makima
Modified Akima piecewise cubic Hermite interpolation
Description
Examples
Akima Interpolation of Cosine Data
Use makima
to interpolate a cosine curve over unevenly spaced sample points.
x = [0 1 2.5 3.6 5 7 8.1 10]; y = cos(x); xq = 0:.25:10; yq = makima(x,y,xq); plot(x,y,'o',xq,yq,'')
With oscillatory functions, the Akima algorithm flattens the curve near local extrema. To compensate for this flattening, you can add more sample points near the local extrema.
Add sample points near $\mathit{x}=6.5$ and $\mathit{x}=9$ and replot the interpolation.
x = [0 1 2.5 3.6 5 6.5 7 8.1 9 10]; y = cos(x); xq = 0:.25:10; yq = makima(x,y,xq); plot(x,y,'o',xq,yq,'')
Data Interpolation with spline
, pchip
, and makima
Compare the interpolation results produced by spline
, pchip
, and makima
for two different data sets. These functions all perform different forms of piecewise cubic Hermite interpolation. Each function differs in how it computes the slopes of the interpolant, leading to different behaviors when the underlying data has flat areas or undulations.
Compare the interpolation results on sample data that connects flat regions. Create vectors of x
values, function values at those points y
, and query points xq
. Compute interpolations at the query points using spline
, pchip
, and makima
. Plot the interpolated function values at the query points for comparison.
x = 3:3; y = [1 1 1 0 1 1 1]; xq1 = 3:.01:3; p = pchip(x,y,xq1); s = spline(x,y,xq1); m = makima(x,y,xq1); plot(x,y,'o',xq1,p,'',xq1,s,'.',xq1,m,'') legend('Sample Points','pchip','spline','makima','Location','SouthEast')
In this case, pchip
and makima
have similar behavior in that they avoid overshoots and can accurately connect the flat regions.
Perform a second comparison using an oscillatory sample function.
x = 0:15; y = besselj(1,x); xq2 = 0:0.01:15; p = pchip(x,y,xq2); s = spline(x,y,xq2); m = makima(x,y,xq2); plot(x,y,'o',xq2,p,'',xq2,s,'.',xq2,m,'') legend('Sample Points','pchip','spline','makima')
When the underlying function is oscillatory, spline
and makima
capture the movement between points better than pchip
, which is aggressively flattened near local extrema.
Akima Interpolation with Piecewise Polynomial Structure
Create vectors for the sample points x
and values at those points y
. Use makima
to construct a piecewise polynomial structure for the data.
x = 5:5; y = [1 1 1 0 0 1 1 2 2 2 2]; pp = makima(x,y)
pp = struct with fields:
form: 'pp'
breaks: [5 4 3 2 1 0 1 2 3 4 5]
coefs: [10x4 double]
pieces: 10
order: 4
dim: 1
The structure contains the information for 10 polynomials of order 4 that span the data. pp.coefs(i,:)
contains the coefficients for the polynomial that is valid in the region defined by the breakpoints [breaks(i) breaks(i+1)]
.
Use the structure with ppval
to evaluate the interpolation at several query points, and then plot the results. In regions with three or more constant points, the Akima algorithm connects the points with a straight line.
xq = 5:0.2:5; m = ppval(pp,xq); plot(x,y,'o',xq,m,'.') ylim([0.2 2.2])
Input Arguments
x
— Sample points
vector
Sample points, specified as a vector. The vector x
specifies the
points at which the data y
is given. The elements of
x
must be unique.
Data Types: single
 double
y
— Function values at sample points
vector  matrix  array
Function values at sample points, specified as a numeric vector, matrix, or array.
x
and y
must have the same length.
If y
is a matrix or array, then the values in the last dimension,
y(:,...,:,j)
, are taken as the values to match with
x
. In that case, the last dimension of y
must be
the same length as x
.
Data Types: single
 double
xq
— Query points
scalar  vector  matrix  array
Query points, specified as a scalar, vector, matrix, or array. The points specified
in xq
are the xcoordinates for the interpolated
function values yq
computed by makima
.
Data Types: single
 double
Output Arguments
yq
— Interpolated values at query points
scalar  vector  matrix  array
Interpolated values at query points, returned as a scalar, vector, matrix, or array.
The size of yq
is related to the sizes of y
and
xq
:
If
y
is a vector, thenyq
has the same size asxq
.If
y
is an array of sizeNy = size(y)
, then these conditions apply:If
xq
is a scalar or vector, thensize(yq)
returns[Ny(1:end1) length(xq)]
.If
xq
is an array, thensize(yq)
returns[Ny(1:end1) size(xq)]
.
pp
— Piecewise polynomial
structure
Piecewise polynomial, returned as a structure. Use this structure with the ppval
function to evaluate the interpolating polynomials at one or more
query points. The structure has these fields.
Field  Description 

form 

breaks  Vector of length 
coefs 

pieces  Number of pieces, 
order  Order of polynomials 
dim  Dimensionality of target 
Because the polynomial coefficients in coefs
are local
coefficients for each interval, you must subtract the lower endpoint of the
corresponding knot interval to use the coefficients in a conventional polynomial
equation. In other words, for the coefficients [a,b,c,d]
on the
interval [x1,x2]
, the corresponding polynomial is
$$f\left(x\right)=a{\left(x{x}_{1}\right)}^{3}+b{\left(x{x}_{1}\right)}^{2}+c\left(x{x}_{1}\right)+d\text{\hspace{0.17em}}.$$
More About
Modified Akima Interpolation
The Akima algorithm for onedimensional interpolation, described in [1] and [2], performs cubic interpolation to produce piecewise polynomials with continuous firstorder derivatives (C1). The algorithm avoids excessive local undulations.
If $${\delta}_{i}=\frac{{v}_{i+1}{v}_{i}}{{x}_{i+1}{x}_{i}}$$ is the slope on interval $$\left[{x}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{i+1}\right)$$, then the value of the derivative $${d}_{i}$$ at the sample point $${x}_{i}$$ is a weighted average of nearby slopes:
$${d}_{i}=\frac{{w}_{1}}{{w}_{1}+{w}_{2}}{\delta}_{i1}+\frac{{w}_{2}}{{w}_{1}+{w}_{2}}{\delta}_{i}.$$
In Akima's original formula, the weights are:
$$\begin{array}{l}{w}_{1}=\left{\delta}_{i+1}{\delta}_{i}\right,\\ {w}_{2}=\left{\delta}_{i1}{\delta}_{i2}\right.\end{array}$$
The original Akima algorithm gives equal weight to the points on both sides, evenly dividing an undulation.
When two flat regions with different slopes meet, the modification made to the original Akima algorithm gives more weight to the side where the slope is closer to zero. This modification gives priority to the side that is closer to horizontal, which is more intuitive and avoids overshoot. In particular, whenever there are three or more consecutive collinear points, the algorithm connects them with a straight line and thus avoids an overshoot.
The weights used in the modified Akima algorithm are:
$$\begin{array}{l}{w}_{1}=\left{\delta}_{i+1}{\delta}_{i}\right+\frac{\left{\delta}_{i+1}+{\delta}_{i}\right}{2},\\ {w}_{2}=\left{\delta}_{i1}{\delta}_{i2}\right+\frac{\left{\delta}_{i1}+{\delta}_{i2}\right}{2}.\end{array}$$
Compared to the spline
algorithm, the Akima algorithm produces
fewer undulations and is better suited to deal with quick changes between flat regions.
Compared to the pchip
algorithm, the Akima algorithm is not as
aggressively flattened and is therefore still able to deal with oscillatory data.
References
[1] Akima, Hiroshi. "A new method of interpolation and smooth curve fitting based on local procedures." Journal of the ACM (JACM) , 17.4, 1970, pp. 589–602.
[2] Akima, Hiroshi. "A method of bivariate interpolation and smooth surface fitting based on local procedures." Communications of the ACM , 17.1, 1974, pp. 18–20.
Extended Capabilities
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
The
makima(
syntax is not supported for distributed arrays.x
,y
)
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
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