Ellips kesalahan .pdf

Ellips kesalahan standar: distribusi normalFungsi padat probabilitas (Probability density function, pdf)µ = 512.431m;yxµ = 1231.273m;f( x)σxy=σσ0.91 ⎧exp⎨−σ 2π⎩2x2y= 1.69mm= 3.24mm( x − µ )=22σ2⎫⎬⎭σ yσ xFebruari 2005gd2212-Hitung Perataan I

Ellips kesalahan standar: distribusi normal dua variableFungsi padat probabilitas (Probability density function, pdf)f( x,y)2πσσβ ==21 − βxσσxyxyσy21 − β⎧⎪exp⎨2 ⎪⎩− 121 − β⎡⎛⎢⎢⎜⎣⎝2( x − µ ) ⎞ ⎛ ( y − µ ) ⎞⎛( y − µ ) ⎞ ⎛ ( y − µ )σxx⎟⎠− 2β⎜⎝σ yσxx⎜⎟⎠⎝σyy⎟ + ⎜⎠ ⎝σyy⎞⎟⎠2⎤⎫⎥⎪⎬⎥⎦⎪⎭µ yFebruari 2005gd2212-Hitung Perataan Iµ xσ x

Ellips kesalahan standar: sumbu-sumbu utamaArah sumbu-sumbu kesalahan maksimum dan minimumYVU2σv: variansi min.; V :arah error minimumσσ uy σ v θµ yTransformasi koordinatσ yσ xX2uµ : mean pada arah X ;xµ : mean pada arahY;yσ : variansi Xσ : variansi maks.; U :arah error maksimum2x2yσ : variansi Xµ xσ x⎡U⎤ ⎡ cosθsinθ⎤⎡X⎤⎢ ⎥ = ⎢⎥⎢⎥⎣V⎦ ⎣−sinθcosθ⎦⎣Y⎦dengan matrik − matrik kovariansi2⎡σx⎢⎢⎣σxyσ ⎤xy2 ⎥σy ⎥⎦dan2⎡σu⎢⎣ 00σ2v⎤⎥⎦Februari 2005gd2212-Hitung Perataan I

Februari 2005gd2212-Hitung Perataan IEllips kesalahan standar: sumbu-sumbu utamaTransformasi koordinat⎥⎦⎤⎢⎣⎡−⎥⎥⎦⎤⎢⎢⎣⎡⎥⎦⎤⎢⎣⎡−=⎥⎦⎤⎢⎣⎡θθθθσσσσθθθθσσcossinsincoscossinsincos002222yxyxyxvu( ) ( )θθσθθσσθσθθσθσσθσθθσθσσ22222222222222sincoscossin0sincossin2cossincossin2cos−+−=+−=++=xyxyyxyxvyxyxu( ) θθ 2cos;2sin21222tan 2yxxyσσσθ−=( )( )2122222222122222224242⎥⎥⎦⎤⎢⎢⎣⎡+−−+=⎥⎥⎦⎤⎢⎢⎣⎡+−++=xyxyyxyxvyxyxuσσσσσσσσσσσσσσEliminasi θMenggunakan dalil perambatan variansi-kovariansi

Ellips kesalahan standar: sumbu-sumbu utamaTransformasi koordinat2σu2σv2= σx2= σxcoscos22θ + 2σθ − 2σxyxy2sinθcosθ+ σ sin2sinθcosθ+ σ sinyy22θθ2 sinsincos22θ cosθ =θ =θ =1 −1 +sincos2cos22θ2θ2θ2 2σ σ2 x+y 1σu= + cos 2θx y+xy22 22 2( σ −σ) 2σtan θcos 2θ=1( 1+tan 2 2θ) 1 2cos 2θ=±tan 2θ=2σFebruari 2005gd2212-Hitung Perataan Iσ2xxy−σ( ( ) )2 2 2 24σ+ σ −σ1 22yxyσ2x−σσσx2u2v2yyσ=σ=2x2x+ σ2+ σ22y2y⎡+ ⎢⎢⎣⎡− ⎢⎢⎣2 2( σ −σ)x42 2( σ −σ)x4yy22+ σ+ σ2xy2xy⎤⎥⎥⎦⎤⎥⎥⎦1 21 2

Ellips kesalahan standar: sumbu-sumbu utamaTransformasi koordinatξ = x cos θ + y sinθ= f1u+ f2vVariansi sebuah titik setelah transformasis2ξ=s22⎡σσcos ⎢2⎢⎣σxyσy⎤ cosθ⎥⎢⎥⎦⎣sinθx xy ⎡ ⎤[ θ sinθ] ⎥ ⎦Akar laten (tersembunyi, eigen value) dari matriks variansi adalah:2σ − λ σ2 2 2 2x xyσ= 0x+ σy( σx−σy)2λσxyσy− λ Dengan solusi: = ±2422+ 4σxyuuuu2( σx− λ1) + vσxy= 02( σx− λ2) + vσxy= 02σxy+ v( σy− λ1) = 02σ + v( σ − λ ) = 0xyy2tanθtanθσλ −σ2σ2xyxy1 x1= =tan 2θ1=22 2λ1−σyσσxyx−σy2λ σ1−σx xy2= =2σxyλ2−σxFebruari 2005gd2212-Hitung Perataan I

Ellips kesalahan standarReferensi:Bjerhammar, Arne, (1973), Theory of errors and generalized matrix inverses, ElsevierScientific Publishing Company, Amsterdam, Hlm 144-147.Mikhail, E. M. dan G. Gracie, (1981), Analysis and adjustment of survey measurements,Van Nostrand Reinhold Company, New York, Hlm.129-136, 220-227.Mikhail, E. M., (1976), Observation and least square, IEP-A Dun-Donnely Publisher,New York, Hlm. 9-35.Turcotte, D. L. dan G. Schubert, (1982), Geodynamics: Application of continuumphysics to Geological problems, John Wiley & Sons, New York, Hlm. 80-83.Wolf, P. R. dan Ghilani, C. D., (1997), Adjustment computations, Statistics and LeastSquares in surveying and GIS, John Wiley & Sons, New York, Hlm. 35-48,357-371.Februari 2005gd2212-Hitung Perataan I