Accepted Manuscript Title: Estimation of the Limit of Detection Using Information Theory Measures Author: Jordi Fonollosa Alexander Vergara Ramon Huerta Santiago MarcoAuthor to whom all correspondence should be addressed: PII: DOI: Reference:
S0003-2670(13)01338-X http://dx.doi.org/doi:10.1016/j.aca.2013.10.030 ACA 232904
To appear in:
Analytica Chimica Acta
Received date: Revised date: Accepted date:
9-8-2013 8-10-2013 11-10-2013
Please cite this article as: J. Fonollosa, A. Vergara, R. Huerta, S. Marco, Estimation of the Limit of Detection Using Information Theory Measures, Analytica Chimica Acta (2013), http://dx.doi.org/10.1016/j.aca.2013.10.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Estimation of the Limit of Detection Using Information Theory Measures
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BioCircuits institute (BCI),
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Jordi Fonollosaa,1, Alexander Vergarab, Ramon Huertaa, Santiago Marcoc,d
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University of California San Diego,
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La Jolla, CA 92093, USA
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Biomolecular Measurement Division, Material Measurement Laboratory,
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National Institute of Standards and Technology,
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Gaithersburg, MD 20899-8362, USA
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Signal and Information Processing for Sensing Systems
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Institute for Bioengineering of Catalonia (IBEC),
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Baldiri Reixac, 4-8, 08028 Barcelona, Spain
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Departament d’Electrònica, Universitat de Barcelona,
Martí i Franqués 1, 08028 Barcelona, Spain
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Author to whom all correspondence should be addressed:
Dr. Jordi Fonollosa Tel.: +1 858 534-6758 Fax: +1 858 534-7664 e-mail:
[email protected] 1
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Abstract
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Definitions of the Limit of Detection (LOD) based on the probability of false positive and/or
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false negative errors have been proposed over the past years. Although such definitions are
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straightforward and valid for any kind of analytical system, the proposed methodologies to
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estimate the LOD are usually simplified to signals with Gaussian noise. Additionally, there is a
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general misconception that two systems with the same LOD provide the same amount of
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information on the source regardless of the prior probability of presenting a blank/analyte
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sample. Based upon an analogy between an analytical system and a binary communication
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channel, in this paper we show that the amount of information that can be extracted from the
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analytical system depends on the probability of presenting the two different possible states. We
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propose a new definition of LOD utilizing Information Theory tools that deals with noise of any
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kind and allows the introduction of prior knowledge easily. Unlike most traditional LOD
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estimation approaches, the new definition is based on the amount of information that the
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chemical instrumentation system provides on the chemical information source. Our findings
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indicate that the benchmark of analytical systems based on the ability to provide information
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about the presence/absence of the analyte (our proposed approach) is a more general and proper
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framework, while converging to the usual values when dealing with Gaussian noise.
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Keywords
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Limit of Detection; Information Theory; Mutual Information; Heteroscedasticity; False
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positive/negative errors; Gas Discrimination and Quantification
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1. Introduction The Limit of Detection (LOD) of an analytical method (or measurement instrument) is a
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fundamental figure of merit. In most settings, it specifies the smallest concentration quantity at
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which the analyte can be detected or distinguished from a blank measurement within a stated
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confidence level, thereby constituting a limiting factor of a chemical detection system. It must be
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remarked that the value of the LOD that is provided in the specifications of any generic system is
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very significant since it may have legal implications and play a relevant role in coordinating
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market regulations by standardization agencies. For example, in 2006 the US Environmental
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Protection Agency (EPA) reestablished the maximum admissible contamination level of arsenic
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in drinking water from 50 nmol/mol to 10 nmol/mol. Based on the LOD of the different
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methodologies to measure the concentration of arsenic, EPA invalidated some of the previously
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accepted techniques to measure arsenic content in water [1, 2]. Therefore, a clear and accurate
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definition of LOD along with a consistent experimental protocol for its estimation is imperative
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for the determination of the LOD provided in the specifications of any chemical system as well
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as for the mutual understanding among technicians, system developers, and policy makers.
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Over the last decades, different definitions of the LOD have been proposed, each leading to
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substantially different estimated values of the LOD [3]. Early definitions of LOD considered
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only the probability of having false positive errors (i.e., the probability of falsely claiming the
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presence of the analyte in a sample, or Type I errors) and not the probability of having false
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negative errors (i.e., the probability of falsely claiming the absence of the target compound, or
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Type II errors), which could lead to 50% the Type II error probability [4]. Similar approaches
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based on the probability of errors were utilized to evaluate the acceptability of analytical
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methods [5]. Subsequent definitions of LOD were adapted to take into account both Type I and 3
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Type II errors [6], recommending to set the LOD at the concentration level that would make the
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probabilities of Type I and Type II errors 5% or less. However, although all these widely
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accepted definitions are generally valid for any kind of analytical system, the proposed
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methodologies utilized to estimate the LOD are usually simplified to signals with Gaussian
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noise.
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A chemical measurement system can be considered in most cases as a black-box with
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input/output signals. The input signal is the concentration of the target analyte, whereas the
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output signal takes the form of an instrument- dependant quantity derived from the instrument
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raw output (e.g., peak-area integral at a certain retention time in a GC-FID configuration). We
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have to note that the LOD is always given in terms of the analyte concentration. Therefore, the
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concentration (input space) uncertainty must be estimated from the variance at the instrument
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output. To do so and build the corresponding calibration model, the systems are frequently
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assumed to have a linear input-output relationship and a constant output variance
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(homoscedasticity) because linear models favor an easy transformation from the output noise
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variance to the input. In some instrumental techniques, however, the input-output relationship
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could be non-linear (e.g. due to competition effects) and the noise distribution can depart from
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the assumed Gaussian distribution. Interestingly, even though some authors have considered
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more complex sources of noise like heteroscedastic stochastic process to solve problems in
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realistic scenarios, they still assume Gaussian noise in their calculations [7, 8].
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The assumption of Gaussian noise in the estimation of the LOD can be too restrictive for
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chemical detection systems because the measured value (i.e., the analyte concentration) is
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necessarily positive. Therefore, the distribution of noise inevitably becomes asymmetric and
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non-Gaussian for very small concentration levels, which corresponds to the concentration range
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explored to estimate the LOD. In attempts to deal with measurements in which the variance is neither normal nor
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homogenous, Fraga et al. introduced a methodology to estimate the LOD of chemical systems
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based on the repetition of measurements with and without the chemical of interest in a test
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sample [9]. Because the presence or absence of the chemical is known by the practitioner, the
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authors were able to estimate the Type I and Type II probability errors from the predictions of
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the chemical system. Then, to estimate the LOD, the concentration of the chemical of interest
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was increased gradually and the error probabilities were evaluated for each concentration level.
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Finally, the LOD was set at the concentration that made the error probabilities lower than a
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defined threshold (10 % for both types of error). The authors ultimately used their methodology
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to optimize the operation of the system by building a Receiver Operating Characteristic (ROC)
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curve changing the relevant parameter.
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All methodologies based on the probability of Type I and Type II errors assume implicitly
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that the two states representing analyte/no-analyte exposure are presented with the same
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probability, i.e. the same a priori probability for both classes. However, in most of the
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applications one of the possible states is expected to be found more often than the other one,
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thereby leaning the probability of the system towards one of the classes. In this paper we show
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that the amount of information that can be extracted about the sample from the analytical system
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depends on the prior probability of presence or absence of the analyte. We propose a new
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definition of LOD based on the amount of information that the chemical system can extract from
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the presented stimulus. This new approach clearly shows that the amount of information
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provided by the analytical method is very dependent on the prior probabilities. In sharp
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comparison to previous definitions, our methodology, which is based on information-theoretic
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tools [10], is sensible to the probability of presenting the analyte and deals with noise of any
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kind. The remainder of the paper is organized as follows. In Section 2 we review previous
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definitions of LOD. Then, we will explore the amount of information that can be extracted by a
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chemical system (Section 3), followed by the proposed methodology to estimate the LOD
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(Section 4), two examples to estimate the LOD of a system (section 5), and the conclusions of
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this work (Section 6).
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2. Limits of Classic Definitions of LOD
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The LOD is a fundamental figure of merit, the definition of which has changed over the
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course of years. An early definition of LOD adopted by the IUPAC in 1975 [4] stated that “the
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limit of detection, expressed as a concentration CL (or amount, qL), is derived from the smallest
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measure, XL, that can be detected with reasonable certainty for a given analytical procedure”,
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determining the actual LOD represented by the following equation:
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= b + σb
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represents the mean of the blank measures,
(1)
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where
is its standard deviation, and k is a
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parameter set according to a defined confidence level [11]. Note that the LOD is expressed in
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units of concentration and that the distribution of the samples is estimated at the input space
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(concentration). However, since the systems are usually simplified to a linear relationship input-
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output, the standard deviation of the blank measurements is often estimated at the output space,
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and the LOD is converted into analyte concentration by utilizing a previously obtained
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calibration model: 6
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y
k σb = b
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(2)
where b is the slope of the linear calibration and
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output space (sensor output).
is the standard deviation calculated at the
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Based on Kaiser’s work, the numerical value k usually adopts a value of 3, so the confidence
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level is set to 99.86%, provided a one-sided normal distribution [12]. In 1995, the IUPAC
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adopted a new definition that considers both the probabilities of Type I and Type II errors [6]. In
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such an approach, the methodology to calculate the value XL included the standard deviation of
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the net concentration when the analyte is not present (σb) and, when the analyte is present, at the
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level of the LOD, σLOD. Assuming that the noise follows a normal distribution, XL can then be
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expressed as:
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= b + 1− σb + 1− σLOD
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(3)
where α and β define the thresholds for the Type I and Type II error probabilities, and z1-α
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and z1-β
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limit the accepted probabilities. If the noise in the system is considered homoscedastic (i.e., with
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a constant variance) and Type I and Type II errors are limited to 5% (as recommended by
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IUPAC), the Eq (3) can be rewritten as [13]:
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are, respectively, the upper percentage points of the two noise distributions that
= b + 2 0.95 σb = b + 3.3σb
(4)
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where the factor 3.3 is the common value to estimate the LOD when the noise is considered
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homoscedastic and its variance is known. If the variance of the noise is unknown, which is the 7
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common scenario, the z-values must be replaced by the equivalent t-values of the t-Student
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distribution [14]. Figure 1 shows a visual comparison between the two classic LOD definitions
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outlined above2. Despite the efforts made by the IUPAC to standardize the more rigorous
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definition adopted in 1995, many investigations are still presenting LOD estimates based on the
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definition that only considers false positive errors. The limitations of the frequent simplifications
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of methodologies to estimate the LOD based on the probabilities of the errors have been
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analyzed previously [7, 15]. They include the assumptions that errors in the blank signal are
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distributed normally, systematic errors are negligible, errors occur only in the y-direction, the y-
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intercept is not significantly different from the measured value, and a good calibration function is
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obtained. Additionally, they do not take into account the a priori probabilities of measuring
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blank samples or the chemical of interest.
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3. Information Theory Applied to Chemical Sensing
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Over six decades ago, Shannon developed the discipline of Information Theory (IT), a
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mathematical model of communication that quantifies the efficiency of data transmission over
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noisy channels by measuring the information content at the source and at the receptor [10].
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Information Theory is today a sound and complete framework for parameter estimation and for
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learning machines in general [16]. In measurement science in general, and, in particular, in
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chemical sensing, Information Theory techniques represent basic tools for algorithm analysis,
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parameter evaluation and optimization, molecular visualization, feature selection, and inference,
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as attested in numerous works [17-27].
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The reader is referred to very complete reviews [13-17] for a more detailed discussion of these definitions.
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The efficiency of a communication system is evaluated by comparing the a posteriori
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probability (i.e., the probability of decoding the original message after its reception) and the a
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priori probability (i.e., the probability of guessing the original message without any additional
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information). Hence, it is possible to establish an analogy between a communication system for
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data transmission and an analytical system. The analyte presence/absence can be seen as a source
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of information —the analytical system is equivalent to the transmission channel— and the output
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of the analytical system (sensor reading) represents the received message. By virtue of this
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analogy, the efficiency of an analytical system can be evaluated by estimating the a posteriori
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probability (the probability assigned to each state of the source knowing the output of the
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analytical system) and the a priori probability (the probability assigned to each state with no
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other information available). The source of information can be a continuous quantity if the goal
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is the estimation of the analyte concentration or a discrete variable if the purpose of the system is
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the determination of presence/absence of the analyte.
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This study is focused on the determination of whether the target compound is present or not
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in a test sample. Hence, the source of information is a random binary variable (present/absent
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analyte). The output of the analytical system, after the definition of a proper threshold, is also a
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random binary variable since it predicts the presence or absence of the target compound. Ideally,
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both variables should present the same state, but, as in the data transmission over a noisy
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channel, there may also be Type I (false positive) and Type II (false negative) errors. Figure 2
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shows a schematic representation of the analytical system with the error probabilities. This
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analogy corresponds to a binary communication channel, where the emitter transmits a bit
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representing the presence/absence of the analyte, and the receiver acquires a bit. The probability
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of flipping the bit during transmission, which is equivalent to a wrong prediction by the
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analytical system, determines the efficiency of the analytical system. Information Theory is based on probability theory and statistics. The two most important
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measures of information of Shannon’s mathematical theory of communication are entropy (S) —
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the information contained in a random variable— and Mutual Information (MI) —the amount of
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information that a second random variable Y yields about the random variable of interest X.
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First, the entropy is the amount of self-contained information in a process and describes the
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level of uncertainty (or disorder) of a system. The entropy of a Discrete Memory-less Source
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(DMS) depends on the probability of presenting the N possible different states (or symbols) xi:
()2 ()
(5)
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=1
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where S is expressed in bits and represents the minimum number of binary bits needed by a
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receiver to reconstruct the original message, and p(xi) is the probability of finding the state xi.
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For a DMS source with two different states (presence or absence of an analyte, see Fig. 2) the
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entropy can be simply expressed as a function of p(x1), the probability of finding the state x1;
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thus
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= −(1 )2 (1 ) − [1 − (1 )]2 [1 − (1 )]
(6)
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where p(x2) = [1-p(x1)]. For the extreme case in which a system has a probability p(x1)=1, the
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system always presents the state x1 and hence there is no uncertainty and S=0. Figure 3 illustrates
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the entropy of a two-state DMS, which is maximized when both states are equally probable i.e.
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p(x1)= p(x2)=0.5, providing a maximum entropy of 1 bit. Second, the Mutual Information is a measure of the information of one random variable
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contained in another random variable. Hence, the MI quantifies the amount of information on the
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state of the variable X contained by the known state of another random variable Y:
,
(, ) 2
( , ) ()()
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(7)
where px(i) and py(j) are the marginal probability distribution functions of variables X and Y and
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p(i,j) is the joint probability distribution function. In our analogy, X is the absence/presence of
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the analyte, while Y is the thresholded binary output of the analytical instrument. Our approach is
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based on the evaluation of the MI given by the observation of Y regarding X. Accordingly, if two
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random variables are statistically independent, the known state of the first variable does not bring
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any information on the unknown state of the second variable and MI=0. Conversely, if both
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variables are coincident, the known state of the first variable makes perfectly ascertainable the
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state of the second variable, and the MI equals S.
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Because Eq. 7 leads to the restriction 0 MI S, the maximum value of MI of an analytical
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system working to discriminate the presence/absence of a compound (Smax = 1 bit, see Fig. 3) is 1
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bit, which corresponds to a system where both states (presence/absence of analyte) are equally
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probable with no Type I or Type II errors. However, the MI of a DMS depends on the a priori
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probability of the states and the probability of Type I and Type II errors. Figure 4 shows the MI
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between the source (presence/absence of analyte) and the analytical prediction of
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presence/absence of analyte for different probabilities of presenting a blank sample in the source 11
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and different Type I and Type II error probabilities3. When the presence/absence of analyte is
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equally probable in the source (p(x1) = pblank =0.5), the maximum value of MI is 1 bit and the
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obtained map is symmetric. However, when the probability of presenting a blank sample
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increases, the maximum value of MI decreases and the Type I error becomes more significant
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because the system is biased towards the probability of presenting a blank sample, which is the
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needed input to obtain a false positive reading from the system.
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4. Definition of LOD based on Mutual Information
Our methodology to estimate the LOD of analytical systems is based on the analogy from a
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binary channel, where the source of information is the random variable representing the
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absence/presence of an analyte (represented by 0/1) and the random variable representing the
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prediction made by the analytical system corresponds to the received message. The maximum of
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MI between these two variables is 1 bit, which would correspond to an ideal analytical system
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with zero errors (Type I or Type II, i.e. α = β = 0, see Fig. 2) and where the prior probability of
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analyte presence is 50 %. Two different contributions impact the amount of information, MI, that
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can be extracted from the system taking into account the knowledge of the output Y. On the one
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hand, the prior probability of analyte presence limits the entropy of the system and the MI. On
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the other hand, the efficiency of the analytical system itself, which is given by Type I and Type
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II error probabilities, misguides the predictions made by the system. Therefore, in order to
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estimate the LOD of an analytical system, it is necessary (i) to set the desired thresholds for the
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We assigned the state ‘blank sample presented’ to x1 and the state ‘analyte presented’ to x2. Therefore, from now on, the probability of presenting a blank sample to an analytical system p(x1)=pblank.
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Type I and Type II probability errors, and (ii) to estimate the prior probability to present
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blank/analyte samples. The thresholds for the Type I and Type II error probabilities can be set to any arbitrary value
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according to the needs and restrictions of the user. The typical IUPAC definition for the LOD
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suggests a probability for both Type I and Type II errors of 5%. In this work we assume that
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there is some a priori knowledge that allows the estimation of the probability of presenting the
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analyte or a blank sample. However, if such information is not available, the results from the
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IUPAC definition are reproduced by setting the probability of presenting the analyte to 50 %.
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Therefore, before estimating the LOD, our methodology needs to define the parameters α, β and
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pblank to determine the MI threshold, MIth. Once MIth is set, the LOD estimation is given by the
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minimum analyte concentration that makes the MI between the source of information and the
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analytical system higher than MIth. Figure 5 shows the MIth for different values of the Type I and
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Type II errors and the probability to present blank samples. The figure also provides a guide to
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determine the MIth defined by the relevant parameters (α, β, and pblank ). For the convenience of
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the reader, the MIth values from Figure 5 for the most common scenarios are provided in Table 1
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(Equation A.1 in the appendix relates the MIth for arbitrary values of α, β, and pblank ).
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In summary, as the concentration rises, the amount of information that the system provides
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from the input is higher, and the MI between the binary input variable (gas present/gas absent)
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and the binary output variable (prediction of the sensor) increases. A MI threshold is defined
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from the accepted tolerances in Type I and Type II errors and the probability to present blank
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samples (see Figure 5). And finally, the LOD is the lowest concentration level that makes the MI
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higher than the threshold.
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It is important to note that MIth depends on three parameters that need to be defined before
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estimating the LOD: Type I and Type II errors and the probability to present blank samples. The
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first two parameters depend exclusively on the accepted tolerances in the errors made by the
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system and can take any value agreed by the practitioner, community, or standardization
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agencies. However, the estimation of the probability to present blank samples may not be
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straightforward. Although in most common scenarios such information is available and can be
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obtained either from previous experiments or in the literature, sometimes the practitioner has to
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face difficulties due to lack of information. If no information on pblank is available, one can
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simply assume pblank = 50 % to reproduce the LOD values proposed by the IUPAC. A more
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accurate solution would include the estimation of pblank from the same set of measurements used
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to calculate the LOD. If the samples to estimate the LOD are obtained from the same
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environment from which the system acquires samples in normal operation, one can expect that
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the obtained pblank from the subset of samples will be a good estimate of the actual pblank. In this
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scenario, the practitioner should collect first all the samples to estimate pblank and then determine
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the MIth to estimate the LOD.
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5. Examples of LOD estimation
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In this section we validate our methodology with two different analytical systems. First, we
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estimate the LOD of a system utilizing synthetic data to show that the noise distribution of the
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measurement, indeed, affects the LOD estimate. Second, we estimate the LOD of an analytical
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system to detect benzene, a compound that has special interest due to its carcinogenic properties
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and presence in industrial environments.
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5.1: LOD estimation under different noise probability density distributions (pdf) 14
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In order to compare our definition of LOD with the methodologies that assume
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homoscedastic Gaussian noise and neglect the probability of presenting the analyte, we
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simulated a system with different noise distributions. Specifically, we built a linear model of a
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system with sensitivity S=0.5 and offset=2: = . +
(8)
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We generated synthetic noise with different probability density function distributions. The
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resulting noise was added to the system output (Y). In particular, we studied four different
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distributions of noise4: a) a Gaussian noise with mean 0 and standard deviation σ = 1, b) a
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uniform distribution in the interval (-1.73,+1.73), c) a discrete binary noise distribution where the
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values ±1 are equally probable, and d) a Gaussian noise distribution with mean 0 and standard
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deviation increasing linearly with the input signal according to σ = 1 +0.1 X. The latter
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distribution reproduces a system, the variability of which is proportional to the input
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concentration and which is a common attribute of analytical systems [28]. Note that the four
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abovementioned distributions have been designed to show the same standard deviation at zero.
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Therefore, the measured standard deviation of the blank samples (at the output space Y) is sn=0.5
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for all the noise distributions, and the estimated LOD coincides for the methodologies that only
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consider the dispersion of the blank samples. In particular, assuming k=3.3, the estimation of the
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LOD corresponds to 3.3 in all the considered examples.
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The methodology based on the MI estimation, however, is sensitive to the different noise
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distributions and the probability to present blank samples. Figure 6 illustrates that MI increases
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differently for different noise distributions, which is the origin of providing different MI 4
The noise distributions are referred to the input space (in units of concentration).
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estimates for the LOD. Figure 7 shows the LOD estimations for the different noise distributions
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and the probability of blank samples (pblank). From Figure 7 we can conclude that i) the LOD
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changes for different noise distributions if we want to obtain the same amount of information
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from the analytical system and ii) the LOD needs to be more restrictive (higher analyte
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concentration) when the probability of presenting blank samples increases. Although the effect
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of pblank becomes only significant when it is expected to measure blank samples most of the time,
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it is a scenario faced by many applications. Additionally, the variation of the LOD caused by the
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different values of pblank or Type I and Type II error probabilities is comparable to the parameter
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divergence of other methodologies and definitions. For example a change in the parameter k
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from 3 to 3.3 represents a variation of the LOD estimation of the same order of magnitude as the
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variations shown by the different noise distributions and pblank. Therefore, the methodology to
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estimate the LOD based on the MI can provide sound and consistent LOD estimations, can be
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adapted to any arbitrary value of Type I and Type II errors, and, in contrast to any methodology
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presented before, give the possibility to introduce a priori knowledge on the probability of
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presenting blank/analyte samples.
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5.2 LOD of a chemosensory system
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5.2.1 Data Collection
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To illustrate the methodology proposed to estimate the LOD, we studied an analytical
334
system to detect benzene, which has been identified as carcinogenic to humans and can be
335
present in the atmosphere from natural sources such as oil seeps and wild fires or originated in
336
industrial environments, gasoline filling stations and automobile combustion engines [29]. In 16
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order to minimize the risks of the individuals exposed to benzene, the Occupational Safety and
338
Health Administration (OSHA) defined permissible exposure limits (PEL) according to the time-
339
weighted average (TWA) and short-term exposure limits (STEL) in different industrial scenarios
340
[30]. Therefore, the measurement of the levels of benzene is important for worker safety, but
341
such measurements need to be performed reliably with instrumentation that has the LOD clearly
342
defined.
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For illustration purposes, we consider a detection system based on a metal oxide (MOX) gas
344
sensor, the conductivity of which changes when the sensing layer is exposed to
345
reducing/oxidizing volatiles. MOX sensors are a common choice due to their cost-effective
346
design, native cross-reactivity (i.e., the vast number of volatiles sensors can detect), sensitivity,
347
and ease of operation [31-33]. In particular, we utilized a TGS2610 MOX gas sensor
348
(Figaro[34]) placed in a 60 ml volume Teflon/stainless steel air-tight gas chamber into which
349
benzene could be injected at different concentrations. In order to ensure a sensor response as
350
reproducible as possible, the sensor was pre-heated for several days before starting the set of
351
measurements while flowing 200 ml/min of clean air in the test chamber. Briefly, the vapor
352
delivery system was composed of two fluidic branches that met each other in the injection point
353
before bringing the resulting gas mixture to the test chamber. On the one hand, the solvent
354
branch included a high-pressure cylinder containing the carrier gas (medical-grade dry-air
355
supplied by Airgas [35]) connected in series to a mass flow controller (supplied by Bronkhorst
356
High-Tech B.V.[36]) with a maximum flow of 200 ml/min. On the other hand, the solute branch
357
was based on a high-pressure calibrated cylinder of benzene at 500 nmol/mol in air (provided
358
and certified by Airgas), the flow of which was regulated by a mass flow controller (Bronkhorst
359
High-Tech B.V.) with a maximum flow rate of 100 ml/min. A computer platform equipped with
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a National Instruments acquisition board (PCI-6014) and LabView software (ver. 6) was adapted
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to command the full set of experiments, which required control over several parameters that can
362
be defined by the user. First, the sensor’s operating temperature was controlled by the voltage
363
applied on the built-in heating element of the sensor, which was kept constant at 5 V during the
364
whole set of experiments. Second, the concentration of the gas sample (benzene) was controlled
365
by the flow of the two fluidic branches in such a way that the total flow was kept constant to 200
366
ml/min. And finally, the resistance of the sensor was acquired at a sampling frequency of 100 Hz
367
and stored in a computer for further processing.
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The sensor was exposed to eight different levels of benzene concentration (12.5, 18.75, 25,
369
31.25, 37.5, 43.75, 50, and 56.25 nmol/mol), each repeated 13 times in a random order for a total
370
of 104 measurements. The experimental procedure for each of the measurements was composed
371
of three different steps. First, clean air was circulated in the test chamber for 5 minutes to capture
372
the signal baseline of the sensor. Second, the sensor was exposed to the mixture of benzene at a
373
concentration level randomly selected from the list of concentrations for 5 minutes. And finally,
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clean air was re-circulated for 5 minutes to purge out the gas sample cell.
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In order to consider only the steady-state portion of the sensor signal, we selected the
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samples of the time series just before air conditions were changed. In particular, for each of the
377
measurements, we selected the 4000 samples comprised from 250 s to 290 s to capture the
378
system response to a blank stimulus and the 4000 samples comprised from 550 s to 590 s to
379
acquire the response to the benzene presentation. Therefore, in total, we have 52000 samples
380
(4000 samples × 13 repetitions) for each of the concentration levels and 416000 samples
381
acquired from blank responses (we extracted the blank response from each of the 104
382
measurements).
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Usually, the baseline of the sensor is removed to improve the performance of the system.
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Baseline is estimated from the value of the transitory response just before the gas exposure [37].
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Therefore, we subtracted the mean of the sensor response during the 40 seconds before the
386
beginning of the gas exposure5.
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5.2.2 LOD Estimation
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In order to estimate the LOD it is necessary to define the accepted limits for the Type I and
389
Type II errors and estimate the probability to present blank samples. For the following example
390
we set the limits for the Type I and Type II errors to 5%, as suggested by the IUPAC. However,
391
in order to illustrate better our methodology, we estimated the LOD for two cases: i) assuming
392
pblank = 50%, which corresponds to the scenario with no a priori knowledge, and ii) assuming
393
pblank = 90%. Therefore, the MI threshold for the LOD estimation is 0.7136 and 0.2978,
394
respectively (see Table 1). Finally, it is necessary to compute the MI between the information
395
source and the sensor prediction for different benzene concentrations. The LOD will be defined
396
by the minimum benzene concentration level that makes the MI larger than the corresponding
397
threshold value.
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To estimate the LOD of a system given pblank, it is necessary to create a dataset with the
399
same ratio of blank measurements versus analyte exposures than pblank. Therefore, to estimate the
400
LOD for pblank = 50%, we built a dataset of sensor predictions with a balanced number of
401
blank/analyte exposures, whereas the dataset to estimate the LOD for pblank = 90% was balanced
402
accordingly. The MI between the source of information (analyte presented/absent) and the sensor
5
The collection of the total set of experiments took about 32 hours, and the measurements were acquired during day and night. Although uncontrolled variables such us room temperature and pressure can be considered constant during the length of a single experiment (15 minutes), they affect the baseline of the sensor during the whole duration of the experiment set.
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403
prediction increases for higher levels of benzene concentration (see Figure 8). The MI threshold
404
to define the LOD is set to 0.7136 (pblank = 50%) and 0.2978 (pblank = 90%). According the proposed definition in this work, the LOD is given by the smallest input
406
concentration that makes the MI higher than the corresponding threshold. Therefore, from Fig. 8
407
we adjusted a 4th degree polynomial function for each pblank to determine the concentration level
408
at which the MI reaches the corresponding threshold. The obtained LOD estimations are 22.4
409
nmol/mol and 25.1 nmol/mol for pblank = 50% and pblank = 90%, respectively. We also estimated
410
the LOD with the definition proposed by the IUPAC. The standard deviation of the blank
411
measures (at the output space) is 7.8 Ω/Ω. In order to convert the LOD to the concentration
412
space, we utilized the Clifford-Tuma model that relates the resistance of a MOX sensor when it
413
is exposed to pure air to the resistance of the sensor when it is exposed to different gas
414
concentrations [38, 39]. The estimated LOD value based on the standard deviation (k=3.3) of the
415
blank samples would be 21.1 nmol/mol, which is an optimistic value (especially for low
416
probabilities of presenting the analyte) compared to the LOD estimations based on Information
417
Theory. From Figure 8 we can conclude that the LOD needs to be shifted towards higher
418
concentration levels for different prior probabilities if the thresholds for the Type I and Type II
419
error probabilities, α and β, are set to be constant.
421
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6. Conclusions
422
In this work, we showed that the amount of information on the absence/presence of a target
423
analyte that can be extracted using a particular instrumental technique depends not only on the
424
prior probabilities of presenting the analyte or a blank sample, but also on the noise distribution. 20
Page 20 of 44
In other words, in order to have the same information on the chemical source, it is necessary to
426
set the limit of detection at different concentration levels if the noise distribution is different.
427
This outcome is an interesting result because analytical systems are usually compared based on
428
the misconception that if the error probabilities (false and/or negative positives) of two systems
429
are the same, the systems are equivalent regardless of the noise distribution present in the
430
system. Based on the amount of information, we proposed a methodology to estimate the LOD
431
that deals with noise of any kind and can provide more accurate comparisons between systems.
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Using synthetic and experimental data with different distributions of noise, we showed that
433
our methodology captures better the information contained in the analytical signals and is
434
sensitive to the noise distribution and the prior probabilities, whereas classic methodologies
435
always provide the same LOD estimate. We showed that a limit of detection estimated from
436
actual information that can be extracted from an analytical system is more informative than
437
simply estimating the error probabilities, thereby providing better comparisons between systems.
438
Therefore, we believe that our methodology can be of special interest to obtain accurate
439
benchmarks of analytical systems, and it can be used to estimate the LOD of a wide variety of
440
analytical systems. Further investigations on the definition and methodologies to estimate the
441
LOD may include maximum likelihood ratio tests [40], which have already been used to evaluate
442
the discrimination ability of analytical systems and can also deal with unbalanced number of
443
samples [41-43]. The Neyman-Pearson lemma [44] shows that, given the Type-I error rate,
444
likelihood ratio testing provides the test with the lowest Type-II error rate (maximum power) for
445
simple-vs-simple hypotheses. Therefore, when performing a hypothesis test between H0: µ=µ0
446
versus H1: µ=µ1, the likelihood ratio test is the most powerful test. However, when designing a
447
test to estimate the LOD of an analytical system, a new sample will have certain probability to
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belong simultaneously to class blank (H0) and class analyte (H1). Therefore, one does not have to
449
reject H0 in favor of H1 since the goal is the optimization of a combined hypothesis of the form
450
ƞH0 + (1-ƞ)H1. Additionally, one needs to build a model for class blank and class analyte from
451
experimental data, and the parameters of the models will unavoidably have some uncertainty that
452
may limit the power of the likelihood ratio test. Hence, it remains as further work to adapt
453
maximum likelihood ratio tests to estimate LOD of analytical systems.
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Appendix:
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Mutual Information for arbitrary values of α (Type I error probability), β (Type II error
462
probability) , and P1(probability of presenting a blank example):
463 464 465 466
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p1 1 1 p1 MI p1 1 log 2 1 p1 log 2 p1 p1 1 1 p1 1 p1 p1 1 1 p1 1 p1 1 p1 p1 log 2 1 p1 1 log 2 p1 p1 1 p1 1 1 p1 p1 1 p1 1 22
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Acknowledgements:
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This work has been supported by the Jet Propulsion Laboratory under the contract number 2013-
470
1479652 and partially funded by the Spanish Ministerio de Economía y Competitividad under
471
the project TEC2011-26143. Alex Vergara was financially supported by the NIST/NIH Research
472
Associateship program administered by the National Research Council and partially financed by
473
NATO under the Science for Peace & Security Program under grant no. SPS-984511. Santiago
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Marco is member of the consolidated research group SGR2009-0753 by the Generalitat de
475
Catalunya. The authors also thank Joanna Zytkowicz for proofreading and revising the
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manuscript. The suppliers and methodological tools identified herein are only specified for the
477
experimental procedures presented in this manuscript. Their mentioning in no way implies
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recommendation or endorsement by the National Institute of Standards and Technology.
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Vitae:
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Jordi Fonollosa (Ph.D., 2009 – University of Barcelona) is a Postdoctoral Researcher at the
482
BioCircuits Institute, UC San Diego. His research is focused on gas sensor array robustness and
483
optimization, support vector machines, and Information Theory applied to chemical sensing.
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Other strong interests include biologically inspired algorithms, signal recovery systems, and
485
infrared sensing technologies.
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Alexander Vergara (Ph.D., 2006 – Universitat Rovira i Virgili) is a NRC research associate
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jointly working at the National Institute of Standards and Technology (NIST) and the National
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Institutes of Health (NIH) and a Visiting Research Scholar at the BioCircuits Institute, UC San
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Diego, where he was a postdoctoral researcher until summer 2012. His work mainly focuses on
490
the use of dynamic methods and information-theoretic formalisms for the optimization of micro
491
gas-sensory systems and on the building of autonomous vehicles that can localize odor sources
492
through a process resembling the biological olfactory processing. His areas of interest also
493
include information theory, signal processing, pattern recognition, feature extraction, chemical
494
sensor arrays, and machine olfaction.
495
Ramón Huerta (Ph.D., 1994 – Universidad Autónoma de Madrid) is a research scientist at the
496
BioCircuits Institute, UC San Diego. Prior his current appointment, he was associate professor at
497
the Universidad Autónoma de Madrid (Spain). His areas of expertise include dynamic systems,
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artificial intelligence, and neuroscience. His work deals with the development algorithms for the
499
discrimination and quantification of complex multidimensional time series, model building to
500
understand the information processing in the brain, and chemical sensing and machine olfaction
501
applications based on bio-inspired technology. Dr. Huerta's research work gathers in a
502
publication record of over 90 articles in peer-reviewed journals at the intersection of computer
503
science, physics, and biology.
504
Santiago Marco (Ph.D., 1993 – University of Barcelona) is Associate Professor at the
505
University of Barcelona and head of the Signal and Information Processing for Sensor Systems
506
Lab at the Institute for Bioengineering of Catalonia, Barcelona, Spain. His research concerns the
507
development of signal/data processing algorithmic solutions for smart chemical sensing based in
508
sensor arrays or microspectrometers integrated typically using microsystem technologies. Dr.
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Marco research has produced over 100 articles in peer-reviewed archival journals. More at
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http://www.ibecbarcelona.eu/artificial_olfaction
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References:
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[1] K.A. James, J.R. Meliker, J.A. Marshall, J.E. Hokanson, G.O. Zerbe, T.E. Byers, Journal of Exposure Science and Environmental Epidemiology (2013). [2] Analytical Feasibility Support Document for the Second Six-Year Review of Existing National Primary Drinking Water Regulations US Environmental Protection Agency, 2009. [3] L.A. Currie, Analytica Chimica Acta, 391 (1999). [4] Pure and Applied Chemistry, 45 (1976). [5] E.F. McFarren, R.J. Lishka, J.H. Parker, Analytical Chemistry, 42 (1970) 358. [6] L.A. Currie, Analytica Chimica Acta, 391 (1999). [7] E. Desimoni, B. Brunetti, Analytica Chimica Acta, 655 (2009). [8] E. Voigtman, K.T. Abraham, Spectrochimica Acta Part B-Atomic Spectroscopy, 66 (2011). [9] C.G. Fraga, A.M. Melville, B.W. Wright, Analyst, 132 (2007) 230. [10] C.E. Shannon, Bell System Technical Journal, 27 (1948) 379. [11] H.M.N.H. Irving, H. Freiser, T.S. West, IUPAC, Compendium of Analytical Nomenclature - The orange book. Definitive rules, Oxford : Pergamon, 1978. [12] H. Kaiser, Analytical Chemistry, 42 (1970) 24A. [13] R. Boque, A. Maroto, J. Riu, F.X. Rius, Grasas Y Aceites, 53 (2002) 128. [14] M.C. Ortiz, L.A. Sarabia, M.S. Sanchez, Analytica Chimica Acta, 674 (2010). [15] E. Desimoni, B. Brunetti, R. Cattaneo, Annali Di Chimica, 94 (2004). [16] J. Principe, Information Theoretic Learning, Springer, 2010. [17] F. Dupuis, A. Dijkstra, Analytical Chemistry, 47 (1975) 379. [18] K. Eckschlager, Information theory in analytical chemistry, John Wiley & Sons, 1994. [19] V. David, A. Medvedovici, Journal of Chemical Information and Computer Sciences, 40 (2000) 976. [20] T.K. Alkasab, J. White, J.S. Kauer, Chemical Senses, 27 (2002) 261. [21] T. Pearce, A. Sanchez-Montañes, Handbook of Artificial Olfaction Machines, WileyVCH, Weinheim, 2002. [22] P.P. Vazquez, M. Feixas, M. Sbert, A. Llobet, Computers & Graphics-Uk, 30 (2006) 98. [23] A. Vergara, M.K. Muezzinoglu, N. Rulkov, R. Huerta, Sensors and Actuators BChemical, 148 (2010) 298. [24] M. Trincavelli, A. Loutfi, Ieee, IEEE International Conference on Robotics and Automation (ICRA), Anchorage, AK, 2010, p. 2852. [25] J. Fonollosa, A. Gutierrez-Galvez, S. Marco, Plos One, 7 (2012). [26] A. Vergara, E. Llobet, Talanta, 88 (2012) 95. [27] J. Fonollosa, L. Fernández, R. Huerta, A. Gutiérrez-Gálvez, S. Marco, Sensors and Actuators B: Chemical, 187 (2013) 331. [28] D.T. O'Neill, E.A. Rochette, P.J. Ramsey, Analytical Chemistry, 74 (2002) 5907.
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[29] P. National Toxicology, Report on carcinogens : carcinogen profiles / U.S. Dept. of Health and Human Services, Public Health Service, National Toxicology Program, 12 (2011) iii. [30] Toxic and Hazardous Substances: Benzene. Occupational Safety and Health Administration. [31] Y.K. Min, H.L. Tuller, S. Palzer, J. Wollenstein, H. Bottner, Sensors and Actuators BChemical, 93 (2003) 435. [32] N. Barsan, D. Koziej, U. Weimar, Sensors and Actuators B-Chemical, 121 (2007) 18. [33] G.F. Fine, L.M. Cavanagh, A. Afonja, R. Binions, Sensors, 10 (2010) 5469. [34] Figaro USA, Inc. [35] Airgas, Inc. [36] Bronkhorst High-Tech B.V. [37] J.W. Gardner, P.N. Bartlett, Oxford University Press, New York, 1999. [38] P.K. Clifford, D.T. Tuma, Sensors and Actuators, 3 (1983) 233. [39] P.K. Clifford, D.T. Tuma, Sensors and Actuators, 3 (1983) 255. [40] A.P. Dempster, Statistics and Computing, 7 (1997) 247. [41] A. Vexler, A. Liu, E. Eliseeva, E.F. Schisterman, Biometrics, 64 (2008) 895. [42] R. Thiebaut, H. Jacqmin-Gadda, Computer Methods and Programs in Biomedicine, 74 (2004) 255. [43] H.S. Lynn, Statistics in Medicine, 20 (2001) 33. [44] J. Neyman, E.S. Pearson, On the problem of the most efficient tests of statistical hypotheses, Springer, 1992.
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Tables:
574
Table 1: Mutual Information between the source of information and the prediction of the system
576
for different values of the probability to present blank samples and error Type I and Type II
577
probabilities. This table sets the threshold to define the MI for different configurations of the
578
system. The case MI (α = 0%; β = 0%) corresponds to the entropy of the system.
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Pblank=0.67
Pblank=0.75
Pblank=0.9
Pblank=0.98
MI (α = 0%; β = 0%)
1
0.9149
0.8113
0.4690
0.1414
MI (α = 5%; β = 5%)
0.7136
0.6450
0.5622
0.2978
0.0720
MI (α = 5%; β = 10%)
0.6205
0.5688
0.4984
0.2663
0.0646
MI (α = 10%; β = 5%)
0.6205
0.5497
0.4727
0.2402
0.0553
MI (α = 10%; β = 10%)
0.5310
0.477
0.4123
0.2111
0.0488
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Figure captions
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Figure 1: Representation of the classic definitions of LOD for a linear homoscedastic system.
581
The first definition has no control over the probability of Type I errors (left). The second LOD
582
definition considers the probability of Type I and Type II errors (right).
583
Figure 2: An analytical system shows an analogy to a communication channel. The presence (or
584
not) of an analyte represents the message in the source (information to be transmitted). The
585
analytical system is equivalent to a noisy channel where the message is transmitted. The readings
586
of the sensory system determine if the analyte is present (or not) at the source. The probabilities
587
of Type I errors (false positive) and Type II errors (false negative) are given by α and β
588
respectively.
589
Figure 3: Entropy of a two-state discrete memoryless source as a function of the probability of
590
finding the state x1. The entropy takes its maximum value S=1bit when both states (analyte
591
present/analyte not present) are equally probable (p(x1)= p(x2)=0.5).
592
Figure 4: Mutual Information (in bits) between the source (analyte presented/absent) and the
593
prediction of the analytical system for different values of the probability of presenting blank
594
samples. Type I error becomes more significant when the probability of presenting blank
595
samples increases.
596
Figure 5: The entropy of an analytical system aimed at the discrimination between
597
absence/presence of an analyte is limited to 1 bit. The MI between the source of information
598
(analyte presented or not) and the prediction of the system depends on the probability of
599
presenting a blank sample (pblank ) and the defined thresholds for Type I and Type II errors (α and
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β respectively). The LOD can be defined by the amount of information that can be extracted
601
from the analytical system. Once the parameters α, β, and pblank are set, this figure provides a
602
guide to determine MIth. Then, the LOD is the lowest concentration level that makes the MI
603
between the source and the output higher than MIth.
604
Figure 6: MI across the concentration of X for a system with homoscedastic σ = 1 Gaussian
605
noise (blue), uniform distribution (green), discrete binary noise (red), and Gaussian noise with
606
standard deviation increasing linearly with the input signal according to σ = 1 +0.1 X (black).
607
The probability of presenting a blank sample (pblank) is 50%, so the threshold that defines the
608
LOD is MIth=0.7136 (see Table 1). The obtained LOD utilizing our methodology for the three
609
systems is 3.30 , 3.12 , 2.0 , and 4.98 respectively. Our methodology to estimate the LOD is
610
sensitive to the noise distribution, whereas the IUPAC methodology provides the same
611
estimation for the three systems (3.3, assuming k=3.3) since the standard deviation of the blank
612
samples is the same for all the simulated systems.
613
Figure 7: LOD for an analytical system with different noise distributions. The LOD is estimated
614
in such a way that the amount of information provided by the system remains constant. In
615
contrast to classical definitions that would estimate LOD = 3.3 . for all the cases, the
616
methodology based on the MI is sensitive to the noise distribution and the probability to present
617
blank samples. Gaussian noise (dark blue), uniform distribution (light blue), discrete binary noise
618
(yellow), and Gaussian noise with increasing standard deviation (dark red).
619
Figure 8: Mutual Information between the source of information (presence/absence of analyte)
620
and the sensor prediction. We estimated the LOD for two different a priori probabilities pblank =
621
50 % (top) and pblank = 90 % (bottom). The red line shows the MI threshold, MIth, which
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determines the concentration for the LOD. MI increases for higher values of benzene
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concentration before it reaches the corresponding maximum value set by the entropy.
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Highlights
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We propose a definition of Limit of Detection (LOD) based on Information Theory.
Analytical systems are compared based on their ability to provide information.
The methodology to estimate the LOD deals with noise distributions of any kind.
Our methodology converges to the same LOD values than traditional methods.
We show different examples to estimate the LOD with our methodology.
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