Interpolation and extrapolation are the statistical tool that is used for estimating the hypothetical values for some particular variable that is based on the observations. On the basis of the overall trend that is observed from the data set, there are various methods of interpolation and extrapolation. Linear interpolation and exploration is a very simple form of interpolation which renders straight line between two or more than two points. Linear is the implication of the fact that the points connected forms the straight lines. In the earlier times, interpolation method was used to predict and study the movement and positions of the celestial bodies and in the today’s world, they are commonly used in the computer graphics. It is required to look at the prefixes inter and extra for identifying the difference between interpolation and extrapolation. In addition to this, there are a few things to be assumed in both the methods along with the assumption that the model has been formulated using the data.
Interpolation is considered to be a useful statistical and mathematical tool which is used for estimating the values between two points. It is the process that is used for ascertaining a value between two points on a curve and any line. The usefulness and application of interpolation has been found not only in statistics but also in any business and science for prediction of any value within the points of two existing data. Linear interpolation is the simplest method that helps the user in estimating the value at positions in between the points on the given data set and the straight line segments is formed by joining the points. Interpolation of each segment can be done independently. However, at each point, linear interpolation results in the discontinuation and it is often desirable to have a smother interpolation function. Interpolation is used for predicting the dependent variable value from the independent variable which is outside the data range.
Extrapolation on other hand is the process by which the value can be found outside the data set that is they help in the prediction of future value. The extrapolation function is used for predicting the future dependant variable for the independent variable that is outside the data range.
Interpolation helps in estimating unknown value at the unknown points by using the points with known sample values and points. For any geographic point, data such as rainfall, elevation, noise level and chemical concentrations and on so on, methods of interpolation can be used to predict such unknown values. There are several methods of interpolation and the description are listed below.
Inverse distance method- In the inverse distance interpolator, it is assumed that each point has a local influence, which diminishes with increase in the distance. All the points that is closer to the processing cell that is greater than all the cells that is farther from the cell. The value of output for each location can be determined by using a specified number of points within particular radius. Further, it has also been assumed in this method that the influence of the variable decreases with the distance from its location on the sample. This method is regarded as an effective moving average interpolator that is applicable to high variable data. Using this data, variable can return to the collection site by recording new value within the general area trend. The interpolated surface is greater than the local minimum value and is less than the local maximum value. This particular method of interpolation comes with several advantages such as influencing the cell values by decreasing or increasing amount of sample points along with estimating the extreme changes such as fault lines and cliffs.
Natural neighbour inverse distance weighted- There are various positive features of natural neighbour inverse distance weighted and the working is mainly based on the clustered scatter points. The datasets of large input points are efficiently handled by this method and the amount of influence caused by the scattered points is determined by the local coordinates of the natural neighbour method. Accurate surface models from the sets of data are created by this method with the data set being vary spatial in linear distribution and sparsely distributed. This particular method is considered to be appropriate when there is uneven density distributions of the sample points. It is not required to specify the parameters such as number of weights or neighbours or radius.
Kriging- This is a technique of geostatistical that consider the degree of variation and the distance between the points on the data for estimation of the values in area unknown. There is a specified number of points to which the mathematic function is fitted by the Kriging tool.
Trend- This statistical method of interpolation that helps in ascertaining surface fitting the sample points using the method of least square regression fit. For every input point, the surface is constructed in such a way that helps in minimizing the difference between the estimated values and actual values. Trend in the sample data is detected using this method of interpolation and is similar to the natural phenomenon. Surface trend are considered good for identifying the interpolated surface that rarely passes through the sample points and identifying the scale patterns in the data.
For the initial condition in terms of estimating the values at cell nodes, two methods are used which include step interpolation method and continuous interpolation method. Variation of distribution for the initial condition that continuously vary in the domain is done by the step interpolation method. There are only closed contour involved in the step interpolation method and ignores the line of open contour. On other hand, for the continuous variations in the conditions within domain should be done by adopting the method of continuous interpolation method.
A new value is estimated using the interpolation method that involves connecting two adjacent points with a straight line. The missing value can be found using interpolation by using different sets of formula representing the first and second set of data points of the values that has been observed. The formula helps in finding out the unknown value on any given point. Comparing the formula of Lagrange’s interpolation, there should be the availability of the n number of sets with the method used for finding the new value. The following formula given below depicts the formula of linear interpolation.
m= m1 + (n-n1) / (n2-n1) * (m2- m1)
(m - m1) = (n-n1) / (n2-n1) * (m2- m1)’
Where, m1 and n1 are considered to be the first coordinate and m2 and n2 are the second coordinate. n is the point that is to perform the interpolation and n is the interpolated value. Therefore, it represents a function that is going through the paired orders
The usage of this formula can be explored by identifying the estimates between the points on the given data set. The estimates between the two set of points can be found out by (m1, n1) and (m2, n2). The example of interpolation can be used to understand the application of the method in detail and how such method is used for finding out the value on given datasets. In addition to this, there can be direct implementation of the linear interpolation in the Microsoft excel provided the fact that the tabulated values are monotonic in m that is no two equal values are equal and the m values are sorted. The direct implementation of the equation of linear interpolation works the same way as the inbuilt equation in the Microsoft excel. For the simple approach that has been described, it is required to have only two loop up functions as against the direct implementation approach which requires six loop up function.
The way interpolation works can be explained by producing a regression line using the data with m between 0 and 10. Value of n corresponding to the given value of m can be found or estimated by using the line of best fit and the value of m is resting at 8. Let us consider the linear equation n =5m + 7 and then the value of m can be estimated using the best fit line. The given equation is plugged with the value of n and it is viewed that the value of n is equal to n= 5 * 8+ 7= 40 + 7= 47. It is considered to be the example of interpolation because the value of m is not among the range of values for making the line of best fit.
The payment made for the particular period of calculation is determined by using linear interpolation. This is so because the calculation period is regarded as the interval between the end dates of two periods. The adjustment of days to maturity that are used in the interpolation is done using the same business days that is used for the end dates period.
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