Truncation

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For other uses, see Truncation (disambiguation).

In mathematics, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.

For example, consider the real numbers

5.6341432543653654

32.438191288

6.3444444444444

To truncate these numbers to 4 decimal digits, we only consider the 4 digits to the right of the decimal point.

The result would be:

5.6341

32.4381

6.3444

Note that in some cases, truncating would yield the same result as rounding, but truncation does not round up or round down the digits; it merely cuts off at the specified digit. The truncation error can be twice the maximum error in rounding.

In statistics, the term truncation is used differently and can be thought of in the same context as censoring. Usually the values that insurance adjusters receive are either left truncated, right censored or both. For example, if policyholders are subject to a policy limit, u, then and loss amounts that are actually above u are reported to the insurance company as being exactly u because u is the amount the insurance companies pay. The insurance company knows that the actual loss is greater than u but they don't know what is is. On the other hand, left truncation occurs when policyholders are subject to a deductible. If policyholders are subject to a deductible, d, any loss amount that is less than d will not even be reported to the insurance company. Any loss amount that is greater than d will be reported to the insurance company is loss - d because that is the amount the insurance company has to pay. Therefore insurance loss data is left truncated because the insurance company doesn't know if there are values below a specific amount. They don't know how many losses occur or how much each loss is.

[edit] Truncation and floor function

Truncation can be done using the floor function. Given a number $x \in \mathbb{R}_+$ to be truncated and $n \in \mathbb{N}_0$ , the number of elements to be kept behind the decimal point, the truncated value of x is

$\mbox{trunc} \left(x,n \right) = \frac{\lfloor 10^n \cdot x \rfloor}{10^n}.$

For negative numbers truncation does not round in the same direction as the floor function: truncation rounds toward zero, the floor function rounds down.