Adjustment Computations
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Transcript of Adjustment Computations
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Introduction to Introduction to Adjustment Computations Adjustment Computations
&&Theory of ErrorsTheory of Errors
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Introduction to Adjustment Computations & Theory of Errors
Introduction
Theory of Errors: To understand, classify and minimize the Errors
Adjustment computations: To adjust the data for Parameter Estimation
Statistical Analysis & Testing: To analyze & validate the results
Importance of Theory of Errors & Statistics in Engg:
- Quantitative Modeling, Analysis & Evaluation
- Decisions based on Insufficient, Incomplete and Inaccurate data
Examples:
(i). Dam safety Analysis
(ii). Earth quake Hazard Analysis
(iii). Design of Traffic Intersection
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Fundamental Concepts
True value Parameters: Not known
Error = Observed Value True Value
Correction + Observation = True (corrected) Value
Ex: A length is measured 3 times, with True
(corrected) value: l, and errors e1, e2, e3:
l1 = l + e1l2 = l + e2l3 = l + e3
Aim: To obtain best possible estimate of l and e
Introduction to Adjustment Computations & Theory of Errors
Purpose of Adjustment: Obtain unique estimates of parameters Obtain estimates of accuracy & precision Stat. Analysis & Testing To fit observations to the model
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Conceptual Model
ObservationsA Priori Info
NonNon--LinearLinearLinearLinear
LineariseLinearise
Parameters Precision
Stat. Testing
Introduction to Adjustment Computations & Theory of Errors
Math Model
DataData
AdjustmentsAdjustments
Estimator
Estimates
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Theory of Errors & Applied Statistics
MODEL: Theoretical abstractions to which the measurements refer.
MATHAMATICAL MODEL: A theoretical system or an abstract concept, by which one
can mathematically describe a physical situation or a set of events.
(a) Functional model: Describes deterministic properties of events. It is a completely
fictitious construction, used to describe a set of physical events by an intelligible
system, suitable for Analysis:
(i) Geometric Model (ii) Dynamic Model (iii) Kinematic Model.
(b) Stochastic model: Model which designates and describes the non-deterministic or
probabilistic (stochastic) properties of variables involved.
ACCURACY: Measure of closeness of the observed value to the true value, in
absolute terms.
PRECISION: Measure of repeatability of observations, or internal consistency of
observations.
Introduction to Adjustment Computations & Theory of Errors
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RELATIVE ACCURACY: (Error / Measured quantity (true or observed)) - it has no units
ERRORS: (a) Blunders/Gross Errors/Mistakes:- Observational/ recording/ reading
errors, due to carelessness/oversight.
(b) Systematic Errors:- Errors which follow a systematic trend, and can be corrected
through mathematical modeling:
(i) Environmental Errors
(ii) Instrumental Errors
(iii) Personal Errors
(iv) Mathematical model Errors
(c) Random Errors:- Residual errors after removing blunders and systematic errors.
Inherent in most observations, they follow random behavior.
HISTOGRAM: A graphical /empirical description of the variability of experimental
information.
Introduction to Adjustment Computations & Theory of Errors
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MEASURES OF CENTRAL TENDENCY (SAMPLE STATISTICS FOR POSITION MEASURES)
(a) Mean (Average), (for population) or Xm (for sample) = (1/n) Xi :a unique value.(b) Mode: The value corresponding to maximum frequency.
(c) Median: Central value(s).
(d) Range: Largest value Smallest value
(e) Mid-Range : (Maximum value + Minimum value)/2
MEASURES OF DISPERSION (SAMPLE STATISTICS FOR DISPERSION MEASURES)
(a) Mean deviation: (1/n) (Xi Xm)
(b) Sample Variance : Sx2 = (1/( n-1)) ( X i Xm ) 2 (reason for using (n-1): E[Sx2] = x2) (c) Standard Deviation : Sx: Square Root of Variance
(d) Sample Covariance : Sx,y = ( 1/(n-1)) ( Xi Xm ) * ( Yi Y m )(e) Max. Error, Median Error, Mean Error
(f)Corrlation Coefficient: x,y = x, y / x y
Introduction to Adjustment Computations & Theory of Errors
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PROBABILITY: Numerical measure of the likelihood of the occurrence of an event
relative to a set of alternative events. It is a non-negative measure, associated
with every event.
-or-
The limit of the frequency of occurrence of an event, when the event is repeated
a large no. of times. (n )RANDOM VARIABLE: If a stat. event (outcome of a stat. expt.) has several
possible outcomes, we associate with that event a stochastic or random variable
X, which can take on several possible values, with a specific probability
associated with each.
Introduction to Adjustment Computations & Theory of Errors
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RANDOM EVENT: Event for which the relative frequency of occurrence
approaches a stable limit as the no. of observations or repetitions of an
experiment, n, is increased to infinity.
SAMPLE SPACE: The set of all possibilities in a probabilistic problem, where
each of the individual possibilities is a sample point. An event is a subset of
the sample space.
(a) Discrete Sample Spaces: Sample points are individually discrete entities,
and countable.
e.g.-throwing a dice.
(b) Continuous Sample Spaces: Sample points can take infinite no. of values.
e.g.-measuring a distance
Introduction to Adjustment Computations & Theory of Errors
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Covariance Matrix
For Vector
=
n
n
x
xx
X2
1
)1*(
=2
2
2
)*(
1,
2,21,2
,12,11
nn
n
xxx
xxxx
xxxxx
nnX
Covariance matrix
Introduction to Adjustment Computations & Theory of Errors
Ex. For Coordinates of the 3-D position of a point: P (X, Y, Z)Symmetric Matrix, with non-negative diagonal elements
=
ZYX
p
=2
2
2
,,
,
ZZYZX
YYX
X
p
,
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Propagation of Covariance: To estimate variance of Y, knowing var. of X
For
TXY GG **=
y1 = 2 * x1 + 2 * x2 + 2 * x3 + 3 ForEx.
y2 = 3 * x1 - x2 - 5
=3.61.23.11.22.32.13.12.15.4
x
Y = G * X + C
and
and y1, y2yCompute
Introduction to Adjustment Computations & Theory of Errors
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Fundamentals of
Adjustment Computations
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Linear Models(i). Straight line : y = a * x + b
(ii). Triangulation : L * A + L * B + L * C = 1800 +
Fundamentals of Adjustment Computations
)(* SinaSincBCAB =
++==
)()()()( axdxxdfafxf
ax
Non-Linear Models :
(i). Range :
(ii). Triangulation :
Linearization Using Taylors Series:
2
12
2
12
2
1221 )()()( ZZYYXXR ++=
Non-linear terms
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Fundamentals of Adjustment ComputationsFor matrix Y and X, related by : Y = F (x)
Non-linear terms++=
= 0)( 0
XXXFXFY
=
n
n
n
xf
xfn
xf
xf
xf
XF
1
1
2
1
1
1
Thus,
0XX
TXY
XFG
GG
==
=
Ex : Variance of the volume of cuboid, sphere, etc.
= G, = G, JacobianJacobian Matrix/ Design MatrixMatrix/ Design Matrix
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Fundamentals of Adjustment ComputationsWeights & Weighted Means
Weighted Meani
iiP P
lPX =
20
120
=P
12
0 =
nPVV T
xxV ii =
l1,l2..are observations with weights P1,P2
Weight is inversely proportional to Variance
: Variance of unit weight
Weight Matrix :9 For no Correlation : Diagonal 9 For equal weight and no correlation : Identity Matrix, I
A posteriori Variance of unit weight
For residual :
9 n 1 = Degrees of Freedom = No. of Obsns. No. of parameters 9 Number of observations = n
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Fundamentals of Adjustment Computations
222 +=XM ,=
Weights & Weighted Means
Mean Square Errors (MSE) :
Where bias is the true value
Average Error :
eav = 0.7979 *
Probable Error (PE) :
Pe = 0.6745 *
Corresponds to 75 percentile
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Fundamentals of Adjustment ComputationsLeast Squares Estimator
Need of an Estimator :
=
=
=
3
2
1
,3
2
1
,2
1
vvv
Vlll
LXX
X
2iV
02
2
=
xV i0
1
2
=
xVi
Consider a system of 3 linear equations with 2 unknowns
For u unknowns and n observations, Three cases
9 n = u unique solutions9 n < u Indeterminate9 n > u Infinite solutions
For case (iii), additional conditions are required.
The best criteria is : square of residuals is minimum
= min
or,
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Fundamentals of Adjustment Computations
Least Square Estimator is statistically the best estimator, as
Least Squares Estimator
=2ix
9 It is an unbiased estimator, satisfying E[V]=0 Best Linear Unbiased Estimator (B.L.U.E)
9 It is a minimum variance estimator, satisfying min
=LsX Most probable value of X
)( XX Ls =
or Probability
9 It is the unique estimator
9 It is the most probable estimator, i.e.
9 It is to compute stat. parameters of adjustment
= max
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Fundamentals of Adjustment Computations
Methods of Least Square Estimations
(i) Method of Observation Equations
(a) Linear: L = A * X
(b) Non Linear: L = F ( X )
(iii) Method of combination of Observation Equations &
condition equations : (F, X ) = 0
(ii) Method of Condition Equations : F ( L ) = 0
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Fundamentals of Adjustment Computations
(i) Method of Observation Equations
Observations expressed as a function ( linear or non linear ) of parameters
L = F ( X )
)1*()*()1*(*
nunnXAL =
LLV =
Linear Models :
n = observations,
Where u = unknown parameters,
A = coefficient matrix of n * u.
ResidualsDF = n u,
Where = covariance matrix of observation,
PVV T 120 =Observation Equations :
Minimizing Function : or with
12
0
= P
LXAV =
VV T 1=
.
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Fundamentals of Adjustment Computations(i) Method of Observation Equations:
By Minimizing this, i.e. 0=
X
0 0T TA PA X A PL N X U = =
Normal Equations :
Solution :
PLAPAAX TT 1)( =
N is normal matrix : AT * P * A
U is matrix : AT * P * L
We can derive :
1N U=
-
12
0
12
0 )(
== NPAATX
1
2
0
= NX
unPUVT
=2
0
Fundamentals of Adjustment ComputationsEstimate of precision of estimated parameter :
A posteriori
A posteriori variance of unit weight :