1-D data interpolation (table lookup) (2024)

Table of Contents

Syntax Description Examples Interpolation of Coarsely Sampled Sine Function Interpolation Without Specifying Points Interpolation of Complex Values Interpolation of Dates and Times Extrapolation Using Two Different Methods Designate Constant Value for All Queries Outside the Domain of x Interpolate Multiple Sets of Data in One Pass Input Arguments x — Sample points vector v — Sample values vector | matrix | array xq — Query points scalar | vector | matrix | array method — Interpolation method 'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'v5cubic' | 'makima' | 'spline' extrapolation — Extrapolation strategy 'extrap' | scalar value Output Arguments vq — Interpolated values scalar | vector | matrix | array pp — Piecewise polynomial structure More About Akima and Spline Interpolation References Extended Capabilities C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™. GPU Code Generation Generate CUDA® code for NVIDIA® GPUs using GPU Coder™. Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool. GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. Distributed Arrays Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™. Version History R2020b: 'cubic' method of interp1 performs cubic convolution See Also MATLAB Command Americas Europe Asia Pacific FAQs References

1-D data interpolation (table lookup)

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Syntax

vq = interp1(x,v,xq)

vq = interp1(x,v,xq,method)

vq = interp1(x,v,xq,method,extrapolation)

vq = interp1(v,xq)

vq = interp1(v,xq,method)

vq = interp1(v,xq,method,extrapolation)

pp = interp1(x,v,method,'pp')

Description

example

vq = interp1(x,v,xq) returnsinterpolated values of a 1-D function at specific query points usinglinear interpolation. Vector x contains the samplepoints, and v contains the corresponding values, v(x).Vector xq contains the coordinates of the querypoints.

If you have multiple sets of data that are sampled at the samepoint coordinates, then you can pass v as an array.Each column of array v contains a different setof 1-D sample values.

example

vq = interp1(x,v,xq,method) specifies an alternative interpolation method: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'v5cubic', 'makima', or 'spline'. The default method is 'linear'.

example

vq = interp1(x,v,xq,method,extrapolation) specifiesa strategy for evaluating points that lie outside the domain of x.Set extrapolation to 'extrap' whenyou want to use the method algorithm for extrapolation.Alternatively, you can specify a scalar value, in which case, interp1 returnsthat value for all points outside the domain of x.

example

vq = interp1(v,xq) returnsinterpolated values and assumes a default set of sample point coordinates.The default points are the sequence of numbers from 1 to n,where n depends on the shape of v:

When v is a vector, the default points are 1:length(v).
When v is an array, the default points are 1:size(v,1).

Use this syntax when you are not concerned about theabsolute distances between points.

vq = interp1(v,xq,method) specifiesany of the alternative interpolation methods and uses the defaultsample points.

vq = interp1(v,xq,method,extrapolation) specifiesan extrapolation strategy and uses the default sample points.

pp = interp1(x,v,method,'pp') returns the piecewise polynomial form of v(x) using the method algorithm.

Note

This syntax is not recommended. Use griddedInterpolant instead.

Examples

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Interpolation of Coarsely Sampled Sine Function

Open Live Script

Define the sample points, x, and corresponding sample values, v.

x = 0:pi/4:2*pi; v = sin(x);

Define the query points to be a finer sampling over the range of x.

xq = 0:pi/16:2*pi;

Interpolate the function at the query points and plot the result.

figurevq1 = interp1(x,v,xq);plot(x,v,'o',xq,vq1,':.');xlim([0 2*pi]);title('(Default) Linear Interpolation');

Now evaluate v at the same points using the 'spline' method.

figurevq2 = interp1(x,v,xq,'spline');plot(x,v,'o',xq,vq2,':.');xlim([0 2*pi]);title('Spline Interpolation');

Interpolation Without Specifying Points

Open Live Script

Define a set of function values.

v = [0 1.41 2 1.41 0 -1.41 -2 -1.41 0];

Define a set of query points that fall between the default points, 1:9. In this case, the default points are 1:9 because v contains 9 values.

xq = 1.5:8.5;

Evaluate v at xq.

Interpolation of Complex Values

Open Live Script

Define a set of sample points.

x = 1:10;

Define the values of the function, $v (x) = 5 x + x^{2} i$ , at the sample points.

v = (5*x)+(x.^2*1i);

Define the query points to be a finer sampling over the range of x.

xq = 1:0.25:10;

Interpolate v at the query points.

vq = interp1(x,v,xq);

Plot the real part of the result in red and the imaginary part in blue.

figureplot(x,real(v),'*r',xq,real(vq),'-r');hold onplot(x,imag(v),'*b',xq,imag(vq),'-b');

Interpolation of Dates and Times

Open Live Script

Interpolate time-stamped data points.

Consider a data set containing temperature readings that are measured every four hours. Create a table with one day's worth of data and plot the data.

x = (datetime(2016,1,1):hours(4):datetime(2016,1,2))';x.Format = 'MMM dd, HH:mm';T = [31 25 24 41 43 33 31]';WeatherData = table(x,T,'VariableNames',{'Time','Temperature'})

WeatherData=7×2 table Time Temperature _____________ ___________ Jan 01, 00:00 31 Jan 01, 04:00 25 Jan 01, 08:00 24 Jan 01, 12:00 41 Jan 01, 16:00 43 Jan 01, 20:00 33 Jan 02, 00:00 31

plot(WeatherData.Time, WeatherData.Temperature, 'o')

Interpolate the data set to predict the temperature reading during each minute of the day. Since the data is periodic, use the 'spline' interpolation method.

xq = (datetime(2016,1,1):minutes(1):datetime(2016,1,2))';V = interp1(WeatherData.Time, WeatherData.Temperature, xq, 'spline');

Plot the interpolated points.

hold onplot(xq,V,'r')

Extrapolation Using Two Different Methods

Open Live Script

Define the sample points, x, and corresponding sample values, v.

x = [1 2 3 4 5];v = [12 16 31 10 6];

Specify the query points, xq, that extend beyond the domain of x.

xq = [0 0.5 1.5 5.5 6];

Evaluate v at xq using the 'pchip' method.

vq1 = interp1(x,v,xq,'pchip')

vq1 = 1×5 19.3684 13.6316 13.2105 7.4800 12.5600

Next, evaluate v at xq using the 'linear' method.

vq2 = interp1(x,v,xq,'linear')

vq2 = 1×5 NaN NaN 14 NaN NaN

Now, use the 'linear' method with the 'extrap' option.

vq3 = interp1(x,v,xq,'linear','extrap')

vq3 = 1×5 8 10 14 4 2

'pchip' extrapolates by default, but 'linear' does not.

Designate Constant Value for All Queries Outside the Domain of x

Open Live Script

Define the sample points, x, and corresponding sample values, v.

x = [-3 -2 -1 0 1 2 3];v = 3*x.^2;

Specify the query points, xq, that extend beyond the domain of x.

xq = [-4 -2.5 -0.5 0.5 2.5 4];

Now evaluate v at xq using the 'pchip' method and assign any values outside the domain of x to the value, 27.

vq = interp1(x,v,xq,'pchip',27)

vq = 1×6 27.0000 18.6562 0.9375 0.9375 18.6562 27.0000

Interpolate Multiple Sets of Data in One Pass

Open Live Script

Define the sample points.

x = (-5:5)';

Sample three different parabolic functions at the points defined in x.

v1 = x.^2;v2 = 2*x.^2 + 2;v3 = 3*x.^2 + 4;

Create matrix v, whose columns are the vectors, v1, v2, and v3.

v = [v1 v2 v3];

Define a set of query points, xq, to be a finer sampling over the range of x.

xq = -5:0.1:5;

Evaluate all three functions at xq and plot the results.

vq = interp1(x,v,xq,'pchip');figureplot(x,v,'o',xq,vq);h = gca;h.XTick = -5:5;

The circles in the plot represent v, and the solid lines represent vq.

Input Arguments

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`x` — Sample points
vector

Sample points, specified as a row or column vector of real numbers.The values in x must be distinct. The length of x mustconform to one of the following requirements:

If v is a vector, then length(x) mustequal length(v).
If v is an array, then length(x) mustequal size(v,1).

Example: [1 2 3 4 5 6 7 8 9 10]

Example: 1:10

Example: [37 11 15 19 23 27 31]'

Data Types: single | double | duration | datetime

`v` — Sample values
vector | matrix | array

Sample values, specified as a vector, matrix, or array of realor complex numbers. If v is a matrix or an array,then each column contains a separate set of 1-D values.

If v contains complex numbers, then interp1 interpolatesthe real and imaginary parts separately.

Example: rand(1,10)

Example: rand(10,1)

Example: rand(10,3)

Data Types: single | double | duration | datetime
Complex Number Support: Yes

`xq` — Query points
scalar | vector | matrix | array

Query points, specified as a scalar, vector, matrix, or arrayof real numbers.

Example: 5

Example: 1:0.05:10

Example: (1:0.05:10)'

Example: [0 1 2 7.5 10]

Data Types: single | double | duration | datetime

`method` — Interpolation method
`'linear'` (default) | `'nearest'` | `'next'` | `'previous'` | `'pchip'` | `'cubic'` | `'v5cubic'` | `'makima'` | `'spline'`

Interpolation method, specified as one of the options in this table.

Method	Description	Continuity	Comments
`'linear'`	Linear interpolation. The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. This is the default interpolation method.	C⁰	Requires at least 2 points Requires more memory and computation time than nearest neighbor
`'nearest'`	Nearest neighbor interpolation. The interpolated value at a query point is the value at the nearest sample grid point.	Discontinuous	Requires at least 2 points Modest memory requirements Fastest computation time
`'next'`	Next neighbor interpolation. The interpolated value at a query point is the value at the next sample grid point.	Discontinuous	Requires at least 2 points Same memory requirements and computation time as `'nearest'`
`'previous'`	Previous neighbor interpolation. The interpolated value at a query point is the value at the previous sample grid point.	Discontinuous	Requires at least 2 points Same memory requirements and computation time as `'nearest'`
`'pchip'`	Shape-preserving piecewise cubic interpolation. The interpolated value at a query point is based on a shape-preserving piecewise cubic interpolation of the values at neighboring grid points.	C¹	Requires at least 4 points Requires more memory and computation time than `'linear'`
`'cubic'`	Cubic convolution used in MATLAB^® 5.	C¹	Requires at least 3 points Points must be uniformly spaced This method falls back to `'spline'` interpolation for irregularly-spaced data Similar memory requirements and computation time as `'pchip'`
`'v5cubic'`	Same as `'cubic'`.	C¹
`'makima'`	Modified Akima cubic Hermite interpolation. The interpolated value at a query point is based on a piecewise function of polynomials with degree at most three. The Akima formula is modified to avoid overshoots.	C¹	Requires at least 2 points Produces fewer undulations than `'spline'`, but does not flatten as aggressively as `'pchip'` Computation is more expensive than `'pchip'`, but typically less than `'spline'` Memory requirements are similar to those of `'spline'`
`'spline'`	Spline interpolation using not-a-knot end conditions. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension.	C²	Requires at least 4 points Requires more memory and computation time than `'pchip'`

`extrapolation` — Extrapolation strategy
`'extrap'` | scalar value

Extrapolation strategy, specified as 'extrap' ora real scalar value.

Specify 'extrap' when you want interp1 toevaluate points outside the domain using the same method it uses forinterpolation.
Specify a scalar value when you want interp1 toreturn a specific constant value for points outside the domain.

The default behavior depends on the input arguments:

If you specify the 'pchip', 'spline', or 'makima' interpolation methods, then the default behavior is 'extrap'.
All other interpolation methods return NaN bydefault for query points outside the domain.

Example: 'extrap'

Example: 5

Data Types: char | string | single | double

Output Arguments

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`vq` — Interpolated values
scalar | vector | matrix | array

Interpolated values, returned as a scalar, vector, matrix, orarray. The size of vq depends on the shape of v and xq.

Shape of v	Shapeof xq	Size of Vq	Example
Vector	Vector	`size(xq)`	If `size(v) = [1 100]` and `size(xq)= [1 500]`, then `size(vq) = [1 500]`.
Vector	Matrix or N-D Array	`size(xq)`	If `size(v) = [1 100]` and `size(xq)= [50 30]`, then `size(vq) = [5030]`.
Matrix or N-D Array	Vector	`[length(xq) size(v,2),...,size(v,n)]`	If `size(v) = [100 3]` and `size(xq)= [1 500]`, then `size(vq) = [5003]`.
Matrix or N-D Array	Matrix or N-D Array	`[size(xq,1),...,size(xq,n),... size(v,2),...,size(v,m)]`	If `size(v) = [4 5 6]` and `size(xq)= [2 3 7]`, then `size(vq) = [2 37 5 6]`.

`pp` — Piecewise polynomial
structure

Piecewise polynomial, returned as a structure that you can passto the ppval function for evaluation.

More About

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Akima and Spline Interpolation

The Akima algorithm for one-dimensional interpolation, described in [1] and [2], performs cubic interpolation to produce piecewise polynomials with continuous first-order derivatives (C1). The algorithm preserves the slope and avoids undulations in flat regions. A flat region occurs whenever there are three or more consecutive collinear points, which the algorithm connects with a straight line. To ensure that the region between two data points is flat, insert an additional data point between those two points.

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 the overshoot. (The original Akima algorithm gives equal weights to the points on both sides, thus evenly dividing the undulation.)

The spline algorithm, on the other hand, performs cubic interpolation to produce piecewise polynomials with continuous second-order derivatives (C2). The result is comparable to a regular polynomial interpolation, but is less susceptible to heavy oscillation between data points for high degrees. Still, this method can be susceptible to overshoots and oscillations between data points.

Compared to the spline algorithm, the Akima algorithm produces fewer undulations and is better suited to deal with quick changes between flat regions. This difference is illustrated below using test data that connects multiple flat regions.

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

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The input argument x (sample points)must be strictly increasing or strictly decreasing. Indices are notreordered.
If the input argument v (samplevalues) is a variable-length vector (1-by-: or :-by-1),then the shape of the output vq matches the shapein MATLAB.
If the input argument v is variable-size,is not a variable-length vector, and becomes a row vector at run time,then an error occurs.
If the sample values or query points contain Inf or -Inf, the output of the generated code might not match the output in MATLAB.
If the input argument xq (querypoints) is variable-size, is not a variable-length vector, and becomesa row or column vector at run time, then an error occurs.
See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

The input argument x (sample points) must be strictly increasing or strictly decreasing. Indices are not reordered.
If the input argument v (sample values) is a variable-length vector (1-by-: or :-by-1), then the shape of the output vq matches the shape in MATLAB.
If the input argument v is variable-size, is not a variable-length vector, and becomes a row vector at run time, then an error occurs.
If the input argument xq (query points) is variable-size, is not a variable-length vector, and becomes a row or column vector at run time, then an error occurs.
See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

The interpolation methods 'pchip' and 'makima' are not supported.
The 'spline' interpolation method is not supported for sample values v containing NaN.

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™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

expand all

R2020b: `'cubic'` method of `interp1` performs cubic convolution

In R2020b, the 'cubic' interpolation method of interp1 performs cubic convolution. The 'v5cubic' and 'cubic' interpolation methods now perform the same type of interpolation, which is consistent with the behavior of interp2, interp3, and interpn. The cubic convolution interpolation method is intended for uniformly-spaced data, and it falls back to 'spline' interpolation for irregularly-spaced data.

In previous releases, 'cubic' was the same as 'pchip', and only 'v5cubic' performed cubic convolution.

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1-D data interpolation (table lookup) (2024)

FAQs

How do you interpolate data tables? ›

Interpolate data within a data table

In the toolbar, click Interpolate. The Interpolate dialog box appears.
In the Range Item list, select the item that you want to interpolate.
Type the desired value in the Value to be added field. Note: ...
Click Interpolate. ...
If necessary, repeat steps 1–4 to receive the desired result.

Tell Me More ›

How to solve interpolation method? ›

How to interpolate

Organize your data. First, put the data you've collected into a chart that shows your independent and dependent variables. ...
Consider creating a graph. ...
Select your two points. ...
Enter values into the interpolation equation. ...
Solve for the missing variable.

Oct 16, 2023

Know More ›

What is the best way to interpolate data? ›

Linear interpolation is the most straightforward and commonly used interpolation method. It comes naturally when we have two points, we connect them with a straight line to fill out the missing information in between. By doing so, we made our assumption that the points on the line represent the unobserved values.

Get More Info Here ›

What is the best data interpolation method? ›

Radial Basis Function interpolation is a diverse group of data interpolation methods. In terms of the ability to fit your data and produce a smooth surface, the Multiquadric method is considered by many to be the best.

Keep Reading ›

How do you interpolate between two data sets? ›

How to interpolate

Identify your data. Use a table to list your data. ...
Create a line of best fit. After using the values to plot a graph, you can draw a line of best fit. ...
Determine your value for interpolation. ...
Use the linear interpolation equation. ...
Solve the equation.

Sep 30, 2022

Know More ›

How to interpolate data from a table in Excel? ›

Here's how to use the FORECAST function to interpolate in Excel:

Chart your data points. The first step is to chart your data points. ...
Designate your cells for the input and output formula. ...
Input your formula into the interpolated value cell. ...
Assign the variables in the function to your cells. ...
Test the formula.

Show Me More ›

What is an example of interpolation of data? ›

Interpolation definition says that interpolation is to estimate the value of a point between two given points in a data set. For example, if a child's height was measured at age 5 and age 6, interpolation could be used to estimate the child's height at age 5.5.

Read The Full Story ›

What is the simplest method of interpolation? ›

One of the simplest methods, linear interpolation, requires knowledge of two points and the constant rate of change between them. With this information, you may interpolate values anywhere between those two points.

What is the fastest interpolation method? ›

The Nearest Point interpolation method is the fastest of all the interpolation methods when used with point data (fig. 19). If used with line or polygon data it can be slower than the Nearest interpolation especially if many of the object vertices lie outside the grid.

Get More Info ›

Why do we interpolate data? ›

Interpolation is a mathematical technique to estimate the values of unknown data points that fall in between existing, known data points. This process helps fill in the blanks. Technical traders use interpolation to understand how prices have behaved in the past, even when they do not have full information.

Show Me More ›

How do you know when to use interpolation? ›

It is used when there is an exact relationship between the values, but no data points are available. Nearest Neighbor Interpolation: This method uses the closest known value to predict the value between two known values. It is useful when there are limited data points available.

What is the easiest method for solving interpolation? ›

One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.

What is the average interpolation method? ›

The Average Interpolated algorithm calculates an average value based on: The interpolated value for the start of the resample interval. The good quality raw historic values in the resample interval.

How many methods are there for interpolation of data? ›

It is widely used to analyze photos and videos, estimate values between known data points, and decrease the amount of data required to describe curves and surfaces. There are several methods of interpolation, including linear interpolation, polynomial interpolation, and spline interpolation.

Get More Info Here ›

What is interpolation of a data set? ›

Interpolation is the process of using known data values to estimate unknown data values. Various interpolation techniques are often used in the atmospheric sciences. One of the simplest methods, linear interpolation, requires knowledge of two points and the constant rate of change between them.

Explore More ›

What are the methods available for interpolation of data? ›

They are: Linear Interpolation Method – This method applies a distinct linear polynomial between each pair of data points for curves, or within the sets of three points for surfaces. Nearest Neighbour Method – This method inserts the value of an interpolated point to the value of the most adjacent data point.

Get More Info Here ›

What is interpolation of tabulated values? ›

The finding of value between tabulated entries is called interpolation. The ex- tending of tabulated values to find values beyond the limits of the table is called extrapolation.

Read The Full Story ›

Can you interpolate in SQL? ›

A SQL string containing variables to interpolate. Variables must start with a question mark and can be any valid R identifier, i.e. it must start with a letter or . , and be followed by a letter, digit, . or _ . Values (for ... ) or a list (for . dots ) to interpolate into a string.

Show Me More ›