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( Least squares)
The method of least squares, also known as regression analysis, is used to model numerical data obtained from observations by adjusting the parameters of a model so as to get an optimal fit of the data. The best fit is that instance of the model for which the sum of squared residuals has its least value, a residual being the difference between an observed value and the value given by the model. The method was first described by Carl Friedrich Gauss around 1794.[1] Least squares corresponds to the maximum likelihood criterion if the experimental errors have a normal distribution. Regression analysis is available in most statistical software packages. The method of least squares grew out of the fields of astronomy and geodesy as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Exploration. The accurate description of the behavior of celestial bodies was key to enabling ships to sail in open seas where before sailors had to rely on land sightings to determine the positions of their ships. The method was the culmination of several advances that took place during the course of the eighteenth century[2] An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres. On January 1, 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Based on this data, it was desired to determine the location of Ceres after it emerged from behind the sun without solving the complicated Kepler's nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.
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