# Linear and Non-Linear Interpolation

## The In-Betweens

Interpolation can loosely be defined as a process in which you estimate unknown function values that are in between function values that you do know. This is commonly required in situations with non-linear relationships, and a particularly convenient approach when you don't even know the actual function definition.As an example: you own a car that is 5 years old. You know it was worth $10,000 when you bought it, and it is worth$6,500 today. If you need to know what its worth was 1 year ago, and have no other information, you can interpolate the year 4 value between $10,000 at year 0 and$6,500 at year 5.There are several examples of interpolation being practical and necessary in real-world mathematics. Here is an example from Actuarial Science:You are a life insurance actuary working on a problem involving partial-year life expectancies. Your life tables give probabilities of survival at each age, but not in between. For example, you can easily calculate the probability that a random 40 year old is still alive at age 50 or 51 using the table, but you need to interpolate if you are calculating a partial-year probability such as the probability that a 40 year old is still alive at age 50.5.

## Other Methods

Choosing one method versus another is a matter of both practicality and appropriateness. As we saw, linear interpolation uses only standard arithmetic operations, and therefore may be easier to use in certain cases. However, if we have reason to believe that the smoothed curve concavity pattern between points that we get from exponential interpolation is true to our expectation of reality, we may want to use that.There are several other uncommon methods of interpolation: trigonometric, polynomial based, or even logarithmic. For example, if you are interpolating sunrise or sunset times based on annual seasonality patterns, you would probably want to use sine functions (the amount of daylight per day graphed over a year has approximately a sinusoidal pattern!).No matter which method you choose or are asked to use, we should always remember two things aboht interpolation.1. It is an estimate only.2. The smaller the interval, the more credible the guess.

## It is Not Extrapolation

Finally, be warned - it is not alright to apply interpolation methods to estate unknown values that are outside of the values that you know. That is called extrapolation, and it is almost never appropriate as an estimation technique. We simply do not have a reason to believe that patterns will continue outside of an interval that we have known values for, particularly in cases where we do not know exactly what the function is formulaically (which, as we said, tends to be the type of function that we want to interpolate for).In short, interpolation is looking at estimates in between known values, while extrapolation is taking a function or model and projecting its validity outside of the range of known data. Keep them straight by the prefixes (inter vs extra). Extrapolation is almost never studied in rudimentary high school and college courses, even in statstics, because of how futile it tends to be in practicality.
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