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Measurements Input System Output Desirable Qualities of Measurements Effects Related Easy and Inexpensive Rapid Responsive and more Informative (high statistical power)

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Burlington Beach

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Designing Sampling Program for Recreational Water (EC, EPA) Sampling Grid for bathing beach water quality Setting the regulatory limits: Select the indicators; Determine indicators illness association; Select indicators levels That corresponds to acceptable risk level. Sampling Problems

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Sampling Designs Model based Design based Examples of sampling designs 1. Simple random sampling 2. Composite sampling 3. Ranked set sampling

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Composite Sampling

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Efficiency of Composite Sampling

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Efficiency for estimating the mean and variance of the distribution Number of Composite samples = m Number of sub-samples in a single C sample = k Properties of the estimator of Variance: 1. It is an unbiased estimator of regardless of the values taken by k and. The variance of this estimator is given by This expression shows that for:, composite sampling improves the efficiency of as an estimator of regardless of the value of k and in this case the maximum efficiency is obtained for k =1 which corresponds to discrete sampling., the efficiency of composite sampling depends only on m and is completely independent of k., the composite sampling results in higher variance and for fixed m the variance is maximized when k =1. It should be noted that the frequently used models to represent bacterial counts belong to case c above. This implies that the efficiency declines by composite sampling and maximum efficiency occurs when k = 1. Case b corresponds to the normal distribution where the efficiency is completely independent of the number of the discrete samples included in the composite sample.

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Health Survey

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The effects of exposure to contaminated water

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Surface water quality criteria (CFU/100mL) proposed by EPA for primary contact recreational use WaterIndicatorGeometric Mean Single Sample Maximum MarineEnterococci35104 FreshEnterococci E. coli 33 126 61 235 WaterIndicatorGeometric Mean Single Sample Maximum MarineEnterococci35104 FreshEnterococci E. coli 33 126 61 235 Based on not less than 5 samples equally spaced over a 30-day period. The selection of : Indicators Summary statistics, number of samples and the reporting period Control limits

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Approximate expression for probability of compliance with the regulations

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Sample size n=5 and 10 # of simulations =10000

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Ratio of single sample rejection probability to that of the mean rule (n = 5,10 and 20)

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The fish (trout) contamination data: 1. Lake Ontario (n = 171); Lake Superior (61) 2. Measurements (total PCBs in whole fish, age, weight, length, %fat) – fish collected from several locations (representative of the population in the lake because the fish moves allover the lake)

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Let x(t) be a random variable representing the contaminant level in a fish at age t. The expected value of x(t) is frequently represented by the expression where b is the asymptotic accumulation level and λ is the growth parameter. Note that 1 – exp(-λt) is cdf of E(λ ) and so an immediate generalization of this is The expected instantaneous accumulation rate is f(t; λ)/F(t; λ). One possible extension is to use the Weibull cdf

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Modeling: Consider a continuous time systems with a stochastic perturbations with initial condition x(0) = x 0, b(x) is a given function of x and t σ(x) is the amplitude of the perturbation ξ = dw/dt is a white noise assumed to be the time derivative of a Wiener process Examples for σ(x)=0 : 1. b(x) = - λx μ(x) = μ(0) exp(- λt ) (pure decay) 2. b(x) = λ{μ(0) - μ(x)} μ(x) = μ(0) {1- exp(- λt )} Bertalanffy equation When σ(x) > 0, a complete description of the process requires finding the pdf f(t,x) and its moments given f(0,x).

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The density f (t,x) satisfies the Fokker-Planck equation or Kolmogorov forward equation Where. When d = 1 this equation simplifies to Multiplying by x n and integrating we obtain the moments equation Clearly dm 0 /dt = 0 and dm 1 /dt = E(b)

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In the first example with b(x) = -λx and σ(x) = σ, we have In the second example with b(x) = λ{B - μ(x)} and σ(x) = σ, we have

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The Quasi Likelihood Equations and the variance of

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Fraser River (BC)

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Hansard/Red Pass

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Ratio of GEV Distributions

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Example is Canadian Ecological Effects Monitoring (EEM) Program for Pulp Mills Risk Identification Risk Assessment Risk Management

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Objectives of Environmental Effects Monitoring Program: Does effluent cause an effect in the environment? Is effect persistent over time? Does effect warrant correction? What are the causative stressors? From 1992, all new effluent regulations require sites to do EEM. Pulp and Paper Pilot program

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Environmental Effects Monitoring: Canadian Pulp and Paper Industry Structure Data and Objective

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Example of data (daphnia survival and reproduction) No. of neonates produced per replicates and total female adult mortality

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Example of reproduction data (one cycle)

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Some simulation results (MLE) 2510203050 0.100.00060.00040.00020.0001 0.0000 0.500.01900.01080.00580.00300.00200.0012 1.000.10140.04740.02440.01240.00830.0050 2.00-0.24810.10980.05220.03430.0203 3.00-0.88250.29050.12630.08080.0470 n

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Table 2 Skewness MLE has a heavy right tail distribution (skewed to the right)

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Table 3 Kurtosis MLE has heavy tails and sharp central part for kurtosis>0 while tails are lighter and the central part is flatter for kurtosis<0

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UMVU Estimator

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UMVU : Closed form expression for n=2m-1

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UMVU: n even

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Modified Estimator

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Some simulation results (MLE) 2510203050 0.100.00060.00040.00020.0001 0.0000 0.500.01900.01080.00580.00300.00200.0012 1.000.10140.04740.02440.01240.00830.0050 2.00-0.24810.10980.05220.03430.0203 3.00-0.88250.29050.12630.08080.0470 n

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Table 2 Skewness MLE has a heavy right tail distribution (skewed to the right)

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Table 3 Kurtosis MLE has heavy tails and sharp central part for kurtosis>0 while tails are lighter and the central part is flatter for kurtosis<0

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UMVU Estimator

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UMVU : Closed form expression for n=2m-1

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UMVU: n even

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Modified Estimator

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