A.%20H.%20El-Shaarawi

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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
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

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Prior Information
Information
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Modeling

• Data
• Time Space
• Seasonal Trend Input-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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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
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Sampling Designs

• Model based
• Design based
• Examples of sampling designs
• Simple random sampling
• Composite sampling
• 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
Water Indicator Geometric Mean Single Sample Maximum
Marine Enterococci 35 104
Fresh Enterococci E. coli 33 126 61 235
Water Indicator Geometric Mean Single Sample Maximum
Marine Enterococci 35 104
Fresh Enterococci 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 n5 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
• Lake Ontario (n 171) Lake Superior (61)
• 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 ? is the growth parameter.
Note that 1 exp(-?t) is cdf of E(? ) and so an
immediate generalization of this is The
expected instantaneous accumulation rate is
f(t ?)/F(t ?). 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) x0, b(x) is a
given function of x and t s(x) is the amplitude
of the perturbation ? dw/dt is a white noise
assumed to be the time derivative of a Wiener
process Examples for s(x)0 1. b(x) - ?x
µ(x) µ(0) exp(- ?t ) (pure decay) 2. b(x)
?µ(0) - µ(x) µ(x) µ(0) 1- exp(- ?t )
Bertalanffy equation When s(x) gt 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 xn and integrating we obtain the
moments equation
Clearly dm0/dt 0 and dm1 /dt E(b)
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In the first example with b(x) -?x and s(x)
s, we have
In the second example with b(x) ?B - µ(x)
and s(x) s, 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)
n
2 5 10 20 30 50
0.10 0.0006 0.0004 0.0002 0.0001 0.0001 0.0000
0.50 0.0190 0.0108 0.0058 0.0030 0.0020 0.0012
1.00 0.1014 0.0474 0.0244 0.0124 0.0083 0.0050
2.00 - 0.2481 0.1098 0.0522 0.0343 0.0203
3.00 - 0.8825 0.2905 0.1263 0.0808 0.0470
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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
kurtosisgt0 while tails are lighter and the
central part is flatter for kurtosislt0
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UMVU Estimator
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UMVU Closed form expression for n2m-1
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UMVU n even
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Modified Estimator
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Some simulation results (MLE)
n
2 5 10 20 30 50
0.10 0.0006 0.0004 0.0002 0.0001 0.0001 0.0000
0.50 0.0190 0.0108 0.0058 0.0030 0.0020 0.0012
1.00 0.1014 0.0474 0.0244 0.0124 0.0083 0.0050
2.00 - 0.2481 0.1098 0.0522 0.0343 0.0203
3.00 - 0.8825 0.2905 0.1263 0.0808 0.0470
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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
kurtosisgt0 while tails are lighter and the
central part is flatter for kurtosislt0
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UMVU Estimator
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UMVU Closed form expression for n2m-1
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UMVU n even
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Modified Estimator
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