Getting “sampling” right for reliable process monitoring: main challenges and solutions

1 General framework

In today’s scientific and industrial landscape, processes are increasingly monitored through a dense ecosystem of sensors that continuously generate large volumes of data (Ge, 2017). This evolution has been strongly driven by the widespread adoption of Process Analytical Technology (PAT) (Dickens, 2010), a framework originally formalized within the pharmaceutical industry (Bakeev, 2010; Gerzon et al., 2022; Sathiyapriyan et al., 2025) and now broadly embraced across biotechnology (Gerzon et al., 2022), environmental monitoring (Vaneeckhaute, 2019), and food production (Gajdoš Kljusurić et al., 2026). In addition, PAT has progressively transcended the traditional boundaries of industrial manufacturing, expanding into diverse fields such as cultural heritage conservation (García-Diego and Zarzo, 2010; Díaz-Arellano and Zarzo, 2025) and healthcare (Perez-Benitez et al., 2023). At its core, PAT aims to understand, control, and optimize processes in real time by linking timely measurements to critical quality attributes and process parameters, for example, supporting Lean Six Sigma objectives by enabling variability reduction, early deviation detection, and continuous process improvement (Rish et al., 2026). Although different fields of applied science often use distinct terminologies for related concepts, PAT can be regarded as an evolution of Process Analytical Chemistry (PAC), itself a specialized subset of process analysis (Kourti, 2006), extending its analytical foundations through chemometrics, artificial intelligence, and advanced data-driven modelling within the digitalized frameworks of Industry 4.0 and the human-centric, sustainable paradigms of Industry 5.0 to enable smarter, autonomous, and adaptive decision-making systems (Szymańska, 2018).

A process can be broadly defined as a dynamic system in which measurable inputs, materials, energy, or information, are manufactured, processed or transformed over time into measurable outputs under the influence of both controllable and uncontrollable factors (Chatti et al., 2019). Such transformations may involve physical or chemical changes, as in classical unit operations commonly recognized in industrial practice, as well as the temporal evolution of any measurable system, including changes in instrument performance, signal stability, or measurement quality over time. From this perspective, a process encompasses not only material transformation but also the dynamic behaviour of analytical and monitoring systems. Technological, industrial and natural processes are inherently variable, heterogeneous, and often only partially observable [Introduction to “process analytical technology” (PAT), 2025]. PAT tools are designed to make these systems measurable by deploying sensors, predominantly spectroscopic (chemical) i.e., multivariate, but also univariate (physical), directly in, on, or near the process stream, enabling real-time, in situ process and material monitoring while minimizing sampling, reagent consumption, waste generation, and overall environmental impact, in accordance with green analytical chemistry principles (Rogers and Jensen, 2019).

The rapid proliferation of PAT sensors has led to an unprecedented abundance of signals. Hyperspectral, multispectral, and single-point spectroscopic measurements are now routinely combined with temperature, pressure, flow rate, pH, and other physical variables (Shokrani et al., 2024). While this sensor-rich environment promises deeper process understanding, there is a predominant implicit assumption that more sensors and more data automatically translate into better monitoring–but this is increasingly being challenged by evidence of poor model transferability, unstable predictions, and misleading process interpretations. Within this context, sampling refers not only to the act of extracting material from a moving stream or acquiring meaningful spectral information directly - via PAT sensors - but process sampling is also related to the scale of the data acquired, ranging from macroscopic bulk measurements to microscopic or localized scrutiny. The challenge is how to be able to guarantee representativeness of increments, samples or spectra from diverse material systems and such as liquids, powders, and heterogeneous solids in different physical states. Depending on how and where measurements are performed, PAT implementations may operate in-line (directly within the process stream, without sample removal), on-line (with automatic diversion and rapid return of the sample), at-line (close to the process, typically involving minimal manual handling), by remote acquisition sensors, or through off-line laboratory analysis, which requires discrete physical sampling and transport to centralized facilities (Blasco et al., 2025). Each configuration is dependent upon a different level of temporal resolution, invasiveness, and representativeness, all of which critically influence the reliability and interpretability of the resulting data.

In this context, multivariate data analysis is indispensable for extracting actionable information from complex, noisy, and highly correlated signals to support process understanding, calibration modelling and process monitoring and control. Chemometrics provides the mathematical and statistical backbone that transforms raw sensor outputs into potential process information through data cleaning and data modelling (Miller, 2010; Mowbray et al., 2022; Dürauer et al., 2024). However, even the most sophisticated algorithms and data models cannot compensate for fundamental weakness in data quality. There is a whole domain, often forgotten, the ‘before analysis’ domain, where expertise in representative sampling is essential (Esbensen, 2025).

This article addresses the main myths about PAT sampling. Drawing on the principles of the Theory of Sampling (TOS), this study emphasizes the framework for representative data acquisition in process monitoring and highlights that trustworthy PAT models depend first and foremost on sound sampling strategies before analysis, before data modelling. We thoughtfully study critical examples in the existing literature where sampling is discussed and where current chemometric trends aimed at mitigating sampling-related issues are presented, such as data augmentation, advanced preprocessing, sensor fusion, and adaptive modelling, highlighting both their potential and their limitations. Through selected data examples across different application domains, we illustrate where sampling assumptions break down, how their consequences manifest in chemometric models, and why a TOS-driven perspective is essential for building robust, interpretable, and reliable PAT systems.

2 PAT challenge driven datasets and software

2.1 Dataset A: heterogeneity at vertical level

A natural sodalite rock originating from Brazil was analysed using portable Raman spectroscopy. Raman spectra were collected with an InnoRaman portable spectrometer (B&W Tek, Inc., United States) equipped with a 785 nm diode laser and a Peltier-cooled charge-coupled device (CCD) detector. The nominal laser power at the source was 330 mW.

Spectral acquisition was performed over the range 100–1,700 cm⁻¹ with a spectral resolution of approximately ±5 cm⁻¹. The measurement conditions consisted of a 5 s exposure time, five accumulations per spectrum, and 100% laser power. Several scans were accumulated for each spectrum to improve the signal-to-noise ratio. Data acquisition and instrument control were carried out using BWSpec software (version 3.26, B&W Tek, Inc.).

Spectra were collected from multiple locations on the rock surface to illustrate spatial sampling at the interface. Five randomly selected points were analysed on the upper surface (defined according to the largest dimension of the rock), and five random points were measured on the opposite (bottom) surface. In addition, five visually distinct points, intentionally selected based on apparent differences in colour or texture, were also analysed. In total, fifteen Raman spectra were acquired from the sodalite sample.

2.2 Dataset B: heterogeneity at the sample-instrument interface

A short-wave infrared hyperspectral imaging (SWIR-HSI, Specim) data set was acquired from ground coffee samples adulterated with chicory (Dataset B.1) and malt (Dataset B.2) at concentration levels of 1, 5, 10, and 15 wt% (the particle size of the samples was assessed by sieving using a 200 µm mesh). 2 g of fine salt was added to each sample adulterated with chicory and 2 g of gross salt was added to each sample adulterated with malt. Before sample preparation, all materials (coffee, chicory, malt and salt) were dried in a ventilated oven at 50 °C for 72 h to minimize moisture-related spectral variability. The reference coffee material consisted of commercially available L’OR Ristretto 11 capsules, containing a caffeinated medium-roast blend of Arabica and Robusta coffee beans. The capsule content was removed and used as the base matrix for all Dataset B samples. Hyperspectral images were acquired in reflectance mode in the short-wave infrared range (996–2,550 nm) at a frame rate of 100.00 Hz and an exposure time of 4.80 m. Samples were analysed under two conditions: unmixed, in which adulterants and salt were added without further homogenization, and mixed, in which samples were manually well homogenized before image acquisition. This sampling strategy was designed to assess the effects of spatial heterogeneity on the spectral response in hyperspectral images.

2.3 Dataset C: heterogeneity in time

The dataset comprises 377 NIR spectra collected from samples obtained during the production of seven batches of Belgian Pale Ale beer. Spectra were acquired using a Spectrum 400 (Perkin Elmer Inc., United States) in the 800–2,500 nm range, with 8 nm spectral resolution, 32 scans per measurement, and a 10 mm optical path length quartz cuvette. For each sample, soluble solids content (SSC) was also measured using a refractometer. The dataset includes samples collected across the main stages of the brewing process and therefore reflects process variability along beer production. Detailed information regarding the brewing procedure, sampling protocol, experimental setup and spectral measurements can be found in the original publication (França et al., 2021).

2.4 Data analysis and software

Data processing and chemometric analyses were performed using MATLAB R2024b (The MathWorks Inc., Natick, MA, United States). Multivariate data modelling was conducted using PLS Toolbox v9.5 (Eigenvector Research Inc., United States). Hyperspectral data handling and visualization were carried out using Hypertools3 (Mobaraki and Amigo, 2018) and in-house code.

3 False myth: your sample is homogeneous

One of the most persistent misconceptions in process monitoring is the assumption that the analysed sample is intrinsically homogeneous. However, most materials and processes in science, technology and industry are inherently heterogeneous (it is only a matter of to which degree), making representative sampling a non-trivial universal challenge. The Theory of Sampling (TOS) provides a rigorous framework to ensure representativity by identifying, quantifying, and minimizing sampling errors associated with heterogeneous lots (Gy, 1992; Esbensen, 2020).

Within the TOS framework, despite the radical different nature of lots, whether stationary or moving, these are in fact governed by the exact same basic principles leading to appropriate sampling strategies. Achieving defensible representative sampling procedures requires adequate coverage in both time and space, sufficient sampled mass or effective volume, correct sampling location, using a suitable number and type of increments (composite sampling) (ibid; Esbensen, 2015; Esbensen et al., 2018). Neglecting any of these aspects compromises representativity, regardless of the analytical technique subsequently employed, nor data modelling approaches.

To address spatial and compositional variability, several increment selection strategies can be adopted. These include random and systematic sampling to ensure unbiased spatial coverage, stratified sampling to account for known structural or compositional domaining and transect or search-based approaches to explore gradients or localized heterogeneities. In contrast, judgmental (or convenience-based) sampling may be useful for exploratory purposes but does not guarantee statistical representativity. The choice among these strategies directly defines how the sampling interface between the sensor and the material is implemented and operated and therefore influences both the variability structure of the acquired data and the reliability of subsequent chemometric modelling (Ahmed, 2024). The different approaches are schematically illustrated in Figure 1a.

Material heterogeneity is the dominant source of sampling error, arising from multiple, often simultaneous, mechanisms. Constitutional Heterogeneity (CH) refers to variations in composition, particle size, or other intrinsic attributes between constituent lot units (e.g., particles, grains), while Distribution Heterogeneity (DH) results from segregation, clustering, or incomplete mixing giving rise to spatial non-uniformity. In dynamic systems, Temporal Heterogeneity (TH) reflects fluctuations in processes over time, whereas flow-induced heterogeneity arises from velocity gradients inducted flows and/or differences in residence time within vessels or containers (e.g., reaction vessels). Representative sampling can only be achieved by explicitly counteracting the effects from these heterogeneities in relation to the geometry and scale of the lot (Figure 1b). With TOS lots are classified as zero-dimensional (discrete objects such as tablets or particles), one-dimensional (streams or flows such as conveyor belts or pipelines), two-dimensional (surfaces or layers such as sheets or textiles), or three-dimensional systems (volumes such as silos or reactors) (Wagner and Esbensen, 2012). Moreover, as illustrated in Figure 1c, the volumetric (3-D) heterogeneity of a material can further amplify the complexity of obtaining a truly representative sample. In the figure it is shown how the analysed composition of a material can vary significantly depending on the sampled region. Although both containers contain the same types of particles, the relative proportions of the large grey, red, and small green particles differ between the two sampling volumes (23%–41%–36% vs. 32%–47%–21%). This visual example reinforces the need for TOS-compliant sampling strategies capable of accounting for structural variability throughout the entire volume.

From a TOS perspective, several types of sampling errors can occur, including fundamental, grouping and segregation, increment delimitation, increment extraction, preparation, weighting, and analytical errors (Esbensen, 2025); Esbensen, Romanach, Román-Ospino, 2018; EURACHEM/CITAC Guide, 2019). Insufficient sampling frequency may lead to under-sampling of fast process dynamics, whereas excessive sampling without representativity merely amplifies redundant or biased information (Holm-Nielsen et al., 2006; Bertanza et al., 2022). Reliable error suppression is critical in this context. Error effect estimation can be supported by replicating or duplicating complete ‘lot-to-aliquot’ sampling-and-analysis pathway experiments (so-called Replication Experiments (RE), see, e.g., the global standard “Horisontal Sampling—Representative Sampling: DS 3077” [Danish Standardisation Foundation (DS), 2024] or by dedicated heterogeneity tests, which allow quantification of experimental variability, thereby strengthening the statistical validity and robustness of the derived models (Minkkinen, 2004; Engström et al., 2022; Chieregati, 2024).

In industrial practice as well in the associated literature, there is a growing tendency to frame process development and optimization within the Quality by Design (QbD) paradigm across pharmaceutical, food, feed and environmental applications (Vaneeckhaute, 2019; Pérez-Beltrán et al., 2023; Isoko et al., 2024). Within this context, Design of Experiments (DOE) plays a central role in systematically building a basis for process knowledge and understanding variability by efficiently exploring the experimental space, either prospectively through planned studies (Farinini, 2024; Digkas et al., 2025; Leardi and Farinini, 2025) or retrospectively using historical data (Borràs-Ferrís et al., 2025). DOE strategies can accommodate experiments with different costs and unequal sample sizes; however, their effectiveness ultimately depends on the representativeness and quality of the collected data. Consequently, minimizing sampling-related errors is essential to ensure that the resulting models capture true process effects rather than being compromised by variability introduced by biased or non-representative sampling.

Nowadays, in certain regulated sectors such as the pharmaceutical industry, process development and control are strongly guided by regulatory frameworks such as Quality by Design (QbD) and Process Analytical Technology (PAT). Within this context, the definition of the Design Space, as established in ICH Q8–Q11 (ICH, 2012), is fundamentally based on multivariate models constructed from experimental and process data. Since these models are only as reliable as the data used for their development, the representativeness of sampling becomes a critical determinant of the validity of the resulting Design Space. In this sense, adherence to the Theory of Sampling (TOS) is not merely a best practice, but a prerequisite for ensuring that the input data used for model building accurately reflects the true process variability. In practice, these strategies are implemented across multiple operational units within an industrial plant, spanning the entire production chain from raw material inspection and monitoring, through intermediate processing and unit operations, to final chemical transformations and product quality control, thereby enabling a holistic and integrated understanding of performance (Bordawekar et al., 2015; Conway et al., 2023; Esbensen and Swarbrick (2018).

4 False myth: PAT solves sampling issues

The transition from Process Analytical Chemistry (PAC) to Process Analytical Technology (PAT) extends the traditional focus on material composition to encompass process dynamics and flow behaviour (Workman et al., 2011). The main aim of PAT is to implement, across different operational units, a sufficient and rigorous number of problem-relevant sensors to prevent non-compliance, facilitate immediate fault detection, and to support monitoring and control of critical process parameters, for example, to ensure consistent process robustness and product quality (Simon et al., 2015). In the PAT framework, measurement representativity is not an attribute related to the sample itself, but is dominantly influenced by specific sensor characteristics, positioning, and the design of process sampling interfaces (PSI) (Doherty and Lange, 2006; Kessler, 2013; Esbensen (2025). Consequently, planning PAT measurements requires careful competence and consideration of several questions: What do you need to know about your sample? What is the objective of the measurement? Monitoring and fault detection, classification, end-of-process detection, or quality prediction? What sampling frequency is required? Where and when does heterogeneity occur in the process? What is the sensitivity, selectivity and limits of detection required? (Kadijevic, 2025). Eventually, the fraction of the process effectively “sampled” depends on several interrelated factors: e.g., sensor type, positioning, design of the process sampling interface (PSI), field of view (FOV), penetration depth (Kessler et al., 2016; Esbensen, 2025). The critical role of the Process Sampling Interface (PSI) in determining the effective sampling of a heterogeneous process stream is illustrated schematically in Figure 2.

The sensor type determines the measurement principle and sensitivity, thereby defining which physical or chemical information can be captured and at what spatial or temporal resolution. Sensor selection requires considering the range of technologies commonly available, including optical, electrochemical, thermal, and spectroscopic sensors, each offering distinct advantages depending on the target analyte and process conditions (Gerzon et al., 2022), while detailed technical descriptions can be found elsewhere according to the type of process under study (Madsen et al., 2011; Biechele et al., 2015; Kessler et al., 2016; Panikuttira and O’Donnell, 2018; Jerome et al., 2019). Comparing their performance under identical operational constraints is essential to assess sensitivity, stability, and suitability for reliable decision-making. Moreover, the choice between off-line, at-line, on-line and in-line configurations directly affects maintenance demands, calibration frequency, environmental robustness, and ultimately the representativity and quality of the collected data.

Geometrical and instrumental aspects are central to ensuring representative measurements. The size, distribution, and modification of sample elements directly influence the fraction of the material measured. Similarly, the suitability, sensitivity, and positioning of spectroscopic probes determine the quality of the data. For instance, insufficient penetration depth can miss critical heterogeneity. Figure 3 provides a clear illustration of the three-dimensional geometrical and size-related issues of the sampling interface when dealing with heterogeneous materials, as exemplified by Dataset A. The sample was analysed 15 times at different points on the surface. RGB photos from different angles, the raw and pre-processed (baseline correction and smoothing p = 2, w = 13) spectra are represented respectively in Figures 3a–c, d–f.

Individual spectra, as well as average spectra with standard deviation, are represented. The spectra show characteristic peaks of sodalite framework at approximately 465 cm⁻¹ and 986 cm⁻¹. Peaks that could correspond to calcite, a typical sodalite inclusion, are observed around 442 cm⁻¹ and 532 cm⁻¹ (Zahoransky et al., 2016). In Figure 2a,d,g), calcite is absent, whereas in the other parts of the figure there is a clear contribution of calcite to the spectra, including at the specifically defined spot. These results provide an analogy for highlighting how geometrical sampling and probe positioning can influence the detection of different phases within a heterogeneous material. While measurements on a single surface are straightforward, different layers within the material can produce varying effects. In an industrial setting, a single surface measurement may result in loss of information about the material heterogeneity at greater depth.

The design of the Process Sampling Interface (PSI) governs how the material is interacting with the analysing sensor radiation, impacting flow stability, reproducibility, and the avoidance of segregation or dead zones. The literature reports numerous studies focused on process measurement and monitoring at laboratory, pilot plant, and industrial scales; however, while sensor implementation is implicitly embedded within these works, it is rarely addressed explicitly or treated as a dedicated topic of investigation. The optimization of sampling strategies is rarely discussed explicitly, although most contributions report the final or optimal sensor configuration. For spectroscopic technologies such as Raman, NIR, and NMR, it is especially apparent that most studies emphasise improvements in data treatment and process evaluation while largely overlooking the optimization of critical sampling interface parameters and the fundamental role of representative sampling. However recently three fundamental publications begin to remedy this situation (Esbensen et al., 2018; Esbensen and Romañach, 2021; Esbensen, 2025) and a limited number of laboratory-scale investigations have begun to address these aspects more explicitly (Gorla et al., 2022a; Schorn-García et al., 2023; Cavallini et al., 2025), reflecting a growing awareness of the need to integrate sampling design into analytical performance.

Some practical insights can be extracted from the literature regarding typical optimal process sampling interface configurations. In many application domains the analytical measurement is still preceded by a discrete sampling step followed by laboratory-based analysis, where primary sampling, sub-sampling and sample preparation dominate the overall uncertainty budget. Typical examples include food and pharmaceutical quality control, where grab or composite samples are transported to the laboratory for chromatographic or spectroscopic characterization, often with limited evaluation of sampling representativeness and time influences (Karnachoriti et al., 2025; Phuong et al., 2025). At-line measurements, often based on a sampling loop equipped with a pump and a degassing step to remove bubbles or volatile compounds, are also commonly used (Phuong et al., 2025). Other options include the use of specific autosamplers equipment (Ji et al., 2023). Depending on the spectroscopic technology and field of application, measurements are commonly performed online by a bypass setup, as for instance can be seen in (Zientek et al., 2016). On the other hand, using in-line or immersion probes, on-stream sampling systems, or chute-mounted sensors, often coupled with quartz or sapphire windows selected based on the required optical and chemical resistance (Kirsch and Drennen, 1996; Reis et al., 2007; Amigo et al., 2008; Grassi et al., 2023; Rosas et al., 2023; Rangel-Gil et al., 2024; Movilla-Meza et al., 2025). Sometimes, practical schematic implementations of probes in complex condition such as a pan coater for coating operation are available in literature Gendre et al., 2011).

Regarding the actual sampling that occurs at the process–sensor interface (PSI), practical results often involve estimating the effective sampling volume based on the sensor spot size and its spatial location within the process (Wikström et al., 2005; Wu et al., 2015). For NIR spectroscopy, specific sampling strategies and practical guidelines have been proposed (Esbensen and Abu-Khalaf, 2022a; Esbensen and Abu-Khalaf, 2022b) and examples of multi-sensor implementations can be found in the literature to address representativeness challenges (Wu et al., 2012; Vargas et al., 2017). Moreover, an interesting inline implementation discussion is presented in (Wahl et al., 2013), where the authors demonstrate that in-line NIR spectroscopy for melt extrusion must be treated as a true sampling operation rather than a purely analytical measurement. By quantifying the effective optical sample volume and showing how laminar flow, limited penetration depth, and probe positioning near the barrel wall can lead to non-representative measurements, it highlights the risk of sampling bias in PAT applications ((Esbensen et al., 2018; Esbensen and Romañach, 2021). Importantly, it shows that representativeness can be significantly improved through process and geometrical modifications, thereby integrating TOS principles directly into sensor and process design. Although not explicitly framed within TOS, (Glader et al., 2025), implicitly demonstrates that nanoparticle size monitoring in manufacturing is highly sensitive to sample conditioning and measurement configuration, and that improper dilution, viscosity effects, or flow conditions can compromise measurement reliability and representativeness. In situations like mining and mineral processing, industrial implementation of XRF and HSI is often carried out with little consideration for the field of view (FOV) or penetration depth. This is particularly evident in large-scale remote sensing applications in mining (Koerting et al., 2024) and in online XRF systems used for real-time material characterization (Sokolov et al., 2025), where operational efficiency tends to be prioritized over a rigorous evaluation of spatial representativeness and analytical depth.

In (Wikström et al., 2005), a comparison of two different Raman measurement approaches is presented, while (A. Gibbons et al., 2022) compares online and offline analytical strategies. Comparisons of results from different spectroscopic approaches highlight how sensor selection, measurement configuration, and sampling strategy influence both the quality of the data and the observability of key process variables, emphasizing that methodological choices directly impact the representativeness and interpretability of the collected information (Meyer et al., 2017; Sagmeister et al., 2019). Interestingly, some studies indicate that appropriately positioning the sensor according to the process step may provide sufficient process information, potentially reducing the need for multiple sensors (Sagmeister et al., 2021; Kadieveich (2026)). A recent review also discusses sampling issues in the food and feed industry in the context of PAT technologies (Gajdoš Kljusurić et al., 2026).

Sensor placement itself is a critical aspect and, in other disciplines, has been addressed using multicriteria optimization approaches (Brentan et al., 2021; Jia et al., 2025). Moreover, ensuring ease of maintenance (Peris and Escuder-Gilabert, 2013) are essential consideration for robust and sustainable sensor deployment. Hot of the press (2026), the 4^th issue of SAMPLING SCIENCE and TECHNOLOGY has the theme title: “Process Analytical Technology”, and is fully dedicated to an in-depth survey of the fundamental interrelationships between TOS, PAT and chemometrics (Ritchie, 2026).

The analytical performance of a measurement system strongly depends on the match between the sensor’s spectral response and the physicochemical properties of the target analytes. Beyond sensitivity, the concept of observability becomes essential: a process or analyte is considered observable only if the sensor configuration allows its presence and variation to be reliably detected and quantified. An example of the importance of the observability principle is provided by the analysis of DATASET B (B1 and B2). False color RGB images reconstructed from pre-processed hyperspectral data (pseudo-absorbance plus standard normal variate) using the wavelengths 1,511, 1945, and 2,216 nm assigned to the red, green, and blue channels, respectively, together with selected raw and preprocessed spectra from each analysed compound, are shown in Figure 4. These wavelengths were selected because they capture relevant chemical contrast among the analysed compounds since they correspond to absorption features associated with overtones and combination bands of molecular groups such as C–H, O–H, and N–H. Selecting three spectrally informative bands allows the hyperspectral cube to be visualized as a false-colour image that highlights compositional differences between materials, facilitating qualitative interpretation of spatial heterogeneity.

Data compression is one of the main approaches to understanding what information is contained in the data. For samples such as those in DATASET B, performing principal component analysis (PCA) on the dataset is expected to reveal common underlying patterns. To verify this, a PCA model was built using a mosaic containing both unmixed and mixed samples with increasing percentages of chicory as an adulterant. All samples contained 2 g of fine table salt. It should also be noted that although the components were initially dried, they were analysed for approximately 15 min under normal laboratory humidity, allowing partial reabsorption of environmental moisture. The results are presented in Figure 5. From the scores of the first two principal components, it is evident that the detection of salt and adulterant is straightforward in the unmixed samples but becomes less clear when the components are mixed. This effect is related to both spatial and spectral resolution. Salt does not exhibit a distinct SWIR signal; instead, the recorded spectrum is mainly associated with water of hydration. Consequently, once mixed with the other components, it becomes extremely difficult to discriminate the salt moisture contribution from that of the surrounding materials. The signal observed in PC1 is mainly associated with differences in water content among the grouped regions as shown in the loadings. In PC2, the regions containing the adulterant are clearly visible in the unmixed samples and remain identifiable, although less distinct, in the mixed ones. This behaviour may be attributed mostly to differences in spectral characteristics, but also to differences in particle size between coffee and the adulterant materials.

These results highlight the importance of selecting sensors with adequate sensitivity to the specific compounds of interest. Consequently, a thorough understanding of the strengths and limitations of each sensor technology is crucial, as these directly determine not only the type and quality of information that can be extracted, such as compound identification and achievable limits of detection, but also whether the system can effectively monitor and capture the relevant process dynamics (Orantes et al., 2008).

The field of view (FOV) controls the lateral area over which the signal is collected, directly affecting the proportion of material contributing to the measurement. Penetration depth defines the volume probed beneath the surface or through the medium, which is particularly critical for optically dense or heterogeneous matrices. Evaluating the adequacy of the FOV, and more generally the heterogeneity captured, can be aided by statistical tools such as variograms or homogeneity curves (Petersen and Esbensen, 2005; Minkkinen and Esbensen, 2014; Dehaine et al., 2016; da Silva et al., 2018; Rocha De Oliveira and De Juan, 2020), which quantify spatial variability and help assess whether the observed area is representative of the bulk material. These analyses provide insight into the scale and extent of heterogeneity, guiding both sensor placement and sampling strategy to ensure reliable and representative measurements.

The analysis of Dataset B2 provides an example of the important of FOV; the impact of different FOVs can be easily determined by comparing results of different ROIs in hyperspectral images. Homogeneity distribution curves (da Silva et al., 2018) have been calculated from the first principal component resulting from PCA models for different ROI regions (highlighted in Figure 6a) in dataset B2 with malt adulteration. A false RGB with the highlighted ROIs (in green and red) are reported in Figure 6.

The results clearly show that, when selecting a central ROI, the degree of mixing can be readily discriminated according to the distribution of components within the samples: in both Figures 6b,c, the upper curves correspond to lower homogeneity, whereas the lower curves indicate a higher degree of mixing. Furthermore, both the theoretical and experimental estimation curves improve with an increasing number of data points, highlighting the importance of adequate sampling density. Even more striking, however, is the strong influence of ROI selection on the actual resulting homogeneity curves. As illustrated in the last column of Figures 6b,c, while the unmixed samples exhibit comparable behaviour, the mixed sample appears markedly more inhomogeneous when a smaller ROI is considered, underscoring the critical role of spatial scale in homogeneity assessment. When assessing distribution homogeneity, the field of view is a critical factor, and studies have shown that, in certain cases, has been reported that analysing a single representative point may provide sufficient information without the need to capture the entire image (Tanzilli et al., 2024).

Overall, sensor positioning influences the representativeness of the measurement, as different locations within the process stream may experience distinct concentration gradients, flow patterns, or heterogeneity. In dynamic PAT contexts, such as mixing, grinding, fermentation, or chemical reactions, both spatial and temporal sampling are critical to achieve the proposed aim. The timing of measurements dictates whether heterogeneity is captured or overlooked. Acquisition frequency is therefore critical, as insufficient temporal resolution may fail to follow rapid process dynamics (Mercier et al., 2013; Ge, 2017). One of the main considerations for modern sensing technologies is the ability to perform fast, or even real-time, measurements while maintaining high data quality, without substantially compromising signal-to-noise ratio, accuracy, or reliability. An example of the technical capabilities of high-speed acquisition has been specifically evaluated for a Raman system, where dedicated studies were performed to demonstrate how quickly reliable spectra can be collected under practical operating conditions (Lintvedt et al., 2022). Moreover, current instrumentation increasingly enables multigranularity analysis, allowing data to be collected and then analysed at different temporal resolutions depending on sensor capabilities and process requirements. While this flexibility offers clear advantages by capturing both slow trends and rapid dynamics, it also introduces challenges related to data integration, synchronization, and interpretation across heterogeneous sources (Reis, 2019).

When analysing Dataset C, an important aspect concerns the sampling strategy and the associated measurement conditions. As commonly observed in many process monitoring studies, also in this case, measurements are performed across multiple processing steps characterised by different levels of heterogeneity, temperature, and physical consistency, while the specific influence of the sampling procedure on the collected spectra is not discussed in detail. Samples of 10 mL were collected at each stage at different time points and measured using a portable NIR instrument; however, aspects such as the specific sampling technique (e.g., manual extraction, valve sampling, or other methods) are not explicitly described.

The fermentation step is the most extensively studied stage in beer production and provides valuable insights for process monitoring (Grassi et al., 2014; Cozzolino, 2016; Horváth et al., 2026). It is also one of the most challenging stages due to the dynamic nature of fermentation, including changes in sugar content, yeast activity, and CO₂ evolution, this stage produces pronounced spectral variability that, if rightly managed, can be exploited to track process evolution, detect deviations, and evaluate the effectiveness of control strategies. A particularly valuable contribution that consider and state these challenges is represented by (Vann et al., 2017; Trojanowski et al., 2024; Goffi et al., 2025). For the fermentation stage, the original study of Dataset C reports a double sampling strategy, with samples taken from both the top and bottom of the vessel at five different time points, with three replicates collected for each condition. In the context of the control charts proposed in the work, it is reasonable to assume that the effects of turbidity and the presence of suspended particles, given their potential impact on spectral variability, were mitigated through spectral preprocessing. This highlights that there is no absolute rule for data treatment or sampling design; rather, both should be defined according to the specific objective of the study and the type of information that needs to be extracted from the data.

Here, we selected batches 1–3 and 5–6 as clear examples for illustrating how sampling frequency can strongly influence both fault detection capability and the discrimination between different process phases, highlighting the importance of considering sampling design alongside data preprocessing in process monitoring applications. A PCA model was performed using batches 1–3 under normal operating conditions (NOC) to construct the model, while batches 5–6 were projected onto this model for evaluation.

As shown in Figure 7, even a sampling frequency of once per day allows changes across the fermentation process to be observed. The scores plot of PC1 (83.89% explained variance) versus PC2 (5.04% explained variance) coloured by batch and time points shows that even a sampling frequency of once per day is sufficient to observe changes during fermentation. PC1 mainly explains the time evolution, and each fermentation exhibits a slight drift over the course of the process and variability at the endpoint. In addition, the two fermentations carried out under abnormal operating conditions (AOC) display a different temporal pattern compared to those under normal operating conditions (NOC), as reflected in the spectral signal. Inspection of the loadings suggests that the anomalies cause a faster transformation and a deviation in the direction of PC2, resulting in greater dispersion along this component. PC2 highlights differences between fermentations, clearly showing that the two AOC fermentations exhibit greater variability from time point 2 onward, as evidenced by the larger dispersion of points compared to the three NOC fermentations along PC2. In this case, the residuals obtained considering five PCs are not helpful for identifying the exact moment at which the change occurs. Of course, in this context, if the goal is to detect anomalies as early as possible, a higher sampling frequency is required, and the temporal resolution limits of the model in terms of anomaly detection need to be systematically evaluated.

Within a quality control framework, the information of interest may concern the distinction between final product quality and/or the characteristics of the starting material. Here, a PCA model was constructed using mashing (M), Circulation (C), Boiling (B), and Carbonatation (P) steps for three batches, onto which the same steps of two AOC batches were projected. As shown in the scores and loadings plots in Figure 8, samples are grouped according to the initial and final stages of production along PC1 and PC2 according to different chemical information, see (França et al., 2021) for more interpretative information.

An interesting thing to notice is that, for the final product, a clear difference can be observed between NOC and AOC samples, whereas the variability during the initial stages is so large that no clear trends can be identified among the different batches. The mashing step appears to be the most variable phase, however, one cannot discriminate whether this effect is due to sampling, sample composition, or simply to the chemical information contained in the acquired spectra.

Poorly planned sampling, whether in timing, location, or sensor selection, can distort model performance and interpretation in exploration, classification, or prediction tasks, the well-known formidable empowering features of chemometrics notwithstanding. Additionally, measurement systems often introduce a time delay between the actual process event and the recorded signal (e.g., due to transport lags, sensor response time, or data processing), which must be considered to correctly interpret process dynamics and avoid misleading cause–effect relationships (mismatch between X- and Y-space samples). While batch processes are typically characterized by time-evolving trajectories monitored at fixed locations, continuous processes require synchronized spatial–temporal monitoring along the process line to track steady-state stability and detect local disturbances with the highest precision required (Cattaldo et al., 2024).

5 False myth: data analysis can solve data issues (chemometric strategies to mitigate sampling bias and emerging trends)

A persistent misconception in Process Analytical Technology (PAT) is that advanced chemometric modelling can compensate for inadequate sampling or poor-quality measurements. In reality, representative sampling and robust measurement protocols constitute the foundation of reliable process monitoring and understanding as well as the success (or futility) of predictive model development. Data analysis can only extract the information that is physically present in the measurements; it cannot reconstruct variability that was never observed.

Beyond the sampling stage, the measurement workflow also includes the sensor spectral acquisition step and the subsequent data modelling stage, both of which may differ substantially depending on whether the system is deployed offline, at-line, online or inline. The concept of Measurement Uncertainty (MU) provides a structured framework to formalize and quantify the contributions of sources of variability by integrating the specific error contributions arising from sampling, analytical measurement, and data modelling respectively according to the relationship (Esbensen (2025); Minnitt et al., 2007; Esbensen and Wagner, 2014):

MU total = TSE + TAE + TDME

Here, the contributions from the Total Sampling Error (TSE) often dominates, exceeding the Total Analytical Error (TAE) by factors of 5–10, or even up to 25 (depending of the heterogeneity encountered), while the Total Data Modelling Error (TDME) is typically smaller (there is a limit to how wrongly one can establish a chemometric model). This hierarchy underscores a fundamental principle: the greatest improvements in reliability are often achieved by improving sampling representativity rather than evermore refined algorithms. It is important to note that, in this context, the term Total Data Modelling Error (TDME) is introduced to emphasise that uncertainty in chemometric models is not solely driven by sampling or analytical variability, but, even if to a smaller extent, is also introduced by the modelling and data-processing stage itself. TDME is not part of the formal Theory of Sampling (TOS) terminology but is proposed to emphasize the existence of errors that account for uncertainties introduced during data processing and modelling, such as preprocessing artefacts, model misspecification, overfitting, algorithmic limitations, and parameter estimation uncertainty. Conceptually, μ can therefore be understood as the aggregated impact of all error sources contributing to the discrepancy between the measured/modelled response and the true process state. Within this framework, TOS-related errors are considered a dominant upstream contribution because lack of representativeness propagates through all subsequent analytical and modelling steps.

Measurement uncertainty can be characterized using estimated statistical descriptors such as ranges, standard deviations, variances, root-mean-squared errors, or multivariate metrics. In complex chemometric applications, multivariate error estimation is particularly informative because it accounts for correlations among variables and provides a holistic representation of variability across high-dimensional datasets. This approach is especially useful when studying different types of replicates (technical, analytical, or process replicates) allowing one to distinguish true process variability from instrument or modelling contributions (Wentzell, 2014; Gorla et al., 2022b). It is important to acknowledge the essential difference between duplicated/replicated analysis (repeated processual operations) and ditto analytical results (numerical data); an attempt to clean up the enormous confusion regarding replication/replicates met with in the specialised analytical and chemometric literature can be found in (Esbensen and Swarbrick, 2018).

A critical consideration in chemometrics is ensuring that both the right variables and fully documentable representative samples are used (Kjeldahl and Bro, 2010). Temporal effects such as measurement delays, residence times, or misalignment with reference data most certainly affect model relevance and interpretability, highlighting the importance of rigorous model validation, ongoing monitoring, and maintenance throughout the model lifecycle (Gouveia et al., 2018; McDermott, 2023).

Recently the advent of Interactive Optimization Technologies (IOT) has caught much attention as “Lean chemometrics”. This has been hailed as the current frontline breakthrough, especially within pharmaceutical manufacturing, e.g., when monitoring powder blending, where the current chemometric choices including building PLS models remains a major drain on time, material, and budget. With IOT, one can instead use spectra of the raw materials (‘pure components’) for example, to track blend homogeneity and relative API/excipient levels in real time, notably without heavy DoE or large reference sample campaign efforts. It is always true that PLS delivers optimal quantitative results, but at a heavy cost of a high calibration burden: extensive sampling campaigns, many reference samples, often several calibration batches, and ongoing model maintenance. IOT changes this situation: Now there is minimal calibration using a pure component optimization model, which drastically reducing time, material, and cost versus the classical PLS approach. This is claimed to be a lean PAT approach, ideal, for example, when an API (Active Pharmaceutical Ingredient) is scarce, or expensive (or both) and therefore extensive PLS calibration are simply not realistic. IOT is these days claimed to be an almost universal solution (Petrusch et al., 2019; Sharma et al., 2020; Rish et al., 2026).

But not so fast. Upon closer scrutiny IOT looks almost like a throwback to the beginning of chemometrics, to the times of Multiple Linear Regression (MLR) on pure spectra - which, however, was quickly replaced by PLS because of the distinctly unrealistic basis for prediction models: artificial samples/pure components compared to real-world industrial or natural samples ores, commodities, biological resources, environmental samples a. o., which most definitely are not identical. Upon reflection IOT only has a positive impact in the technological sphere, in industrial manufacturing or processing, and especially in the pharmaceutical sector, where the nature of the components in use are pure components for obvious pharmacopeia demands. The verdict for IOT in general is that it is not a new panacea regarding calibration and validation. The world still very much needs the all-powerful PLS empowerment, and the world’s reflecting chemometricians accept this readily.

A growing trend in chemometrics and PAT-oriented applications is the use of artificial data augmentation techniques, such as Synthetic Minority Over-sampling Technique (SMOTE) and related synthetic oversampling methods, to address class imbalance or sparsely populated regions of the experimental space. The effect of artificial data augmentation in spectral monitoring models can be clearly illustrated through the comparison presented in Figure 9. The figure compares three PCA-based Multivariate Statistical Process Control (PCA-MSPC) of the previous beer example (Dataset C), where three batches are considered as NOC and two are considered OFC. The models were constructed under different training conditions, allowing the geometric consequences of synthetic data generation to be visualized directly.

The left panel shows the reference PCA-MSPC model built exclusively from real Normal Operating Condition (NOC) spectra. The monitoring chart represents Hotelling’s T^2 versus Q residual statistics. Blue points correspond to NOC samples used to build the model, while orange points correspond to spectra measured under Out-of-Control (OFC) conditions. As expected, the OFC samples appear clearly separated from the NOC region, exceeding the monitoring limits and therefore indicating abnormal process behaviour. The upper-right panel shows a model constructed by artificially augmenting the NOC dataset with SMOTE. In this case, synthetic spectra generated by interpolation between existing NOC observations were added to the calibration dataset. These synthetic samples appear clustered around the original NOC region. Despite the significant increase in the number of NOC observations, the position of the OFC samples relative to the monitoring limits remains essentially unchanged. The OFC samples still fall outside the control region, and the limits themselves exhibit only negligible variations. The lower right panel further illustrates that even when synthetic samples are generated for the OFC class, the geometric structure of the monitoring space remains largely unaffected. The augmented OFC points simply densify the existing OFC trajectory without expanding the monitoring boundaries or introducing new directions of variance. The comparison presented in Figure 9 demonstrates that artificial data augmentation does not modify the geometric structure of spectral monitoring models. While the number of samples increases, the underlying covariance structure and latent projection space remain essentially unchanged. The behaviour observed in the figure arises from the geometric nature of projection-based spectral models.

While such techniques may improve numerical performance indicators (e.g., classification accuracy or balanced accuracy), they do not generate new physical information and therefore do not correct inadequate sampling. Synthetic samples are typically constructed as linear or nonlinear interpolations between existing data points and thus remain constrained by the sampling variability already present in the dataset. Thus, from a sampling perspective, this practice risks reinforcing existing sampling bias rather than mitigating it. Regions of the process space that were never measured remain unobserved, and heterogeneity not captured at the sensor–process interface most definitely cannot be reconstructed algorithmically. In PAT environments, often characterized by multiphase flows, segregation phenomena, and nonlinear dynamics, models trained on artificially densified datasets may appear statistically robust while remaining blind to critical process states, transitions, or rare but consequential events.

Moreover, artificial augmentation can blur the distinction between genuine process variability and modelling artefacts. Interpolation between neighbouring samples implicitly assumes that the underlying process space has been adequately sampled, an assumption not automatically satisfied in real-world industrial settings. Synthetic data may well populate physically implausible regions of the variable space, apparently masking sampling gap intervals, but this provides but a false sense of coverage. This distortion is particularly problematic for model validation and uncertainty estimation: cross-validation or test-set performance may improve simply because the training space has been artificially densified. However, the model’s ability to extrapolate to truly new process conditions remain alarmingly untested. In terms of uncertainty budgeting, data augmentation does not reduce Total Sampling Error; it merely reshapes the statistical distribution of the training set without increasing process observability.

This does not mean that these efforts are without scope. In recent years, increasing research has focused on overcoming the limitations of traditional data augmentation by explicitly accounting for the physical origins of measurement variability (Flanagan et al., 2025; Ekwaro-Osire et al., 2025). In this context, emerging generative and physics-informed modelling approaches are being explored to produce synthetic data that better reflect realistic error structures, including instrumental noise, signal distortions, and variability patterns consistent with real analytical systems (Schiemer et al., 2024). These strategies aim to move beyond purely statistical manipulation towards data generation frameworks that incorporate mechanisms of error formation. Within this perspective, understanding how measurement errors propagate in both univariate and multivariate settings is becoming increasingly important, as multivariate models may amplify or redistribute error structures in ways that are not directly observable at the univariate level (Wentzell, 2014). Therefore, also elucidating multivariate error propagation remains an open challenge and represents a key step toward improving the realism of synthetic data and enhancing model robustness in PAT applications.

Although this is true, it remains the case that strategies focused on increasing the informational content of primary measurements, rather than artificially inflating datasets, are still the most effective option. First, experimental and sampling design should be revisited. Targeted acquisition of new samples in poorly represented regions of the process space, such as rare compositions, transient states, spatial extremes, or early and late process stages, directly addresses representativity and relevance intervals/gaps. Adaptive or active sampling strategies, in which exploratory models guide when and where new measurements should be collected, are particularly promising in PAT because they explicitly link model uncertainty to data acquisition. Second, multi-scale and multi-sensor approaches can enhance observability by combining complementary measurement principles (e.g., spectroscopic and physical sensors; surface- and bulk-sensitive probes). Data fusion strategies offer avenue to enhance informational content by integrating complementary datasets from multiple sensors. However, their effectiveness critically depends on the primary sensors being informative, as simply merging large amounts of highly collinear data adds complexity without improving insight or reliability (de Oliveira et al., 2020; Casian et al., 2022; Strani et al., 2024). Third, rigorous replicate designs, including process, sampling, and analytical replications, combined with direct or indirect heterogeneity testing allow quantitative estimation of sampling uncertainty and facilitate separation of true process variability from noise. This grounds model development in measurement realism rather than dataset size. When physical sample or spectral acquisition is constrained, modelling strategies that explicitly acknowledge this limitation are preferable to synthetic oversampling. Domain-aware validation schemes (e.g., leave-one-batch-out or leave-one-condition-out validation), conservative uncertainty quantification, and hierarchical or multilevel models reflecting the nested structure of process variability (batch, location, time) provide more honest assessments of model performance.

Digital twins and process simulators are powerful modern enablers of smart manufacturing, supporting hypothesis testing, scenario exploration, real-time optimization, and predictive decision-making. When embedded in Industry 4.0 frameworks, they enhance process understanding, facilitate virtual experimentation, and reduce development time and costs. However, they can never replace representative primary observation, relevant measurement and intimate process knowledge gained from empirical studies. At best, they help identify where additional real sampling is most needed. Moreover, the effective use of digital twins cannot take place without process knowledge and benefits from hybrid modelling approaches, where first-principles understanding is integrated with data-driven methods. Only by combining sound sampling, mechanistic insight, and advanced analytics can robust, reliable, interpretable and transferable models be achieved (Mihai et al., 2022). Consistent with a modicum of TOS and chemometric competence, trustworthy PAT models emerge only from well-designed measurement and sampling strategies, not from post hoc numerical densification of biased datasets.

6 Recommendations and best practices

The following recommendation summarizes our remedies against the main myths outlined above, emphasizes the many practical issues involved, and gives best practice guidance for technological and industrial monitoring. Table 1 provides a concise synthesis of common misconceptions, the associated technical or operational problems, and the corresponding recommended actions to support robust process monitoring. For practical implementation, it is advisable to use this summary as a reference when designing monitoring architectures.

TABLE 1. Myths, practical issues, and best practice.

For reliable industrial process monitoring, it is recommended to adopt a science-driven strategy in which sampling design is treated as a primary element of system development. Best practice begins with thorough process understanding supported by historical knowledge and empirical data analysis. Monitoring strategies should be developed by first characterizing process behaviour, including flow regime, mixing dynamics, particle size distribution, and potential heterogeneity sources. Sensor location, process sampling interface measurement geometry, and sampling frequency should be optimized to ensure that acquired signals are representative of the true process state while minimizing systematic bias related to limited analysing interaction volume or matrix effects. The integration of experimental design methods and multivariate exploratory analysis is strongly recommended to assess sampling adequacy before predictive model deployment. Sampling should be embedded as a critical parameter within Quality by Design workflows and explicitly evaluated during risk analysis and method development to reveal potential monitoring gaps and ensure alignment with critical quality attributes. Recently the Theory of Sampling (TOS) has been inducted into a formal Risk Analysis/Risk Management framework (Esbensen and Paoletti, 2022).

From a broader perspective, regulatory and standardization frameworks are increasingly recognizing the importance of representative PAT measurements for data integrity and model validity. Future guidelines will likely require clearer documentation and justification of sampling strategies, reinforcing the need for systematic and transparent approaches (Ritchie, 2026).

In practice, robust monitoring systems should prioritize representativeness from the earliest development stages, recognizing that advanced chemometric models and high-performance sensors cannot compensate for poor sampling design. Reliable process monitoring and control is only achieved when monitoring architectures are built on sound sampling and analytical science, ensuring that the subsequent chemometric predictions and related decisions are meaningful, reproducible, and applicable to real industrial conditions.

7 Conclusion

Sampling remains one of the most underestimated and frequently misunderstood sources of error in Process Analytical Technology (PAT) implementations. Continuous or high-frequency measurements are often assumed to be inherently representative of the process simply because of the volume of collected data. However, this assumption is misleading: sensors interrogate only a limited fraction of the material system, frequently under physical, spatial, or temporal constraints. When sampling fails to counteract undue contributions to observable process heterogeneity, the resulting datasets may exhibit high precision yet remain systematically biased, creating false confidence in model robustness and ultimately compromising process control and decision-making. Within PAT frameworks, representativeness must therefore be treated as a primary design criterion rather than a secondary consideration. Reliable monitoring depends on the ability of the PAT sampling strategy to reflect the true spatial distribution of materials and the temporal dynamics of the process, lest being compromised by unnecessary measurement system error manifestations. Without careful consideration of process sampling in compliance with the TOS, advanced chemometric or machine learning models risk modelling or learning only apparent non-causal correlations rather than physically meaningful relationships, leading to unreliable predictions and poor transferability. In this context, data quality originates upstream, before analysis. Analysis sensu stricto and data modelling (chemometrics) can refine spectral data variability, but they cannot recover inherent informational variability that was never measured.

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Footnotes

Edited by: Huan-Tsung Chang, Chang Gung University, Taiwan

Reviewed by: Stefano Fornasaro, University of Trieste, Italy

Ricard Boqué, University of Rovira i Virgili, Spain