Variance of Linear MMSE estimator from three measurements.
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I am here presenting a question on calculating the variance of an Estimator. This is a problem in localization of vehicle using ranging from Vehicular Ad hoc Network and Change in location from Inertial Navigation System:
So I have three measurements as follows
beginequation*
Delta hatx_i,i-1 = Delta x_i,i-1 + epsilon_x_INS = Delta x_i,i-1 + mathcalN(0,sigma_x_INS^2)
endequation*
beginequation*
Delta haty_i,i-1 = Delta y_i,i-1 + epsilon_y_INS = Delta y_i,i-1 + mathcalN(0,sigma_y_INS^2)
endequation*
$(Delta haty_i,i-1, Delta hatx_i,i-1)$ are the estimated change in position, which is shown as sum of actual position and an error(a normal distribution). This is data from an Inertial navigation system
beginequation*
hatr_i = r_i + epsilon_r = r_i + mathcalN(0,sigma_r^2)
endequation*
$hatr_i$ is the estimated range measurement, which is represented as sum of actual range and an error in ranging(a normal distribution again). This is data from ranging between vehicle and a Road side unit $RSU_j$ that is located at $(h_j,k_j)$
After collecting the above three data for $i=1,2,cdots,m$ number of time instances, the Minimum Mean Square Error (MMSE) estimator for $m^th$ instance location of vehicle:
beginequation
hatvecx = (A^T A)^-1 A^T vecb
endequation
Where $A$ is of size $(m-1 times 2)$beginequation*
A
=
beginbmatrix
-2sum_a = 1^m-1 Delta hatx_a+1,a & -2sum_a = 1^m-1 Delta haty_a+1,a \
vdots & vdots\
-2sum_a = i^m-1 Delta hatx_a+1,a & -2sum_a = i^m-1 Delta haty_a+1,a \
vdots & vdots\
-2Delta hatx_m,m-1 & -2 Delta haty_m,m-1
endbmatrix
endequation*
$vecx$ represents the most current two dimensional location of the vehicle:
beginequation*
vecx = beginbmatrix
x_m \
y_m
endbmatrix
endequation*
$vecb$ is of size $(m-1 times 1)$ as below:
begineqnarray*
beginbmatrix
hatr_1^2 - hatr_m^2 - 2 h_j sum_a=1^m-1 Delta hatx_a+1,a - 2 k_j sum_a=1^m-1 Delta haty_a+1,a -sum_a=1^m-1Delta hatx_a+1,a ^2 - sum_a=1^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_i^2 - hatr_m^2 - 2 h_j sum_a=i^m-1 Delta hatx_a+1,a - 2 k_j sum_a=i^m-1 Delta haty_a+1,a -sum_a=i^m-1Delta hatx_a+1,a ^2 - sum_a=i^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_m-1^2 - hatr_m^2 - 2 h_j Delta hatx_m,m-1 - 2 k_j Delta haty_m,m-1 -Delta hatx_m,m-1^2 - Delta haty_m,m-1 ^2\
endbmatrix
endeqnarray*
My question:'' How does one determine the variance of $hatvecx$ ". My idea of finding this variance is to understand how error in three types of input data $epsilon_x_INS,epsilon_y_INS,epsilon_r$ is effecting the error in $hatvecx$ estimate. Looking forward to all thoughts from the community.
linear-algebra covariance variance estimation-theory
asked Aug 27 at 12:36
prithvi shenoy
1
add a comment |Â
up vote
0
down vote
favorite
I am here presenting a question on calculating the variance of an Estimator. This is a problem in localization of vehicle using ranging from Vehicular Ad hoc Network and Change in location from Inertial Navigation System:
So I have three measurements as follows
beginequation*
Delta hatx_i,i-1 = Delta x_i,i-1 + epsilon_x_INS = Delta x_i,i-1 + mathcalN(0,sigma_x_INS^2)
endequation*
beginequation*
Delta haty_i,i-1 = Delta y_i,i-1 + epsilon_y_INS = Delta y_i,i-1 + mathcalN(0,sigma_y_INS^2)
endequation*
$(Delta haty_i,i-1, Delta hatx_i,i-1)$ are the estimated change in position, which is shown as sum of actual position and an error(a normal distribution). This is data from an Inertial navigation system
beginequation*
hatr_i = r_i + epsilon_r = r_i + mathcalN(0,sigma_r^2)
endequation*
$hatr_i$ is the estimated range measurement, which is represented as sum of actual range and an error in ranging(a normal distribution again). This is data from ranging between vehicle and a Road side unit $RSU_j$ that is located at $(h_j,k_j)$
After collecting the above three data for $i=1,2,cdots,m$ number of time instances, the Minimum Mean Square Error (MMSE) estimator for $m^th$ instance location of vehicle:
beginequation
hatvecx = (A^T A)^-1 A^T vecb
endequation
Where $A$ is of size $(m-1 times 2)$beginequation*
A
=
beginbmatrix
-2sum_a = 1^m-1 Delta hatx_a+1,a & -2sum_a = 1^m-1 Delta haty_a+1,a \
vdots & vdots\
-2sum_a = i^m-1 Delta hatx_a+1,a & -2sum_a = i^m-1 Delta haty_a+1,a \
vdots & vdots\
-2Delta hatx_m,m-1 & -2 Delta haty_m,m-1
endbmatrix
endequation*
$vecx$ represents the most current two dimensional location of the vehicle:
beginequation*
vecx = beginbmatrix
x_m \
y_m
endbmatrix
endequation*
$vecb$ is of size $(m-1 times 1)$ as below:
begineqnarray*
beginbmatrix
hatr_1^2 - hatr_m^2 - 2 h_j sum_a=1^m-1 Delta hatx_a+1,a - 2 k_j sum_a=1^m-1 Delta haty_a+1,a -sum_a=1^m-1Delta hatx_a+1,a ^2 - sum_a=1^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_i^2 - hatr_m^2 - 2 h_j sum_a=i^m-1 Delta hatx_a+1,a - 2 k_j sum_a=i^m-1 Delta haty_a+1,a -sum_a=i^m-1Delta hatx_a+1,a ^2 - sum_a=i^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_m-1^2 - hatr_m^2 - 2 h_j Delta hatx_m,m-1 - 2 k_j Delta haty_m,m-1 -Delta hatx_m,m-1^2 - Delta haty_m,m-1 ^2\
endbmatrix
endeqnarray*
My question:'' How does one determine the variance of $hatvecx$ ". My idea of finding this variance is to understand how error in three types of input data $epsilon_x_INS,epsilon_y_INS,epsilon_r$ is effecting the error in $hatvecx$ estimate. Looking forward to all thoughts from the community.
linear-algebra covariance variance estimation-theory
asked Aug 27 at 12:36
prithvi shenoy
1
what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am here presenting a question on calculating the variance of an Estimator. This is a problem in localization of vehicle using ranging from Vehicular Ad hoc Network and Change in location from Inertial Navigation System:
So I have three measurements as follows
beginequation*
Delta hatx_i,i-1 = Delta x_i,i-1 + epsilon_x_INS = Delta x_i,i-1 + mathcalN(0,sigma_x_INS^2)
endequation*
beginequation*
Delta haty_i,i-1 = Delta y_i,i-1 + epsilon_y_INS = Delta y_i,i-1 + mathcalN(0,sigma_y_INS^2)
endequation*
$(Delta haty_i,i-1, Delta hatx_i,i-1)$ are the estimated change in position, which is shown as sum of actual position and an error(a normal distribution). This is data from an Inertial navigation system
beginequation*
hatr_i = r_i + epsilon_r = r_i + mathcalN(0,sigma_r^2)
endequation*
$hatr_i$ is the estimated range measurement, which is represented as sum of actual range and an error in ranging(a normal distribution again). This is data from ranging between vehicle and a Road side unit $RSU_j$ that is located at $(h_j,k_j)$
After collecting the above three data for $i=1,2,cdots,m$ number of time instances, the Minimum Mean Square Error (MMSE) estimator for $m^th$ instance location of vehicle:
beginequation
hatvecx = (A^T A)^-1 A^T vecb
endequation
Where $A$ is of size $(m-1 times 2)$beginequation*
A
=
beginbmatrix
-2sum_a = 1^m-1 Delta hatx_a+1,a & -2sum_a = 1^m-1 Delta haty_a+1,a \
vdots & vdots\
-2sum_a = i^m-1 Delta hatx_a+1,a & -2sum_a = i^m-1 Delta haty_a+1,a \
vdots & vdots\
-2Delta hatx_m,m-1 & -2 Delta haty_m,m-1
endbmatrix
endequation*
$vecx$ represents the most current two dimensional location of the vehicle:
beginequation*
vecx = beginbmatrix
x_m \
y_m
endbmatrix
endequation*
$vecb$ is of size $(m-1 times 1)$ as below:
begineqnarray*
beginbmatrix
hatr_1^2 - hatr_m^2 - 2 h_j sum_a=1^m-1 Delta hatx_a+1,a - 2 k_j sum_a=1^m-1 Delta haty_a+1,a -sum_a=1^m-1Delta hatx_a+1,a ^2 - sum_a=1^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_i^2 - hatr_m^2 - 2 h_j sum_a=i^m-1 Delta hatx_a+1,a - 2 k_j sum_a=i^m-1 Delta haty_a+1,a -sum_a=i^m-1Delta hatx_a+1,a ^2 - sum_a=i^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_m-1^2 - hatr_m^2 - 2 h_j Delta hatx_m,m-1 - 2 k_j Delta haty_m,m-1 -Delta hatx_m,m-1^2 - Delta haty_m,m-1 ^2\
endbmatrix
endeqnarray*
My question:'' How does one determine the variance of $hatvecx$ ". My idea of finding this variance is to understand how error in three types of input data $epsilon_x_INS,epsilon_y_INS,epsilon_r$ is effecting the error in $hatvecx$ estimate. Looking forward to all thoughts from the community.
linear-algebra covariance variance estimation-theory
asked Aug 27 at 12:36
prithvi shenoy
1
I am here presenting a question on calculating the variance of an Estimator. This is a problem in localization of vehicle using ranging from Vehicular Ad hoc Network and Change in location from Inertial Navigation System:
So I have three measurements as follows
beginequation*
Delta hatx_i,i-1 = Delta x_i,i-1 + epsilon_x_INS = Delta x_i,i-1 + mathcalN(0,sigma_x_INS^2)
endequation*
beginequation*
Delta haty_i,i-1 = Delta y_i,i-1 + epsilon_y_INS = Delta y_i,i-1 + mathcalN(0,sigma_y_INS^2)
endequation*
$(Delta haty_i,i-1, Delta hatx_i,i-1)$ are the estimated change in position, which is shown as sum of actual position and an error(a normal distribution). This is data from an Inertial navigation system
beginequation*
hatr_i = r_i + epsilon_r = r_i + mathcalN(0,sigma_r^2)
endequation*
$hatr_i$ is the estimated range measurement, which is represented as sum of actual range and an error in ranging(a normal distribution again). This is data from ranging between vehicle and a Road side unit $RSU_j$ that is located at $(h_j,k_j)$
After collecting the above three data for $i=1,2,cdots,m$ number of time instances, the Minimum Mean Square Error (MMSE) estimator for $m^th$ instance location of vehicle:
beginequation
hatvecx = (A^T A)^-1 A^T vecb
endequation
Where $A$ is of size $(m-1 times 2)$beginequation*
A
=
beginbmatrix
-2sum_a = 1^m-1 Delta hatx_a+1,a & -2sum_a = 1^m-1 Delta haty_a+1,a \
vdots & vdots\
-2sum_a = i^m-1 Delta hatx_a+1,a & -2sum_a = i^m-1 Delta haty_a+1,a \
vdots & vdots\
-2Delta hatx_m,m-1 & -2 Delta haty_m,m-1
endbmatrix
endequation*
$vecx$ represents the most current two dimensional location of the vehicle:
beginequation*
vecx = beginbmatrix
x_m \
y_m
endbmatrix
endequation*
$vecb$ is of size $(m-1 times 1)$ as below:
begineqnarray*
beginbmatrix
hatr_1^2 - hatr_m^2 - 2 h_j sum_a=1^m-1 Delta hatx_a+1,a - 2 k_j sum_a=1^m-1 Delta haty_a+1,a -sum_a=1^m-1Delta hatx_a+1,a ^2 - sum_a=1^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_i^2 - hatr_m^2 - 2 h_j sum_a=i^m-1 Delta hatx_a+1,a - 2 k_j sum_a=i^m-1 Delta haty_a+1,a -sum_a=i^m-1Delta hatx_a+1,a ^2 - sum_a=i^m-1Delta haty_a+1,a ^2 \
vdots\
hatr_m-1^2 - hatr_m^2 - 2 h_j Delta hatx_m,m-1 - 2 k_j Delta haty_m,m-1 -Delta hatx_m,m-1^2 - Delta haty_m,m-1 ^2\
endbmatrix
endeqnarray*
My question:'' How does one determine the variance of $hatvecx$ ". My idea of finding this variance is to understand how error in three types of input data $epsilon_x_INS,epsilon_y_INS,epsilon_r$ is effecting the error in $hatvecx$ estimate. Looking forward to all thoughts from the community.
linear-algebra covariance variance estimation-theory
asked Aug 27 at 12:36
prithvi shenoy
1
asked Aug 27 at 12:36
prithvi shenoy
1
asked Aug 27 at 12:36
prithvi shenoy
1
asked Aug 27 at 12:36
prithvi shenoy
1
1
what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28
add a comment |Â
what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28
what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28
add a comment |Â
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what have you worked on so far ? any thoughts of how to tackle this problem ?
– Ahmad Bazzi
Aug 27 at 12:46
I could utilize the relation for var (X.Y). Follow up Question: Since the time series data, say $Delta x$ are from same sensor, at different time instances, should I treat them as independent random variables?
– prithvi shenoy
Aug 27 at 15:08
Correct me if I am wrong with calculating the variance of $A$ beginequation* sigma_A = beginbmatrix 4times(m-1) times sigma_x_INS^2 & 4times(m-1) times sigma_y_INS^2 \ vdots & vdots\ 4times(m-i) times sigma_x_INS^2 & 4times(m-i) times sigma_y_INS^2\ vdots & vdots\ 4times sigma_x_INS^2 & 4timessigma_y_INS^2 endbmatrix endequation* I think it is fair to assume that a sensor reading is independent of sensor reading at a previous instance.
– prithvi shenoy
Aug 27 at 15:28