| A conventional model of adjustment usually assumes that the observational vector is random but the design matrix is non-random or deterministic. For this type of models, we can use the least squares (LS) method to estimate the model parameters, which is known to have all important statistical properties such as unbiasedness and minimum variance if the model is linear. However, in practice, we often have to deal with a model in which both the observational vector and (some or all of the elements of) the design matrix may contain random errors. This type of models has been known as errors-in-variables (EIV) models.The EIV model could be scatterly found in statistical literature. The interest in the EIV model has been revitalized as a result of the publication by Golub and van Loan (1980) on total least squares, even though Golub and van Loan (1980) actually used the modern language of singular value decomposition to re-account the method invented by Pearson (1901). Nevertheless, the EIV model has since become a basic mathematical model (see e.g., van Huffel and Vandewalle1991) and has found a wide variety of applications in different areas of science and engineering, including geodesy, signal and image processing, computer vision, communication engineering, for example.The simplest method to estimate the unknown parameters of an EIV model is to ignore the randomness of design matrix as if it were deterministic and then apply the LS method to estimate the parameters. However, the LS estimate is not optimal in this case, since the random errors in the design matrix have not been taken into account. In order to properly account for all the random errors araising in the system of observation/measurement, the total least squares (TLS) method was invented to handle both the random observational vector and the random elements of the design matrix. From this point of view, the TLS method is statistically more rigorous than the LS method to estimate the parameters in an EIV model. Nevertheless, the TLS method is computationally much more complicated than the LS method.While the EIV model has been widely applied in practice, we realize that the TLS and LS methods may lead to negligible differences in the estimated parameters for some problems. This is particularly true in geodetic coordinate transformation. Actually, for such problems, we do not need the computationally heavy TLS method; the computationally cheap and easy LS method is sufficient. Thus a natural question is under what condition we can simply use the cheap LS method. In such cases, we may further want to know whether or not the correctness and accuracy of the estimated parameters have been sacrified when compared with those by using the exact TLS method; and if any, to what extent. The above-posed questions can be partly reformulated as the mathematical question of how the random errors in the design matrix affect the LS estimate. Unfortunately, almost nothing has ever been done to answer this important question, except for two statistical publications by Hodges and Moore (1972) and Davies and Hutton (1975), who made some rough estimates based on some over-simplified assumptions. Thus as one of the major motivations of this dissertation, we will systematically investigate the effect of the random errors of the design matrix on the LS estimate.Reliability has been one of the important topics in geodesy since Baarda (1968) published his report in the series of the Netherlands Geodetic Commission. Although it has been substantially investigated in the conventional geodetic adjustment model, little has been done in association with the EIV model. Schaffrin and Uzun (2012) assumed a simple stochastic model and discussed the reliability of the EIV model. The problem is that their reliability measure contains the TLS estimate of parameters themselves, which are actually not available practically. Even worse, the TLS estimate itself can be significantly affected by outliers, which may nullify the reliability analysis. Proszynski (2013) made a naive application of the reliability theory to the EIV model. Nevertheless, no new insight into the unique feature of an EIV model can be demonstrated from the published paper, to our surprise. Thus as the second group of major motivations of this dissertation, we will systematically develop the reliability theory for the EIV model on a rather general assumption on the stochastic model. More specifically, we will investigate the effect of random errors of the design matrix on the conventional reliability measures and then develop a systematic theory of reliability for the EIV model.The major contributions of this dissertation can be summarized as follows:(1) Starting from a general stochastic model for the EIV model, we systematically investigate the effect of the random errors of the design matrix on the estimated quantities of geodetic interest, in particular, the model parameters, the variance-covariance matrix of the estimated parameters and the variance of unit weight. By taking the bias and the variance-covariance matrix into account, we can further compute the mean squared error matrix for the LS estimate. By simplifying our bias formulae, we can then readily show that the corresponding statistical results obtained by Hodges and Moore (1972) and Davies and Hutton (1975) are actually the special cases of our study. As a by-product, we also find and correct the over-simplified formulae (and/or mistakes) reported in the above publications. The theoretical analysis of bias has shown that the effect of random matrix on adjustment depends on the design matrix itself, the variance-covariance matrix of its elements and the model parameters. The relative bias of the parameters is shown to decrease with the increase of the square of the signal-to-noise ratio of the design matrix, but the relative bias of the standard deviation of the parameters only decreases linearly with the increase of the same signal-to-noise ratio.(2) By using the derived formulae of bias, we can then attempt to remove the effect of the random matrix from the LS estimate and accordingly obtain the bias-corrected LS estimate for the EIV model. Nevertheless, the random errors of the design matrix can make the LS estimate become much less efficient when compared with the TLS estimate.(3) We derive the bias of the LS estimate of the variance of unit weight. It is easy to see that the random errors of the design matrix can significantly affect the LS estimate of the variance of unit weight. The theoretical analysis can be used to successfully all the anomalously large estimates of the variance of unit weight reported in the geodetic literature (see e.g., Schaffrin and Felus2008; Cai and Grafarend2009). We then propose a bias-corrected estimate for the variance of unit weight in this dissertation.(4) We investigate how random errors in the design matrix would affect the measures of classic reliability. We derive the formulae to numerically compute the biases of classic reliability measures (internal and external reliability) due to the random errors in the design matrix, which are roughly inversely proportional to the square of the signal-to-noise ratio of the design matrix. We also correct these biases in order to construct the corresponding reliability measures for the observation vector of EIV model.(5) We develop the theory and method of reliability of EIV model in the general weighted case, starting from a partial EIV model. We derive the formulae to compute the internal and external reliability measures both for the observation vector and the random design matrix. The formulae of reliability have shown that the redundant numbers in the EIV model are basically dependent on the relative magntitude between the variances of the observation vector and the elements of the random design matrix. Therefore, if the variances of the observation vector are small relative to those of the elements of the random design matrix and/or if the model parameters are sufficiently large, gross errors in the observations will be hard to detect. |
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