star netsunday - greenfeld (60)

Introduction to Least Squares Adjustments with STAR*NETBy Dr. Joshua Greenfeld Surveying Program Coordinator NJ Institute of Technology.

Intro to LS with Star*Net

(c) Dr. Joshua Greenfeld

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ObjectiveLeast Squares is known to be the most appropriate method for adjusting and analyzing surveying measurements. This statement is correct only if the measurements and their accuracies have been examined and assessed carefully prior to performing the adjustments. STAR*NET is a prevalent Least Squares adjustment software. As with any software, one has to understand what the program does, how it does it and how to evaluate or assess the quality of the results it produces.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 2

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ObjectiveThe objective of the seminar is to provide an in-depth understanding of the theoretical and practical aspect of measurement adjustment with STAR*NET. It will introduce the Least Squares concept, error theory, error estimation terminology and statistical analysis of the results as it pertains to STAR*NET. The seminar will include an overview of the software, problem solving examples and a discussion on how it should be used to obtain honest results. Seminar attendants are encouraged to bring their own laptops loaded with STAR*NET. A demo copy of STAR*NET can be downloaded from http://www-ec.njit.edu/surveying/goodies.htm Intro to LS with Star*Net(c) Dr. Joshua Greenfeld 3

Outline Introduction to Star*Net Program manus and options Data input issues What How Why

Understanding the output report Results Quality parameters or indicators Statistics Analysis(c) Dr. Joshua Greenfeld 4


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STAR*NET The STAR*NET program is a general purpose, rigorous least squares analysis program designed to adjust 2D and 3D survey networks. Its 2D inputs include horizontal angles, horizontal distances, directions, azimuths or bearings, and station coordinates. In a 3D adjustment, slope distances, zenith angles, elevation differences and station elevations can also be input. In addition, GPS vectors can be input together with traditional surveying measurements.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 5

STAR*NETThe output consists of: A file of adjusted station coordinates and a statistical analysis of the adjustment. Graphical facilities are provided to allow the user to plot the network, including error ellipses of the adjusted points and relative error ellipses between stations.



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STAR*NET One of the main features of STAR*NET is the capacity to weight all input data both independently or by category. This means that data can be defined as being FREE or FIXED and anything in between. Giving accurately known measurements more weight than those measurements known to be less accurate is a fundamental principle on which Least Squares is based. The ability to control the weighting of data provides the user with a powerful adjustment tool.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 7

STAR*NET - Menus

o ott P Pl ott or or Pllo ellp lle He or Fii e o xt Fiille ) ) tt F Te on on a T ex ex uttt ta bu on) iint a T a o b atiion) ott !! a P e Pr w Pll d p ew r P Fille ed rropag F vie ork re p o re ar ng y a e rrorr Pr ettw iistii by tes er b c s Ne Ls d dicat tt L w ew tpu catte ind s pu e V Vii e es ica (e O Ou iind t ( Fiill F a ew rs m n ata ew rs ( me V o Vi D D rro ust Er d u putt t E djj npu n L s an Liis an A diitt II s Ed ns n on ct or tiio Ru Ru e o Opt rojje tte o Op P ea c P ea ctt C Cr oe ojje tiing ctt Pr ec xiis ojje tt P Ex Pro Se Se an E Pr a n ew w en en N e O ea Op tte a a a re CrIntro to LS with Star*Net (c) Dr. Joshua Greenfeld

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4

STAR*NET menus



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STAR*NET Options

When to apply?



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Reducing Measured Horizontal Distance to EllispoidDh H N

S

Topo grap hy Sea L e Ellips vel - Geo id oid

R

S' D = R R+h

R ) S' = D ( R+h S' = D ( R ) R+H+N11



Reducing Measured Horizontal Distance to Grid DistanceFormula:

S=D(

R ) K12 R+H+N

Where: S- Grid Distance D- Horizontal (Measured) Distance H- Mean Elevation (Above Mean Sea Level) N- Mean Geoid Height (About -32m in NJ) R- Mean Radius of the Earth (About 6,372,000m) K12-Grid Scale factor of the Line.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 12

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STAR*NET Options

When to apply?

Grayed out?Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 13

STAR*NET Options

Grayed out? Why to apply?Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 14

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STAR*NET General Options



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Definition: Given independent variables each with an uncertainty, error propagation is the method of determining an uncertainty in a function of these variables. Error propagation is the evaluation of errors in computed quantities as a function of the errors in our measurements or known errors from previous computations. Examples: Computed errors Measurement or given Errors Ex, Ey Angular and distance Earea CoordinatesIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 16

Error Propagation

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Error Propagation Error propagation is a way of combining two or more random errors together to get a third. It can be used when you need to measure more than one quantity to get at your final result. For example, an angle and a distance to compute coordinates Error propagation can also be used to combine several independent sources of random error on the same measurement.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 17

Examples:(a) Random error of a sum If y = x1 + x2 + x3 +2

...

+ xn2 2

Then y = x1 + x2 + x3 + + xn2

A leveling loop was measured with the following accuracies: H1 = 12.34 0.01 H2 = -8.72 0.02 H3 = 4.93 0.005 H4 = -8.53 0.01Intro to LS with Star*Net

The closure is 0.02 The accuracy is of the loop: 0.012+0.022+0.0052+0.012 =0.02518


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STAR*NET General Options



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Trigonometric Leveling For distances larger than 1000 ft.e Level Lin

Refraction

{

E D Rod (r) V B

m Z Zm C

tC tal a rizon Ho

Level Line G

}

F

Earth Curvature

Level Line Hhi

AIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 20

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LEVELING METHODSTRIGONOMETRIC LEVELING For distances larger than 1000 ft. (Curvature) C = 0.0239 F 2 (Refraction) R = 0.0033 F2 The combined correction is: h = C R = 0.0206 F 2 (F = 1000ft.) = 0.0675 K 2 (K = 1000 Meters) elevb = eleva + S x cosZ + h + HI - HTIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 21

LEVELING METHODSTRIGONOMETRIC LEVELING H > 1000 ft. Example: The slope distance and zenith angle between points A and B were measured with a total station instrument as 9585.26 and 814220. Assume HI = HT. If the elevation of A is 1238.42 ft, what is the elevation of B? V = 9585.26 cos 814220 = 1382.77 ft h = 0.0206 x (9585.26 x sin 814220 / 1000)2 = 1.85 ft elevB = 1238.42 + 1382.77 + 1.85 = 2623.04 ftIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 22

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STAR*NET Instrument Options



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Error Propagation in Distance MeasurementsThe estimated error in an EDM observed Distance is:

D = i2 + t2 + a 2 + ( D b ppm ) 2Where the EDM specifications are: (a + bppm)



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Error Propagation in Distance MeasurementsExample: D = i2 + t2 + a 2 + ( D b ppm ) 2

A distance of 372.69 was measured with a total station. The EDM specifications of the total station is 3mm +2ppm. Assuming that the instrument and target centering accuracy is 0.01 ft., what is the accuracy of the observed Distance is:

D = 0.012 + 0.012 + 0.00962 + (372.69 0.000002)2 = 0.017Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 25

Error Propagation in Angle MeasurementsThe estimated error in an observed angle is:2 2 2 2 = + + + pr i t l

Where: - Error due to pointing and readingpr

i - Error due to Instrument centering t - Error due to Target centeringIntro to LS with Star*Net

l - Error due to Leveling error(c) Dr. Joshua Greenfeld 26

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Error Propagation in Angle MeasurementsDIN 18723 STANDARD: Based on n direction measured with both faces of the instrument.

=pr

EXAMPLE

2 DIN n

An angle measured 6 times by an observer with a total station having a DIN18273 value of 5". What is the estimated error in the angle due to pointing and reading?

=prIntro to LS with Star*Net (c) Dr. Joshua Greenfeld

2 x 5" = 4.1" 6

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Error Propagation in Angle MeasurementsTarget Miscenteringt D d

= tt

2 D12 + D2

D1 D2



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Error Propagation in Angle MeasurementsTarget Miscentering: EXAMPLE Hand held targets are centered over a station to within 0.02 ft. What is the error in the angle due to target centering if the backsight distance is 250 ft and the foresight distance is 150 ft?

= 0.02tIntro to LS with Star*Net

2502 + 1502 250 x150

= 32.1"


Where: = 206265/radians 29

Error Propagation in Angle MeasurementsInstrument Miscenteringi D1 D2 P22 D3 = D12 + D2 2 D1 D2 cos Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 30

P1

D3

= ii

D3 D1 D2 2

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Error Propagation in Angle MeasurementsInstrument Miscentering: EXAMPLE What is the error in a 50 angle due to instrument miscentering, if the setup is within 0.005 ft and the sight distances are 150 and 250 ft?

= 0.005i

191.81 = 3.7" 150 x 250 2(c) Dr. Joshua Greenfeld 31


S

Error Propagation in Angle MeasurementsEffects of MislevelingI D

fd P P

= l

( f d tan(vb ))2 + ( f d tan(v f ))2 n

Where: V is the vertical angle fd is the fractional division of the bubble is the sensitivity of the bubbleIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 32

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Error Propagation in Angle MeasurementsEffects of Misleveling : EXAMPLEHow much error in the horizontal angle can be expected in a sun shot, if the instrument has a bubble with a sensitivity of 30"/div and is leveled to within 0.5 divisions. Assume a backsight zenith angle of 9130'45" and a foresight zenith 91 angle to the sun of 5515'30"? (Assume angle turned 3DR.) 55

= l

[0.5 x 30 x tan(1 30' 45" )]2 + [0.5 x 30 x tan(34 44' 30" )]2 6

=

0.4 2 + 10.4 2 = 4.2" l 6 Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

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Estimated Error in Angle MeasurementsThe estimated error in an observed angle is:2 2 2 2 = + + + pr i t

l

or

= 4.12 + 32.12 + 3.7 2 + 4.22 = 32.8"Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 34

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Estimated Error in Angle MeasurementsExample: running the Trav2D sample7 8 3 2 9 4 5

10

6

1



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Estimated Error in Angle MeasurementsExample: running the Trav2D sample Project Default Instrument Distances (Constant) (ft) Distances (PPM) Angles (Seconds) Directions (Seconds) Azimuths & Bearings (Seconds) Centering Error Instrument (ft) Centering Error Target (ft)Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

Run1 0.02 5.00 3.00 2.00 4.00 0.00 0.00

Run2 0.02 5.00 3.00 2.00 4.00 0.01 0.0136

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Computed Error in Angle MeasurementsAt 1 2 3 4 5 6 3 7 8 9 9 10 6 From 6 1 2 3 4 5 2 3 7 8 8 9 10 To 2 3 4 5 6 1 7 8 9 5 10 6 1 Angle 99-47-25 99- 47115-10-00 115- 1094-51-53 94- 51216-46-09 216- 46106-26-42 106- 2686-57-49 86- 57225-47-02 225- 4797-31-36 97- 31115-14-57 115- 1483-45-28 83- 45156-15-44 156- 15106-12-32 106- 12164-00-42 164- 00(c) Dr. Joshua Greenfeld

StdErr R1 StdErr R2 3.00 17.99 3.00 28.56 3.00 36.03 3.00 38.74 3.00 27.53 3.00 20.45 3.00 38.78 3.00 28.05 3.00 20.31 3.00 23.14 3.00 28.10 3.00 27.97 3.00 25.4537


Computed Error in Distance MeasurementsFrom To 1 2 2 3 3 4 4 5 5 6 6 1 3 7 7 8 8 9 9 5 9 10 10 6Intro to LS with Star*Net

Distance 205.03 134.19 105.44 161.57 160.71 308.30 115.41 284.40 191.66 161.95 166.90 151.34

StdErr R1 0.0210 0.0207 0.0205 0.0208 0.0208 0.0215 0.0206 0.0214 0.0210 0.0208 0.0208 0.0208(c) Dr. Joshua Greenfeld

StdErr R2 0.0253 0.0250 0.0249 0.0252 0.0252 0.0258 0.0250 0.0257 0.0253 0.0252 0.0252 0.025138

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Estimated errors and the final resultsStn 1 2 3 4 5 6 7 8 9 N run1 E run1 N run2 E run2 5045.5720 5495.3380 5250.5510 5500.3384 5310.5600 5380.3163 5220.5812 5325.3307 5160.5781 5175.3272 5000.5703 5190.3308 5420.5707 5345.3097 5370.5741 5065.3303 5185.5816 5015.3310 5045.5720 5495.3380 5250.5473 5500.3122 5310.5450 5380.2827 5220.5632 5325.3146 5160.5303 5175.3141 5000.5271 5190.3420 5420.5317 5345.2803 5370.5367 5065.3095 5185.5248 5015.3120 5020.5119 5040.3234(c) Dr. Joshua Greenfeld

10 5020.5729 5040.3258Intro to LS with Star*Net

N E 0.000 0.000 0.004 0.026 0.015 0.034 0.018 0.016 0.048 0.013 0.043 -0.011 0.039 0.029 0.037 0.021 0.057 0.019 0.061 0.00239

STAR*NET Output Listing



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STAR*NET GPS Options



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GPS Vector WeightingWeighting can be in the form of covariances, (i.e. Trimble and Leica vectors) The G1, G2 and G3 lines contain this information: G1 From-To DX DY DZ (vector components) G2 CvXX CvYY CvZZ (vector covariances) G3 CvXY CvXZ CvYZ (vector covariances) Example:G0 'V1 Day125(1) 14:14 00120013.DAT G1 0012-0013 -507.727507 -5749.936110 -8484.248757 0012G2 6.289353856E-008 2.066252796E-007 7.586683470E-008 6.289353856E2.066252796E7.586683470EG3 7.061187387E-008 -1.869271092E-008 -6.316526460E-008 7.061187387E1.869271092E6.316526460EIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 42

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GPS Vector WeightingWeighting can be in the form of standard errors and correlations, (i.e. Ashtech vectors) The G1, G2 and G3 lines will contain this information: G1 From-To DX DY DZ (vector components) G2 SDX SDY SDZ (vector standard errors) G3 CrXY CrXZ CrYZ (vector correlations) Example:G1 0012-0013 -507.727507 -5749.936110 -8484.248757 0012G2 6.289353856E-08 2.066252796E-07 7.586683470E-08 6.289353856E2.066252796E7.586683470EG3 6.194168205E-01 -2.706096579E-01 -5.044993490E-01 6.194168205E2.706096579E5.044993490EIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 43

What is a Covariance?A covariance is a measure of dependency or correlation between two measurements (or computed parameters).

n 1 Where: sxy - Covariance between x and y x,y - A set of two measurementsIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 44

s xy =

[( x x) ( yi

i

y )]

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Covariance If the covariance is equal to 0, the two variables (i.e. measurements) are independent. If the covariance is not equal to 0, the variables have some degree of dependency between them.

How dependent are these variables?



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Correlation CoefficientThe dependency is measured with a Correlation Coefficient . xy s xy = or = x y sx s y The Correlation Coefficient can have values between 1 and -1 ( -1 1 ) If = 0, The variables are independent If = 1 or (-1), The variable are fully dependent.



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Examplex y vx = vy = xi - xaver yi - yaver vx2 vy2 Vx x v y 174172.55 212161.94 -111.31 -160.20 12390.03 25662.68 17831.47 174397.18 212489.34 113.32 167.21 12841.76 27957.60 18947.95 174171.08 212489.21 -112.78 167.07 12720.12 27912.80 -18842.88 174394.63 212148.06 110.77 -174.08 12270.55 30304.11 -19283.36 174283.86 212322.14 0.00 0.00 50222.45 111837.18 -1346.82

Sx = 129.39 Sy = 193.08 Sxy = -448.94Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

=-0.02

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Examplex y vx = vy = xi - xaver yi - yaver vx2 vy2 Vx x v y

174172.55 224172.55 -150.00 -225.00 22500.00 50625.00 33750.00 174272.55 224322.55 -50.00 -75.00 2500.00 5625.00 3750.00 174372.55 224472.55 50.00 75.00 2500.00 5625.00 3750.00 174472.55 224622.55 150.00 225.00 22500.00 50625.00 33750.00 174322.55 224397.55 0.00 0.00 50000.00 112500.00 75000.00

Sx = 129.10 Sy = 193.65 Sxy = 25000.00Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

= 1.00

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STAR*NET Modeling Options

Calculations on the grid are on the ellipsoid! Calculations on the grid require correction of Angles azimuths for DoV



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Surfaces in Geodesyh=H+NEllip s oid

H hNormal to Ge o id50

Deflection of the verticalIntro to LS with Star*Net (c) Dr. Joshua Greenfeld

Nor mal to

ID GEO SO I D LLIP E

HY AP GR PO TO

OCEAN

N

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DoV applicationNP Computing corrected horizontal direction d: d = D ( sin cos) tan (90 z) - Where: 90- DoV Za 90D is the measured direction Zg 90 is the azimuth of the line is North-South component of the DoV is East-West component of the DoV z is the Zenith angle Computing corrected Zenith angle z: z = Z + ( cos + sin) Where Z is the observed Zenith angle Intro to LS with Star*Net(c) Dr. Joshua Greenfeld 51

STAR*NET Input

Be aware which files are checked?



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9090 -

90-

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Sample of input data# Triangulation/Trilateration 2D Network C 1 5102.502 5793.197 ! ! # Angle and Distance Observations at Perimeter M 3-2-4 87-07-04 1364.40 M 3-2-6 40-58-15 951.92 4.0 .03 # Angles at Center Point A 6-1-5 43-35-22 A 6-2-1 57-28-12 # Distance Cross-ties D 1-4 2070.76 D 2-5 2034.47 # Elevation E 0013 205.450 ! 'BM-9331 # Bearings BIntro to LS B Star*Net -23.7 ! (c) Dr. Joshua Greenfeld A-with 0-06

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STAR*NETData Type Codes

Code# C E P A D V DV B M

MeaningRemainder of line is a comment and is ignored Coordinate values for a station Elevation value for a station Geodetic Position for a station (Grid jobs only) Turned Angle Distance Zenith Angle or Elevation Difference (3D data only) Distance and Vertical (3D Data only) Bearing or Azimuth Measure (Observations to another network point)(c) Dr. Joshua Greenfeld 54


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STAR*NETData Type Codes

CodeM SS TB T TE DB L

MeaningMeasure (Observations to another network point) Sideshot (Observations to a sideshot point) Traverse Begin Traverse (All observations to next network point) Traverse End Begin Direction Set Differential Level Measurement



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Data Code FormatCode Station C Station A At-From-To D From-To B From-To M At-From-To M At-From-To T V At From-To Observations North East Angle Horiz Distance Bearing (or Az) Angle Distance Angle Slope Dist Zenith Angle Distance Zenith(c) Dr. Joshua Greenfeld

[Std Errors] [Std Errors] [Std Error] [Std Error] [Std Error] [Std Errs] [Std Errs] [Std Errs] [Std Error]

[HI/HT]

[HI/HT]

[HI/HT]56


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Standard Error SymbolsEntryNumeric Value Nothing Entered The & Symbol The ! Symbol The * SymbolIntro to LS with Star*Net

ExplanationAn actual Standard Error value (i.e. 0.03) Defaults to the value defined in the instrument options Defaults to the value defined in the instrument options The observation is FIXED The observation is FREE(c) Dr. Joshua Greenfeld 57

STAR*NETFull adjustment of data Compute approximate coordinates from data, calculate observations from inverses and compare them with actual observations Compute the differences between computed and actual values of the observations. Indicates largest differences. Effective mainly for large redundancies.(c) Dr. Joshua Greenfeld 58


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STAR*NET preanalysisAnalyzes the geometric strength of the network using the approximate layout and the instrument accuracies. No actual observations are used.C C C C # D D D D D A A A A A # B 1 2 3 4 51002 51005 51328 51416 101009 ! ! 101343 # Approximate coordinates 101291 # for proposed survey 101073 4 # Proposed Observations3

1-2 12-3 23-4 34-1 41-3 12-1-3 23-2-4 34-3-1 41-4-2 11-4-3 1-

# 1-2 ? ! 1- to LS with Star*Net Fixed Bearing Intro

159

2


STAR*NET - Output



60

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Summary of Files Used and Option Settings ========================================= Project Folder and Data Files Project Name TRAV2D Project Folder C:\...\STARPLUS\STARNET\EXAMPLES C:\ ...\ STARPLUS\ STARNET\ Data File List Trav2D.dat Project Option Settings STAR*NET Run Mode : Adjust with Error Propagation Type of Adjustment : 2D Project Units : FeetUS; DMS FeetUS; Coordinate System : LOCAL Default Project Elevation : 0.0000 FeetUS Apply Average Scale Factor : 1.0000000000 Input/Output Coordinate Order : North-East NorthAngle Data Station Order : At-From-To At- FromDistance/Vertical Data Type : Slope/Zenith Convergence Limit; Max Iterations : 0.010000; 10 Default Coefficient of Refraction : 0.070000 Earth Radius : 6372000.00 m Create Coordinate File : Yes Create Ground Scale Coordinate File : No Create Dump File : No Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 61

Output Explained!

Instrument Standard Error Settings Project Default Instrument Distances (Constant) Distances (PPM) Angles Directions Azimuths & Bearings Centering Error Instrument Centering Error Target

Output Explained!: : : : : : : 0.020000 5.000000 3.000000 2.000000 4.000000 0.000000 0.000000 FeetUS Seconds Seconds Seconds FeetUS FeetUS

Summary of Unadjusted Input Observations ======================================== Number of Entered Stations (FeetUS) = 2 (FeetUS) Fixed Stations 1 Free Stations 6Intro to LS with Star*Net

N 5045.5720 N 5000.0000

E Description 5495.3380 E Description 5190.000062


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Output Explained!Number of Angle Observations (DMS) = 13 At From To Angle StdErr 1 6 2 99-47-25.00 3.00 99- 472 1 3 115-10-00.00 3.00 115- 103 2 4 94-51-53.00 3.00 94- 514 3 5 216-46-09.00 3.00 216- 465 4 6 106-26-42.00 3.00 106- 266 5 1 86-57-49.00 3.00 86- 573 2 7 225-47-02.00 3.00 225- 477 3 8 97-31-36.00 - 313.00 97 8 7 9 115-14-57.00 3.00 115- 143.00 9 8 5 83-45-28.00 83- 459 8 10 156-15-44.00 3.00 156- 1510 9 6 106-12-32.00 3.00 106- 126 10 1 164-00-42.00 3.00 164- 00Intro to LS with Star*Net


63

Output Explained!Number From 1 2 3 4 5 6 3 7 8 9 9 10 Number From 8 of Distance Observations (FeetUS) = 12 (FeetUS) To Distance StdErr 2 205.0300 0.0210 3 134.1900 0.0207 4 105.4400 0.0205 5 161.5700 0.0208 6 160.7100 0.0208 1 308.3000 0.0215 7 115.4100 0.0206 8 284.4000 0.0214 9 191.6600 0.0210 5 161.9500 0.0208 10 166.9000 0.0208 6 151.3400 0.0208 of Azimuth/Bearing Observations (DMS) = 1 To Bearing StdErr 7 N79-52-31.00E - 52FIXED N79(c) Dr. Joshua Greenfeld 64


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Output Explained!Adjustment Statistical Summary ============================== Convergence Iterations Number of Stations = = 3 10 26 18 8

Number of Observations = Number of Unknowns = Number of Redundant Obs = Observation Angles Distances Az/Bearings Az/Bearings Total Count 13 12 1 26

Obs unkError Factor 0.308 1.354 0.000 0.945

Sum Squares of StdRes 0.379 6.767 0.000 7.146

The Chi-Square Test at 5.00% Level Passed ChiLower/Upper Bounds (0.522/1.480)Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 65

CASE I4

About Redundancy 4 CASE II3 3

Two cases with the same redundancy.1 2

9 Measurements: Angle Dist 4-1-2 1-2 13-4-1 2-1 23-4-1 1-2 11-2-3 3-2 32-3-4Intro to LS with Star*Net

Both have a redundancy of 9-5 = 4

1

2

9 Measurements: Angle Dist 4-1-2 3-2 33-4-1 2-1 22-3-4 1-3 11-2-3 4-1-3 2-3-166


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About RedundancyAdjustment Statistical Summary: CASE I CASE II Convergence Iterations = 2 2 Number of Stations = 4 4 Number of Observations = 10 10 Number of Unknowns = 6 6 Number of Redundant Obs = 4 4 Observation Count Sum Squares of StdRes Angles 6 5 0.676 0.393 Distances 3 4 2.194 2.000 Az/Bearings 1 1 0.000 0.000 Total 10 10 2.870 2.393 The Chi-Square TestIntro to LS with Star*Net (c) Dr. Joshua Greenfeld

Error Factor 0.531 0.443 1.352 1.118 0.000 0.000 0.847 0.77367

Passed Passed

About RedundancyDifferences in adjusted coordinates of CASE I and CASE IIStn N E N E N E 1 51002.00 101009.00 51002.00 101009.00 0.00 0.00 2 51005.00 101343.00 51005.00 101343.05 0.00 0.05 3 51328.00 101291.01 51328.03 101291.04 0.03 0.03 4 51416.01 101073.00 51416.05 101073.01 0.04 0.01



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STAR*NET StatisticsResidual StdRes = StdErr Where k is the number of observation of a specific type (Angle, Distance, Bearings, ) Ideally should be equal to 1! A StdRes larger than 3 is flagged with * to indicate potential blunder in observation. The observation with the largest StdRes is not necessarily the erroneous observation!Intro to LS with Star*Net


69

STAR*NET StatisticsWhere k the number of observation of a specific type (Angle, Distance, Bearings, ) n the total number of observations r the redundancy ErrorFactorType= (StdResType)2 / r * n/k

Error Factors should be roughly equal, and should all approximately be within a range of 0.5 to 1.5. If for example, the Error Factor for angles is equal to 15.7 and that for distances is equal to 2.3, then there is almost certainly a problem the angles.

In our sample output ErrorFactorAngles =Intro to LS with Star*Net

0.379 / 8 * 26/13 = 0.307870


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STAR*NET StatisticsThe Total Error Factor =

(StdRes)2 / r

= So

Where r is the redundancy So is the standard deviation of unit weight or the posteriori reference standard deviation. The total error factor is the value used in the 2 test to evaluate the overall quality of the adjustment. It should statistically not be different than 1. In our sample output The total Error Factor =Intro to LS with Star*Net

7.146 / 8 = 0.94571


Output Explained!Adjusted Coordinates (FeetUS) (FeetUS) ============================= Station 1 2 3 4 5 6 7 8 9 10 N 5045.5720 5250.5510 5310.5600 5220.5812 5160.5781 5000.5703 5420.5707 5370.5741 5185.5816 5020.5729 E 5495.3380 5500.3384 5380.3163 5325.3307 5175.3272 5190.3308 5345.3097 5065.3303 5015.3310 5040.3258 Description

Existing post Iron pipe



72

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Output Explained!Adjusted Observations and Residuals =================================== Adjusted Angle Observations (DMS) At From To 1 6 2 2 1 3 3 2 4 4 3 5 5 4 6 6 5 1 3 2 7 7 3 8 8 7 9 9 8 5 9 8 10 10 9 6 6 10 1Intro to LS with Star*Net

Angle 99-47-25.73 99- 47115-10-00.58 115- 1094-51-52.62 94- 51216-46-08.95 216- 46106-26-42.45 106- 2686-57-49.67 86- 57225-47-02.80 225- 4797-31-36.84 97- 31115-14-56.80 115- 1483-45-27.60 83- 45156-15-43.77 156- 15106-12-31.68 106- 12164-00-41.79 164- 00-

Residual StdErr StdRes 0-00-00.73 3.00 0.2 0- 000-00-00.58 3.00 0.2 0- 00-0-00-00.38 3.00 0.1 00-0-00-00.05 3.00 0.0 000-00-00.45 3.00 0.2 0- 000-00-00.67 3.00 0.2 0- 000-00-00.80 3.00 0.3 0- 000-00-00.84 3.00 0.3 0- 00-0-00-00.20 3.00 0.1 00-0-00-00.40 3.00 0.1 00-0-00-00.23 3.00 0.1 00-0-00-00.32 3.00 0.1 00-0-00-00.21 3.00 0.1 0073


Adjusted Angle Observations OutputAt Fr To 1 6 2 2 1 3 3 2 4 4 3 5 5 4 6 6 5 1 3 2 7 7 3 8 8 7 9 9 8 5 9 8 10 10 9 6 6 10 1Intro to LS with Star*Net

Explained!2

Res StdEr StdRes Res/StdEr (Res/StdEr) 0.2 0.24333 0.05921 0.73 3.0 0.19333 0.03738 0.58 3.0 0.2 0.38 3.0 0.1 0.12667 0.01604 0.01667 0.00028 0.05 3.0 0.0 0.45 3.0 0.2 0.15000 0.02250 0.22333 0.04988 0.67 3.0 0.2 0.80 3.0 0.3 0.26667 0.07111 0.84 3.0 0.3 0.28000 0.07840 0.06667 0.00444 0.20 3.0 0.1 0.40 3.0 0.1 0.13333 0.01778 0.07667 0.00588 0.23 3.0 0.1 0.32 3.0 0.1 0.10667 0.01138 0.07000 0.00490 0.21 3.0 0.1 Total = 0.37918(c) Dr. Joshua Greenfeld 74

37

Output Explained!Adjusted Distance Observations (FeetUS) (FeetUS) From To Distance Residual StdErr StdRes 1 2 205.0400 0.0100 0.0210 0.5 2 3 134.1879 -0.0021 0.0207 0.1 3 4 105.4495 0.0095 0.0205 0.5 4 5 161.5593 -0.0107 0.0208 0.5 5 6 160.7097 -0.0003 0.0208 0.0 6 1 308.3091 0.0091 0.0215 0.4 3 7 115.4461 0.0361 0.0206 1.8 7 8 284.4084 0.0084 0.0214 0.4 8 9 191.6303 -0.0297 0.0210 1.4 9 5 161.9382 -0.0118 0.0208 0.6 9 10 166.8910 -0.0090 0.0208 0.4 10 6 151.3327 -0.0073 0.0208 0.3 Adjusted Azimuth/Bearing Observations (DMS) From To Bearing Residual StdErr StdRes 8 7 N79-52-31.00E 0-00-00.00 FIXED 0.0 N79- 520- 00Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 75

Output Explained!Adjusted Bearings (DMS) and Horizontal Distances (FeetUS) FeetUS) ======================================================= (Relative Confidence of Bearing is in Seconds) From To 1 1 2 3 3 4 5 5 7 8 10 10 2 6 3 4 7 5 6 9 8 9 6 9 Bearing N01-23-50.77E N01- 23S81-36-25.05W S81- 36N63-26-08.64W N63- 26S31-25-43.98W S31- 25N17-39-05.84W N17- 39S68-11-52.93W S68- 11S05-21-24.62E S05- 21N81-07-04.60W N81- 07S79-52-31.00W S79- 52S15-07-27.80W S15- 07S82-24-16.75E S82- 24N08-36-48.43W N08- 36Distance 205.0400 308.3091 134.1879 105.4495 115.4461 161.5593 160.7097 161.9382 284.4084 191.6303 151.3327 166.8910 95% RelConfidence Brg Dist PPM 9.90 0.0382 186.1428 9.75 0.0398 129.2249 8.74 0.0386 287.5566 10.41 0.0384 363.9420 6.77 0.0370 320.3780 11.04 0.0386 238.9558 10.86 0.0317 197.5552 9.94 0.0316 194.9982 0.00 0.0390 137.1992 6.79 0.0397 207.1534 9.66 0.0320 211.1315 8.78 0.0326 195.374176



38

Relative Confidence StatisticsThe relative confidence measure is based on relative ellipses. The relative confidence for the distance is equal to a The relative confidence for the bearing is: Brg = tan-1(b/Distance)*206265 ppm = 106*RelConfDist/DistanceIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 77

b

a

Output Explained!Traverse Closures of Unadjusted Observations============================================ (Beginning and Ending on Adjusted Stations) TRAVERSE 1 Error Angular = -2.00 Sec, 6 Angles, -0.33 Sec/Angle Error Linear = 0.0066 S, 0.0171 W Horiz Precision = 0.0183 Error in 1075.2400, 1:58661, 17.05 PPM

From 1 1 2 3 4 5 6

To 6 2 3 4 5 6 1

Unadj Bearing S81-36-25.05W BS S81- 36N01-23-50.38E N01- 23N63-26-09.29W N63- 26S31-25-44.05W S31- 25S68-11-53.38W S68- 11S05-21-24.29E S05- 21N81-36-25.05E N81- 36(c) Dr. Joshua Greenfeld

Unadj Dist 205.0300 134.1900 105.4400 161.5700 160.7100 308.300078


39

Error Propagation ================= Station Coordinate Standard Deviations (FeetUS) (FeetUS) Station 1 2 3 4 5 6 7 8 9 10 N 0.000000 0.015585 0.014735 0.014460 0.014470 0.006333 0.019066 0.019439 0.015109 0.008732 E 0.000000 0.004052 0.014852 0.017007 0.016275 0.016133 0.015505 0.019109 0.019168 0.019052

Output Explained!

D = 1

D

X 2 2 X + Y 2 2 Y + 2 X Y X Y

STAR*NET does not list covariances!Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 79

Output Explained!Station Coordinate Error Ellipses (FeetUS) (FeetUS) Confidence Region = 95% Station 1 2 3 4 5 6 7 8 9 10 Semi-Major SemiAxis 0.000000 0.038169 0.036572 0.043750 0.040942 0.039841 0.047415 0.048011 0.046924 0.046690 Semi-Minor SemiAxis 0.000000 0.009835 0.035845 0.032735 0.034134 0.014570 0.037014 0.046333 0.036976 0.021252 Azimuth of Major Axis 0-00 1-59 1123-21 12362-23 6265-17 6581-48 81163-34 16330-37 3088-40 8886-51 86-



80

40

Output Explained!Relative Error Ellipses (FeetUS) (FeetUS) Confidence Region = 95% Stations Semi-Major Semi-Minor SemiSemiFrom To Axis Axis 1 2 0.038169 0.009835 1 6 0.039841 0.014570 2 3 0.038587 0.005684 3 4 0.038379 0.005315 3 7 0.036988 0.003775 4 5 0.038612 0.008619 5 6 0.031749 0.008462 5 9 0.031578 0.007804 7 8 0.039021 0.000000 8 9 0.039700 0.006288 10 6 0.031951 0.007090 10 9 0.032612 0.007081Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

Azimuth of Major Axis 1-59 181-48 81116-39 11631-53 31162-53 16267-09 67174-50 17498-59 9879-53 7914-27 1497-38 97170-18 170-

Elapsed Time = 00:00:00 81

Locating the Source of ErrorsSome of the major causes of completely failed or poor adjustments are as follows: 1. Incorrect data entry 2. Options set incorrectly 3. For grid jobs, the zone set incorrectly or geoid height not entered properly. 4. Field data collection blunders 5. Incorrect assignment of standard errors to observations 6. Systematic errors in field data - EDM out of calibration, etc. 7. Invalid geometry of the network - 3D point with no vertical observation, etc. 8. Weak network geometryIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 82

41

Locating the Source of ErrorsIndications of poor or failed adjustment1. Adjustment fails the Chi Square test. 2. Adjustment does not converge within 5 to 10 iterations. 3. Large residuals on observations. 4. Warning messages about Singularities and Geometric Weakness in the solution. 5. Extremely large error ellipses on adjusted points. 6. Fixed stations are showing coordinate changes.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 83

Locating the Source of ErrorsSuggested steps to determine the source of adjustment problems.1. Review the Error File, and resolve all errors and warnings that may be present. 2. Check all your option settings. 3. Review the values for any obvious blunders and their standard errors to see if they look reasonable. 4. Run Data Check Only. Review the network graphically with the plot file looking for obvious gaps and blunders in the data.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 84

42

Locating the Source of ErrorsSuggested steps to determine the source of adjustment problems.5. Check that the Angle Station Order and Coordinate Order options are set Correctly 6. Review the Statistical Summary carefully. See whether there is one data type that has a much worse Error Factor than the others. 7. Review the listing of observations and residuals. Look for single residuals that have standardized residuals larger than 3.0 8. Check your input standard errors. Are they reasonable?Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 85

Locating the Source of ErrorsSuggested steps to determine the source of adjustment problems.9. Run a minimally constrained adjustment. 10.Run the adjustment but restrict it to a single iteration. Look at the changes in coordinates after the first iteration to see which stations seem to be acting strangely. 11.Run the adjustment with the Coordinate Changes each Iteration



86

43

Locating the Source of ErrorsSuggested steps to determine the source of adjustment problems.12.Use the .DATA OFF and .DATA ON inline options to turn off sections of your input data file. 13.Unweight observations that you suspect by using a standard error of *. 14.Run Blunder Detect.



87

Error ellipsesWhen we compute Sx and Sy for a point we actually get a box such as: +Sy -Sx +Sx -Sy This box defined a 68% probability for the error to fall in. Intro to LS with Star*Net(c) Dr. Joshua Greenfeld

88

44

Error ellipsesA more rigorous way to do this is using an ellipse. The ellipse provides us with a Bivariate normal distribution with x = y = 0. The density function of this BivariateNormal distribution is:f ( x, y ) = 1 2 x y2 x y 1 x 2 exp 2 2 1 x y 2(1 ) x

y + y

2

Where: x and y are the standard deviation of the 2 axis is the correlation coefficient x,y are the points for which f (x,y ) is evaluated.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 89

Error ellipsesf(x,y) f(y)

f(x)

y x

y

x If we cur the bellat a particular f (x,y ) we get an ellipse called error ellipseIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 90

45

Special Cases of Error Ellipses

Sx = Sy xy=0

Sx < Sy xy=0

Sx > Sy xy=0

when xy=0 (which is the same as xy=0) the axis of the ellipse coincides with the axis of x,y.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 91

Special Cases of Error Ellipses

Sx = Sy xy= 0.5

Sx = Sy xy=-0.5

Sx = Sy xy=0.9

When xy0 (or xy0) the axis of the ellipse are rotated with respect to x,y.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 92

46

The General Case of Error EllipsesSx V Sy t Sv Su

t is rotation angle from Y axis to axis of largest error. U Su is the semi-major axis of semiellipse. (Largest error) u Sv is the semi-minor axis of semiX ellipse. (Least error) v Sx is the standard deviation in X of coordinate x Sy is the standard deviation in Y of coordinate y

The transformation equation between U,V and X,Y is:

U cos t V = sin t Intro to LS with Star*Net

sin t X cos t Y 93


Computing Error EllipsesVtan 2t = 2 S xy2 2 S X SY

Sx Su t Sv

U X

Sy

K=

2 ( S X SY2 ) 2 2 + S XY 4

Su2 =

2 2 S X SY +K 2

Su2 =

2 S X SY2 K 294



47

Probability and the size of the error ellipseFrom the 2 distribution table with v = 2 we get the following values: C Probability 1.000 39.4% 2.000 86.5% 3.000 98.9% 3.500 99.8%Intro to LS with Star*Net

C Probability 1.177 50% 2.146 90% 2.447 95% 3.035 99%95


Error Ellipse Example1 4 2 3

From the adjustment: o = 0.02X2 X3 1.199 -0.099 Qxx = N-1 = -0.099 0.583 -1.660 0.193 -1.402 0.460Intro to LS with Star*Net (c) Dr. Joshua Greenfeld

Y2 -1.660 0.193 2.634 2.725

Y3 -1.402 0.460 2.725 3.96296

48

Error Ellipse Examplexx = o2 Qxx 0.0004796 -0.0000396 -0.0006640 -0.00056082 SX2

-0.0000396 0.0002332 0.0000772 0.0001840

-0.0006640 0.0000772 0.0010536 0.0010900

-0.0005608 0.0001840 0.0010900 0.0015848

SX2 X3 2 SX3

SX2 Y 2 SX3 Y 2 2 SY2

SX2 Y 3 SX3 Y 3 SY2 Y3 2 SY397



Error Ellipse Examplep 2 0.0004796 -0.0006640 -0.0006640 0.0010536 22 SX2 2 SX 2 Y 2

SY 2

2Sxy= -0.00133 Chart for 2t 2t 2 -S 2 = -0.00057 Sx y + + - + tan(2 t) = 2.31358885 +360 246.6246 2 t dd = + - +180 t= 1231844 +180 K= 0.0007234 Su =0.0386 S2u = 0.00149 2 = 4.32E-05 Sv =0.0066 Sv Intro to LS with Star*Net(c) Dr. Joshua Greenfeld

98

49

Error Ellipse ExampleThe size of the error ellipse with 95% probability that the true point falls inside the error ellipse, we have to multiply Su and Sv by 2.447 or: Su =0.0386 x 2.447 = 0.0945 and Sv =0.0066 x 2.447 = 0.0162



99

Relative Error EllipseRelative error ellipses are computed between two points. X = Xi - Xj Y = Yi - Yj Propagating the variances and covariances of the coordinates of points i and j we get: S2X = S2Xi + S2Xj 2SXi Xj S2Y = S2Yi + S2Yj 2SYi Yj SX y = SXi Yi + SXj Yj SXiYj SXjYi The relative error ellipses are computed as before using the above variances/covarianceIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 100

50

Example of Relative Error EllipseUsing the above exampleX2 Y2 X3 Y3

0.0004796 -0.0000396 -0.0006640 -0.0005608

-0.0000396 0.0002332 0.0000772 0.0001840

-0.0006640 0.0000772 0.0010536 0.0010900

-0.0005608 0.0001840 0.0010900 0.0015848

2Sxy = 0.0000072 S2X = 0.000792 S2x - S2y = 0.0003336 2 S Y = 0.000458 t = 03706 SX y = 0.0000036 S2 = 0.000792 S =0.0281432 u u 2 = 0.0004584 S =0.0214094 Sv vIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 101

The Chi Square Distribution 2 ,v = 2 ,v vS 22

=

x2 v 2

1 v 2 2

0

u

v2 2

e

u 2

du



102

51

The Chi Square Distribution1. Distribution of sample variances 2. Used to build confidence intervals for population variance. 3. Starts at 0 and goes to 4. Distribution is NOT symmetric 5. Based on degrees of freedom, v, from sample. 6. Table gives critical 2,v value which delineates area in upper tail.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 103

The Chi Square Distributionchi square as a function of the degree of freedom 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0Intro to LS with Star*Net

2 4 6 8 10

5

10(c) Dr. Joshua Greenfeld

15

20104

52

The 95% Confidence Interval (5% Level of Significance.) 2 Distribution (2.025 and 2.975) (DoF(v) DoF(v1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2v,a

2v,b

DoF (v)17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60

2v,a

2v,b

.001 .051 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.23 6.91

5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8

7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8 24.4 40.5

30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0 59.3 83.3105



x and S2 and covariances are point estimationsbecause they provide a single value for these parameters

__

Confidence Interval

Confidence Interval is an interval-based estimator. The Need Because we want to know how good these estimations are and how much can we rely on them. In other words, we want to define how much deviation from the estimator is likely, given that the value for and are still unknown.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 106

53

Statistical TestingStatistical testing is used to compare results with previous ones or with given standards. In testing we seek a judgment whether x or S2 are consistent with the assumption (hypothesis) that the sample was drawn from a population with specific parameter values such as N(,). We test a hypothesis about the probability distribution of a random variable.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 107

Statistical TestingSimple hypothesis Test when the probability distribution is known i.e. N(,), 2, t, F, etc. Composite hypothesis test when the probability distribution is NOT known or not completely known. Ho Null hypothesis hypothesis of no difference H1 Alternative hypothesis hypothesis of differenceIntro to LS with Star*Net (c) Dr. Joshua Greenfeld 108

54

Statistical TestingEXAMPLE This is a null hypothesis that the Ho : 1 = o mean 1 from which one sample was drawn is equal (or is not different from) the mean o of a population having a specific mean. H1 : 1 o This is the alternative hypothesis.



109

Statistical TestingThe test statistics (computed from the sample)

=2

vS

2

__

2

t=

y

S/ n

S12 F= 2 S2



110

55

Statistical TestingThe Rejection Region. The value for the statistic that indicates rejection of Ho. This is equivalent to the test statistic being outside the confidence interval when constructing confidence intervals.2 > 2 , v t > t , v F > F , v1, v2

Once we have a hypothesis we can make a decision to accept it or to reject it. This decision may be correct or incorrect.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 111

Statistical TestingThere are 4 possible outcomes to a hypothesis test Hypothesis is Hypothesis is correct incorrect Correct decision Type II error P = 1- P= (Confidence level) Type I error Correct decision P= P = 1- (Significance (Power of test) level)(c) Dr. Joshua Greenfeld 112

Hypothesis is accepted

Hypothesis is rejectedIntro to LS with Star*Net

56

Statistical TestingHo distribution Type II error Ha distribution Type I error

Reject Ha

Critical Value

Reject Ha

- the size of the type I error or the probability of a type I error to occur. = P [reject Ho when Ho is true]Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 113

Statistical Testing is expressed in % which is also called Significance Level of the Test. Common values for are: 5% - significant 1% - highly significant of 5% means that we are willing to risk rejecting ho, when it is correct, 5 times out of 100 trials.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 114

57

Statistical Testing Errors Example1. 2. 3. 4. Population of 10,000 people. Test for flu virus that has 95% confidence level. 9200 test negative for flu 800 test positive. Type I error: people who test positive for flu but do not have it: 800 0.05 = 40 Type II error: people who have flu but test negative for it are: 9200 0.05 = 460 Thus = 460/10,000 = 0.046 or 4.6% Summary: 460 + 40 = 500 people incorrectly test for flu. or 5% of the population.Intro to LS with Star*Net (c) Dr. Joshua Greenfeld 115

Using Two Tail 2 Tablesa /2 confidence region 2 1-/2, v /2,

/2 2 /2, v /2, 2

Interval 90% 95% 99%Intro to LS with Star*Net

2v,a 2.05 2.025 2.005

2v,b 2.95 2.975 2.995116


58

Testing the Probability DistributionTesting the Variance (The 2 Test) Ho : S2 = o2 H1 : S2 o2 v2,a o2 v2,b o2 2

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