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using | FormType = Form |
| | JacobianForm or RootForm.
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using | OutputType = GaussianParams<OutputSpace, Scalar> |
| | The output's distribution.
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using | OutputVector = typename OutputType::VectorType |
| | An output value.
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using | OutputMatrix = typename OutputType::MatrixType |
| | A covariance over the output's differences.
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using | CrossMatrix = internal::CrossMatrixOf<InputSpace, OutputSpace, Scalar> |
| | The cross-covariance's type, and \( \Psi \)'s: input differences by output differences.
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| using | JacobianMatrix |
| | The Jacobian's type: output differences by input differences.
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| using | RootMatrix |
| | The root's type: over the input's differences.
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using | FactorMatrix = std::conditional_t<IsRoot, RootMatrix, JacobianMatrix> |
| | What the form holds beside its cross-covariance: \( H \) in the Jacobian form, \( S \) in the root form.
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| template<MatrixExpression Mean, MatrixExpression Factor, MatrixExpression Cross, MatrixExpression Explained, MatrixExpression Inputs, MatrixExpression Unexplained> |
| | OutputWithCross (OutputSpace space, Mean &&mean, Factor &&factor, Cross &&cross, Explained &&explained, Inputs &&inputs, Unexplained &&unexplained) |
| | The prediction in space with mean mean, normalized, and the parts of its linearization: factor, \( H \) or \( S \) as the form says, cross, \( \Sigma_{xy} \) or \( \Psi \), and the symmetric explained, inputs, and unexplained, whose sum is the output covariance.
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| OutputType const & | output () const & |
| OutputType | output () && |
| OutputMatrix const & | explainedCovariance () const |
| OutputMatrix const & | inputsCovariance () const |
| OutputMatrix const & | unexplainedCovariance () const |
| CrossMatrix const & | crossCovariance () const |
| CrossMatrix | crossCovariance () const |
| JacobianMatrix const & | jacobian () const |
| JacobianMatrix | jacobian () const |
| RootMatrix const & | root () const |
| CrossMatrix const & | whitenedCross () const |
template<typename InputSpace, typename
OutputSpace, typename Form, typename Scalar = double>
class Emrys::OutputWithCross< InputSpace, OutputSpace, Form, Scalar >
The result of transform() with an input marked by withCross(): the output's Gaussian and its linearization in that input, in the Form the method computes it, JacobianForm or RootForm (decisions.md [linearization-forms]).
condition() conditions the input on an observation of the output.
The output covariance is the sum of three parts, each computed as a product of quantities of its own size, so that none is recovered by subtraction: \( S_y = H P H^T + N + \Omega \), the part the marked input explains (explainedCovariance()), the part the other uncertain inputs explain (inputsCovariance()), such as noise passed to the function, and the part neither explains (unexplainedCovariance()), the function's curvature over the input's spread as the method sees it, which is zero for the Jacobian form. Uncertain inputs are independent, so the joint root is block-diagonal and the root form's \( \Psi \) splits by input: \( H P H^T = \Psi^T \Psi \) over the marked input's rows, and \( N \) the same sum over the other inputs' rows.
Each form holds what its method computes and derives the rest, as the table below says: the Jacobian form holds \( H \) and \( \Sigma_{xy} \), and the root form \( S \) and \( \Psi \), from which \( \Sigma_{xy} = S \Psi \) is a product and \( H = \Psi^T S^{-1} \) a solve.
| Asked for | Jacobian form | Root form |
| \( H P H^T \), \( N \), \( \Omega \), and their sum | held | held |
| \( \Sigma_{xy} \), for updates::Standard and updates::Symmetric | held | \( S \Psi \), a product |
| a full state's \( P_{xS} H_S^T \) from the blocks read | a product | \( P_{xS} S^{-T} \Psi \), a solve |
| \( H \): information forms, iterated updates, part-state Joseph | held | \( \Psi^T S^{-1} \), a solve |
updates::Joseph over the whole input is \( (I - KH) P (I - KH)^T + K R K^T \) in the Jacobian form and, since \( H S = \Psi^T \), \( (S - K \Psi^T)(S - K \Psi^T)^T + K R K^T \) in the root form: one product of the input's size cubed and no solve, still a sum of two positive semidefinite terms.
- Template Parameters
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template<typename InputSpace, typename
OutputSpace, typename Form, typename Scalar = double>
template<MatrixExpression Mean, MatrixExpression Factor, MatrixExpression Cross, MatrixExpression Explained, MatrixExpression Inputs, MatrixExpression Unexplained>
| Emrys::OutputWithCross< InputSpace, OutputSpace, Form, Scalar >::OutputWithCross |
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OutputSpace | space, |
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Mean && | mean, |
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Factor && | factor, |
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Cross && | cross, |
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Explained && | explained, |
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Inputs && | inputs, |
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Unexplained && | unexplained ) |
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The prediction in space with mean mean, normalized, and the parts of its linearization: factor, \( H \) or \( S \) as the form says, cross, \( \Sigma_{xy} \) or \( \Psi \), and the symmetric explained, inputs, and unexplained, whose sum is the output covariance.
Each matrix passed as an rvalue of its stored type is moved. For the library's transforms, which every prediction comes from.
1.18.0