Objective Function
The objective function can be written as
since (I – H) is also symmetric and idempotent. It can be shown from this, that the expected value of S is m-n. Note, however, that this is true only if the weights have been assigned correctly. If unit weights are assumed, the expected value of S is, where is the variance of an observation.
If it is assumed that the residuals belong to a Normal distribution, the objective function, being a sum of weighted squared residuals, will belong to a Chi-squared distribution with m-n degrees of freedom. Some illustrative percentile values of are given in the following table.
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m-n 10 9.34 18.3 23.2 25 24.3 37.7 44.3 100 99.3 124 136
These values can be used for a statistical criterion as to the goodness-of-fit. When unit weights are used, the numbers should be divided by the variance of an observation.
Read more about this topic: Linear Least Squares (mathematics)
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