Inverse estimation is a classical and well-known problem in regression. In simple terms, it involves the use of an observed value of the response to make inference on the corresponding unknown value of the explanatory variable. To our knowledge, however, statistical software is somewhat lacking the capabilities for analyzing these types of problems. In this paper, we introduce investr (which stands for inverse estimation in R), a package for solving inverse estimation problems in both linear and nonlinear regression models. 1Running the above block of code produces Figure 1 and the following output to the R console: estimate lower upper 2.9314 2.6035 3.2587where estimate is the ML estimate (1), and lower/upper correspond to the lower/upper bounds of the 90% inversion interval for x 0 (2). If instead the new water sample was subjected to the field test three times, thereby producing three response values corresponding to x 0 , say 3.17, 3.09, and 3.16 µg/ml, we would simply supply calibrate with a vector of these values as in calibrate(mod, y0 = c(3.17, 3.09, 3.16), interval = "inversion", level = 0.9) If interval = "inversion", and the slope of the model is not ignificant at the specified α level, then finite confidence limits for x 0 will not be produced. For example, suppose badfit is an "lm" object for which the slope is not significant at the α = 0.1 level. Then, as illustrated in Figure 2, calibrate(badfit, y0 = 10, level = 0.9) will either produce two semi-infinite intervals, e.g., Error: The calibration line is not well determined. The resulting confidence region is the union of two semi-infinite intervals:
The United States Air Force currently relies on schedule-based inspections using nondestructive evaluation methods for ensuring airframe integrity. The sensitivity of a nondestructive evaluation method is quantified statistically using a probability of detection process. The purpose of the probability of detection process is to generate a [Formula: see text] metric for a given nondestructive evaluation technique and corresponding defect (e.g. crack). This process could be conducted under various inspection conditions and defect sizes. The set of factors varied in the process is controlled to allow each nondestructive evaluation inspection to be treated as statistically independent. Current United States Air Force structural inspections are performed at time intervals that adhere to the independence assumption. However, the United States Air Force plans to service airframes based on their actual condition instead of the current schedule-based approach. Accordingly, there is emphasis on developing advanced health management technologies, such as structural health monitoring systems, which provide an automated and real-time assessment of a structure’s ability to serve its intended purpose. Therefore, structural health monitoring is considered to be equivalent to an in situ nondestructive evaluation structural inspection device. With a structural health monitoring system, the time interval between inspections will be much smaller than the time intervals between nondestructive evaluation inspections. Since structural health monitoring measurements are from the same sensors, in the same location, the independent measurement assumption used to analyze nondestructive evaluation methods is invalid. In this article, we present a statistical method consistent with current probability of detection process, yet designed to appropriately analyze dependent data. We demonstrate this method first with simulated data and then with experimental data from three test specimens of a representative aircraft structural component. This method leverages the advantages of a structural health monitoring system through its frequent measurements while maintaining its usefulness through appropriately computed probability of detection values. Furthermore, we present a numerical method for estimating the number of test specimens needed to achieve a desired [Formula: see text] value.
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