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Principles of Least Squares

Introduction

- In surveying, we often have geometric constraints

for our measurements - Differential leveling loop closure 0
- Sum of interior angles of a polygon (n-2)180
- Closed traverse Slats Sdeps 0
- Because of measurement errors, these constraints

are generally not met exactly, so an adjustment

should be performed

Random Error Adjustment

- We assume (hope?) that all systematic errors have

been removed so only random error remains - Random error conforms to the laws of probability
- Should adjust the measurements accordingly
- Why?

Definition of a Residual

If M represents the most probable value of a

measured quantity, and zi represents the ith

measurement, then the ith residual, vi

is vi M zi

Fundamental Principle of Least Squares

In order to obtain most probable values (MPVs),

the sum of squares of the residuals must be

minimized. (See book for derivation.) In the

weighted case, the weighted squares of the

residuals must be minimized.

Technically the weighted form shown assumes that

the measurements are independent, but we can

handle the general case involving covariance.

Stochastic Model

- The covariances (including variances) and hence

the weights as well, form the stochastic model - Even an unweighted adjustment assumes that all

observations have equal weight which is also a

stochastic model - The stochastic model is different from the

mathematical model - Stochastic models may be determined through

sample statistics and error propagation, but are

often a priori estimates.

Mathematical Model

- The mathematical model is a set of one or more

equations that define an adjustment condition - Examples are the constraints mentioned earlier
- Models also include collinearity equations in

photogrammetry and the equation of a line in

linear regression - It is important that the model properly

represents reality for example the angles of a

plane triangle should total 180, but if the

triangle is large, spherical excess cause a

systematic error so a more elaborate model is

needed.

Types of ModelsConditional and Parametric

- A conditional model enforces geometric conditions

on the measurements and their residuals - A parametric model expresses equations in terms

of unknowns that were not directly measured, but

relate to the measurements (e.g. a distance

expressed by coordinate inverse) - Parametric models are more commonly used because

it can be difficult to express all of the

conditions in a complicated measurement network

Observation Equations

- Observation equations are written for the

parametric model - One equation is written for each observation
- The equation is generally expressed as a function

of unknown variables (such as coordinates) equals

a measurement plus a residual - We want more measurements than unknowns which

gives a redundant adjustment

Elementary Example

Consider the following three equations involving

two unknowns. If Equations (1) and (2) are

solved, x 1.5 and y 1.5. However, if

Equations (2) and (3) are solved, x 1.3 and y

1.1 and if Equations (1) and (3) are solved, x

1.6 and y 1.4. (1) x y 3.0 (2)

2x y 1.5 (3) x y 0.2 If we

consider the right side terms to be measurements,

they have errors and residual terms must be

included for consistency.

Example - Continued

x y 3.0 v1 2x y 1.5

v2 x y 0.2 v3 To find the MPVs for

x and y we use a least squares solution by

minimizing the sum of squares of residuals.

Example - Continued

To minimize, we take partial derivatives with

respect to each of the variables and set them

equal to zero. Then solve the two equations.

These equations simplify to the following normal

equations. 6x 2y 6.2 -2x 3y

1.3

Example - Continued

Solve by matrix methods.

We should also compute residuals v1 1.514

1.443 3.0 -0.044 v2 2(1.514)

1.443 1.5 0.086 v3 1.514 1.443

0.2 -0.128

Systematic Formation of Normal Equations

Resultant Equations

Following derivation in the book results in

Example Systematic Approach

Now lets try the systematic approach to the

example. (1) x y 3.0 v1 (2) 2x

y 1.5 v2 (3) x y 0.2

v3 Create a table

Note that this yields the same normal equations.

Matrix Method

Matrix form for linear observation

equations AX L V Where

Note m is the number of observations and n is

the number of unknowns. For a redundant solution,

m gt n .

Least Squares Solution

Applying the condition of minimizing the sum of

squared residuals ATAX ATL or NX

ATL Solution is X (ATA)-1ATL N

-1ATL and residuals are computed from V

AX L

Example Matrix Approach

Matrix Form With Weights

Weighted linear observation equations WAX

WL WV Normal equations ATWAX NX

ATWL

Matrix Form Nonlinear System

We use a Taylor series approximation. We will

need the Jacobian matrix and a set of initial

approximations. The observation equations

are JX K V Where J is the Jacobian

matrix (partial derivatives) X contains

corrections for the approximations K has

observed minus computed values V has the

residuals The least squares solution is X

(JTJ)-1JTK N-1JTK

Weighted Form Nonlinear System

The observation equations are WJX WK WV

The least squares solution is X

(JTWJ)-1JTWK N-1JTWK

Example 10.2

Determine the least squares solution for the

following F(x,y) x y 2y2

-4 G(x,y) x2 y2 8 H(x,y)

3x2 y2 7.7 Use x0 2, and y0 2 for

initial approximations.

Example - Continued

Take partial derivatives and form the Jacobian

matrix.

Example - Continued

Form K matrix and set up least squares solution.

Example - Continued

Add the corrections to get new approximations and

repeat. x0 2.00 0.02125 1.97875 y0 2.00

0.00458 2.00458

Add the new corrections to get better

approximations. x0 1.97875 0.00168

1.98043 y0 2.00458 0.01004 2.01462 Further

iterations give negligible corrections so the

final solution is x 1.98 y 2.01

Linear Regression

Fitting x,y data points to a straight line y

mx b

Observation Equations

In matrix form AX L V

Example 10.3

Fit a straight line to the points in the table.

Compute m and b by least squares. In matrix

form

Example - Continued

Standard Deviation of Unit Weight

Where m is the number of observations and n is

the number of unknowns Question What about

x-values? Are they observations?

Fitting a Parabola to a Set of Points

Equation Ax2 Bx C y This is still a

linear problem in terms of the unknowns A, B, and

C. Need more than 3 points for a redundant

solution.

Example - Parabola

Parabola Fit Solution - 1

Set up matrices for observation equations

Parabola Fit Solution - 2

Solve by unweighted least squares solution

Compute residuals

Condition Equations

- Establish all independent, redundant conditions
- Residual terms are treated as unknowns in the

problem - Method is suitable for simple problems where

there is only one condition (e.g. interior angles

of a polygon, horizon closure)

Condition Equation Example

Condition Example - Continued

Condition Example - Continued

Condition Example - Continued

Note that the angle with the smallest standard

deviation has the smallest residual and the

largest SD has the largest residual

Example Using Observation Equations

Observation Example - Continued

Observation Example - Continued

Note that the answer is the same as that obtained

with condition equations.

Simple Method for Angular Closure

Given a set of angles and associated variances

and a misclosure, C, residuals can be computed by

the following

Angular Closure Simple Method

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