Adjustment Computations

Introduction to Introduction to Adjustment Computations Adjustment Computations

&&Theory of ErrorsTheory of Errors

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

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


Purpose of Adjustment: Obtain unique estimates of parameters Obtain estimates of accuracy & precision Stat. Analysis & Testing To fit observations to the model

Conceptual Model

ObservationsA Priori Info

NonNon--LinearLinearLinearLinear

LineariseLinearise

Parameters Precision

Stat. Testing


Math Model

DataData

AdjustmentsAdjustments

Estimator

Estimates

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.


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.


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


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.


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


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


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

,

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


Fundamentals of

Adjustment Computations

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

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

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


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

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,


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


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


(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=

.

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 :

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Adjustment Computations