# Interpolation and Extrapolation

• skp338
This involves fitting a straight line to the data points and using it to estimate the value of z at the unknown point (x1, y1). This method is less prone to error compared to interpolation and extrapolation, and the degree of polynomial does not need to be chosen. The error in the estimate can be calculated using the difference between the actual and predicted values of z. In summary, for finding z at an unknown point (x1, y1), it is recommended to use linear regression as it is more accurate and does not require choosing a degree for the polynomial. The error in the estimate can also be calculated.
skp338
Hi,
I've a table of data in the following format:
x--> 0.1 3.767 4.395 5.0223
y
0.1 | 1 1.5 2.0 2.0---->z
0.6764 | 1 2.0 2.2 3.599-->z
1.10146 | 2 2.2 2.5 3.686-->z
1.3855 | 2.5 2.618 2.673 2.718-->z

Now using the above table, i wish to find z at unknown point say (x1,y1). As far as i know, I've two options:
(1) Interpolation and extrapolation (2) Curve Fitting.
I mention my problems with each:
(1) Interpolation and extrapolation: I can do bilinear interpolation to some satisfaction but what if the interpolation point is out of range. That is I'm not sure about the best method to do extrapolation. Please suggest.
(2)Curve Fitting: I'm thinking of Polynomial curve fitting. I'm not sure about the degree of polynomial. Is there any criteria do choose the degree of polynomial? Now suppose I fix the degree to three, what is the best menthod to obtain the coefficients of the polynomial? Please help. The last question related to curve fitting is How do I know the error I'm making in getting z for an unknow point (x1, y1). Thanks in advance.

I would recommend a linear regression.

## 1. What is interpolation and extrapolation?

Interpolation and extrapolation are mathematical techniques used to estimate values between known data points or outside the range of known data points, respectively. Interpolation is used to estimate values within the range of known data points, while extrapolation is used to estimate values outside the range of known data points.

## 2. What is the difference between interpolation and extrapolation?

The main difference between interpolation and extrapolation is the range of data points used. Interpolation uses known data points to estimate values within that range, while extrapolation uses known data points to estimate values outside that range. In other words, interpolation is used to fill in the gaps between known data points, whereas extrapolation is used to extend the known data points beyond their range.

## 3. What are the common applications of interpolation and extrapolation?

Interpolation and extrapolation are commonly used in fields such as statistics, engineering, and science to estimate values in between or outside known data points. They can be used in data analysis, curve fitting, and prediction models, among other applications.

## 4. What are some limitations of interpolation and extrapolation?

One limitation of interpolation and extrapolation is that they rely heavily on the quality and quantity of the known data points. If the data is sparse or inaccurate, the estimated values may also be inaccurate. Additionally, extrapolation is considered less reliable than interpolation as it involves making assumptions about data points outside the known range.

## 5. How can I determine the accuracy of an interpolation or extrapolation?

The accuracy of an interpolation or extrapolation can be determined by comparing the estimated values to the actual values. If the estimated values are close to the actual values, then the interpolation or extrapolation is considered accurate. However, it is important to note that there may be a margin of error due to the limitations mentioned earlier.

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