# DataInterpolations.jl

DataInterpolations.jl is a library for performing interpolations of one-dimensional data. By "data interpolations" we mean techniques for interpolating possibly noisy data, and thus some methods are mixtures of regressions with interpolations (i.e. do not hit the data points exactly, smoothing out the lines). This library can be used to fill in intermediate data points in applications like timeseries data.

## API

All interpolation objects act as functions. Thus for example, using an interpolation looks like:

u = rand(5)
t = 0:4
interp = LinearInterpolation(u, t)
interp(3.5) # Gives the linear interpolation value at t=3.5


We can efficiently interpolate onto a vector of new t values:

t′ = 0.5:1.0:3.5
interp(t′)


In-place interpolation also works:

u′ = similar(u, length(t′))
interp(u′, t′)


## Available Interpolations

In all cases, u an AbstractVector of values and t is an AbstractVector of timepoints corresponding to (u,t) pairs.

• ConstantInterpolation(u,t) - A piecewise constant interpolation.

• LinearInterpolation(u,t) - A linear interpolation.

• QuadraticInterpolation(u,t) - A quadratic interpolation.

• LagrangeInterpolation(u,t,n) - A Lagrange interpolation of order n.

• QuadraticSpline(u,t) - A quadratic spline interpolation.

• CubicSpline(u,t) - A cubic spline interpolation.

• AkimaInterpolation(u, t) - Akima spline interpolation provides a smoothing effect and is computationally efficient.

• BSplineInterpolation(u,t,d,pVec,knotVec) - An interpolation B-spline. This is a B-spline which hits each of the data points. The argument choices are:

• d - degree of B-spline
• pVec - Symbol to Parameters Vector, pVec = :Uniform for uniform spaced parameters and pVec = :ArcLen for parameters generated by chord length method.
• knotVec - Symbol to Knot Vector, knotVec = :Uniform for uniform knot vector, knotVec = :Average for average spaced knot vector.
• BSplineApprox(u,t,d,h,pVec,knotVec) - A regression B-spline which smooths the fitting curve. The argument choices are the same as the BSplineInterpolation, with the additional parameter h<length(t) which is the number of control points to use, with smaller h indicating more smoothing.

• CubicHermiteSpline(du, u, t) - A third order Hermite interpolation, which matches the values and first (du) order derivatives in the data points exactly.

• PCHIPInterpolation(u, t) - a type of CubicHermiteSpline where the derivative values du are derived from the input data in such a way that the interpolation never overshoots the data.

• QuinticHermiteSpline(ddu, du, u, t) - A fifth order Hermite interpolation, which matches the values and first (du) and second (ddu) order derivatives in the data points exactly.

## Extension Methods

The follow methods require extra dependencies and will be loaded as package extensions.

• Curvefit(u,t,m,p,alg) - An interpolation which is done by fitting a user-given functional form m(t,p) where p is the vector of parameters. The user's input p is a an initial value for a least-square fitting, alg is the algorithm choice to use for optimize the cost function (sum of squared deviations) via Optim.jl and optimal ps are used in the interpolation. Requires using Optim.
• RegularizationSmooth(u,t,d;λ,alg) - A regularization algorithm (ridge regression) which is done by minimizing an objective function (l2 loss + derivatives of order d) integrated in the time span. It is a global method and creates a smooth curve. Requires using RegularizationTools.

## Plotting

DataInterpolations.jl is tied into the Plots.jl ecosystem, by way of RecipesBase. Any interpolation can be plotted using the plot command (or any other), since they have type recipes associated with them.

For convenience, and to allow keyword arguments to propagate properly, DataInterpolations.jl also defines several series types, corresponding to different interpolations.

The series types defined are:

• :linear_interp
• :quadratic_interp
• :lagrange_interp
• :quadratic_spline
• :cubic_spline
• :akima_interp
• :bspline_interp
• :bspline_approx
• :cubic_hermite_spline
• :pchip_interp
• :quintic_hermite_spline

By and large, these accept the same keywords as their function counterparts.