The principle of least squares states that the sum of the squares of the residuals— multiplied by their appropriate weights—will be a minimum for that set of answers (parameters) that has the greatest probability of being correct. The concept is simultaneously simple and complex because it applies with equal validity to computing a simple mean of two equally weighted measurements as well as adjustment of the most complex problem that can be described with functional model equations. Although it is correct to say that there is no known method proven to be better than a least squares solution, it is also true to say, within reason, that least squares can be used to obtain any desired answer. The difference lies in the selection of weights, and that is the responsibility of the user. The least squares procedure is specific and proven, but least squares can also be abused, sometimes unwittingly, to the point where a solution has questionable, marginal, little, or no value. The challenge in using least squares is to select the appropriate model, to write the equations (observation and/or condition) correctly, and to assign legitimate weights to the observations. Generating the solution could be a challenge (if done longhand), but computers are programmed to handle the matrices and to crunch the numbers needed to find the most probable solution to the problem. Having done all that, one could still argue that the most challenging part of using least squares is interpreting the results—such as using the covariance matrices to track three-dimensional spatial data accuracy.
Given one measurement of a distance, there is no basis for adjustment. In order to use least squares, there must be “extra” measurements. When tying in sideshots to a survey traverse, there is no “extra” measurement to the radial point, and no adjustment of that point is possible. But when computing around a closed loop traverse, the coordinates of the endpoint must be the same as those used for the beginning point. Once coordinates of each traverse point are computed, least squares is an appropriate procedure by which to find the most probable coordinate values of the surveyed points. Of course, if the sideshot points are tied in a second time from a separate survey point, redundancy does then exist and an adjustment of such redundant positions is possible.
In order to use least squares competently, the user must decide upon the appropriate model (is the coordinate system local, state plane, UTM, or geodetic) and write equations for the computations that utilize the measurements. As stated above, if there is no redundancy in the measurements, no adjustment is possible. But, given a loop traverse and redundant measurements, the equations used in the computations must be consistent with the model being used. In most cases, a slope distance must be reduced to horizontal, and the horizontal distance must be reduced to the ellipsoid if using geodetic coordinates or to the state plane coordinate grid (in the appropriate zone) if using state plane coordinates. Least squares cannot be used to correct errors caused by using the wrong model. Often, the model is implicit and defined by the context. For example, when measuring a distance with a plumb bob and steel tape, the implied model is horizontal distance, and everyone knows what that is. But, if the distance is measured with EDM or with GPS, it becomes more important to be specific about the definition of horizontal.It is the user’s responsibility to assure compatibility between the measurements, the model, and the solution obtained from a least squares adjustment.
In addition to providing the best possible geometrical answer to a network of redundant observations, a least squares adjustment can also be used to determine the statistical properties (standard deviations) of the answers obtained from the adjustment. This provides the spatial data analyst with valuable tools (error propagation) for making decisions about how the answers are used or interpreted. For example, elevations of points on a building are determined very carefully and compared with elevations determined earlier (say, six months). Did the building move during that time interval? If the building did move, how much did it move? If the differences are small, does that mean the wall really did move? Or, is it possible that the observed difference is the result of accumulation of random errors in the measurements? Least squares, error propagation, error ellipses, and hypothesis testing are tools that can be used to make statements based upon inferences having a rigorous statistical foundation at a level of confidence chosen by the user.
Linearization
Taken by itself, the concept of least squares represents too much computational effort to be practical. However, if the least squares process is combined with matrices (for computational compactness) and with computers (for processing speed), a least squares solution becomes feasible for a greater variety of problems. A second drawback to using least squares is that matrices are valid only for systems of linear equations and many spatial data computations involve nonlinear geometrical relationships. That obstacle is overcome using a process called linearization in which nonlinear equations are replaced by their Taylor series approximation.
When using least squares to solve a linear problem, the solution is obtained on the basis of a single iteration. However, successive iteration is required when using least squares to solve nonlinear problems. Since only the first two terms of the Taylor series are used in the matrix formulation (point-slope form of a line), the solution (a set of corrections to the previously adopted values) of a set of equations will be only an approximation. Stated differently, a consequence of linearization is that the answer being sought changes from “the actual numerical value” to “What correction(s) to the previous approximate value(s) will provide a better solution?” (For linear problems, the correction is the value between zero and the correct answer.) The most important question is “What is an acceptable answer?” An acceptable answer is one that fulfills the original (nonlinear) conditions within some tolerance selected by the user. That puts the user in control.
A corollary to the previous question might be “How small must the corrections be to be acceptable?” Answering this question also keeps the user in control. But, achieving the goal of finding the right answer or making the corrections go to zero (within some tolerance) typically requires an enormous amount of number crunching—feasible only when done on a computer. Very briefly, the overall process for solving a nonlinear problem is as follows:
1. Identify the geometry of the problem and write the appropriate equations.
2. Linearize the equations and take partial derivatives to be used in the matrix formulation.
3. Establish some initial value for each unknown parameter as being reasonably close to the final answer.
4. Run the least squares adjustment to find corrections to the initial estimates.
5. Look at the results. Are the corrections small enough to quit? If so, do.
If not, update the previous estimate using current corrections and run the adjustment again. Are the corrections smaller, and are they small enough?
Two important concepts described above are iteration and convergence. Iteration is the process of using results from a previous solution to solve the problem again. Convergence is the desirable condition realized when each successive correction is smaller than the previous one. If a solution converges slowly, it may take many iterations to solve a problem. Given that computers are programmed to do the number crunching, the time and effort required may or may not be an issue. If solutions are being generated in real time, rapid convergence is preferable and linear models that do not require iteration are even more desirable. When using the GSDM, network adjustments can be formulated as a linear model.
