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| In the [[mathematics|mathematical]] field of [[numerical analysis]], '''interpolation''' is a method of constructing new data points within the range of a [[discrete set]] of known data points.
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| In [[engineering]] and [[science]], one often has a number of data points, obtained by [[sampling (statistics)|sampling]] or [[experimentation]], which represent the values of a function for a limited number of values of the independent variable. It is often required to '''interpolate''' (i.e. estimate) the value of that function for an intermediate value of the independent variable. This may be achieved by [[curve fitting]] or [[regression analysis]].
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| A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently. A few known data points from the original function can be used to create an interpolation based on a simpler function. Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in accuracy.
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| There is also another very different kind of interpolation in mathematics, namely the "interpolation of operators". The classical results about interpolation of operators are the [[Riesz–Thorin theorem]] and the [[Marcinkiewicz theorem]]. There are also many other subsequent results.
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| [[Image:Splined epitrochoid.svg|300px|thumb|An interpolation of a finite set of points on an [[epitrochoid]]. Points through which curve is [[spline (mathematics)|splined]] are red; the blue curve connecting them is interpolation.]]
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| ==Example==
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| For example, suppose we have a table like this, which gives some values of an unknown function ''f''.
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| [[Image:Interpolation Data.svg|right|thumb|230px|Plot of the data points as given in the table.]] | |
| {| cellpadding=0 cellspacing=0
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| |width="20px"|
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| ! ''x''
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| |width="10px"|
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| !colspan=3 align=center| ''f''(''x'')
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| |-
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| | || 0 || ||align=right| 0
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| | || 1 || ||align=right| 0 || . || 8415
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| | || 2 || ||align=right| 0 || . || 9093
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| |-
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| | || 3 || ||align=right| 0 || . ||1411
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| | || 4 || ||align=right| −0 || . || 7568
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| |-
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| | || 5 || ||align=right| −0 || . || 9589
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| |-
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| | || 6 || ||align=right| −0 || . || 2794
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| |}
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| Interpolation provides a means of estimating the function at intermediate points, such as ''x'' = 2.5.
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| There are many different interpolation methods, some of which are described below. Some of the concerns to take into account when choosing an appropriate [[algorithm]] are: How accurate is the method? How expensive is it? How [[smooth function|smooth]] is the interpolant? How many data points are needed?
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| <br clear="all" /> | |
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| ===Piecewise constant interpolation===
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| [[Image:Piecewise constant.svg|thumb|right|Piecewise constant interpolation, or [[nearest-neighbor interpolation]].]]
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| {{details|Nearest-neighbor interpolation}}
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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.
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| <br clear="all" /> | |
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| ===Linear interpolation===
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| [[Image:Interpolation example linear.svg|right|thumb|230px|Plot of the data with linear interpolation superimposed]]
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| {{Main|Linear interpolation}}
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| 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.
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| Generally, linear interpolation takes two data points, say (''x''<sub>''a''</sub>,''y''<sub>''a''</sub>) and (''x''<sub>''b''</sub>,''y''<sub>''b''</sub>), and the interpolant is given by:
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| :<math> y = y_a + \left( y_b-y_a \right) \frac{x-x_a}{x_b-x_a} \text{ at the point } \left( x,y \right) </math>
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| Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not [[derivative|differentiable]] at the point ''x''<sub>''k''</sub>.
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| 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 ''x''<sub>''a''</sub> and ''x''<sub>''b''</sub> and that ''g'' is twice continuously differentiable. Then the linear interpolation error is | |
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| :<math> |f(x)-g(x)| \le C(x_b-x_a)^2 \quad\text{where}\quad C = \frac18 \max_{y\in[x_a,x_b]} |g''(y)|. </math>
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| 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.
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| <br clear="all" /> | |
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| ===Polynomial interpolation===
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| [[Image:Interpolation example polynomial.svg|right|thumb|230px|Plot of the data with polynomial interpolation applied]]
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| {{Main|Polynomial interpolation}}
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| Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a [[linear function]]. We now replace this interpolant with a [[polynomial]] of higher [[degree of a polynomial|degree]].
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| Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
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| :<math> f(x) = -0.0001521 x^6 - 0.003130 x^5 + 0.07321 x^4 - 0.3577 x^3 + 0.2255 x^2 + 0.9038 x. </math>
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| <!-- Coefficients are 0, 0.903803333333334, 0.22549749999997, -0.35772291666664, 0.07321458333332, -0.00313041666667, -0.00015208333333. -->
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| Substituting ''x'' = 2.5, we find that ''f''(2.5) = 0.5965.
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| Generally, if we have ''n'' data points, there is exactly one polynomial of degree at most ''n''−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power ''n''. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation.
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| 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, i.e. 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]].
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| <br clear="all" /> | |
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| ===Spline interpolation===
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| [[Image:Interpolation example spline.svg|right|thumb|230px|Plot of the data with spline interpolation applied]]
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| {{Main|Spline interpolation}}
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| Remember that linear interpolation uses a linear function for each of intervals [''x''<sub>''k''</sub>,''x''<sub>''k+1''</sub>]. 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 (mathematics)|spline]].
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| For instance, the [[natural cubic spline]] is [[piecewise]] cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by
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| : <math> f(x) = \begin{cases}
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| -0.1522 x^3 + 0.9937 x, & \text{if } x \in [0,1], \\
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| -0.01258 x^3 - 0.4189 x^2 + 1.4126 x - 0.1396, & \text{if } x \in [1,2], \\
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| 0.1403 x^3 - 1.3359 x^2 + 3.2467 x - 1.3623, & \text{if } x \in [2,3], \\
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| 0.1579 x^3 - 1.4945 x^2 + 3.7225 x - 1.8381, & \text{if } x \in [3,4], \\
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| 0.05375 x^3 -0.2450 x^2 - 1.2756 x + 4.8259, & \text{if } x \in [4,5], \\
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| -0.1871 x^3 + 3.3673 x^2 - 19.3370 x + 34.9282, & \text{if } x \in [5,6].
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| \end{cases} </math>
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| In this case we get ''f''(2.5) = 0.5972. | |
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| Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother. However, the interpolant is easier to evaluate than the high-degree polynomials used in polynomial interpolation. It also does not suffer from [[Runge's phenomenon]].
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| {{Clear}}
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| ==Interpolation via Gaussian processes==
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| [[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, i.e., for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as [[Kriging]].
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| ==Other forms of interpolation==
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| Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is '''interpolation''' by [[rational function]]s, and [[trigonometric interpolation]] is interpolation by [[trigonometric polynomial]]s. Another possibility is to use [[wavelet]]s.
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| The [[Whittaker–Shannon interpolation formula]] can be used if the number of data points is infinite.
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| [[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.
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| Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to [[Hermite interpolation]] problems.
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| 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 (mathematics)|transportation theory]].
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| ==Interpolation in digital signal processing==
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| 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 a higher sampling rate ([[Upsampling]]) using various digital filtering techniques (e.g., 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 (i.e., 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''.<ref>[http://www.amazon.com/Multirate-Digital-Signal-Processing-Crochiere/dp/0136051626 R.E. Crochiere and L.R. Rabiner. (1983). Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice–Hall.]</ref>
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| ==Related concepts==
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| The term ''[[extrapolation]]'' is used if we want to find data points outside the range of known data points.
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| In [[curve fitting]] problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to [[least squares]] approximation.
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| [[Approximation theory]] studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function.
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| ==See also==
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| {{Div col|cols=2}}
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| * [[Barycentric_coordinate_system_%28mathematics%29|Barycentric coordinates – for interpolating within on a triangle or tetrahedron]]
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| * [[Bilinear interpolation]]
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| * [[Brahmagupta's interpolation formula]]
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| * [[Extrapolation]]
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| * [[Imputation (statistics)]]
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| * [[Missing data]]
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| * [[Multivariate interpolation]]
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| * [[Newton–Cotes formulas]]
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| * [[Polynomial interpolation]]
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| * [[Simple rational approximation]]
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| {{Div col end}}
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| ==References==
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| <references />
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| ==External links==
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| * [http://sol.gfxile.net/interpolation/index.html Sol Tutorials - Interpolation Tricks]
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| [[Category:Interpolation| ]]
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| [[Category:Video]]
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| [[Category:Video signal]]
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2013 has been so far so good for almost all of the brands yet even the brand-new ones are creating their mark. The best thus far have been AVG, avast, Norton, Bitdefender plus Kaspersky. Their all modern packages have set guidelines for the number one software with the best services.
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