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A. H. El-Shaarawi National Water Research Institute and McMaster University

A. H. El-Shaarawi National Water Research Institute and McMaster University

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## A. H. El-Shaarawi National Water Research Institute and McMaster University

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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**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**What are the Sources of the foundations?**• Concepts and abstraction. • Schematization == Models • Models and reality (deficiency in theory leads to revision of models)**What are the Tools?**• Philosophy “different schools of statistical inference”. • Mathematics. • Science and technology.**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.**Environmental Problem**Hazards Exposure Control • Trend Analysis • Regulations • Improving Sampling Network • Estimation of Loading • Spatial & Temporal Change • Tools for: • Data Acquisition • Analysis & Interpretation • Modeling • Model Assessment • E Canada • H Canada • DFO • INAC • Provincial • EPA • International**Prior Information**Information**Modeling**Data Time Space Seasonal Trend Input-output Net-work Error +Covariates**Measurements**Input System Output Desirable Qualities of Measurements • Effects Related • Easy and Inexpensive • Rapid • Responsive and more Informative (high statistical power)**Sampling Problems**Setting the regulatory limits: Select the indicators; Determine indicators illness association; Select indicators levels That corresponds to acceptable risk level. Designing Sampling Program for Recreational Water (EC, EPA) Sampling Grid for bathing beach water quality**Sampling Designs**• Model based • Design based • Examples of sampling designs • Simple random sampling • Composite sampling • Ranked set sampling**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.**Surface water quality criteria (CFU/100mL) proposed by EPA**for primary contact recreational use 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**Approximate expression for probability of compliance with**the regulations**Ratio of single sample rejection probability to that of the**mean rule (n = 5,10 and 20)**The fish (trout) contamination data:**• Lake Ontario (n = 171); Lake Superior (61) • Measurements (total PCBs in whole fish, age, weight, length, %fat) – fish collected from several locations (representative of thepopulation in the lake because the fish moves allover the lake)**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 λis 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**Modeling: Consider a continuous time systems with a**stochastic perturbations with initial condition x(0) = x0, 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).**The density f (t,x) satisfies the Fokker-Planck equation or**Kolmogorov forward equation Where . When d = 1 this equation simplifies to Multiplying by xnand integrating we obtain the moments equation Clearly dm0/dt = 0 and dm1 /dt = E(b)**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**The Quasi Likelihood Equations**and the variance of**Example is Canadian Ecological Effects Monitoring (EEM)**Program for Pulp Mills • Risk Identification • Risk Assessment • Risk Management