## Interpolation with Curve Fitting Toolbox

*Interpolation* is a process for estimating values that lie
between known data points.

Interpolation involves creating of a function *f * that matches
given data values *y _{i}* at given data sites

*x*where

_{i}*f*(

*x*) =

_{i}*y*, for all

_{i}*i*.

Most interpolation methods create the interpolant *f* as the unique
function of the formula

$$f(x)={\displaystyle \sum _{j}{f}_{j}(x){a}_{j}},$$

where the form of the functions *f _{j}* depends
on the interpolation method.

For spline interpolation, the *f _{j}* are the

*n*consecutive B-splines

*B*(

_{j}*x*) =

*B*(

*x*|

*t*,...,

_{j}*t*

_{j+k}),

*j*= 1:

*n*, of order

*k*for a knot sequence

*t*

_{1}≤

*t*

_{2}≤ ... ≤

*t*

_{n + k}.

### About Interpolation Methods

Curve Fitting Toolbox™ supports the interpolation methods described in the following table.

Method | Description |
---|---|

Nearest neighbor | Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point. |

Linear | Linear interpolation. This method fits a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces. |

Natural neighbor | Natural neighbor interpolation. This method sets the value of an interpolated point to a weighted average of the nearest data points. The interpolating surface is C1 continuous, except at the sample points. |

Shape-preserving (PCHIP) | Piecewise cubic Hermite interpolation (PCHIP). This method preserves monotonicity and the shape of the data (for curves only). |

Cubic spline | Cubic spline interpolation. This method fits a different cubic polynomial between each pair of data points for curves, or between sets of three points for surfaces. |

Biharmonic (v4) | MATLAB |

Thin-plate spline | Thin-plate spline interpolation. This method fits smooth surfaces that also extrapolate well (for surfaces only). |

Interpolant surface fits use the MATLAB function `scatteredInterpolant`

for the linear,
nearest neighbor, and natural neighbor methods, and the MATLAB function `griddata`

for the cubic spline and
biharmonic methods. The thin-plate spline method uses the `tpaps`

function.

The interpolant method you use depends on several factors, including the characteristics of the data being fit, the required smoothness of the curve, speed considerations, and post-fit analysis requirements. The linear and nearest neighbor methods fit models efficiently, and the resulting curves are not very smooth. The natural neighbor, cubic spline, shape-preserving, and biharmonic methods take longer to fit models, and the resulting curves are very smooth.

For example, the following plot shows a nearest neighbor interpolant fit and a
shape-preserving (PCHIP) interpolant fit for the nuclear reaction data from the
`carbon12alpha.mat`

sample data set. The nearest neighbor
interpolant is not as smooth as the shape-preserving interpolant.

**Note**

Goodness-of-fit statistics, prediction bounds, and weights are not defined for interpolants. Additionally, the fit residuals are always 0 (within computer precision) because interpolants pass through the data points.

Biharmonic interpolant fits consist of radial basis function interpolants. All
other interpolants supported by Curve Fitting Toolbox are *piecewise polynomials* and consist of multiple
polynomials defined between data points. For cubic spline and PCHIP interpolation,
four coefficients describe each piece. Curve Fitting Toolbox uses a cubic (third-degree) polynomial to calculate the four
coefficients. Refer to the following for more information:

`spline`

for cubic spline interpolation`pchip`

for shape-preserving (PCHIP) interpolation, and for a comparison of PCHIP and cubic spline interpolation`scatteredInterpolant`

,`griddata`

, and`tpaps`

for surface interpolationIt is possible to fit a single polynomial interpolant to data, with a degree one less than the number of data points. However, the behavior of such fits is unpredictable between data points. Piecewise polynomials with lower-order segments do not diverge significantly from the fitting data domain, so they are useful for analyzing a wider range of data sets.

### Selecting an Interpolant Fit

#### Select Interpolant Fit Interactively

Open the Curve Fitter app by entering `curveFitter`

at the
MATLAB command line. Alternatively, on the **Apps** tab, in the **Math, Statistics and
Optimization** group, click Curve Fitter.

On the **Curve Fitter** tab, in the **Fit Type** section, select an
**Interpolant** fit. The app fits an interpolating curve or
surface that passes through every data point.

In the **Fit Options** pane, you can specify the
**Interpolation method** value.

For curve data, you can set **Interpolation method** to
`Linear`

, `Nearest neighbor`

,
`Cubic spline`

, or ```
Shape-preserving
(PCHIP)
```

. For surface data, you can set **Interpolation
method** to `Linear`

, ```
Nearest
neighbor
```

, `Natural neighbor`

, ```
Cubic
spline
```

, `Biharmonic (v4)`

, or ```
Thin-plate
spline
```

.

For surfaces, the Interpolant fit uses the `scatteredInterpolant`

function for
the `Linear`

, `Nearest neighbor`

, and
`Natural neighbor`

methods, the `griddata`

function for the
`Cubic Spline`

and `Biharmonic (v4)`

methods, and the `tpaps`

function for the
`Thin-plate spline`

method. Try the ```
Thin-plate
spline
```

method when you require both smooth surface interpolation
and good extrapolation properties.

**Tip**

If your data variables have very different scales, clear the
**Center and scale** check box to see the
difference in the fit. Normalizing the inputs might influence the
results of the piecewise `Linear`

and ```
Cubic
Spline
```

interpolation methods, and the ```
Nearest
neighbor
```

and `Natural neighbor`

surface
interpolation methods.

#### Fit Linear Interpolant Model Using the `fit`

Function

Load the `census`

sample data set.

`load census`

The variables `pop`

and `cdate`

contain data for the population size and the year the census was taken, respectively.

You can use the `fit`

function to fit any of the interpolant models described in Interpolant Model Names. In this case, fit a linear interpolant model using the `'linearinterp'`

option, and then plot the result.

```
f = fit(cdate,pop,'linearinterp');
plot(f,cdate,pop);
```

**Compare Linear Interpolant Models**

Load the `carbon12alpha`

sample data set. Create both nearest neighbor and PCHIP interpolant fits using the `'nearestinterp'`

and `'pchip'`

options.

load carbon12alpha f1 = fit(angle,counts,'nearestinterp'); f2 = fit(angle,counts,'pchip');

Compare the fitted curves `f1`

and `f2`

by plotting them in the same figure.

p1 = plot(f1,angle,counts); xlim([min(angle),max(angle)]) hold on p2 = plot(f2,'b'); hold off legend([p1;p2],'Counts per Angle','Nearest Neighbor','PCHIP',... 'Location','northwest')