Data.Manifold.Shade

Shades

data Shade x

A Shade is a very crude description of a region within a manifold. It can be interpreted as either an ellipsoid shape, or as the Gaussian peak of a normal distribution (use http://hackage.haskell.org/package/manifold-random for actually sampling from that distribution).

For a precise description of an arbitrarily-shaped connected subset of a manifold, there is Region, whose implementation is vastly more complex.

Constructors

Shade

Fields

_shadeCtr :: !(Interior x)

_shadeExpanse :: !(Metric' x)

Instances

ImpliesMetric Shade

IsShade Shade

(Show (Interior x), Show (Metric' x), WithField ℝ PseudoAffine x) => Show (Shade x)

PseudoAffine x => Semimanifold (Shade x)

(WithField ℝ PseudoAffine x, Geodesic (Interior x), SimpleSpace (Needle x)) => Geodesic (Shade x)

type MetricRequirement Shade x = (Manifold x, SimpleSpace (Needle x))

type Interior (Shade x) = Shade x

type Needle (Shade x) = Needle x

pattern (:±) :: () => (WithField ℝ Manifold x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x infixl 6

Span a Shade from a center point and multiple deviation-vectors.

data Shade' x

A “co-shade” can describe ellipsoid regions as well, but unlike Shade it can be unlimited / infinitely wide in some directions. It does OTOH need to have nonzero thickness, which Shade needs not.

Constructors

Shade'

Fields

_shade'Ctr :: !(Interior x)

_shade'Narrowness :: !(Metric x)

Instances

ImpliesMetric Shade'

IsShade Shade'

LtdErrorShow x => Show (Shade' x)

AffineManifold x => Semimanifold (Shade' x)

(WithField ℝ AffineManifold x, Geodesic x, SimpleSpace (Needle x)) => Geodesic (Shade' x)

type MetricRequirement Shade' x = (Manifold x, SimpleSpace (Needle x))

type Interior (Shade' x) = Shade' x

type Needle (Shade' x) = Needle x

(|±|) :: forall x. WithField ℝ EuclidSpace x => x -> [Needle x] -> Shade' x infixl 6

Similar to ':±', but instead of expanding the shade, each vector restricts it. Iff these form a orthogonal basis (in whatever sense applicable), then both methods will be equivalent.

Note that '|±|' is only possible, as such, in an inner-product space; in general you need reciprocal vectors (Needle') to define a Shade'.

class IsShade shade where

Methods

shadeCtr :: Lens' (shade x) (Interior x)

Access the center of a Shade or a Shade'.

occlusion :: (PseudoAffine x, SimpleSpace (Needle x), s ~ Scalar (Needle x), RealFloat' s) => shade x -> x -> s

Check the statistical likelihood-density of a point being within a shade. This is taken as a normal distribution.

factoriseShade :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => shade (x, y) -> (shade x, shade y)

coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y) => shade x -> shade y

orthoShades :: (PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => shade x -> shade y -> shade (x, y)

ASCII version of '✠'.

linIsoTransformShade :: (SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y, Num' (Scalar x)) => (x +> y) -> shade x -> shade y

projectShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade y -> shade x

Squash a shade down into a lower dimensional space.

embedShade :: (Semimanifold x, Semimanifold y, Object (Affine s) (Interior x), Object (Affine s) (Interior y), SemiInner (Needle x), SemiInner (Needle y)) => Embedding (Affine s) (Interior x) (Interior y) -> shade x -> shade y

Include a shade in a higher-dimensional space. Notice that this behaves fundamentally different for Shade and Shade'. For Shade, it gives a “flat image” of the region, whereas for Shade' it gives an “extrusion pillar” pointing in the projection's orthogonal complement.

Instances

IsShade Shade'

IsShade Shade

Lenses

shadeExpanse :: Lens' (Shade x) (Metric' x)

shadeNarrowness :: Lens' (Shade' x) (Metric x)

Construction

fullShade :: WithField ℝ PseudoAffine x => Interior x -> Metric' x -> Shade x

fullShade' :: WithField ℝ PseudoAffine x => Interior x -> Metric x -> Shade' x

pointsShades :: (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x]

Attempt to find a Shade that describes the distribution of given points. At least in an affine space (and thus locally in any manifold), this can be used to estimate the parameters of a normal distribution from which some points were sampled. Note that some points will be “outside” of the shade, as happens for a normal distribution with some statistical likelyhood. (Use pointsCovers if you need to prevent that.)

For nonconnected manifolds it will be necessary to yield separate shades for each connected component. And for an empty input list, there is no shade! Hence the result type is a list.

pointsShade's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x]

pointsCovers :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade x]

Like pointsShades, but ensure that all points are actually in the shade, i.e. if [Shade x₀ ex] is the result then metric (recipMetric ex) (p-x₀) ≤ 1 for all p in the list.

pointsCover's :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => [Interior x] -> [Shade' x]

coverAllAround :: forall x s. (Fractional' s, WithField s PseudoAffine x, SimpleSpace (Needle x)) => Interior x -> [Needle x] -> Shade x

Evaluation

prettyShowShade' :: LtdErrorShow x => Shade' x -> String

class Refinable m => LtdErrorShow m where

Minimal complete definition

showsPrecShade'_errorLtdC

Methods

prettyShowsPrecShade' :: Int -> Shade' m -> ShowS

Instances

LtdErrorShow ℝ⁰

LtdErrorShow ℝ

LtdErrorShow ℝ⁴

LtdErrorShow ℝ³

LtdErrorShow ℝ²

(LtdErrorShow x, LtdErrorShow y, (~) * (Scalar (DualVector (Needle' x))) (Scalar (DualVector (Needle' y)))) => LtdErrorShow (x, y)

Misc

(✠) :: (IsShade shade, PseudoAffine x, SimpleSpace (Needle x), PseudoAffine y, SimpleSpace (Needle y), Scalar (Needle x) ~ Scalar (Needle y)) => shade x -> shade y -> shade (x, y) infixl 5

Combine two shades on independent subspaces to a shade with the same properties on the subspaces (see factoriseShade) and no covariance.

intersectShade's :: forall y. Refinable y => NonEmpty (Shade' y) -> Maybe (Shade' y)

class (WithField ℝ PseudoAffine y, SimpleSpace (Needle y)) => Refinable y where

Class of manifolds which can use Shade' as a basic set type. This is easily possible for vector spaces with the default implementations.

Minimal complete definition

Nothing

Methods

subShade' :: Shade' y -> Shade' y -> Bool

a subShade' b ≡ True means a is fully contained in b, i.e. from minusLogOcclusion' a p < 1 follows also minusLogOcclusion' b p < 1.

refineShade' :: Shade' y -> Shade' y -> Maybe (Shade' y)

Intersection between two shades.

convolveMetric :: Functor p => p y -> Metric y -> Metric y -> Metric y

If p is in a (red) and δ is in b (green), then p.+~^δ is in convolveShade' a b (blue).

Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/ShadeCombinations.ipynb#shadeConvolutions

convolveShade' :: Shade' y -> Shade' (Needle y) -> Shade' y

Instances

Refinable ℝ⁰

Refinable ℝ

Refinable ℝ⁴

Refinable ℝ³

Refinable ℝ²

Refinable ℝ¹

(Refinable a, Refinable b, (~) * (Scalar (DualVector (DualVector (Needle b)))) (Scalar (DualVector (DualVector (Needle a))))) => Refinable (a, b)

(SimpleSpace a, SimpleSpace b, Refinable a, Refinable b, (~) * (Scalar a) ℝ, (~) * (Scalar b) ℝ, (~) * (Scalar (DualVector a)) ℝ, (~) * (Scalar (DualVector b)) ℝ, (~) * (Scalar (DualVector (DualVector a))) ℝ, (~) * (Scalar (DualVector (DualVector b))) ℝ) => Refinable (LinearMap ℝ a b)

mixShade's :: forall y. (WithField ℝ Manifold y, SimpleSpace (Needle y)) => NonEmpty (Shade' y) -> Maybe (Shade' y)

Weakened version of intersectShade's. What this function calculates is rather the weighted mean of ellipsoid regions. If you interpret the shades as uncertain physical measurements with normal distribution, it gives the maximum-likelyhood result for multiple measurements of the same quantity.

dualShade :: forall x. (PseudoAffine x, SimpleSpace (Needle x)) => Shade x -> Shade' x

Misc

Shades

shadesMerge

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x))

=> ℝ

How near (inverse normalised distance, relative to shade expanse) two shades must be to be merged. If this is zero, any shades in the same connected region of a manifold are merged.

-> [Shade x]

A list of n shades.

-> [Shade x]

m ≤ n shades which cover at least the same area.

Attempt to reduce the number of shades to fewer (ideally, a single one). In the simplest cases these should guaranteed cover the same area; for non-flat manifolds it only works in a heuristic sense.

pointsShades' :: forall x. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x)) => Metric' x -> [x] -> [([x], Shade x)]

pseudoECM :: forall x p. (WithField ℝ PseudoAffine x, SimpleSpace (Needle x), Functor p) => p x -> NonEmpty x -> (x, ([x], [x]))

data WithAny x y

Essentially the same as (x,y), but not considered as a product topology. The Semimanifold etc. instances just copy the topology of x, ignoring y.

Constructors

WithAny

Fields

_untopological :: y

_topological :: !x

Instances

AdditiveGroup x => Monad (WithAny x)

Functor (WithAny x)

AdditiveGroup x => Applicative (WithAny x)

(Show x, Show y) => Show (WithAny x y)

Generic (WithAny x y)

(NFData x, NFData y) => NFData (WithAny x y)

(VectorSpace x, Monoid y) => VectorSpace (WithAny x y)

AffineSpace x => AffineSpace (WithAny x y)

(AdditiveGroup x, Monoid y) => AdditiveGroup (WithAny x y)

PseudoAffine x => PseudoAffine (WithAny x y)

Semimanifold x => Semimanifold (WithAny x y)

type Rep (WithAny x y)

type Diff (WithAny x y) = Diff x

type Scalar (WithAny x y) = Scalar x

type Interior (WithAny x y) = WithAny (Interior x) y

type Needle (WithAny x y) = Needle x

shadeWithAny :: y -> Shade x -> Shade (x `WithAny` y)

shadeWithoutAnything :: Shade (x `WithAny` y) -> Shade x

Local data fit models

estimateLocalJacobian :: forall x y. (WithField ℝ Manifold x, Refinable y, SimpleSpace (Needle x), SimpleSpace (Needle y)) => Metric x -> [(Local x, Shade' y)] -> Maybe (Shade' (LocalLinear x y))

estimateLocalHessian :: forall x y. (WithField ℝ Manifold x, Refinable y, Geodesic y, FlatSpace (Needle x), FlatSpace (Needle y)) => NonEmpty (Local x, Shade' y) -> QuadraticModel x y

data QuadraticModel x y

Constructors

QuadraticModel

Fields

_quadraticModelOffset :: Interior y

_quadraticModel :: Shade (Needle y, (Needle x +> Needle y, Needle x `⊗〃+>` Needle y))

Differential equations

type DifferentialEqn x ð y = Shade (x, y) -> LocalDifferentialEqn x ð y

data LocalDifferentialEqn x ð y

Constructors

LocalDifferentialEqn

Fields

_predictDerivatives :: Shade' ð -> Maybe (Shade' (LocalLinear x y))

_rescanDerivatives :: Shade' y -> Shade' (LocalLinear x y) -> Shade' (LocalBilinear x y) -> (Maybe (Shade' y), Maybe (Shade' ð))

propagateDEqnSolution_loc :: forall x y ð. (WithField ℝ Manifold x, Refinable y, Geodesic (Interior y), WithField ℝ AffineManifold ð, Geodesic ð, SimpleSpace (Needle x), SimpleSpace (Needle ð)) => DifferentialEqn x ð y -> LocalDataPropPlan x (Shade' y, Shade' ð) (Shade' y) -> Maybe (Shade' y)

data LocalDataPropPlan x ym yr

Constructors

LocalDataPropPlan

Fields

_sourcePosition :: !(Interior x)

_targetPosOffset :: !(Needle x)

_sourceData, _targetAPrioriData :: !ym

_relatedData :: [(Needle x, yr)]

Instances

(Show (Interior x), Show ym, Show yr, Show (Needle x)) => Show (LocalDataPropPlan x ym yr)

Range interpolation

rangeOnGeodesic :: forall i m. (WithField ℝ PseudoAffine m, Geodesic m, SimpleSpace (Needle m), WithField ℝ IntervalLike i, SimpleSpace (Needle i)) => m -> m -> Maybe (Shade i -> Shade m)

Warning: This function never worked properly. Use rangeWithinVertices.

rangeWithinVertices :: forall s i m t. (RealFrac' s, WithField s PseudoAffine i, WithField s PseudoAffine m, Geodesic i, Geodesic m, SimpleSpace (Needle i), SimpleSpace (Needle m), AffineManifold (Interior i), AffineManifold (Interior m), Object (Affine s) (Interior i), Object (Affine s) (Interior m), Traversable t) => (Interior i, Interior m) -> t (i, m) -> Maybe (Shade i -> Shade m)