regression analysis control of built engineering objects, comparing to the plan surveying...
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Regression analysis
Control of built engineering objects, comparing to the plan
Surveying observations – position of points
Linear regression
Regression plan
Regression curve
Least squares method
Deformation analysis
Linear regression I.
Correlation coefficient, tightness of relationof x and y coordinates
1. Only the y coordinates are supposed to have error
1
)()(,
,,
n
yyxxc
cr ii
yxyx
yxyx
02)(min! ***2 dvvvvdvvvbxmvy iiii
dxAdvlxAv
y
y
y
b
m
x
x
x
v
v
v
nnn
..
1
....
1
1
..2
1
2
1
2
1
x
y ?? bmbxmy
Linear regression I. cont.
lAAAx
lAxAAlxAAvA
lxAvas
vAgtransposinafterAv
dxAvdvv
*1*
****
**
**
0
00
022
xsmysbmsmxs
ysxsmsbs
ys
xscoordsintweightpo
y
yxlA
nx
xxAA
i
ii
i
i
i
ii
i
ii
2
*2
*
0
0
0
1
.2
,
n
yycoeffncorrelatiothefromrm i
yx
yyx
Linear regression II.
x
y
)cos()sin( iyiixi vvvv
bvxvy
mbvxmvy
iiii
xiiyii
sintancos
tan
cossincostancos
tantansincos
iiiiii
iiii
yxbvyxbv
yxbvv
0sincossincossincos2
0coscossincos2
min!cossincos22
iiii
ii
iii
yxbyxb
yxbb
yxbv
vi – distance from the line
Linear regression II. cont.
Moving the origin of the coordinate system to the weight point:
0cossincossin
0cossincos2222
iiiiii
ii
yxyxyx
yxbn
tan2
arctan21
02cos2sin21
00cos
22
22
mysxs
ysxs
ysxsysxs
pointweightthethroughgoesbbn
ii
ii
iiii
xsmysb
Regression plan
iiii
iiiii
zcybxav
vcybxavz
cybxaz
min!2
nnnn z
z
z
c
b
a
yx
yx
yx
v
v
v
..
1
......
1
1
..2
1
22
11
2
1
0
0
2
2
*2
2
*
ii
ii
iii
iii
i
ii
ii
ii
iiii
iiii
zsys
zsxs
b
a
ysysxs
ysxsxs
cscoordinatepointweightfor
z
zy
zx
lA
nyx
yyyx
xyxx
AA
Regression polynomial
iinn xaxaxaxaay ...2
210
niniiii xaxaxaavy ...2
210
lAAAxlAxv
y
y
y
a
a
a
xx
xx
xx
v
v
v
mnnmm
n
n
m
*1*2
1
1
0
22
11
2
1
....
..1
........
..1
..1
..
Badly conditioned equation system
Distance calculation
Point-line distance
0
22
cybxa
linetheofequationba
cybxat iii
0
222
dzcybxa
planetheofequationcba
dzcybxat iiii
Point-plane distance
x
yt
t
x
z
y
Coordinate transformationHelmert (orthogonal)
ABPABPAP
ABPABPAp
kbkaxx
kbkayy
sincos
cossin
rbmaxx
mbrayy
PPAP
PPAp
Solution using least squares method
rbmaxvx
mbrayvy
PPAxpP
PPAypp
PPPAxp
pPPAyp
xrbmaxv
ymbrayv
unknowns:yA, xA, r, m
Matrix form:
pn
p
p
A
A
nnxpn
xp
yp
x
x
y
m
r
x
y
ab
ab
ba
v
v
v
...
10
............
10
01
...1
1
11
11
1
1
lAAAx *1* lAxv
22
22*
0
0
0
0
iiii
iiii
ii
ii
baab
baba
abn
ban
AA
iiii
iiii
i
i
xayb
xbya
x
y
lA*
Weight point coordinates:
iiiiii
iiiiii
A
A
xaybmba
xbyarba
xn
yn
22
22
0
0
Transformation using three parameters
ABABAP
ABABAp
kbkaxx
kbkayy
sincos
cossin
ABABAP
ABABAp
baxx
bayy
sincos
cossin
PABABAxp
pABABAyp
xbaxv
ybayv
sincos
cossin
Only rotation and offset (k = 1)
Unknowns: , yA, xA
Correction equation is not linear,Series development
...,...),(,...),( 2
021
01201021
dxx
fdx
x
fxxfxxf
...
sincos
cossin
.........
cossin10
sincos01
101010
101010
0101
0101
xbax
ybay
lba
ba
A A
A
Affine transformation
FbEaxx
DbCayy
PPAP
PPAp
............
1
1
1
...3
2
1
33
22
11
3
2
1
y
y
y
D
C
y
ba
ba
ba
v
v
vA
y
y
y
Different scale along the coordinate axis
Two independent equation system of three unknown
ii
ii
iA
iiii
iiii
ii
yb
ya
y
D
C
y
bbab
baaa
ban
2
2
Weight point coordinates simplify the equation system
Polynomial transformation
n
i
ii
nn
n
i
ii
nn
xbxbxbxbbX
yayayayaaY
0
2210
0
2210
...
...
Used for large areas
3rd power polynomials 20 unknowns, min. 10 common points
4th power polynomials 30 unknowns, min. 15 common points
5th power polynomials 42 unknowns, min. 21 common points
Weight point coordinates reduce the effect of rounding errors
InterpolationLinear interpolation
Lagrange interpolation (polynomial)
Spline interpolation
Low order polynomials between the pointsCubic spline 3
32
210 xaxaxaay iiiii
1 2…
n
Continuous curves 1st and 2nd derivates are the same at the points
(n-1) * 4 unknowns
(n-1) * 2 equation (through the points)
(n-2) equation for the 1st derivates
(n-2) equation for the 2nd derivates
+2 boundary condition