Covariance & Covariance Propagation
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Precision
degree of closeness among repeated measurements.
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Accuracy
degree of closeness of a measurement to its true value.
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Standard deviation
measure of dispersion
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smaller standard deviation
higher precision.
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Probable error
measure of the uncertainty or reliability of an observation.
6
Uncertainty
degree of doubt associated with a measured or computed value.
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Joint distribution
probability distribution describing two or more random variables together.
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Observation Model
mathematical representation of survey observations.
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Functional + Stochastic
Observation =
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Functional model
mathematical relationship between observations and unknown parameters.
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Stochastic model
describes the statistical properties and uncertainties of survey observations.
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Stochastic model
statistical description of uncertainties in survey observations.
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Random errors
unpredictable errors that cannot be completely eliminated.
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Systematic errors
consistent errors caused by identifiable factors.
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Blunders
mistakes caused by human or procedural errors.
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Adjustment
process of determining the most probable values of unknown parameters from observations.
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Least Squares
adjustment method that minimizes the sum of squared residuals.
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Redundancy
additional observations beyond the minimum required.
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Degrees of Freedom
number of independent redundant observations available for adjustment.
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Independent Events
events where the occurrence of one does not affect the probability of the other.
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P(X ≤ x, Y ≤ y) = P(X ≤ x) P(Y ≤ y)
Or:
F(x, y) = F(x) F(y)
Or:
F(x, y) = F(x) F(y)
Condition for Independence:
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E[XY] = E[X] E[Y] = μx μy
Expected Value of Independent Variables:
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Covariance
measure of the degree and direction of relationship between two random variables.
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cov(X, Y) = σxy
σxy = E[(X − μx)(Y − μy)]
σxy = E[(X − μx)(Y − μy)]
Covariance
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Positive
positively correlated variables.
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Zero
uncorrelated variables.
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Negative
negatively correlated variables.
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Zero covariance
indicates no linear correlation, but does not necessarily mean no relationship.
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Non-zero covariance
indicates a linear relationship between variables.
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Pearson Product-Moment Correlation Coefficient (ρ)
measures the strength and direction of linear correlation between two variables.
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ρxy = σxy / (σx × σy)
Pearson Product-Moment Correlation Coefficient (ρ)
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−1 ≤ ρxy ≤ +1
Range of Correlation Coefficient:
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+1
perfect positive correlation.
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0
no linear correlation.
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-1
perfect negative correlation.
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Covariance
indicates direction of relationship; depends on measurement units.
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Correlation
indicates strength and direction; unitless and ranges from −1 to +1.
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Uniformly distributed X and Y
X + Y may have a triangular distribution.
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Normally distributed X and Y
X + Y is normally distributed.
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Variance-Covariance Matrix (Σ)
square matrix containing variances and covariances of random variables.
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Diagonal Elements
variances of individual observations.
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Off-Diagonal Elements
covariances between different observations.
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Symmetric Matrix
covariance matrix where σij = σji.
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X
vector consisting of observables.
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μ
vector of true or expected values.
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ε (epsilon)
vector of true errors.
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Covariance Propagation
process of determining how uncertainties in input variables affect the uncertainties of computed variables.
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Variance
expected squared deviation from the mean.
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Covariance
expected product of deviations of two variables from their means.
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y = Ax + b
Where:
• x — input random vector
• y — output random vector
• A — transformation matrix
• b — constant vector
Where:
• x — input random vector
• y — output random vector
• A — transformation matrix
• b — constant vector
General Linear Transformation:
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Y1 = a11X1 + a12X2 + b1
Y2 = a21X1 + a22X2 + b2
Y2 = a21X1 + a22X2 + b2
Two-Variable Transformation:
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Σyy = A Σxx Aᵀ
Where:
• Σxx — covariance matrix of input variables
• Σyy — covariance matrix of output variables
• A — transformation matrix
• Aᵀ — transpose of matrix A
Where:
• Σxx — covariance matrix of input variables
• Σyy — covariance matrix of output variables
• A — transformation matrix
• Aᵀ — transpose of matrix A
Covariance Propagation Formula:
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Jacobian Matrix (Jyx)
matrix containing the partial derivatives of output variables with respect to input variables.
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Σyy = Jyx Σxx Jyxᵀ
Nonlinear Covariance Propagation:
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Linear Function
function where output variables are linear combinations of input variables.
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Nonlinear Function
function where output variables are not linear combinations of input variables.
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σy² = a1²σx1² + a2²σx2² + ... + an²σxn²
Variance of a Linear Combination for uncorrelated input variables:
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σy² = (∂Y/∂X1)²σx1² + (∂Y/∂X2)²σx2² + ...
Variance of a Nonlinear Function:
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σx+y² = σx² + 2σxy + σy²
Variance of X + Y
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σxy = 0
Therefore:
σx+y² = σx² + σy²
Therefore:
σx+y² = σx² + σy²
Variance of X + Y For Independent Variables:
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μx+y = μx + μy
Mean of the Sum:
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Covariance Matrix (C)
represents the absolute variances and covariances of observations.
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Cofactor Matrix (Q)
represents relative variances and covariances of observations.
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C = kQ
Where:
• C — covariance matrix
• Q — cofactor matrix
• k — arbitrary scale constant or variance factor
Where:
• C — covariance matrix
• Q — cofactor matrix
• k — arbitrary scale constant or variance factor
Relationship Between C and Q:
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Diagonal Elements of Q
cofactor variances.
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Off-Diagonal Elements of Q
cofactor covariances.
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Weight Matrix (W)
matrix representing the relative importance or reliability of observations.
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W = Q⁻¹ = kC⁻¹
Weight Matrix (W)
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Inverse Cofactor Matrix
used to determine observation weights.
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Uncorrelated Measurements
measurements whose covariances are zero.
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Wii = k / sii²
Where:
• Wii — weight of the ith observation
• k — arbitrary scale constant
• sii² — variance of the ith observation
Where:
• Wii — weight of the ith observation
• k — arbitrary scale constant
• sii² — variance of the ith observation
Weight Matrix For Uncorrelated Observations:
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Unity Weights
all observations have equal weights.
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k = sii² = σ₀²
Variance Factor / Factor of Unit Weight:
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σ₀²
prior variance or variance factor of unit weight.
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