This is easy if the target set is an aggregation of the source set, but more difficult, if the boundaries of the target set are independent of the source set Bracken and Martin, This implicitly assumes that the data are uniformly distributed throughout the polygon. This is usually not the case for population related data. Many researches have taken place and have looked at ways of better population interpolation, those include improves in areal interpolation which weights data values for a partial polygon proportionally to the ratio of partial polygon area to complete polygon area Shepard, et al.

For point-based, the typical examples are conditions based on geostatistical concepts Kriging , locality nearest neighbor and finite element methods, IDW, TIN , smoothness and tension spline , or ad hoc functional forms polynomials, multi-quadrics and have been discussed by several researchers including Mitas, et al.

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These areal and point based interpolation techniques, however does not satisfy wholly the needs of a planner who what to generate demographic surfaces Wadembere, from micro population data. There is need to generate demographic boundaries for demarcating the demographic spatial extent, prediction, simulation, zoning, and for extending demographic spatial extent MODC extrapolation as in most cases the demographic points do not cover the whole study and do not reach the boundaries. It also requires being in position to define a variable shape, handle demographic holes, breaks, and boundaries; and extend the demographic surface to any desired geographical area and boundary to help to incorporate knowledge of the data and study area.

Thus, the need to look for ways how those techniques can be improved or what can be done so that we generate demographic surfaces. Hence, in this paper a procedure called Geo-Demographic Interpolation and Extrapolation is being coined and adopted for population interpolation and extrapolation. Using the modules included in GIS packages, the spatial extent of surface generated from discrete points, is also always up to the extent of data points if using TIN and if use IDW, it will be covering an area in form of a rectangle.

However, this has its shortcoming in that the grid is just clipped and the points do not carry with them the respective MODC. Thus, for demographic interpolation with the aim of surface and 3D representation Wadembere, , more than the traditional interpolation and extrapolation is needed and this can be achieved through the stages as outlined in figure 1.

Here, we divided the process of achieving demographic spatial interpolation and extrapolation into uni-demographic dealing with interpolation and extrapolation of one DC at a time and is designed for representing one DC. Both lead to geo-demographic interpolation and extrapolation, which is being developed to fulfill the requirements of demographics representation and is accomplished through the following steps. Although conceptual surfaces are usually considered smooth, discontinuities can be present sometimes, for example in case of physical, political, social, and cultural barriers e.

Demographic holes are defined and used to refer to spatial extents that do not have population or a particular DC being considered in analysis. It can be of any shape but should be a planar polygon with more than two sides.

## Interpolation and Extrapolation, Volume 2

Demographic breaks are similar to holes expect that they have only two sides i. They can be used at the end of study area or DC.

Demographic boundaries are linear demarcation at the end of the DC, study area, breaks or the edges of the demographic holes. In an effort to get boundaries some factors have to considered: 1 pixels will have neighbors outside the study area and therefore without values. Some decision must be made about the behavior of the surface outside the study area e. After identifying the boundaries, the next step is generation of points at boundaries point get coordinates of points along the boundary.

To ensure that points extrapolated follow Tobler's Law of Geography Martin, , they take on the resolution of nearby points. This is accomplished by starting at the beginning of the boundary. At that point, search for the nearby three points as those are the points which will be nodes in the building TIN find the shortest distance between any two points. It is this distance, which is added to the first point at the beginning of the boundary to get the location of the next point along the boundary. For location of the rest of points, the preceding boundary point must be part of the three points from which the shortest distance is searched in order to locate the next boundary point.

The process continues to complete the boundary. The extrapolated boundary do not carry with them the Magnitude of Demographic characteristic MODC ; for extrapolating the MODC at boundaries we proceed by looking for all data points neighboring each interpolated location with a predefined spatial extent, that the spatial sparsity and quantity of existing points is reflected in the new interpolated point. In order to obtain a surface that reflects a change in DC, the MODC for new point will be obtained from neighborhood by an iteration process by employing IDW and after an appropriate MODC is given to the point; then go on the next points.

This means the number of new points introduced do not alter the appearance of the surface, thus being able to use in demographic surfaces in prediction and simulation like obtaining the optimum number of persons per certain spatial extent.

## Interpolation and Extrapolation, Volume 2

After establishing points along boundaries, then proceed to surface building; since the location of demographic data points is seldom regular or even rectangular and boundaries nearly never follow the grid lines, to avoid problems in representation the TIN is used. If need arises, algorithms and programs have been developed to interpolate irregular points, lines and polygons to a regular grid Abdul-Rahman, et al.

TIN allows the preservation of the original data points in the model of a surface Peucker, et al. The data points are the nodes of the irregular grid, the nodes are connected by arcs or edge s, forming a mesh of triangles of different size and orientation. It can be adapted to varying resolution requirements in the same network. The formula was first published by Waring , rediscovered by Euler in , and published by Lagrange in Jeffreys and Jeffreys Lagrange interpolating polynomials are implemented in the Wolfram Language as InterpolatingPolynomial [ data , var ].

They are used, for example, in the construction of Newton-Cotes formulas. When constructing interpolating polynomials, there is a tradeoff between having a better fit and having a smooth well-behaved fitting function. The more data points that are used in the interpolation, the higher the degree of the resulting polynomial, and therefore the greater oscillation it will exhibit between the data points.

### Table of Contents

Therefore, a high-degree interpolation may be a poor predictor of the function between points, although the accuracy at the data points will be "perfect. For points,. Note that the function passes through the points , as can be seen for the case ,. Generalizing to arbitrary ,. Then define the fundamental polynomials by. Now let , More generally, let be an arbitrary distribution on the interval , the associated orthogonal polynomials , and , Lagrange interpolating polynomials give no error estimate.

Various Methods Used. 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 - Wikipedia

The interpolation which results from joining the points on the graph with a straight line is called Linear Interpolation. This is called Polynomial Interpolation. 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 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.

go here Commonly used extrapolation techniques are:. Just like with Linear Interpolation, this method is built on the fact that the given data points have a linear relationship with each other. 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.

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This method, wherein conic section templates are used for extrapolation, is mostly implemented using software. The historic set of curves that were standardized by renowned mathematician, Euler, can also be used in the extrapolation of certain kinds of data. 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. Examples of Real World Applications. 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.

Answer: Interpolation. Airplane, or Aerospace Engineering. Readings of these values are taken at regular intervals. Thermodynamics and Fluid Mechanics. 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. Here are a few real examples where extrapolation is used to forecast future trends. 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.