Interpolation Meaning !!INSTALL!!

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.[1][2]

A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.

Interpolation Meaning

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The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.

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.

The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, and suppose that x lies between xa and xb and that g is twice continuously differentiable. Then the linear interpolation error is

In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomial interpolation and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.

However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon).

More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials.

Linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress.[3]

Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).

A key feature of mimetic interpolation is that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem. As a result, mimetic interpolation conserves line, area and volume integrals.[4] Conservation of line integrals might be desirable when interpolating the electric field, for instance, since the line integral gives the electric potential difference at the endpoints of the integration path.[5] Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path.

Linear, bilinear and trilinear interpolation are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.[6]

Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as Kriging.

Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Pad approximant, and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series. Another possibility is to use wavelets.

When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to the displacement interpolation problem used in transportation theory.

Multivariate interpolation is the interpolation of functions of more than one variable. Methods include bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions.They can be applied to gridded or scattered data. Mimetic interpolation generalizes to n {\displaystyle n} dimensional spaces where n > 3 {\displaystyle n>3} .[8][9]

In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (Upsampling) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of the original signal above the original Nyquist limit of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing.[10]

Both extrapolation and interpolation are useful methods to determine or estimate the hypothetical values for an unknown variable based on the observation of other datapoints. However, it can be hard to distinguish between these methods and understand how they differ from each other.

One of the easiest ways to understand these differences is to understand the prefix of each term. Extra- refers to "in addition to," while inter- means "in between." Thus, extrapolation indicates a user is trying to find a value in addition to existing values, while interpolation means that they want to determine a new value in between existing values.

In data science or mathematics, interpolation is about calculating a function's value based on the value of other datapoints in a given sequence. This function may be represented as f(x), and the known x values may range from x0 to xn.

Linear interpolation is among the simplest interpolation methods. Here, a straight line is drawn between two points on a graph to determine the other unknown values. The simple method frequently results in inaccurate estimates.

In polynomial interpolation, polynomial functions are used on a graph to estimate the missing values in a data set. It is a more precise, accurate method. The polynomial's graph fills in the curve between known points to find data between those points.

The Newton method is also known as Newton's divided differences interpolation polynomial. The Lagrange and Newton interpolation methods result in the smallest polynomial function, i.e., the polynomial of the lowest possible degree that goes through the data points in the data set. Both methods give the same result but use different computations to arrive at the result.

In spline interpolation, piecewise functions are used to estimate the missing values and fill the gaps in a data set. Instead of estimating one polynomial for the entire data set as occurs in the Lagrange and Newton methods, spline interpolation defines multiple simpler polynomials for subsets of the data. For this reason, it usually provides more accurate results and is considered a more reliable method.

Extrapolation is about predicting hypothetical values that fall outside a particular data set. The predictive quality of extrapolation means the method is usually used to predict unknown future values, unlike interpolation, which is usually about estimating past values.

Interpolation is a method of fitting the data points to represent the value of a function. It has a various number of applications in engineering and science, that are used to construct new data points within the range of a discrete data set of known data points or can be used for determining a formula of the function that will pass from the given set of points (x,y). In this article, we are going to discuss the meaning of interpolation in Statistics, its formulas, and uses in detail.

Interpolation is a method of deriving a simple function from the given discrete data set such that the function passes through the provided data points. This helps to determine the data points in between the given data ones. This method is always needed to compute the value of a function for an intermediate value of the independent function. In short, interpolation is a process of determining the unknown values that lie in between the known data points. It is mostly used to predict the unknown values for any geographical related data points such as noise level, rainfall, elevation, and so on. 2351a5e196