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Curve Fitting, a numerical method of statistical analysis is a very good example of both interpolation as well as extrapolation. In it, a few measured data points are used to plot a mathematical function, and then, a known curve that fits best to that function is constructed. Since the shape of the fitted curve is known, it can even be extended beyond the range of given values.

(def.) The numerical method of interpolation refers to the calculation of values that lie somewhere in the middle of the given discrete set of data points. ✏ A simple explanation of this concept would be to consider the graph of a mathematical function where only a few discrete plotted points are available. ✏ Drawing straight or curved lines that seem to be a good fit between each set of given points would naturally help give the graph a definitive shape resembling that of a known function. ✏ Doing this is nothing short of interpolation because the corresponding values of x and y [or f(x)] for any point, even if it has not been defined (by explicitly plotting it on the graph), can be obtained.

(def.) The numerical method of extrapolation is used to calculate points that are outside the range of the given set of discrete data points by using relevant methods of assumption. ✏ As in the case of interpolation, a graph of a mathematical function with a handful of discrete plotted points can be considered. ✏ Just like in the previous analogy, a larger set of values can be obtained by interpolating the function, either linearly or by taking help of the shape of a predictable nonlinear function. ✏ Since the shape of the graph has been defined based on a known function, the corresponding values for any point that has not only been included in the given discrete set of data points but also is beyond the scope or range of the given data, can be predicted.

Interpolation is generally done on mathematical functions by making use of curve fitting or regression techniques (the analysis of the relationship between variables). Some methods of interpolation that are generally used are: ♦ Linear Interpolation The interpolation which results from joining the points on the graph with a straight line is called Linear Interpolation. ♦ Nonlinear Interpolation • Polynomial Interpolation If the values plotted on the graph represent a polynomial function (an equation whose degree is more than 1), interpolation to obtain unknown values is done by substituting the coordinates of the required point in the equation of the polynomial function. This is called Polynomial Interpolation. • Spline Interpolation Polynomial Interpolation has its limitations, and can yield unreliable results for higher degrees unless certain special sets of polynomials are used in calculations. Spline Interpolation is an alternative technique that is more efficient in which a combination of different low-degree polynomials for different segments of the graph are used for interpolation. ♦ Multivariate Interpolation Multivariate Interpolation simply means that interpolation is done on functions having more than one variable, and hence, span over more than one spatial dimension. Interpolation using Gaussian processes is a good example of Multivariate Interpolation.

It is impossible to extrapolate a set of finite data without using some method of interpolation to figure out which mathematical function can be applied as the basis to predict additional data. Interpolation using Lagrange Polynomials, or by trying to find the Newton's Series for the data, are two common methods of interpolation that generally precede extrapolation. Commonly used extrapolation techniques are: ♦ Linear Extrapolation Just like with Linear Interpolation, this method is built on the fact that the given data points have a linear relationship with each other. ♦ Polynomial Extrapolation Using any method of interpolation will not yield accurate results upon extrapolation; iterative numerical methods wherein finite distances are calculated are generally used in this technique. A befitting polynomial function that can be used for extrapolation needs to be deduced. ♦ Conic Extrapolation This method, wherein conic section templates are used for extrapolation, is mostly implemented using software. ♦ French Curve Extrapolation The historic set of curves that were standardized by renowned mathematician, Euler, can also be used in the extrapolation of certain kinds of data. ♦ Extrapolation using Fast Fourier Transform Fast Fourier Transform is a mathematical technique where the data used is convoluted. Using Fast Fourier Transform for extrapolation saves a lot of computation time.

The various techniques of interpolation see their widest application in the field of engineering. Here are a few examples of real scenarios where interpolation is truly a godsend. Telecommunication ✏ When an analog signal (such as what you speak on the phone) is transmitted over a long distance, it needs to be converted to a digital format. ✏ Digital data is discrete rather than continuous, and hence, the signal needs to be sampled at regular intervals. ✏ The sampled function is optimized and sent across the world through physical cables or wires. ✏ At the recipient end, once the sampled version of the signal is obtained, the original signal, obviously, needs to be reconstructed. ✏ How does one obtain the entire original signal from the bits and pieces that have been transmitted? Answer: Interpolation. Airplane, or Aerospace Engineering ✏ For a rocket that is sent into outer space, the value of each of its parameters, like speed, acceleration, temperature, and so on, can be represented in the form of mathematical functions. Readings of these values are taken at regular intervals. ✏ In case of any unforeseen event, for example: To avert a bigger crisis of shrapnel from some explosion in space whizzing about in the rocket's trajectory, scientists would need to have access to the values of each parameter at various specific instants of time. ✏ To obtain the values for those instants where readings were not explicitly taken, interpolation can be used. Thermodynamics and Fluid Mechanics ✏ In various industries pertaining to chemicals, mechanical engineering, or material sciences, knowing the thermal and physical properties of various fluid substances is a must. ✏ Although the calculation and computation of many of these values is done with the help of software and databases, on a very high scale, the sheer volume of numbers to be crunched makes it a tedious, time-consuming process. ✏ The techniques of interpolation eradicate the tedious task of calculating many values physically which in turn saves a LOT of time.

Extrapolation and prediction go hand in hand, and hence, in almost every industry whenever anything needs to be planned for the future, or with reference to projected values, extrapolation is always used. Forecasting Here are a few real examples where extrapolation is used to forecast future trends. 1. It is possible to predict what the population of the world will be in any future year by extrapolating the data from the past few decades that we currently have. 2. Physical data about our planet, solar system, and in fact, the entire universe can be extrapolated to predict major events including catastrophes and natural disasters. A simple example would be: studying the temperature, width, depth, and other parameters of a fault line with respect to time, and extrapolating the data to predict when an earthquake could occur. 3. Virologists can extrapolate the data regarding the outbreak of a disease in a particular area, and predict the extent to which it will spread over a period of time. This extrapolated data can prove to be invaluable while designing strategies to contain the disease. 4. Corporate firms use extrapolation on almost a daily basis to predict growth, profit, and other similar trends. In fact, they design their business strategies for the entire year on the basis of this extrapolated data. 5. Another statistical application of extrapolation methods is the prediction of stock market trends. However, this is not recommended at all, as the extrapolated data sometimes may not match the actual situation, and can end up misleading anyone who makes decisions on its basis. Engineering and Mathematical Applications. 1. When audio signals are transmitted over long distances, they are susceptible to disturbance and noise. Using extrapolation techniques to predict the signal during its reconstruction, at the recipient end, can help obtain a much clearer signal. 2. The field of image processing has benefited greatly because of extrapolation. As a result of nonlinear extrapolation of data, HDTVs have the simplicity of computation.