Measurement Analysis:

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Lecture 8

• Measurement Analysis
• Probability Distributions,Errors and their
  Analysis
• Least Squares Theories
• Analysis of Least Squares Output
• Advanced Least Squares

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Probability and probability distributions

• Random variable
• Is a numerical outcome of a random phenomenon.
  May be discrete or continuous
• Discrete random variable
• a random variable that can take on a finite
  collection of values.
• Example consider an experiment that consists of
  flipping a coin twice. If H indicates a head and
  T tail the possible outcomes for this experiment
  as (H, H) (H, T), (T, H), (T, T) The
  number of heads (or tails) occurring in this
  experiment can be 0, 1 or 2. Therefore, the
  number of heads (or tails) is a random variable
  in that it can assumes values of 0, 1, or 2.
• Continuous random variable
• a random variable that takes all values in some
  interval of real numbers.
• Example Generating a random number between 0
  1nd 1.

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Probability and probability distributions

• Probability
• The numerical measure of the chance or likelyhood
  that a particular event will occur.
• Probability function
• f(x), gives the probability that a particular
  random variable will assume a particular value.
• Probability distribution
• is a table, graph or mathematical formula that
  shows all possible values of the random variable,
  x, and the associated probability function, f(x).
• Different random variables have different
  probability distributions which have different
  properties.

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Discrete probability distribution
The instructor of a large class gives 15 each of
As and Ds, 30 each of Bs and Cs and 10 Fs.
If a student is selected at random from his
course,his grade on a 4 point scale (A 4) is a
discrete random variable X having the distribution
P(Xgt3) P(X3) P(X 4) .3
.15 .45
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Continuous probability distribution

• The continuous uniform probability distribution
  is used in all situations in which all values of
  the random variable are equally likely eg random
  number generator on a calculator can be used to
  generate random numbers between 0 and 1.
• Each number has an equal probability of being
  generated.
• A plot of the probability density function, f(x)
  looks like
• For each possible outcome the probability density
  function is the same.The probability density
  function, f(x), is the function that represents
  the probability distribution for continuous
  random variables.

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Continuous probability distribution

• The value of the probability density function
  does NOT represent probability.
• The probability that a continuous random variable
  will assume a value between given limits a and b
  is given by the area under the graph of the
  probability density function between a and b.
• Therefore we treat continuous and discrete random
  variables and probability distributions
  differently
• 1. Discrete compute probability of random
  variable taking a particular value. Continuous
  compute probability of random variable taking a
  value in a particular interval.
• 2. The probability of a random variable taking on
  a value within some given interval is given by
  the area under the graph of the probability
  density function.

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Area as a measure of probability
x1
x2
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Describing probability distribution

• central tendency, dispersion, skewness, kurtosis
• mean point at which the density curve would
  balance if it were made out of solid material.

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Physical interpretation of E(x)
f(x)
Consider shape of distribution cut from plate of
constant density k Small section shown will
have weight f(xi )Dx k The moment of this
section about a fulcrum at x m is (xi - m)
f(xi) Dx k The shape will balance at x m ,
when the sum of moments 0
Dx
fxi
x
xi
m
xi - m
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Physical interpretation of E(x)
Sum of moments 0
1
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nth moment of a distribution about its origin is
defined
nth moment of a distribution about its mean is
defined
1st moment of a distribution about its mean is
defined
2nd moment of a distribution about its mean is
defined
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Rectangular Distribution
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Moment generating function
Distribution can also be defined in terms of the
moment generating function of a random variable X
Moments of a distribution can be deduced directly
from the moment generating function
1st moment
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Moment generating function
2nd moment
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Moment generating function for the normal
distribution
The exponent can be written as
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Moment generating function for the normal
distribution

• mean
• 1st moment
• expected value

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Joint Random Variables
bivariate data - each individual in the
population have two variables associated with it.
A pair of jointly recurring random variables
(x,y) has a PDF f(x,y) so that
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Mean of multivariate frequency functions
Concept can be extended into n-dimensional joint
random variable
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Covariance of multivariate frequency functions
If joint random variable (x1,x2) had PDF f
(x1,x2), m1 Ex1 and m2 Ex2. The second
moment about the mean forms the covariance
Hence Cov 0 x1 and x2 are independent variables
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Correlation coefficient
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Example
N 19
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Propagation of Errors

• In many instances in geomatics measurements are
  combined to compute further quantities. Although
  we are aware of the errors which exists in the
  measured quantities (random errors - we assumes
  gross errors have been removed) we require a
  method to determine the resultant error in the
  computed quantity.
• We will consider
• Propagation of means
• Propagation of variances and covariances

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Propagation of means
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Propagation of Variances
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Propagation of Variances
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d1
d2
a1
a2
a3
d3
d4
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Error Analysis

• Recall The steps to be followed in making a
  measurement for use in geomatics are
• Select a mathematical model to simplify physical
  reality
• Follow a measurement procedure
• Transform measurements via the mathematical model
  to the values required, use stochastic model
• eg mathematical model can be linear or multiple
  regression.

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Errors

• All measurements have errors.
• An absolute error ,e, is defined as
• e M - T
• where M is the measured value
• T is the true value (usually unknown)

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Definitions

• accuracy the nearness of the measurement to the
  absolute truth. To have some idea of accuracy
  you must have a good idea of the truth
• eg if you measure the speed of light, you have a
  good idea of the accuracy of your measurement
  because the speed of light is already well
  determined.
• precision the reliability or repeatability of a
  measurement. Precision has no bearing on
  accuracy.
• eg if you measured the distance between 2 points
  10 times and found the reading agreed to a
  standard deviation of 0.1mm you have very precise
  measurements but still no estimate of accuracy.

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Difference between Accuracy and Precision
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Types of Errors

• Gross
• Systematic
• Random

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Gross Errors

• Mistakes or blunders. Often factors of 10, 100,
  1000 out. Therefore, relatively easy to spot eg
  look on a scatterplot.
• Gross errors arise from inattention or
  carelessness of the observer (eg students!) in
  handling instrument, reading scales or booking
  results. Therefore gross errors are usually
  caused by people.
• eg, when measuring the length of a line, booking
  down wrong values or transposing digits.
• Gross errors are detected by making repeat,
  independent observations (eg use different
  observer to take the same measurement).
• Gross errors must be eliminated from a data set
  prior to any statistical analysis.

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Systematic Errors

• Usually caused by defects in equipment. Equipment
  may give constant offset to true measured value
  throughout its lifetime, eg faulty graduation of
  a scale, or may drift giving different erroneous
  measurement with time eg gravity meters.
• The actual systematic error may be small, but
  given certain measurement conditions may
  accumulate.
• eg, Measure a straight line distance along the
  ground with a tape shorter than the total length.
  If the tape is in error e cm and the distance
  measured comprises n tape lengths, the cumulative
  systematic error is n.e cm.

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Reduction of Systematic Errors

• Systematic errors can be removed by calibrating
  equipment against some known control eg EDMs are
  calibrated over known baselines in a controlled
  thermal and atmospheric environment.
• Alternatively, if a functional relationship for
  the systematic is known , a term can be added
  into the mathematical model to reduce the effect
  to an insignificant magnitude
• Eg e I sec, h is the error in the horizontal
  circle reading of a theodolite having a
  collimation error I at an elevation h. If I has
  been found by testing the theodolite an
  appropriate correction could be applied, or
  alternatively use the principle of reversal (ie
  observe on both faces of the theodolite)
• Systematic errors are not random and are not
  independent. Therefore their presence severely
  inhibits meaningful statistical analysis and
  every effort must be made to remove them prior to
  analysis.

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Random Errors

• For a prolonged series of measurements of the
  same object, observations tend to cluster about
  the mean with a symmetric bell-shaped
  distribution
• The deviations of these observations from the
  true observation or population mean are caused
  by random errors. These have the following
  properties
• 1. Small errors are more frequent than large
  ones
• 2. Positive and negative errors are equally
  frequent
• 3. Very large errors do not occur.
• These errors occur by chance, and are independent
  of each other.

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The Normal Distribution

• The normal distribution is the frequency
  distribution for random errors.
• Two parameters define this distribution mean and
  standard deviation
• The mean represents the most probable value of
  the property being measured.
• The standard deviation indicates the spread of
  the measurements or the width of the normal
  curve.

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The Normal Curve

• Curve representing the normal distribution is
  called a normal curve.
• Characteristics
• symmetry bell-shaped, ie same shape on either
  side of the centre point, with the largest
  frequency of values located near the centre
• unimodal mean, median and mode all coincide at
  the same value, ie, the mean is the most
  frequently occuring value (the mode) and lies at
  the point that divide the curve exactly in half
  (the median)
• asymptoticextends from the mean toward infinity
  in both directions