Current factor analysis receptor models are ill-posed |
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| Institution: | 1. Electrical Engineering Department, Khaje Nasir Toosi University, Tehran, Iran;2. Electrical Engineering Department, Shahid Beheshti University, Tehran, Iran;3. School of New Technologies, Iran University of Science and Tech., Tehran, Iran;4. Electrical and Electronics Engineering Department, University of Qom, Qom, Iran;1. School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;2. Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;3. Information & Telecommunication Company, State Grid Jibei Electric Power Co., Ltd., Beijing 100053, China;1. Stantec, Calgary, AB, T2A 7H8 Canada;2. Cumulative Environmental Management Association, Air Working Group, Fort McMurray, AB, T9H 4A4 Canada;3. Government of Alberta, Alberta Environment and Parks, Edmonton, AB, T6H 4T8, Canada |
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| Abstract: | Current factor models lack sufficient physical constraints to guarantee a unique, physically valid solution; in this sense they are ill-posed. Any realistic factor model must obey certain natural physical constraints, for example, the predicted source contributions and elemental compositions must be non-negative. Five such constraints are given in the paper. As shown by a simple example with only two sources and three elements, these natural constraints are insufficient to define a unique factor model. The same is shown to be true for a more complex example with seven sources and 10 elements. Since the examples use simulated data without observational or other errors, they prove that current factor models are, in general, biased in the statistical sense. The examples also show that the bias, or systematic error, can be very large. Thus, while factor analysis continues to be a valuable screening tool for unexpected sources, in the hands of the inexperienced it could lead to serious errors in source apportionment and derived source compositions. |
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