SplineLibrary provides the functionality for building, calibrating, and evaluating different kinds of splines for use in latent state representation. It implements the functionality behind spline design, spline constructions, customization, calibration, and evaluation of a wide variety of spline types and basis functions.
SplineLibrary achieves its design goals by implementing its functionality over several packages the perform the following:
· Univariate Function Package: The univariate function package implements the individual univariate functions, their convolutions, and reflections.
· Univariate Calculus Package: The univariate calculus package implements univariate difference based arbitrary order derivative, implements differential control settings, implements several integrand routines, and multivariate Wengert Jacobian.
· Spline Parameters Package: The spline parameters package implements the segment and stretch level construction, design, penalty, and shape control parameters.
· Splinbe Basis Function Set Package: The spline basis function set package implements the basis set, parameters for the different basis functions, parameters for basis set construction, and parameters for B Spline sequence construction.
· Spline Segment Package: The spline segment package implements the segment’s inelastic state, the segment basis evaluator, the segment flexure penalizer, computes the segment monotonicity behavior, and implements the segment’s complete constitutive state.
· Spline Stretch Package: The spline stretch package provides single segment and multi segment interfaces, builders, and implementations, along with custom boundary settings.
· Spline Grid/Span Package: The spline grid/span package provides the multistretch spanning functionality. It specifies the span interface, and provides implementations of the overlapping and the nonoverlapping span instances. It also implements the transition splines with custom transition zones.
· Spline PCHIP Package: The spline PCHIP package implements most variants of the local piecewise cubic Hermite interpolating polynomial smoothing functionality. It provides a number of tweaks for smoothing customization, as well as providing enhanced implementations of Akima, Preuss, and HaganWest smoothing interpolators.
·
Spline B Spline Package: The spline B Spline
package implements the raw and the processed basis B Spline hat functions. It
provides the standard implementations for the monic and the multic B Spline
Segments. It also exports functionality to generate higher order B Spline
Sequences.
·
Tension Spline Package: The tension spline
package implements closed form family of cubic tension splines laid out in
the basic framework outlined in Koch and Lyche (1989), Koch and Lyche (1993),
and Kvasov (2000).
1. Predictor Ordinates: The segment independent/input values.
2. Response Values: The segment dependent/output values.
3. C^{0}, C^{1}, and C^{2} Continuity: C^{0} refers to base function continuity. C^{1} refers to the continuity in the first derivative, and C^{2} refers to continuity in the second.
4. Local Piecewise Parameterized Splines: Here the space formulation is in the local variate space that spans 0 to 1 within the given segment – this is also referred to as piecewise parameterization.
5. Bias: This is the first term in the Spline Objective Function – essentially measures the exactness of fit.
6. Variance: This refers to the second and the subsequent terms in the Spline Objective Function – essentially measures the curvature/roughness.
1. Definition: “Spline is a sufficiently smooth polynomial function that is piecewisedefined, and possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as knots).^{“} [Spline (Wiki), Judd (1998), Chen (2009)]
2. Drivers:
a. Lower degree, gets rid of oscillation associated with the higher degrees [Runge’s phenomenon (Wiki)]
b. Easy, accurate higher degree smoothness specification
3. Basic Spline: Covered in [Spline (Wiki), Bartels, Beatty, and Barsky (1987), Judd (1998), De Boor (2001), Fan and Yao (2005), Chen (2009), Katz (2011)].
4. History: Schoenberg (1946), Ferguson (1964), Epperson (1998).
1. Overview
2. Calibration Framework SKU
3. Spline Builder SKU
4. B Splines
5. Polynomial Spline Stretches
6. Local Spline Stretches
7. Spline Calibration
8. Spline Jacobian
9. Shape Preserving Splines
10. KochLycheKvasov Tension Splines
11. Multidimensional Splines
12. References
13. Figures
14. Spline Library Software Components
1. Definition: Calibration is the process of inferring the latent state elastic properties from the specified inputs.
2. Classes of static fields: Elastic and inelastic
3. Calibrator Creation: On creation, objects acquire specific values for the constitutive inelastic fields. Volatile Latent State elastic fields may as yet be undefined.
4. Calibration is Inference: Since calibrated parameters are used for eventual prediction, calibration is essentially inference. Bayesian classification (an alternate, generalized calibration exercise) is inference too.
5. Calibration and entityvariate focus:
6. Latent State Construction off of hard/soft signals: Hard Signals are typically the truthness signals. Typically reduce to one calibration parameter per hard observation, and they include the following:
8. Truthness/Smoothness vs. Information Propagation: In segmentbysegment calibration challenges associated with inferring a composite Latent State, if the inference is based purely off of the truthness measurements, the information directionality/propagation/flow is irrelevant. Notions such as are important primarily owing to the smoothness axioms.
1. Latent State Specification: The latentstate here refers to the state whose dependent response values we wish to calibrate/infer as a function of the predictor ordinate. For e.g., in fixed income finance, the Prevailing Interest Rate, the survival, the forward rate, FX spot/forward would each constitute a potentially separate latent state that needs to be inferred.
2. Latent State Quantification Metric: A given latent state may be described by one/more alternate, mutually exclusive quantification metric. Again, the discounting latent state may be quantified using a discount factor, zero rate etc:
3. Manifest Measure: The Latent State may be inferred using a variety of external experimental measurements each of which produces a convolved signal of the latent state. Each such signal is referred to as a manifest metric. Again, the discounting latent state may be inferred from the cash instrument manifest metric, the swap rate manifest metric etc:
1. Symbols and Definition:
2. Monotonicity: increases with increase in (and vice versa).
3. Convexity: should also be convex wherever is convex (and vice versa).
4. Smoothness: Smoothness (also called shapepreserving) corresponds to the least curvature. Even C^{0} can be “smooth”, and so is C^{k}.
5. Locality: Locality means that the dependence of is primarily only on and . This is advantageous to schemes that locally modify/insert the points.
6. Approximation Order: Approximation Order indicates the smallest polynomial degree by which departs from as the density of increases. More formally, it is the m in , where .
7. Other Desired Criteria:
8. Assessment of Monotonicity and Convexity: An individual segment can be assessed to be monotone/convex etc:, but from the data PoV, peaks, valleys, and inflection occur only at the knots. These can be assessed only at the span level.
Base Formulation
BSplines
Introduction
1. Motivation: As postulated by De Boor et. al. (see De Boor (2001)), B Splines have a geometric interpolant context – thereby with the correspondingly strong CADG/curve/surface construction focus. Smoothening occurs as a natural part of this.
2. k^{th} Order BSpline Interpolant: Higher order BSplines are defined by the recurrence where and if , and otherwise.
3. Recursive Interpolant Scheme: B Spline formulation is recursively interpolant, i.e., the order k spline is interpolant over the order splines on nodes and  this formulation automatically ensures nodal continuity.
4. BSpline Order Relationships: Assuming no coincident knots, the following statements are all EQUIVALENT/TRUE:
5. Expository Formulation:
6. Spline Coefficient Partition of Unity: Using the earlier formulation , it is easy to show that . This simply follows from the recursive nodal interpolation property.
7. Smoothness Multiplicity Order Linker: # smoothness conditions at knot + the multiplicity at the knot = BSpline Order.
8. Starting Node debiasing: Left node is always weighted by in the interpolation scheme, but the left node asymmetry is maintained because the denominator in , i.e., increases in length.
9. Other Single BSpline Properties:
10. Formulation off of Starting Node and Starting Order: Given the starting node and the starting order , the contribution to the node (i.e., m nodes after the start) and the order (i.e., n nodes after start) can be “series”ed as
11. Cardinal BSpline Knot Sequence: Knot sequence => Uniformly spaced knots, simplifying the interpolant/recursive analysis significantly  .
Range 











· Cardinal BSpline Order 3:
Range 











12. Noncoinciding B Spline Segment Relations:
Range 















13. Bernstein BSpline Knot Sequence: Knot sequence  occurs times, and occurs times.
· for .
· has derivatives at , and derivatives at  this is also referred to as smoothness conditions at , and smoothness conditions at .
14. BSpline vs. Spline: BSpline is just a single basis polynomial that is valid across a set of knots. “Spline” is a linear combination of such basis B Splines – i.e., the set of all the ’s.
15. Spline Definition: where . ’s are the coefficients – or nodal points  that can be interpolated.
B Spline Derivatives
1. BSpline Derivative Formulation:
·
2. B Spline Order 3 Nodal Slopes: The slopes match across the left and the right segment, as shown below, thereby making continuous.
Range 
Left
Slope 
Right
Slope 

 
0 








 
3. B Spline Continuity Condition: From the B Spline derivative formulation it is clear that if both and are continuous, then will be continuous. Given that B Spline order 3 is continuous, by induction, is continuous.
Polynomial Spline Basis
Function
1. Most direct polynomial spline fit is the Lagrange polynomial that passes through the sequence of given points.
2. Young (1971) was one of first to apply shapepreserving polynomial using diminished Lagrange Polynomials (Lagrange Polynomials (Wiki)), showing that comonotone interpolant with an upper bound on the polynomial degree exists (Raymon (1981)).
3. Knot Insertion and Control Techniques: Careful knot insertion can produce:
1. Bernstein Polynomial of degree n, and term : where .
2. Derivative of the Bernstein Polynomial: .
3. Bernstein Recurrence: .
4. Reduction of BSplines to Bernstein’s Polynomial: From the recurrence relation, it is clear that this is exactly the same recurrence as that for Bsplines, except that it happens over repeating knots at and .
Local Spline Stretches
Local Interpolating/Smoothing Spline Stretches
1. Hermite Cubic Splines: The “local information” here takes the form of user specified left/right slopes + calibration points.
a. 2 User Specified local slopes + 2 points => 4 sets of equations. Solve for the coefficients.
b. C^{1} continuity is maintained, and C^{2} continuity is not.
c. Segment control is completely local. Both the head and nonhead calibration are identical/analogous for this reason.
2. C^{1} Hermite Formal Definition: For C^{1}, the Hermite polynomial of degree is given as , where and are expressed in terms of the j^{th} Lagrange coefficient of degree , as
·
·
3. CatmullRom Cubic Splines (Catmull and Rom (1974)): Instead of explicitly specifying the left/right segment slopes, they are inferred from the “averages” of the prior and the subsequent points, i.e., , and . Here refers to the slope vector, and to the point vector.
a. Again, C^{1} continuity is maintained, and C^{2} continuity is not.
b. Segment control is not completely local, but still local enough – it only depends on the neighborhood of 3 points.
4. Cardinal Cubic Splines: This is a generalization of the CatmullRom spline with a tightener coefficient , i.e., , and . corresponds to tightening, and corresponds to loosening.
a. Again, C^{1} continuity is maintained, and C^{2} continuity is not.
b. Segment control is “local” in the CatmullRom sense  it only depends on the neighborhood of 3 points.
5. CatmullRom/Cardinal Splines as Interpolation Splines: As interpolating splines, both CatmullRom and Cardinal are primarily useful in heuristic knotinsertions – CatmullRom as linear in the gaps, and Cardinal as tense linear gap knots.
· The local knot point insertion may be generalized as follows: The targeted knot insertions follow the formulation paradigm , where is the set of the neighborhood points. Similar formulation (with potentially different function forms, of course) may be used for each of the derivatives. CatmullRom and cardinal use 1D, strictly neighboring adjacencies, as well as tense linear averaging.
6. HermiteBessel Splines: These splines use 4 basis functions per segment, therefore they are cubic polynomial, but . The first are set at each node as the first derivative of the quadratic that passes through , , and (the edges are handled slightly differently, as shown below). Specifically:
·
·
· for
7. Hyman’s Monotone Preserving Cubic Spline:
· Hyman (1983) applies the stringent conditions to preserve monotonicity by applying the de BoorSchwarz criterion.
· Define . If locally monotone (i.e., ), then set . If nonmonotone (i.e., ), then set .
· Put another way (Iwashita (2013)): For cubic polynomial splines, the first derivative should be in the range where .
· Adjustment for Spurious Extrema => To ensure that no spurious extrema is introduced in the interpolant, if , and if .
8. Hyman89 Extension to Hyman83: Doherty, Edelman, and Hyman (1989) relax the Hyman83 stringency posited for monotonicity preservation. Define the following:
·
·
·
·
· If , , , , and all have the same sign, then .
· If , , , , and all have the same sign, then .
· Finally, set if , and otherwise.
9. Hyman’s Monotone Preserving Quintic Spline:
· For quintic polynomial splines, the first derivative should be in the range where .
· The constraint on the second derivative is: .
· Monotonicity Preserving Quintic Spline => Enhancement of the criterion established by de Boor and Schwartz (1977) (Hyman (1983), Doherty, Edelman, and Hyman (1989)).
o Set if , and otherwise. Then
o If , then , AND:
o If , then .
· Second Derivative Tests for Monotonicity Preserving Quintic Spline =>
o Define the following constants:
§ .
§ .
§ if , and otherwise.
§ if , and otherwise.
o Define Ranges and as:
§ , AND
§ .
If and overlap, then should lie in their common range. If they do not overlap, i.e., , reset as . Setting this ensures that and overlap, so the other tests aspects may now continue.
10. Harmonic Splines: Introduced by Fritsch and Butland (1984) as:
· if , if . Boundary Conditions are:
·
·
· Harmonic Spline Monotonicity Filter =>
o if AND AND
o if
o if
o if AND AND
o if
o if
· Continuous Limiters => For harmonic splines, as the predictor ordinate widths become identical , setting , we get . This is the Van Leer limit (Van Leer (1974)). Huynh (1993) reviews several such limiters.
· Shortcoming of these Limiters => Since they rely on min/max/modulus functions, by definition they are not smooth close to transition edge. This is rectified by Le Floc’h (2013), who defines a new limiter based on rational functions: for , and otherwise. This produces a stable interpolator.
11. Akima Cubic Interpolator (Akima (1970)):
13. ShapePreserving Knotbased Cubic: Ideas are taken from the awesome paper by Pruess (1993). The basic idea is to take the interval , and partition it into 3 parts by locating the two knots at , and . Evolve the criteria for the selection of and (and, of course, their corresponding responses) such that the local spline has shapepreserving feature, and avoids being global (i.e., preserves locality).
a. Using the above notations, the basic equations are:
· for .
· for .
· for .
b. The corresponding maintenance solution then becomes (in terms of , , , and , whose specification will then complete the inference):
· .
· .
· .
· .
· .
· .
· .
· .
· .
c. Choice of and : and may be generated using typical generation schemes (e.g., using the Fritsch and Butland (1984) algorithm).
d. The Preuss Inequalities: It is specified as follows. Set
· ; ;
· If and , you are done, since the chosen and also preserve convexity – you can go and set the second derivatives, and set and .
e. Mismatch in the Preuss Inequalities: If the Preuss inequalities are not met, and need to be modified such that where are obtained using the double sweep algorithm below.
f. Preuss (1993) Double Sweep: First find and from the following regimes:
· If , ; .
· If , ; .
· If , ; .
· Finally the and initializations are set from:
· If , ; .
· If , ; .
· Preuss (1993) Backward Sweep for : First set , then set using the following:
o If , then , and .
o If , then , and .
o If , then , and .
· Preuss (1993) Setting 2^{nd} Derivatives: .
· Preuss (1993) Final Step – Setting and : Setting , verify the following inequalities:
o
o
o If these inequalities are satisfied, then you have your , , , and . Otherwise, reduce and till they are satisfied.
Space Curves and Loops
Spline Segment Calibration
Smoothing Best Fit Splines
1. Definition: Here the treatment is limited to within a segment. In this, the segment coefficients are calibrated to the following inputs:
2. Nomenclature:
3. Spline Set Setup:
4. Best Fit Penalizer Setup:
5. Curvature Penalizer Setup:
6. Second Derivative:
a. !
7. Joint Linearized Minimizer Setup:
8. Number of Equations/Unknowns Review:
a. For intermediate segments, the following equations determine the unknowns:
i. Number of Continuity Constraints =>
ii. Number of Left/Right Node Values => 2
iii. Number of Roughness Penalizer Constraint => at least 1
iv. Thus, minimal number of degrees of freedom on a persegment basis: . This will be the number of “free” parameters we will use for to extract for each segment.
b. For left most segments, the following equations determine the unknowns:
i. Number of Left/Right Node Values => 2
ii. Number of Roughness Penalizer Constraint => at least 1
iii. Thus, for the set of parameters, the number of undetermined parameters:
c. For the span as a whole, the number of degrees of freedom/undetermined parameters is . You may determine:
i. The rightmost second derivative, AND
ii. Possibly, the leftmost second derivative
iii. For , this will complete the set of undetermined coefficients.
Segment Best Fit Response with Constraint
Matching
1. Purpose: Here we assume that a linear transform exists the hidden state quantification metric and the measurement manifest metric.
2. Caveat with the Segmentwise Representation: Optimizing on certain constraints (such as multisegment constraints) now ends up producing a highly nonsparse, dense matrix. This is simply a reflection on the multisegment spanning nature of the constraint and the eventual optimization.
3. Constraint and LeastSquare Spec: where and . Note that when the hidden state quantification metric is identically the same as the measurement manifest metric, and . This corresponds to computing the leastsquare minimization over the observations.
4. Constraint Formulation Development: , or , where . The parallel between this and the original leastsquares formulation can now be extended in a straightforward manner.
5. Weighted Constrained LSM:
6. Optimization of :
Spline Jacobian
1. Segment Quote Jacobian: Formulation for quote Jacobian is different than those for the coefficient edge value Jacobian, since the former automatically figures in the design matrix in the sensitivity matrix extractor pseudocalibration stage. Thus, the quote sensitivities are effectively external sensitivity constraints transmitted via the design matrix quote sensitivities.
2. Quote Jacobian Matrix: The Quote Sensitivity coefficients are calibrated identically to that of the base coefficient sensitivities. This simply follows from the linearity of the quote sensitivity formulation. The only area where there is nonlinearity is in the product term , and that appears only at the constraint equations. Others are identically the same.
3. Latent State Quote Sensitivity: Spline Formulation of the Latent State automatically implies that the quote sensitivity of the latent state is restricted by the above, and is therefore also a spline in itself. This further implies that the boundary formulation is subject to the similar edge conditions as before.
4. Terminology and Nomenclature:
Shape Preserving Spline
Shape Preserving Tension Spline
1. Integrated vs. Partitioned Shape Controller: Integrated Shape Controllers apply shape control on a basis functionbybasis function basis (certain basis functions such as flat/linear polynomial functions need no shape controller applied on them). Partitioned shape controllers apply shape control on a segmentbysegment basis.
2. Shape
Controller Parameter Types:
3. Shape
Control as part of Basis Function formulation:
4. Drawbacks
of Shape Control as part of Basis Function formulation:
5. Shape
Control using overdetermined Basis Function Set:
6. Drawbacks
of Shape Control using Overdetermined Basis Function formulation:
7. Potentially
Best of Both – Partitioned Basis and Shape Control:
8. Drawbacks
of Using Partitioned Basis Functions:
9. Partitioned vs. Integrated Tension Splines: Partitioned splines are designed such that the interpolant functional and the shape control functional are separated by formulation (e.g., rational splines). Integrated tension splines are formulated such that the shape preservation is an inherent consequence of the formulation, and there is no separation between the interpolant and the shape control functionality.
10. Explicit Shape Preservation Control in Partitioned Splines: , where is the interpolant, and is the shape controller. Typically is determined (among other things) by the continuity criterion , and contains an explicit design parameter for shape control (for e.g., in the case of rational splines).
11. Shape Control Design: Asymptotically, depending on the shape design parameter , should switch between linear and polynomial (i.e., typically cubic – Qu and Sarfraz (1997)). Further, design such that , so that and .
12. Rational Cubic Spline Formulation:
13. Rational Cubic Spline Coefficients:
14. Rational Cubic Spline Derivatives:
15. Designing for the Segment Inflection/Extrema Control:
16. Coconvex
choice for : A similar analysis can be done to make the spline coconvex,
but the corresponding formulation requires a nonlinear solution for .
17. Generalized
Shape Controlling Interpolator: Given a pair of points , a spline , and a spline , we define a shape controlling interpolator spline by , with the constraints .
18. Generically
Partitioned Spline Derivative:
19. Partitioned
Interpolating Spline Coefficient: Given ,
20. Interpolating Polynomial
Splines of Degree n: Given ,
i. Mathematical simplicity
ii. Completeness.
21. “Derivative
Completeness” Nature of Polynomial Basis Function: One big advantage for
polynomial basis functions is that they are “derivative complete” in the local
as well as global sense, i.e., the basis polynomials are
sufficient to uniquely determine the continuity
constraint. This is not true of nonpolynomial basis functions (exponential basis
functions, for e.g., need an infinite number of derivatives for completely
derivative coefficient determination), therefore their shape needs global
determination.
22. Polynomial
Interpolating Spline Coefficient microJack:
·
,,
·
,,
·
,,
23. Curvature
Design in Integrated Tension Splines: Cubic spline is interpolant on across the nodes, and
linear spline is interpolant on y. Thus, (the tension spline
interpolant) offers the tightness vs. curvature smoothness tradeoff.
24. Basis Function Interpolant:
25. Integrated Tension Spline Types: Sets containing both exponential and hyperbolic basis splines and a linear spline satisfy .
26. Exponential Basis Functions:
27. Hyperbolic Basis Functions:
28. Alternate specifications of the segment interpolation (Trojand (2011)). Renka (1987) provides techniques for setting σ under several circumstances:
o
Finding σ when f is
bound.
§
To get the minimum
tension factor required we need to find the zeros of f’ (Renka (1987)).
o
Finding σ when f’ is
bound.
§
To get the minimum
tension factor required we need to find the zeros of f’ (Renka (1987)).
o
Finding σ from the bound
values of convexity/concavity (Renka
(1987)).
29. Problems with Hyperbolic/Tension Splines:
14. Advantage of Basis Curve Optimizing Formulation: This formulation can readily/easily incorporate linearized constraints in an automatic manner – as long as the explicit constraints are recast to be specified with the current segment.
KochLycheKvasov Tension Splines
1. Exponential BSpline Specification: Expounded in detail in Cline (1974), Koch and Lyche (1989), Koch and Lyche (1993), and Kvasov (2000). First extend the knot set with 6 new points , ¸¸¸, and such that and , but arbitrary otherwise.
2. Exponential Hat Functions:
3. Properties of the Splines: as defined above is the basis on top of which all the higher order splines are built. is nonzero only for , where .
4. Layout of Base Monic Setup: With reference to figure 9, the monic basis function may be estimated from the corresponding primitive hat functions and (referred to as A and B respectively in Figure 9) as:
5. Monic BSpline Normalizer:
6. Monic BSpline Cumulative Normalized Integrand:
7. Monic BSpline Scaled Integrand:
8. Monic BSpline Scaled Integrand: .
9. Quadratic and Cubic Exponential Tension Splines: Higher order splines are recursively defined from where:
Here . Further, and correspond to quadratic and cubic tension splines, respectively.
10. Similarities between Exponential Tension BSplines and Polynomial BSplines: Notice the similarities, the iterative higherorder definitions, and the partition of unity as well.
11. Cubic Exponential Tension BSpline: This corresponds to the case, i.e., , with validity in the interval .
12. Piecewise Cubic Interpolant Expansion: Remember that, no matter what the basis tension functions are, for piecewise tension continuity, they are expected to satisfy for , i.e., this entity varies linearly across the segment.
13. Tension Spline Curvature Penalizing Norm: The pure curvature penalizer may now be altered to become a curvature + length penalizer. Thus, . Notice that both and (i.e., separated squares) are included individually in the set up.
14. Constrained Optimizer Estimate for : If the RMS bestfit error is to be limited to where is an extraneously specified closeness of fit metric, the constraint may be expressed as .
15. Drawbacks of the above method: This involves yet another nonlinear root extractor. Other nonlinear root extraction parts in curve building are:
Thus, the stability of the precision norm technique outlined above is riddled with challenges.
16. Parallel with HaganWest Forward Interpolator: In HaganWest (Hagan and West (2006)) minimalist quadratic interpolator, the segment length is incorporated in a slightly different way – as a minimizer of the jumps at the knots (i.e., if , minimizing at the knots).
17. The other Tension Splines: They all have the property that the tension parameter moves smoothly from cubic (low tension) to linear (high tension), and have different forms for and . These forms may make them computationally less expensive too.
o Nonuniform Rational Cubic Tension Spline with linear Denominator =>
§
§
o Nonuniform Rational Cubic Tension Spline with Quadratic Denominator =>
§
§
o Nonuniform Exponential Rational Spline =>
§
§
18. Tension Implied by the Basis Function Set: Given the tension interpolant relation inside a given segment , we can infer from .
19. Caveat for using KLKtype Splines for Local State Shape Proxying: Often (and this may be true for other B Splines as a whole too), the B Spline basis choice may produce segment node edge values and their corresponding derivatives of zero. In this case, you may have a singular calibration matrix that does not calibrate. In particular, this is the case for iterated B Splines that are constructed to vanish and fade rapidly at the edges. This poses for problems for segmentlocal splines (that may span between 0 and 1 within a given segment).
1. Penalty Minimizer Estimator Metric: Choice of the “normalized curvature area” shown in figures 5) and 6) are two possible penalty estimator choices. Obviously, closer the area is to zero, the better the penalizing match is.
2. Dimensionless Penalizing Fit Metric: Choosing the representation in 5), and recognizing that the segment is set in the flat base , we can derive the representation in 7).
3. Dimensionless Penalty Estimator (DPE): Using Figure 7), we now define DPE as
4. Pros/Cons of the above Representation of DPE: If the basis functions have neardelta functional forms (Figure 8), DPE will still remain , and the metric is not very meaningful in that case. Fortunately, such deltatype basis functions are rare.
5. Aggregate DPE Measure: Need a consolidated DPE metric that spans across all the segments in a span, i.e., the span DPE.
Least Squares Best Fit + Curvature + Segment Length Penalty Formulation
1. Nomenclature:
2. The Formulation: where .
3. SegmentLevel Decomposition: Segmentlevel decomposition ensures optimal segment coefficient formulation to within the boundaries of a segment (from the least squares fit point of view) – however, not necessarily global optimum. Further, these optimal constraints provide an extra degree of freedom at the segment level, and not necessarily at the stretch/span level.
4. SegmentLevel Decomposition Formulation: where , , and .
5. Least Squares Minimization Review: From earlier,
6. Curvature Penalty Minimization: Again, from earlier,
7. Segment Length Penalty Minimization: Similar to the curvature penalty, we get,
8. Combining it all: .
2. Bipolynomial
2D Spline: For the 2D Segment Range and , working in the local variate space and , we transform the spline basis on to the local
representation basis as .
3. Bilinear
2D Spline: This produces a surface. Here , therefore the first derivatives (and on) are discontinuous
on the grid boundaries. From the observation set , we get from the boundary match the following values for :
·
·
·
·
4. BiCubic
Interpolation: This produces a surface. Here , therefore the first, second, and the first cross
derivatives (i.e., , , , , , and ) are continuous across the grid boundaries. From the
observation set , their first derivatives, and their cross derivatives, we
get from the boundary match the following values for as before. The common
way is to cast these as a sequence of 2D relations by unraveling the continuity
constraints, and thereby linearizing the formulation.