text:
Journal topic
Atmos. Meas. Tech., 11, 5687–5699, 2018
https://doi.org/10.5194/amt-11-5687-2018
Atmos. Meas. Tech., 11, 5687–5699, 2018
https://doi.org/10.5194/amt-11-5687-2018

Research article 18 Oct 2018

Research article | 18 Oct 2018

# A machine learning approach to aerosol classification for single-particle mass spectrometry

A machine learning approach to aerosol classification for single-particle mass spectrometry
Costa D. Christopoulos1, Sarvesh Garimella1,a, Maria A. Zawadowicz1,b, Ottmar Möhler2, and Daniel J. Cziczo1,3 Costa D. Christopoulos et al.
• 1Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA
• 2Institute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, Germany
• 3Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
• anow at: ACME AtronOmatic, LLC, Portland, OR, USA
• bnow at: Atmospheric Sciences and Global Change Division, Pacific Northwest National Laboratory, Richland, WA, USA

Correspondence: Daniel J. Cziczo (djcziczo@mit.edu)

Abstract

Compositional analysis of atmospheric and laboratory aerosols is often conducted via single-particle mass spectrometry (SPMS), an in situ and real-time analytical technique that produces mass spectra on a single-particle basis. In this study, classifiers are created using a data set of SPMS spectra to automatically differentiate particles on the basis of chemistry and size. Machine learning algorithms build a predictive model from a training set for which the aerosol type associated with each mass spectrum is known a priori. Our primary focus surrounds the growing of random forests using feature selection to reduce dimensionality and the evaluation of trained models with confusion matrices. In addition to classifying ∼20 unique, but chemically similar, aerosol types, models were also created to differentiate aerosol within four broader categories: fertile soils, mineral/metallic particles, biological particles, and all other aerosols. Differentiation was accomplished using ∼40 positive and negative spectral features. For the broad categorization, machine learning resulted in a classification accuracy of ∼93 %. Classification of aerosols by specific type resulted in a classification accuracy of ∼87 %. The “trained” model was then applied to a “blind” mixture of aerosols which was known to be a subset of the training set. Model agreement was found on the presence of secondary organic aerosol, coated and uncoated mineral dust, and fertile soil.

1 Introduction

Following the introduction of random forests in the 1990s, recent developments in deep learning and neural networks have helped to trigger a renewed interest in machine learning. This has led to the development of numerous easy-to-use, freely available open-source packages in popular programming languages like Python, and these tools are increasingly being used in academia and industry. While random forests have been used for complex classification and regression analysis in various fields, studies that employ random forests in aerosol mass spectrometry remain sparse. Utilizing these tools, the primary purpose of our study is to introduce a framework for growing random forests, reducing dimensionality, ranking chemical features, and evaluating performance using confusion matrices. Such properties are desirable for SPMS studies, where input variables can become redundant and interpretability is more limited with more advanced methods such as neural networks. Neural networks rely on a series of variable transformations rectified by nonlinear activation functions, making details of a given classification notoriously difficult to follow. The interpretability and explainability of these models remains an active area of research. Overall, analysis techniques such as those coming out of recent artificial intelligence research can prove useful for helping to tease out the subtle yet significant impact that aerosol chemistry has on the climate system.

Atmospheric aerosols impact clouds and the Earth's radiative budget. A lack of understanding of aerosol composition therefore contributes to uncertainty in determination of both anthropogenic and natural climate forcing (Boucher et al., 2013; Lohmann and Feichter, 2005). Aerosols directly affect atmospheric radiation by scattering and absorption of radiation from both solar and terrestrial sources. The radiative forcing from particulates in the atmosphere depends on optical properties that vary significantly among different aerosol types (Lesins et al., 2002). Aerosols also indirectly affect climate via their role in the development and maintenance of clouds (Vogelmann et al., 2012; Lubin and Vogelmann, 2006). Ultimately, the formation, appearance, and lifetime of clouds are sensitive to aerosol properties like shape, chemistry, and morphology (Lohmann and Feichter, 2005; Andreae and Rosenfeld, 2008). Characterization of aerosol properties plays a vital role in understanding weather and climate.

The chemical composition and size of aerosols have been analyzed on a single-particle basis in situ and in real time using single-particle mass spectrometry (SPMS; Murphy, 2007). First developed ∼2 decades ago, SPMS permits the analysis of aerosol particles in the ∼150–3000 nm size range, while differentiating internal and external aerosol mixtures and characterizing both semi-volatile (e.g., organics and sulfates) and refractory (e.g., crystalline salts, elemental carbon, and mineral dusts) particle components. Particles are typically desorbed and ionized with an ultraviolet (UV) laser, and resultant ions are detected using time-of-flight mass spectrometry (Murphy, 2007). A complete mass spectrum of chemical components is normally produced from each analyzed aerosol particle (Coe and Allan, 2006). Despite almost universal detection of components found in atmospheric aerosols, SPMS is not normally considered quantitative without specific laboratory calibration (Cziczo et al., 2001).

Chemical composition of an individual atmospheric aerosol particle is a complex interplay between its primary composition at the source (e.g. dust, biogenic organic, anthropogenic organic, soot) and its atmospheric processing up to the time of detection. Atmospheric processing can include a combination of coating with secondary material, coagulation, and cloud processing. Even different primary aerosol types can have similar mass spectral markers. For example, fly ash, mineral dust, and bioaerosol can all contain strong phosphate signal (Zawadowicz et al., 2017). Secondary material is often difficult to differentiate from primary material, but even minor compositional changes can be atmospherically important (Hoose and Möhler, 2012). As one example, mineral dusts are known to be effective at nucleating ice clouds; however, despite minor addition of mass, atmospherically processed mineral dust is less suitable for ice formation (Cziczo et al., 2013). As a second example, ice nucleation in mixed-phase clouds has been suggested to be predominantly influenced by feldspar, a single component among the diverse mineralogy of atmospheric dust (Atkinson et al., 2013). Using current SPMS data analysis approaches, it is difficult to detect these minor yet important compositional differences, and new robust and generalizable analysis techniques are critical.

We show that supervised training with random forests can differentiate aerosols in SPMS data more accurately than simpler approaches. Various clustering methods have been used to group aerosol types (Murphy et al., 2003; Gross et al., 2008), but these algorithms are known to combine chemically similar aerosols as they do not incorporate known particle labels in the training process. Another limitation encountered is the need to manually reduce the number of final clusters due to grouping of mathematically similar yet chemically distinct aerosols (Murphy et al., 2003). Such “unsupervised” clustering algorithms automatically group unlabeled data points in feature space, in this case mass spectral signals. For the purposes of setting broad aerosol categories, which are chemically distinct and easily separable in feature space, clustering is the simpler tool, and the data are easier to interpret. For identification of new or potentially unexpected atmospheric aerosols, such properties are desirable; however, the advantages of clustering greatly diminish when considering similar particle types that overlap in feature space. Fertile soils, for instance, are often grouped into a single category despite different sources and atmospheric histories.

Clustering algorithms should be considered as a tool to use alongside supervised classification. The latter may be used to further explore unique aerosol types or verify manually labeled clusters with higher precision. Furthermore, the ensemble approach presented here also produces interpretable variable rankings and probabilistic predictions that assist in characterizing measurement uncertainty. Uncertainties associated with mass spectrometry include the determination of mass peak areas, internal mixing of aerosols during the experiment, and transmission efficiency. Additionally, the classification method itself introduces and quantifies uncertainty in aerosol identification as a result of imperfect class separation and parameter uncertainty. The choice of supervised or unsupervised machine learning will depend on the researcher's use case, and each method has unique advantages and disadvantages. We note that a limitation of the random forest approach – and for supervised learning in general – is the inability to classify aerosol types outside of the training set. The ability of a random forest to characterize ambient atmospheric data sets, therefore, will strongly depend on which aerosols are contained within the training set. Additionally, it is noted that comparisons between all machine learning models are sensitive to user-defined parameters and algorithm implementation.

In this study, we demonstrate the capabilities of random forests to automatically differentiate particles on the basis of chemistry and size. The resulting model can capture minor compositional differences between aerosol mass spectra. By testing predictions using an independent, or “blind”, data set, we illustrate the feasibility of combining online analysis techniques such as SPMS with machine learning to infer the behavior and origin of aerosols in the laboratory and atmosphere.

2 Methodologies

## 2.1 PALMS

The Particle Analysis by Laser Mass Spectrometry (PALMS) instrument was employed for these studies. PALMS has been described in detail previously (Cziczo et al., 2006). Briefly, the instrument samples aerosol particles in the size range from ∼200 to ∼3000 nm using an aerodynamic lens inlet into a differentially pumped vacuum region. Particle aerodynamic size is acquired by measuring particle transit time between two 532 nm continuous-wave neodymium-doped yttrium aluminum garnet (Nd:YAG) laser beams. A pulsed UV 193 nm excimer laser is used to desorb and ionize the particles, and the resulting ions are extracted using a unipolar time-of-flight mass spectrometer. The resulting mass spectra correspond to single particles. The UV ionization extracts both refractory and semi-volatile components and allows analysis of all chemical components present in atmospheric aerosol particles (Cziczo et al., 2013).

## 2.2 Data set

A set of “training data” was acquired by sampling atmospherically relevant aerosols. The majority of the data set was acquired at the Karlsruhe Institute of Technology (KIT) Aerosol Interactions and Dynamics in the Atmosphere (AIDA) facility during the Fifth Ice Nucleation workshop – part 1 (FIN01). The remainder were acquired at our Aerosol and Cloud Laboratory at MIT. The FIN01 workshop was an intercomparison effort of ∼10 SPMS instruments, including PALMS. The training data correspond to spectra of known particle types that were aerosolized into KIT's main AIDA and a connected auxiliary chamber for sampling by PALMS and the other SPMSs (Table 1). Hereafter we group both chambers with the name “AIDA”. The number of training spectra acquired varied by particle type, ranging from ∼250 for secondary organic aerosol (SOA) to ∼1500 for potassium-rich feldspar (“K-feldspar”). In total, ∼50 000 spectra are considered, with each spectrum containing 512 possible mass peaks and an aerodynamic size (Table 2). The FIN01 workshop included a blind sampling period, where AIDA was filled with an unknown number of aerosol categories known to be from the training set (i.e., for which spectra had already been acquired). Knowledge of size, specific types, and concentrations were not known a priori.

Table 1Description of aerosol types used in the training data set. Rows are grouped by broad aerosol categories in the following order: fertile soil, mineral/metallic, biological, and other. “n/a” stands for not applicable.

Table 2Features rankings for differentiation of particles between labels and between broad categories in positive and negative ion modes. See text for additional details. “n/a” stands for not applicable.

Figure 1 illustrates a simple differentiation of particles using only two mass peaks in one (negative) polarity. Mass peaks represent fractional ion abundance, measured as a total signal (ion current) normalized to allow for spectra-to-spectra comparison (Cziczo et al., 2006). In this example, the normalized areas of negative mass peaks 24 (${\mathrm{C}}_{\mathrm{2}}^{-}$) and 16 (O) are plotted. Distinct aerosol types are differentiated by color with clusters forming in this two-dimensional space. Note that spectra of the same aerosol type form distinct clusters (e.g., Arizona Test Dust, ATD), as do similar aerosol classes (e.g., soil dusts). Co-plotted in Fig. 1 are data from the blind experiment. Distinct clusters of spectra from the blind experiment are noticeable and correlate with known clusters. As described in the next section, machine learning algorithms draw “decision boundaries” that best separate different groups of data points based on a set of rules. Machine learning is not bound by the simplistic two-dimensional space shown in Fig. 1 and instead uses all 512 mass peaks and aerodynamic size.

Figure 1Aerosol training data plotted as feature area 16 (O) versus area 24 (${\mathrm{C}}_{\mathrm{2}}^{-}$). Axes represent peak areas normalized to total signal obtained from PALMS (i.e., 1=100 % of signal). This illustrates simple two-dimensional clustering of aerosols from the training data set by type. Co-plotted are ∼500 randomly drawn spectra from the AIDA blind experiment, which were known to be a subset of the training data aerosols.

Figure 2Schematic of decision tree classification for a single aerosol spectrum. From left to right, a mass spectrum is normalized with respect to total ion current, forming the elements of normalized feature vector X. A trained decision tree then applies a series of tests to a discreet number of peaks in order to arrive at a categorical aerosol prediction (the leaves).

## 2.3 Aerosol classification

A trained classification model maps a continuous input vector X to a discreet output value using a set of parameters “learned” from the data. Figure 2 illustrates the mapping of a mass spectrum to vector space. In contrast to traditional, hard-coded classification methods, machine learning determines parameters that partition the data set. To form X, mass spectra are converted to dimensional vectors normalized to the total ion current (i.e., the total of all mass peaks amounts to 1 in each spectrum). The elements of the vectorized mass spectrum, termed “features”, hold information about the ionization efficiency and relative abundance of chemical species in each aerosol and serve as the variables for the machine learning model.

Machine learning is conducted in two phases: training and testing. During training, a model is constructed and iteratively updated based on data (i.e., mass spectra) from the training set. For this work, the set of known aerosol types sampled by PALMS was converted to dimensional vectors. These data form the basis set for defining each aerosol type. A random forest was used to generate predictions of aerosol type. A single decision tree is a statistical decision model that performs classification based on a series of comparisons relating a variable Xi (in this case a normalized mass peak in X) to a learned threshold value (Breiman, 2001). A random forest is an ensemble of perturbed decision trees, whereby a final classification is made by averaging the predictions across all trees (described below in Sect. 2.4). Represented as an algorithmic tree, a binary decision tree consists of a hierarchy of nodes where each node connects via branches to two other nodes deeper in the tree. At each node, one of the two branches is taken based on whether a normalized peak Xi is greater or less than a threshold value. Each branch leads to another node where a different test is performed. After a series of tests, one at each node, a class is assigned to a given sample; these are the so-called “leaves”. Figure 2 illustrates the classification model for a single decision tree.

Each test in the tree narrows the set of reachable output leaves and thus the sample space of possible aerosol labels. After h tests in this study, where h ranges from 10 to 3000, the set of reachable leaves and possible labels is 1 and the decision tree outputs a prediction. Because PALMS is unipolar – either a positive or negative mass spectrum is produced – simultaneous generation of positive and negative spectra on a particle-by-particle basis is not possible. Two separate classification models, one for each polarity, were generated to classify aerosols. These are hereafter referred to as the “positive” and “negative” classification algorithms.

## 2.4 Random forests

A random forest is an ensemble of decision tree classifiers where each classifier independently labels an unknown spectrum vector X. To make a final prediction of aerosol type, trees within an ensemble “vote” on a classification label. Each vote has equal weight, and the spectrum is assigned to the majority choice. Each tree within an ensemble is independently grown on a subset of the training data so that a commonly voted-for label implies a higher certainty. Adding members to an ensemble increases the robustness of a classification model by providing alternative hypotheses and is therefore preferable to single classifiers.

Before an ensemble method is implemented for classification, trees are independently grown during training. A total of k trees, with k=110, were grown using a bootstrap sample from the training set. In bootstrap sampling, each tree sees an independent sample set of equal size drawn from the full training set by sampling spectra with replacement. On average, each tree is built with ∼63 % of the original data, leaving a portion of the training set unsampled. The unsampled data for each tree, known as “out-of-bag” observations, are recorded and later provide a means to assess classification error for the forest. To determine model error, predictions are made for each point in the data set using only the subset of trees that did not use the point for training. Each training point is left out at least once. This is analogous to making predictions with a separately trained forest that did not observe the point and prevents testing with the same data used for training.

Given a bootstrap sample, a tree is grown by sequentially creating tests that maximize the separation between classes in parameter space. A test is created by defining a comparison that minimizes the information entropy of a possible split, thus minimizing the randomness of prediction labels (Breiman, 1996). To generate variability in the model, only a random set of splits is tested at each node, and only the best split in terms of entropy is chosen (Breiman, 2001). After iteratively defining thresholds for each new node, the tree grows in size until a series of tests ending at some node Sq uniquely characterizes an aerosol as a particle type. A leaf is then appended to node Sqwith the corresponding label. In classification mode, an aerosol spectrum that passes the same tree will undergo the same series of tests and will end in the same leaf, thus being labeled in the same way. For the purposes of this study, each tree had ∼3300 nodes.

The number of variables per split is chosen to be 11, and the number of trees is 110. Using grid search, the optimal model was determined by enumerating combinations of these parameters on a coarse grid and selecting the values that produce the lowest test error, or out-of-bag error. Given several lists of parameters, where each list corresponds to a different model hyperparameter, models are trained one by one until each combination of parameters has been tested. For this study, the grid representing variables per split was spaced by 1, and the grid for number of trees was spaced by 5. The number of nodes in each tree depends on other hyperparameters and cannot be explicitly set. Model behavior is primarily sensitive to the number of variables per split and shows weak dependence on the number of trees and number of input variables beyond small values. As the number of variable splits increases, error decreases exponentially to a local minimum before again rising due to overfitting. Alternatively, as the number of trees is increased, the error converges to some nonzero value, a known characteristic of random forests where test error converges to the generalization error. The models were trained with the Python 2.7 Scikit-learn module on a MacBook Pro with 16 GB 1600 MHz DDR3 memory and a 2.5 GHz Intel Core i7 processor. A typical random forest model took about 5–10 s to train, and we found a linear relationship between runtime and both the number of trees and variables per split.

Overall, the generalizability and robust performance of random forests is owed significantly to the series of random statistical procedures used to construct such models. An ensemble classifier reduces variability by averaging predictions over a series of independently trained models, and bagging introduces additional randomness by producing “perturbed” versions of the original data via random sampling of input data. The randomness used in constructing forests, both in bagging the training set and choosing variable splits, works to decorrelate the output of each tree even as the inputs become correlated (Breiman, 2001). As the number of trees increases, the law of large numbers guarantees a convergence of the out-of-bag error to the generalization error.

## 2.5 Dimensionality reduction and chemical feature selection

Dimensionality reduction is the process of representing data with fewer variables than initially present in the data set, in this case less than the original 512 mass peaks and aerodynamic size. In addition to facilitating data visualization, reducing computation time, and limiting overfitting (Mjolsnes, 2001), dimensionality reduction, in the context of aerosol mass spectra, also indicates the most important chemical markers for differentiation. Feature ranking was algorithmically determined by comparing the performance of trees before and after removing information about peak Xi. The method is that the values of variable Xi are permuted for tree k in the out-of-bag set so that the variable is irrelevant to the final label. The change in misclassification before and after the permutation is calculated and then repeated for all trees so that a variable ranking is obtained (Breimann, 2001). Table 2 ranks mass peaks (features) by polarity in importance using this method. The columns on the left list feature rankings (i.e., most to least important for correct classification) for the entire set of aerosol types. The columns on the right list rankings when aerosol types are grouped into the broad, chemically similar, categories. A final ranking was determined by sequentially adding variables and observing classification performance response. All variables preceding two e-foldings in classification error were maintained in the final model. Both the specific aerosol type and broad aerosol category models were retrained using this subset of the initial variables, listed in Table 2.

## 2.6 Comparison to Euclidean distance classifier

To access relative model performance, we contrast the results with a simple classifier that compares unseen aerosols to a set of class mean vectors. Using the Euclidean distance metric, the unknown aerosol is assigned to the nearest class. This simple baseline classifier helps to put results in the context of machine learning techniques that rely on distance-based metrics such as k-means and hierarchical clustering. K-means clustering attempts to divide the data points into k distinct clusters, representing spectra as vectors. Using Euclidean distance, the standard algorithm assigns points to centroids, or clusters, which are essentially mean vectors representing the average of all points in the cluster. Assuming perfect convergence of k-means clustering, where k is the number of aerosol classes, each cluster represents the mean of aerosol in that class. The random forest results below demonstrate many areas of improvement over the simple classifier.

Figure 3Column-normalized confusion matrices showing fraction of aerosols labeled as j that belong to i, where i and j are row and column indices, respectively. Confusion matrices are determined from training data of known origin and are used to compute probability distributions. Aerosol types (Table 1) are grouped into four broad categories delineated by the bold horizontal and vertical bars. From top to bottom or left to right: fertile soils, mineral/metallic, biological, and other. Classification accuracy, the average probability of a correct aerosol prediction across all labels, is computed by averaging diagonal matrix elements. For all aerosol types, the accuracy is 87 % in positive ion mode and 87 % in negative ion mode.

Figure 4Column-normalized confusion matrices for the broad categorization of aerosols following the convention in Fig. 3. (a, b) For all aerosol categories, the random forest has an accuracy of 93 % in positive ion mode and 91 % in negative ion mode. (c, d) The Euclidean distance classifier has an accuracy of 70 % in positive ion mode and 69 % in negative ion mode.

3 Results

## 3.1 Confusion matrices and probabilistic model performance

A confusion matrix captures misclassification tendencies by pair-wise matching the model prediction with the true aerosol type or broad category (Powers, 2007) and can be understood as a contingency table matching model predictions to true labels. Confusion matrices represent model predictions as columns i and true aerosol type or category as rows j,where class names are mapped to integers i, $j\in \mathit{\left\{}\mathrm{1},\mathrm{2},\mathrm{\dots },y\mathit{\right\}}$. In this study, matrices have been normalized along each column to show the fraction of aerosols labeled as j that actually belong to i (Figs. 3 and 4). For aerosol classification, these matrices can also be interpreted as similarity measures between particle types. Since the basis of classification is separation of physical quantities, misclassifications result from similarity in mass peaks and their ion abundance between aerosol types. This is most easily visualized as overlapping clusters in the simple two-dimensional space in Fig. 1.

Model performance for each aerosol is summarized in the diagonal elements of the confusion matrix P, which represent the fraction of aerosol in column j labeled correctly. The classification accuracy (a) is given by averaging diagonal elements of P. A perfect classification model produces the identity matrix, as all data points are classified correctly 100 % of the time. For example, in the positive confusion matrix, SOA and agar growth medium are correctly labeled in the test set 100 % of the time. Barring element truncation, all columns of P add to 1.

Figures 3 and 4 display confusion matrices as heat maps for the full set of particle labels and broad grouped particle categories, respectively. Broad categories are delineated by bold horizontal and vertical lines in Fig. 3 as fertile soil (Argentinian, Chinese, Ethiopian, Moroccan, and two German soils), pure mineral dust and metallic particles (ATD, illite NX, fly ash, Na-feldspar, and K-feldspar), biological particles (agar growth medium, P. syringae bacteria, cellulose, Snomax, and hazelnut pollen), and other particles (K-feldspar with sulfuric acid (SA) and SOA coatings, soot, and SOA). Some model confusion exists between fertile soils and coated/uncoated feldspars which can be explained by the fact that soils are mineral dust mixed with organic and other materials.

Positive mass spectra appear to hold more information with respect to differentiating aerosols than negative mass spectra. Label-wise classification accuracy for the negative algorithm ranges from 3 to 5 % lower. A large part of this performance discrepancy is due to greater ability of positive spectra to differentiate coated particles within the “other” category.

In addition to quantifying misclassification tendencies between classes, the confusion matrix can be redefined to show confusion for aerosols within broad categories themselves. The precision score (Powers, 2007) captures the classification behavior for some subset of aerosol L by averaging fractions of correctly classified aerosols for labels within that category:

$\begin{array}{}\text{(1)}& \mathrm{precision}\phantom{\rule{0.25em}{0ex}}\mathrm{score}\phantom{\rule{0.125em}{0ex}}\left(L\right)=\frac{\mathrm{1}}{\mathrm{|}L\mathrm{|}}\sum _{i=j}^{\mathrm{|}L\mathrm{|}}P\left(i\in L,j\in L\right).\end{array}$

When applied to Pl, the precision score captures classification performance in a population with only aerosol labels contained in L. The algorithm is expected to correctly label an aerosol in such a population with a probability equal to the precision score. The precision score is valuable when using the classification model as a particle screener, producing probability distributions over a subset of aerosol labels of interest. The confusion characteristics are shown in Table 3 for each category in terms of the precision score and the mean and standard deviation of misclassification within each category. Although both models perform similarly for biological spectra, discrepancies of 2–5 % appear in the remaining categories. For regimes consisting of only mineral/metallic or other particles, the positive algorithm shows intraclass performance advantages not only in terms of the precision score but also, most notably, in terms of fewer mislabeling of mineral/metallic particles. The largest precision discrepancy is observed for fertile soils, where the positive ion algorithm has a 5 % advantage in precision with approximately half the false labeling rate.

Table 3Model performance by category and ion mode in a population consisting entirely of aerosols within that category. (a) Average classification accuracy where 1.0=100 % precision (Powers, 2007). (b) Mean and standard deviations of misclassification.

Across all categories, the random forest shows improvements over the Euclidean classifier in terms of both accuracy and precision. Figure 4 directly compares confusion matrices for the two methods, revealing overall accuracy improvements of at least 20 %. The largest improvements are in the fertile soil and other category, where accuracy rises between 20 % and 39 % with the random forest. Computing the full confusion matrix for the Euclidean technique (as in Fig. 3) reveals similar results, with far more frequent mislabeling between fertile soils and coated/uncoated particles than our approach. These results reinforce the fact that chemically similar aerosols which overlap in feature space will often be grouped together when using a single distance-based classifier. The improvement from random forests is likely a result of (a) the ensemble approach, which is known to produce better generalizability than single classifiers, and (b) the tendency of aerosols with similar chemical properties and atmospheric effect to appear mathematically distinct with a distance metric.

Figure 5Model predictions of ∼5000 aerosols sampled from the AIDA FIN01 blind mixture, which was known to be a subset of the training data. All percentages represent relative number concentrations. Middle left: aerosol types input to the chamber for the blind mixture. Middle right: aerosol types input to the chamber for the blind mixture and above the detection limit for PALMS. Model predictions are shown for negative and positive ion mode on the left and right, respectively. Bottom: broad categories. Top: breakout by aerosol type of the non-SOA categories above the 1 % level. Note that (1) the soot in the blind mixture was known to be below the instrument detection limit and therefore is not expected to be found in the data (Cziczo et al., 2006); (2) coagulation of SOA and mineral dust, which occurred after aerosol input to the chamber, was often categorized as mixed mineral and organic particles or fertile soils (i.e., mixtures of mineral and organic components) considered in the training data set; and (3) the aerosols types reported by AIDA do not account for PALMS transmission efficiency (see text for details).

Beyond classification, the obtained variable rankings alone provide interesting insights into the data set. It is noteworthy that while most of the features are logical differentiators of the aerosol types investigated in FIN01 there were also surprises. One example is 59+ (cobalt), determined to be one of the most important features for differentiation. Further investigation determined this material was associated with tungsten carbide contaminant from dry-powder-dispersion equipment used on some samples. The contamination affected feldspar samples used during the second half of the AIDA measurements in particular. This serves to illustrate the lack of a priori judgment by the algorithm and an unintended benefit of machine learning processes (i.e., contamination identification).

## 3.2 Characterization of blind data

As part of the FIN01 workshop, an a priori unknown number of aerosol types from Table 1 were aerosolized into the AIDA chamber at unknown size and relative concentration. PALMS, one member of the blind intercomparison effort, collected ∼25 000 spectra. After data analysis, the aerosol types and relative abundances were provided to each group (Fig. 5, top center).

The presence or absence of particle types in the blind set was initially diagnosed by choosing particles predicted at or above the 1 % level. We note here that this step was based on the knowledge that (1) a distinct set of particles would be placed in the chamber and (2) particles present at or below the 1 % level were most likely contaminated. We further note that this step is unique to a blind study and would not be applicable to the atmosphere.

Figure 5 illustrates the fractional percentages for each aerosol category. Because SOA was nearly always labeled correctly (Fig. 3), the remaining aerosols are considered separately using the full set of candidate aerosol labels. Both positive and negative models arrived at similar results, with inconsistencies primarily associated with the presence of trace fertile soils and mineral dust/fly ash particles. The positive algorithm identifies Argentinean soil, German soil, ATD, and cellulose as each comprising ∼2–4 % of the AIDA population, whereas the frequency of these aerosols was too low to consider in the negative algorithm. Alternatively, the negative model estimates Na-feldspar at ∼14 % of the total population, a label not identified by the positive algorithm. This discrepancy can partially be explained by the 1 % selection criterion for aerosols present in the population. Fertile soils, ATD, and cellulose frequently accumulate error along rows in the full positive confusion matrix, indicating frequent confusion with other categories (Fig. 3). Furthermore, with the observed misclassification rates ranging ∼1–4 %, it is expected that these aerosol labels are false positives. The negative model offers an alternative hypothesis, suggesting these miscellaneous aerosols are Na-feldspar. Since there is significant model agreement on the percentages of SOA and coated feldspars, this part of the blind mixture population can be characterized with more certainty. For the disputed aerosol labels, more credence is lent to the negative classification algorithm on the basis of improved precision for fertile soils.

The aerosols reported in the blind mixture were soot, mineral dust, and SOA. The soot aerosols used in the blind study were smaller than in the training data experiments and were below the cutoff diameter for PALMS; they were therefore not detected and therefore could not be identified by the algorithms. This bias in transmission efficiency should be noted, whereby aerosols are detected at a rate that depends on their size and aerodynamic properties (Cziczo et al., 2006). The result is that particles with diameters below ∼200 nm or greater than ∼1000 nm are detected with increasing inefficiency, which leads to relative undercounting of small soot or large mineral dust (Cziczo et al., 2006). The specific mineral component was not identified and may have been either a pure mineral or soil dust. Both algorithms robustly labeled SOA with large agreement, consistent with the 100 % accuracy observed in the test set.

SOA-coated mineral dust was identified as a particle type. This material was not directly input to AIDA, but the report is most likely correct, due to coagulation within the AIDA chamber during the course of the blind experiment. Since percentages were reported before particles entered the chamber, it is not possible to directly verify the fraction of SOA-coated aerosols or the extent to which coagulation occurs, as the process is time dependent. This may also explain some indications of fertile soils, which are known to be mixtures of mineral and organic components. The training data set did not contain coagulated SOA and mineral dust but did include SOA-coated K-feldspar, which explains the identification.

While both models identified a variety of fertile soils, and not a single type, these results are largely consistent with the presence of coagulated organics and minerals and the known uncertainties highlighted by the confusion matrices discussed previously. Given the presence of any single mineral dust, some confusion with fertile soils, SA-coated Feldspar, and Na-feldspar is expected (Fig. 3). Moreover, as discussed previously (Gallavardin et al., 2008a, b), AIDA backgrounds are not completely particle free. During the FIN01 study, contaminated particles from previous test aerosol were frequently observed as background, and they could also be the origin of some low-concentration particles matching fertile soil chemistry. Overall, discrepancies between the reported aerosol fractions and model predictions can be accounted for with model and experiential uncertainties.

An additional consideration is experimental bias in the training data, which could result in test errors that underestimate true generalization errors in real aerosol populations. For SPMS, spurious relationships between spectra may arise due to instrumental parameters that are assumed to be constant between the training, test, and blind data. This consideration plagues all SPMS analysis requiring a training set, where correlations may arise as a result of signals that depend on ambient properties like temperature, humidity, and pressure or instrument parameters such as laser power. Although several well-established steps were taken to minimize overfitting – including dimensionality reduction and out-of-bag testing – data set bias may still exist if these quantities vary significantly between aerosol types in the training or blind data.

4 Conclusions and future work

This study lays out a framework for training and implementing random forests on SPMS data, with a focus on dimensionality reduction and the evaluation of model performance with confusion matrices. A key benefit to the proposed method is chemical feature selection, which allows researchers to identify potentially important chemical markers between arbitrary groups of aerosols or identify sources of contamination. In this particular study, the contaminant was identified and removed in the dimensionality reduction step while reasoning through the subset of ranked features. As illustrated by Fig. 2, cobalt is suspiciously identified as the second-most-important variable for classification, but it is a known component of the dry-powder-dispersion equipment used on some samples. The contaminate peak would be present in a cluster analysis, but it would not be obvious to pick out and remove as standard clustering is not typically suited for variable rankings.

For future studies tackling ambient atmospheric data that may contain aerosol types absent from the training set, a form of subspace selection may be used to improve results. The region of parameter space where training data are available can be characterized with a joint probability density function. One such approach is kernel density estimation – a machine learning method that approximates a multidimensional probability density function in a non-parametric manner based on data density. To obtain accurate probability estimates, the method should be fit with a smaller set of important but uncorrelated peaks. The task of classification is then preceded by a filtering step. Spectra residing in the subspace containing the training data should first be identified based on the probability density function. Then, only these particles that are most certain to lie in the training subspace are classified using the classification model as described in this paper. An alternative is to combine the method with clustering by classifying particles in each automatically identified cluster.

Overall, the random forest approach allows for differentiation of aerosols within a SPMS data set, augmenting existing tools and reducing the need for a qualitative comparison between mass spectra. Across a representative sample of possible aerosol types, the behavior of each algorithm predictably allows users to infer the presence or absence of specific aerosols and quantify aerosol abundance. Machine learning is automated, and the output of the model must then be informed by human knowledge of aerosol chemistry. Machine learning should therefore be considered as an additional tool to interpret mass spectra to better distinguish aerosols with unique properties in terms of atmospheric chemistry, biogenic cycles, and population health.

The random forest classification framework described here may be generalized to any instrument, or set of instruments, capable of collecting physical and chemical information that distinguishes particles. Although the method described here is applied to a stand-alone SPMS and tested with a set of “blind” data, ancillary laboratory or field data can be integrated to expand the data set. The success of these algorithms is data dependent, where better performance is expected for instruments that provide more, and more quantitative, analysis of the aerosol properties. Although the algorithms implemented in this study were primarily used to categorize SOA, mineral dust, fertile soil, and biological aerosols, these models can adopt an arbitrarily large set of aerosol data.

Data availability
Data availability.

The labeled aerosol data sets used to train the classifiers in this paper are available at Harvard Dataverse (Christopoulos, 2018).

Author contributions
Author contributions.

CDC wrote code for the models and analysis used in this paper with the direction of SG and DJC. MAZ provided the positive and negative training datasets and details surrounding experiments in which these data were collected. MAZ, SG, and DJC provided knowledge of particle chemistry in the context of PALMS and helped identify important features and contaminants. OM provided AIDA data and details surrounding the FIN01 experiment. CDC, MAZ, SG, and DJC participated in writing the manuscript.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We thank the FIN01 and AIDA teams for logistical support and scientific discussions. We acknowledge funding from NSF, which made our participation possible (grant AGS-1461347). Maria A. Zawadowicz acknowledges the support of the NASA Earth and Space Science Fellowship program, and Daniel J. Cziczo acknowledges the support of the Victor P. Starr Career Development Chair.

Edited by: Francis Pope
Reviewed by: two anonymous referees

References

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Gross, D., Atlas, R., Rzeszotarski, J., Turetsky, E., Christensen, J., Benzaid, S., Olson, J., Smith, T., Steinberg, L., and Sulman, J.: Environmental chemistry through intelligent atmospheric data analysis, Environ. Modell. Softw., 25, 760–769, 2008.

Henning, S., Ziese, M., Kiselev, A., Saathoff, H., Möhler, O., Mentel, T. F., Buchholz, A., Spindler, C., Michaud, V., Monier, M., Sellegri, K., and Stratmann, F.: Hygroscopic growth and droplet activation of soot particles: uncoated, succinic or sulfuric acid coated, Atmos. Chem. Phys., 12, 4525–4537, https://doi.org/10.5194/acp-12-4525-2012, 2012.

Hiranuma, N., Augustin-Bauditz, S., Bingemer, H., Budke, C., Curtius, J., Danielczok, A., Diehl, K., Dreischmeier, K., Ebert, M., Frank, F., Hoffmann, N., Kandler, K., Kiselev, A., Koop, T., Leisner, T., Möhler, O., Nillius, B., Peckhaus, A., Rose, D., Weinbruch, S., Wex, H., Boose, Y., DeMott, P. J., Hader, J. D., Hill, T. C. J., Kanji, Z. A., Kulkarni, G., Levin, E. J. T., McCluskey, C. S., Murakami, M., Murray, B. J., Niedermeier, D., Petters, M. D., O'Sullivan, D., Saito, A., Schill, G. P., Tajiri, T., Tolbert, M. A., Welti, A., Whale, T. F., Wright, T. P., and Yamashita, K.: A comprehensive laboratory study on the immersion freezing behavior of illite NX particles: a comparison of 17 ice nucleation measurement techniques, Atmos. Chem. Phys., 15, 2489–2518, https://doi.org/10.5194/acp-15-2489-2015, 2015a.

Hiranuma, N., Möhler, O., Yamashita, K., Tajiri, T., Saito, A., Kiselev, A., Hoffmann, N., Hoose, C., Jantsch, E., Koop, T., and Murakami, M.: Ice nucleation by cellulose and its potential contribution to ice formation in clouds, Nat. Geosci., 8, 273–277, https://doi.org/10.1038/ngeo2374, 2015b.

Hoose, C. and Möhler, O.: Heterogeneous ice nucleation on atmospheric aerosols: a review of results from laboratory experiments, Atmos. Chem. Phys., 12, 9817–9854, https://doi.org/10.5194/acp-12-9817-2012, 2012.

Lesins, G., Chylek, P., and Lohmann, U.: A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing, J. Geophys. Res.-Atmos., 107, 1–12, https://doi.org/10.1029/2001jd000973, 2002.

Lohmann, U. and Feichter, J.: Global indirect aerosol effects: a review, Atmos. Chem. Phys., 5, 715–737, https://doi.org/10.5194/acp-5-715-2005, 2005.

Lubin, D. and Vogelmann, A.: A climatologically significant aerosol longwave indirect effect in the Arctic, Nature, 439, 453–456, https://doi.org/10.1038/nature04449, 2006.

Mjolsness, E.: Machine Learning for Science: State of the Art and Future Prospects, Science, 293, 2051–2055, https://doi.org/10.1126/science.293.5537.2051, 2001.

Murphy, D. M.: The design of single particle laser mass spectrometers, Mass Spectrom. Rev., 26, 150–165, 2007.

Murphy, D. M , Middlebrook, A. M., and Warshawsky, M.: Cluster Analysis of Data from the Particle Analysis by Laser Mass Spectrometry (PALMS) Instrument, Aerosol Sci. Tech., 37, 382–391, https://doi.org/10.1080/02786820300971, 2003.

Peckhaus, A., Kiselev, A., Hiron, T., Ebert, M., and Leisner, T.: A comparative study of K-rich and Na/Ca-rich feldspar ice-nucleating particles in a nanoliter droplet freezing assay, Atmos. Chem. Phys., 16, 11477–11496, https://doi.org/10.5194/acp-16-11477-2016, 2016.

Powers, D. W.: Evaluation: From Precision, Recall and F-Factor to ROC, Informedness, Markedness and Correlation, Journal of Machine Learning Technologies, 7, 1–24, 2007.

Saathoff, H., Naumann, K.-H., Schnaiter, M., Schöck, W., Möhler, O., Schurath, U., Weingartner, E., Gysel, M., and Baltensperger, U.: Coating of soot and (NH4)2SO4 particles by ozonolysis products of α-pinene, J. Aerosol Sci., 34, 1297–1321, https://doi.org/10.1016/S0021-8502(03)00364-1, 2003.

Steinke, I., Funk, R., Busse, J., Iturri, A., Kirchen, S., Leue, M., Möhler, O., Schwartz,T., Schnaiter, M., Sierau, B., Toprak, E., Ullrich, R., Ulrich, A., Hoose, C., and Leisner, T.: Ice nucleation activity of agricultural soil dust aerosols from Mongolia, Argentina, and Germany, J. Geophys. Res.-Atmos., 121, 13559–13576, https://doi.org/10.1002/2016JD025160, 2016.

Vogelmann, A., McFarquhar, G., Ogren, J., Turner, D., Comstock, J., Feingold, G., Long, C., Jonsson, H., Bucholtz, A., Collins, D., Diskin, G., Gerber, H., Lawson, R., Woods, R., Andrews, E., Yang, H., Chiu, J., Hartsock, D., Hubbe, J., Lo, C.,Marshak, A., Monroe, J., McFarlane, S., Schmid, B., Tomlinson, J., and Toto, T.: Racoro Extended-Term Aircraft Observations of Boundary Layer Clouds, B. Am. Meteorol. Soc., 93, 861–878, 2012.

Zawadowicz, M. A., Froyd, K. D., Murphy, D. M., and Cziczo, D. J.: Improved identification of primary biological aerosol particles using single-particle mass spectrometry, Atmos. Chem. Phys., 17, 7193–7212, https://doi.org/10.5194/acp-17-7193-2017, 2017.


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# Isomer definitions - quiz

HideShow resource information
• Created by: Tasha.L
• Created on: 11-09-16 12:09

## 1. Define 'chain-branching isomer'.

• They have the same molecular formula but interact differently with plane polarised light.
• They have the same molecular formula but a different arrangement of the carbon chain.
• They have the same molecular formula but a different structural formula.j
1 of 7

## 2. Define 'positional isomer'.

• They have the same molecular formula but a different arrangement of atoms in space.
• They have the same structural formula and chain length, but differ in the position of their functional group.
• They have the same molecular formula but different functional groups.

## 3. Define 'functional isomer'.

• They have the same molecular formula but different functional groups.
• They have the same molecular formula but a different structural formula.
• They have the same molecular formula and chain length, but differ by the position of the functional group.

## 4. Define 'optical isomer'.

• They have the same molecular formula and chain length, but differ by the position of the functional group.
• They have the same molecular formula but interact with plane polarised light differently.
• They have the same molecular formula but a different structural formula.

## 5. Define 'geometric isomer'.

• They occur in alkenes only due to the restricted rotation about the double bond, caused by the pi bond.
• They have the same molecular formula but a different arrangement of atoms in space.
• They have the same molecular formula but differ in how they act with plane polarised light.

format: plain


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# Does it really take that long?

5 posts / 0 new
Offline
Joined: 03/29/2012 - 09:40
Does it really take that long?

Hello.

I am running an OpenMx under 32-bit Windows (since as I understand it is not possible to do it under 64-bit?). My data file contains appr. 30000 twin pairs, and I am running a bivariate common sex-spesific ACE model with two moderators. It seems like either I have written a very slow script (although parameter estimates are reasonable) or it is just too many parameters to estimate (110 in a saturated model). It was taking so much time that in order to verify that my script actually works, I took only 1000 twin pairs for the analysis. But even then it takes appr. 2.5 h to run a saturated model! Is it normal with this number of parameters or it sounds like a mistake in the script? If former, is there a way to speed up the process?

Thank you beforehand.
Julia

Offline
Joined: 07/31/2009 - 15:12
It can take this long,

It can take this long, especially with lots and lots of raw data. The easiest speed-up is to use covariance matrices, which will be difficult if you're including moderators via definition variables. You can also turn off the calculation of the Hessian matrix. This is used to generate standard errors at the end of optimization. This is a very time-intensive process, and you can disable it in the mxRun command if all you want is the overall model fit.

Offline
Joined: 03/29/2012 - 09:40
Thank you so so much for this

Thank you so so much for this suggestion! Now it takes only 10-15 min for the whole dataset:) An improvement is still desired, but I will try it in Linux in order to be able to use more RAM. Thanks again!

Offline
Joined: 07/31/2009 - 15:14
Parallel?

The other way to speed things up is parallel processing. I don't know whether this is switched on in your case, or not? This option might speed things up:

mxOption(NULL, "Number of Threads", omxDetectCores() - 1)
• it will assign as many threads as there are available, less one just to keep the system happy.
Offline
Joined: 07/31/2009 - 15:24
To echo Ryne's comment, use

To echo Ryne's comment, use mxOption() to disable standard errors and calculated Hessian:

model <- mxOption(model, "Standard Errors", "No")
model <- mxOption(model, "Calculate Hesian", "No")

In addition, if you model is taking a long time to run, they I would recommend turning on checkpointing: http://openmx.psyc.virginia.edu/docs/OpenMx/latest/File_Checkpointing.html

Log in or register to post comments


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# Math Help - Find Equation of Sphere

1. ## Find Equation of Sphere

Find equation of a sphere with radius r and center P_0.

NOTE: P_0 is read "P sub zero."

r = 1; P_0 = (3, 1, 1)

2. Originally Posted by magentarita
Find equation of a sphere with radius r and center P_0.

NOTE: P_0 is read "P sub zero."

r = 1; P_0 = (3, 1, 1)
You are supposed to know the general equation of a sphere with the center C and the radius r:

$(x-x_C)^2+(y-y_C)^2+(z-z_C)^2=r^2$

All values are known - so where is the problem?

3. ## ok..........

Originally Posted by earboth
You are supposed to know the general equation of a sphere with the center C and the radius r:

$(x-x_C)^2+(y-y_C)^2+(z-z_C)^2=r^2$

All values are known - so where is the problem?
Are you saying to plug and chug with what's been given?

4. Originally Posted by magentarita
Are you saying to plug and chug with what's been given?
Correct - nothing more.

And that's the reason why I can't believe that you have had difficulties with this problem.

5. ## I.....

Originally Posted by earboth
Correct - nothing more.

And that's the reason why I can't believe that you have had difficulties with this problem.
I simply rushed through it. You are right. The question is easy.


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# Abstract characterization of white noise

Let $$\xi$$ be a random variable valued in the space of Schwartz distributions $$\mathcal{S}'(\mathbb{R}^d)$$.

For any open set $$R\subset\mathbb{R}^d$$ let $$\Sigma(R)$$ be the $$\sigma$$-algebra generated by $$\{\xi(f)\,|\,f\in\mathcal{S}(\mathbb{R}^d), \,\textrm{supp}\,f\subset R\}$$.

For any $$\lambda\in(0,\infty)$$ the random variable $$\xi_\lambda$$ valued in $$\mathcal{S}'(\mathbb{R}^d)$$ is defined by $$\xi_\lambda(f) = \xi(f_\lambda)$$ for all $$f\in\mathcal{S}(\mathbb{R}^d)$$, where $$f_\lambda(x) = \lambda^{d/2}\,f(\lambda x)$$.

Assume that:

(1) the $$\sigma$$-algebras $$\Sigma(R_1)$$ and $$\Sigma(R_2)$$ are independent for any disjoint open sets $$R_1,R_2\subset\mathbb{R}^d$$.

(2) for any $$\lambda\in(0,\infty)$$ the law of $$\xi_\lambda$$ coincides with the law of $$\xi$$.

Is it true that under the above assumptions $$\xi$$ must be the white noise?

• What is your definition of the white noise? Oct 6 at 16:03
• Given the adopted point of view is that of random distributions rather than generalized stochastic processes, white noise is the Borel probability measure on the space of distributions whose characteristic function is $f\mapsto\exp(-\frac{1}{2}\|f\|_{L^2}^2)$, for arbitrary Schwartz function $f$. Oct 6 at 17:27
• possibly useful reference sma.epfl.ch/~rdalang/articles/AOP1168.pdf Oct 6 at 17:36

## 1 Answer

These assumptions are not sufficient. Take $$d=1$$ and for simplicity let us work with the antiderivative of $$\xi$$. You are then asking if a process with independent increments and Brownian scaling is necessarily a Brownian motion. Any $$\alpha$$-stable Levy process $$L$$ has independent increments and scaling $$(L_t)_{t\geq 0}\sim\lambda^{1/\alpha}(L_{\lambda t})_{t\geq0}$$, so the process $$X_t=L_{t^{\alpha/2}}$$ is a counterexample.

My guess is that translation adding translation invariance and nontriviality (note that actually $$\xi\equiv 0$$ is also a counterexample) to the assumptions should be sufficient.

• Thanks. So the question is whether $\xi$ satisfying (1), (2) and (3) translational invariance is necessarily proportional to the white noise. Oct 7 at 8:18


text:
Agda-2.6.0: A dependently typed functional programming language and proof assistant

Agda.Interaction.Options.IORefs

Description

Some IORefs to access option values in pure code

Synopsis

# Documentation

In Pretty and String we want to know whether we are allowed to insert unicode characters or not.

Constructors

 UnicodeOk AsciiOnly


text:
# Math Help - Trig - Prove tan by using similar triangles

1. ## Trig - Prove tan by using similar triangles

Hi, the problem i have here isn't with my trig skills but with my ability to use similar triangles to prove that KZ = tan.

***Could someone please tell me which triangles to use.***

I know from wikipedia, that this is true.

Thank you

2. Originally Posted by mx-
Hi, the problem i have here isn't with my trig skills but with my ability to use similar triangles to prove that KZ = tan.

***Could someone please tell me which triangles to use.***

I know from wikipedia, that this is true.

Thank you
Prove that KZ is equal to the tangent of what angle??

-Dan

3. The above diagram is of a unit circle. Where KZ denotes the length of the segment from the point K to the point Z.

I must prove in general that KZ is the tangent using similar triangles with the Information provided in the first diagram.

It might also help to know that LZ = OJ = cosθ and JZ = OL = sin θ.

Thanks.


text:
# 5.6 Minimum and maximum values  (Page 4/6)

 Page 4 / 6

Above argument is valid for all continuous function which may have varying combination of increasing and decreasing trends within the domain of function. The function values at end points of a closed interval are extremums (minimum or maximum) - may not be least or greatest. In the general case, there may be more minimum and maximum values apart from the ones at the ends of closed interval. This generalization, as a matter of fact, is the basis of “extreme value theorem”.

## Extreme value theorem

The extreme value theorem of continuous function guarantees existence of minimum and maximum values in a closed interval. Mathematically, if f(x) is a continuous function in the closed interval [a,b], then there exists f(l) ≤ f(x) and f(g) ≥ f(x) such that f(l) is global minimum and f(g) is global maximum of function.

As discussed earlier, there at least exists a pair of minimum and maximum at the end points. There may be more extremums depending on the nature of graph in the interval.

## Range of function

If a function is continuous, then least i.e. global minimum, “A” and greatest i.e. global maximum, “B”, in the domain of function correspond to the end values specifying the range of function. The range of the function is :

$\left[A,B\right]$

If function is not continuous or if function can not assume certain values, then we need to suitably analyze function and modify the range given above. We shall discuss application of the concept of least and greatest values to determine range of function in a separate module.

## Determining minimum and maximum values

There are three cases for determining minimum and maximum values. However, we should clearly underline that these methods give us relative minimum and relative maximum values – which may or may not be the greatest (global) or least (global) values. We need to interpret minimum and maximum in the context of specified domain to ascertain whether minimum and maximum are least and greatest respectively or not?

(i) function is differentiable in the domain of function.

(ii) function is continuous in the domain of function

(iii) function is discontinuous at certain points in the domain of function.

## Function is differentiable

The derivative of function exists for all values of x in the domain. In this case, we follow the algorithm given here (without proof- its proof is based on Taylor’s expansion) :

1: Determine first derivative.

2: Equate derivative to zero.

3: Solve equation obtained in the step 2 for x.

4: If there is no real solution of equation, then function has no minimum or maximum.

5: If there is real solution of equation, then determine second derivative. Put root values in the expression of second derivative one after another and see whether second derivative is non-zero. If second order derivative is positive non-zero, then function is minimum at that root value. On the other hand, if second order derivative is negative non-zero, then function is maximum at that root value. We should note that these conclusions are valid for all higher even derivatives, which we might need to evaluate.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x


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# Number Series Questions for SSC JE PDF

0
982

## Number Series Questions for SSC JE PDF

Download SSC JE Number Series Questions PDF. Top 15 SSC JE Number Series questions based on asked questions in previous exam papers very important for the SSC exam.

Instructions

In the following five questions, select the missing number from the given responses.

Question 1:

a) 1

b) 30

c) 11

d) 0

Instructions

In the following questions, select the missing number from the given responses.

Question 2:

a) 24

b) 27

c) 29

d) 22

Instructions

In each of the following questions, select the related word/letters/numbers from the given alternatives.

Question 3: $\frac{1}{8}:\frac{1}{64}::\frac{1}{16}:\frac{1}{?}$

a) 128

b) 126

c) 144

d) 132

Instructions

In each of the following questions, which one set of letters when sequentially placed at the gaps in the given letter series shall complete it ?

Question 4: Find out the wrong number in the sequence
102, 101, 98, 93, 86, 74, 66, 53

a) 101

b) 66

c) 74

d) 93

Question 5: 40% of 1620 + 30% of 960 = ? % of 5200

a) 12

b) 24

c) 16

d) 18

Question 6: What will come in place of the question mark (?) in the series?
3, 8, 27, 112, (?), 3396

a) 565

b) 452

c) 560

d) 678

Question 7: Find out the set of numbers amongst the four sets of numbers given in the alternatives which is most like the set given in the question.
Given Set : (8, 56, 72)

a) (7, 56, 63)

b) (3, 15, 24)

c) (6. 42, 54)

d) (5, 30, 35)

Instructions

In each of the following questions a series is given, with one/two term(s) missing. Choose the correct alternative from the given ones that will complete the series.

Question 8: 9, 8, 10, 16, 11, ?, 12, 64

a) 28

b) 36

c) 25

d) 32

Instructions

In each of the following questions, select the missing number from the given responses.

Question 9:

a) 20

b) 25

c) 10

d) 15

Instructions

In each of the following questions, select the missing number from the given responses.

Question 10: 5 1 25
6 2 18
10 4 25
3 3 ?

a) 10

b) 9

c) 3

d) 4

Instructions

In each of the following questions, select the missing number from the given responses :

Question 11: 216 209 202
522 515 508
633 626 ?

a) 620

b) 608

c) 602

d) 619

Question 12: A series is given with one term missing. Choose the correct alternative from the given ones that will complete the series.
144, 256, 400, ?

a) 441

b) 576

c) 625

d) 289

Question 13: A series is given with one term missing. Choose the correct alternative from the given ones that will complete the series.
30, 60, 360, 3600, ?

a) 48500

b) 50500

c) 50400

d) 40800

Question 14: A series is given with one term missing. Choose the correct alternative from the given ones that will complete the series.
17, 27, 42, 62,?

a) 82

b) 87

c) 81

d) 84

Question 15: A series is given with one term missing. Select the correct alternative from the given ones that will complete the series.
2, 7, 16, ? , 46, 67

a) 26

b) 27

c) 29

d) 31

let the missing term be y

here in 1st figure , the pattern is (3 x 2 x 4) = 24

in 2nd figure , (2 x -1 x -2 ) = 4

in 3rd figure also the same thing will be followed ,

y = (6 x 5 x 0 ) = 0

The pattern followed is : Sum of first row and the positive difference between second and third rows.

=> 18 + (6-3) = 21 and 21 + (9-4) = 26

Similarly, 24 + (8-3) = 24 + 5 = 29

=> Ans – (C)

$\frac{1}{8}:\frac{1}{64}$ = 8

let the missing number be y

$\frac{1}{16}:\frac{1}{y}$ = 8

y = 128

here the given pattern is 102, 101, 98, 93, 86, 74, 66, 53

as we ca see that

102 – 101 = 1

101 – 98 = 3

98 – 93 = 5

93 – 86 = 7

as we can see that the difference between consecutive terms is increasing by 2 in every case but when we check

86 – 74 = 12 we see that the pattern is not getting fullfilled in case of 74 so in order to make the same pattern the term should be 77 inspite of 74 .

lets check with 77

86 – 77 = 9

hence it is confirmed that the wrong number is 74

40% of 1620 = $\frac{40 \times 1620}{100}$ = $648$
30% of 960 = $\frac{30 \times 960}{100}$ = $288$
648 + 288 = 936
Let the unknown percentage be $x$.
$936 = x \% of 5200$
$x = \frac{936 \times 100}{5200} = 18$

Hence option D is the correct answer.

Let the missing term be y

here the sequence is 3, 8, 27, 112, y, 3396

here the pattern is :

8 = 3 x 2 + 2

27 = 8 x 3 + 3

112 = 27 x 4 + 4

so from the highlighted pattern , we can say that

y = 112 x 5 + 5 = 565

(8, 56, 72)
follows the pattern
(8 , 8 x 7 , 8 x 9 )
So (6. 42, 54) is similar to (8, 56, 72)
as it follows the pattern
(6 , 6 x 7 , 6 x 9 )

In the given question there are two series
Series 1
8, 8 x 2 = 16 , 16 x 2 =32 , 32 x 2 =64
Series 2
9,10,11,12
So option D is correct

The product of 2 numbers in a sector is equal to the central number in the previous sector.

$3\times5=15$, $8\times3=24$

$2\times7=14$

Similarly, $5\times3=15$

=> Ans – (D)

Each row of the given matrix follows the pattern
$5^{2}/1$ =25
$6^{2}/2$ =18
$10^{2}/4$ =25
$3^{2}/3$ =3

In each row the next term is obtained by subtracting 7 from the previous term
In Row 1
216,216-7 =209,209-7 =202
In Row 2
522,522-7 =515 ,515-7 =508
In Row 3
633, 633-7 =626, 626-7 =619
So ? = 619

The pattern followed is :

Thus, $24^2=576$

=> Ans – (B)

Expression : 30, 60, 360, 3600, ?

Thus, missing number = 50400

=> Ans – (C)

17 + 10 = 27

27 + 15 = 42

42 + 20 = 62

62 + 25 = 87

=> Ans – (B)

Numbers with difference of 4 are added.

2 + 5 = 7

7 + 9 = 9

16 + 13 = 29

29 + 17 = 46

46 + 21 = 67

=> Ans – (C)

We hope this Number Series Questions pdf for SSC JE exam will be highly useful for your Preparation.


text:
# Refraction

Last updated

In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. [1] Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

## Contents

For light, refraction follows Snell's law, which states that, for a given pair of media, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equal to the ratio of phase velocities (v1 / v2) in the two media, or equivalently, to the indices of refraction (n2 / n1) of the two media. [2]

${\displaystyle {\frac {\sin \theta _{1}}{\sin \theta _{2}}}={\frac {v_{1}}{v_{2}}}={\frac {n_{2}}{n_{1}}}}$

Optical prisms and lenses use refraction to redirect light, as does the human eye. The refractive index of materials varies with the wavelength of light, [3] and thus the angle of the refraction also varies correspondingly. This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors. [4]

## Light

Refraction of light can be seen in many places in our everyday life. It makes objects under a water surface appear closer than they really are. It is what optical lenses are based on, allowing for instruments such as glasses, cameras, binoculars, microscopes, and the human eye. Refraction is also responsible for some natural optical phenomena including rainbows and mirages.

### General explanation

A correct explanation of refraction involves two separate parts, both a result of the wave nature of light.

1. Light slows as it travels through a medium other than vacuum (such as air, glass or water). This is not because of scattering or absorption. Rather it is because, as an electromagnetic oscillation, light itself causes other electrically charged particles such as electrons, to oscillate. The oscillating electrons emit their own electromagnetic waves which interact with the original light. The resulting "combined" wave has wave packets that pass an observer at a slower rate. The light has effectively been slowed. When light returns to a vacuum and there are no electrons nearby, this slowing effect ends and its speed returns to c.
2. When light enters, exits or changes the medium it travels in, at an angle, one side or the other of the wavefront is slowed before the other. This asymmetrical slowing of the light causes it to change the angle of its travel. Once light is within the new medium with constant properties, it travels in a straight line again.

#### Explanation for slowing of light in a medium

As described above, the speed of light is slower in a medium other than vacuum. This slowing applies to any medium such as air, water, or glass, and is responsible for phenomena such as refraction. When light leaves the medium and returns to a vacuum, and ignoring any effects of gravity, its speed returns to the usual speed of light in a vacuum, c.

Common explanations for this slowing, based upon the idea of light scattering from, or being absorbed and re-emitted by atoms, are both incorrect. Explanations like these would cause a "blurring" effect in the resulting light, as it would no longer be travelling in just one direction. But this effect is not seen in nature.

A more correct explanation rests on light's nature as an electromagnetic wave. [5] Because light is an oscillating electrical/magnetic wave, light traveling in a medium causes the electrically charged electrons of the material to also oscillate. (The material's protons also oscillate but as they are around 2000 times more massive, their movement and therefore their effect, is far smaller). A moving electrical charge emits electromagnetic waves of its own. The electromagnetic waves emitted by the oscillating electrons, interact with the electromagnetic waves that make up the original light, similar to water waves on a pond, a process known as constructive interference. When two waves interfere in this way, the resulting "combined" wave may have wave packets that pass an observer at a slower rate. The light has effectively been slowed. When the light leaves the material, this interaction with electrons no longer happens, and therefore the wave packet rate (and therefore its speed) return to normal.

#### Explanation for bending of light as it enters and exits a medium

Consider a wave going from one material to another where its speed is slower as in the figure. If it reaches the interface between the materials at an angle one side of the wave will reach the second material first, and therefore slow down earlier. With one side of the wave going slower the whole wave will pivot towards that side. This is why a wave will bend away from the surface or toward the normal when going into a slower material. In the opposite case of a wave reaching a material where the speed is higher, one side of the wave will speed up and the wave will pivot away from that side.

Another way of understanding the same thing is to consider the change in wavelength at the interface. When the wave goes from one material to another where the wave has a different speed v, the frequency f of the wave will stay the same, but the distance between wavefronts or wavelength λ=v/f will change. If the speed is decreased, such as in the figure to the right, the wavelength will also decrease. With an angle between the wave fronts and the interface and change in distance between the wave fronts the angle must change over the interface to keep the wave fronts intact. From these considerations the relationship between the angle of incidence θ1, angle of transmission θ2 and the wave speeds v1 and v2 in the two materials can be derived. This is the law of refraction or Snell's law and can be written as [6]

${\displaystyle {\frac {\sin \theta _{1}}{\sin \theta _{2}}}={\frac {v_{1}}{v_{2}}}}$.

The phenomenon of refraction can in a more fundamental way be derived from the 2 or 3-dimensional wave equation. The boundary condition at the interface will then require the tangential component of the wave vector to be identical on the two sides of the interface. [7] Since the magnitude of the wave vector depend on the wave speed this requires a change in direction of the wave vector.

The relevant wave speed in the discussion above is the phase velocity of the wave. This is typically close to the group velocity which can be seen as the truer speed of a wave, but when they differ it is important to use the phase velocity in all calculations relating to refraction.

A wave traveling perpendicular to a boundary, i.e. having its wavefronts parallel to the boundary, will not change direction even if the speed of the wave changes.

### Law of refraction

For light, the refractive index n of a material is more often used than the wave phase speed v in the material. They are, however, directly related through the speed of light in vacuum c as

${\displaystyle n={\frac {c}{v}}}$.

In optics, therefore, the law of refraction is typically written as

${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}$.

### Refraction in a water surface

Refraction occurs when light goes through a water surface since water has a refractive index of 1.33 and air has a refractive index of about 1. Looking at a straight object, such as a pencil in the figure here, which is placed at a slant, partially in the water, the object appears to bend at the water's surface. This is due to the bending of light rays as they move from the water to the air. Once the rays reach the eye, the eye traces them back as straight lines (lines of sight). The lines of sight (shown as dashed lines) intersect at a higher position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is.

The depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the target fish appear to be in a different place, and the fisher must aim lower to catch the fish. Conversely, an object above the water has a higher apparent height when viewed from below the water. The opposite correction must be made by an archer fish. [8]

For small angles of incidence (measured from the normal, when sin θ is approximately the same as tan θ), the ratio of apparent to real depth is the ratio of the refractive indexes of air to that of water. But, as the angle of incidence approaches 90o, the apparent depth approaches zero, albeit reflection increases, which limits observation at high angles of incidence. Conversely, the apparent height approaches infinity as the angle of incidence (from below) increases, but even earlier, as the angle of total internal reflection is approached, albeit the image also fades from view as this limit is approached.

### Dispersion

Refraction is also responsible for rainbows and for the splitting of white light into a rainbow-spectrum as it passes through a glass prism. Glass has a higher refractive index than air. When a beam of white light passes from air into a material having an index of refraction that varies with frequency, a phenomenon known as dispersion occurs, in which different coloured components of the white light are refracted at different angles, i.e., they bend by different amounts at the interface, so that they become separated. The different colors correspond to different frequencies.

### Atmospheric refraction

The refractive index of air depends on the air density and thus vary with air temperature and pressure. Since the pressure is lower at higher altitudes, the refractive index is also lower, causing light rays to refract towards the earth surface when traveling long distances through the atmosphere. This shifts the apparent positions of stars slightly when they are close to the horizon and makes the sun visible before it geometrically rises above the horizon during a sunrise.

Temperature variations in the air can also cause refraction of light. This can be seen as a heat haze when hot and cold air is mixed e.g. over a fire, in engine exhaust, or when opening a window on a cold day. This makes objects viewed through the mixed air appear to shimmer or move around randomly as the hot and cold air moves. This effect is also visible from normal variations in air temperature during a sunny day when using high magnification telephoto lenses and is often limiting the image quality in these cases. [9] In a similar way, atmospheric turbulence gives rapidly varying distortions in the images of astronomical telescopes limiting the resolution of terrestrial telescopes not using adaptive optics or other techniques for overcoming these atmospheric distortions.

Air temperature variations close to the surface can give rise to other optical phenomena, such as mirages and Fata Morgana. Most commonly, air heated by a hot road on a sunny day deflects light approaching at a shallow angle towards a viewer. This makes the road appear reflecting, giving an illusion of water covering the road.

### Clinical significance

In medicine, particularly optometry, ophthalmology and orthoptics, refraction (also known as refractometry) is a clinical test in which a phoropter may be used by the appropriate eye care professional to determine the eye's refractive error and the best corrective lenses to be prescribed. A series of test lenses in graded optical powers or focal lengths are presented to determine which provides the sharpest, clearest vision. [10]

## Water waves

Water waves travel slower in shallower water. This can be used to demonstrate refraction in ripple tanks and also explains why waves on a shoreline tend to strike the shore close to a perpendicular angle. As the waves travel from deep water into shallower water near the shore, they are refracted from their original direction of travel to an angle more normal to the shoreline. [11]

## Acoustics

In underwater acoustics, refraction is the bending or curving of a sound ray that results when the ray passes through a sound speed gradient from a region of one sound speed to a region of a different speed. The amount of ray bending is dependent on the amount of difference between sound speeds, that is, the variation in temperature, salinity, and pressure of the water. [12] Similar acoustics effects are also found in the Earth's atmosphere. The phenomenon of refraction of sound in the atmosphere has been known for centuries; [13] however, beginning in the early 1970s, widespread analysis of this effect came into vogue through the designing of urban highways and noise barriers to address the meteorological effects of bending of sound rays in the lower atmosphere. [14]

## Related Research Articles

In astronomy, aberration is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. It causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period, T, — the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

In optics, the refractive index of a material is a dimensionless number that describes how fast light travels through the material. It is defined as

Total internal reflection (TIR) is the optical phenomenon in which the surface of the water in a fish-tank when viewed from below the water level, reflects the underwater scene like a mirror, with no loss of brightness (Fig. 1). In general, TIR occurs when waves in one medium reach the boundary with another medium at a sufficiently slanting angle, provided that the second ("external") medium is transparent to the waves and allows them to travel faster than in the first ("internal") medium. TIR occurs not only with electromagnetic waves such as light and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam (Fig. 2), we tend to describe the reflection in terms of "rays" rather than waves. In a medium whose properties are independent of direction, such as air, water, or glass, each "ray" is perpendicular to the associated wavefronts.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circular birefringence and circular dichroism are the manifestations of optical activity. Optical activity occurs only in chiral materials, those lacking microscopic mirror symmetry. Unlike other sources of birefringence which alter a beam's state of polarization, optical activity can be observed in fluids. This can include gases or solutions of chiral molecules such as sugars, molecules with helical secondary structure such as some proteins, and also chiral liquid crystals. It can also be observed in chiral solids such as certain crystals with a rotation between adjacent crystal planes or metamaterials.

Brewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. Media having this common property may be termed dispersive media. Sometimes the term chromatic dispersion is used for specificity. Although the term is used in the field of optics to describe light and other electromagnetic waves, dispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, in gravity waves, and for telecommunication signals along transmission lines or optical fiber.

An optical prism is a transparent optical element with flat, polished surfaces that refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape of an optical prism is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms. Prisms can be made from any material that is transparent to the wavelengths for which they are designed. Typical materials include glass, plastic, and fluorite.

In physics, Bragg's law, or Wulff–Bragg's condition, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move, as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering. The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible.

Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.

Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

Etendue or étendue is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics.

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other. Since the wave cannot be reflected, it is completely transmitted instead. When the thickness is a multiple of a half-wavelength of the light, the two reflected waves reinforce each other, increasing the reflection and reducing the transmission. Thus when white light, which consists of a range of wavelengths, is incident on the film, certain wavelengths (colors) are intensified while others are attenuated. Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water. It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses.

## References

1. The Editors of Encyclopaedia Britannica. "Refraction". Encyclopaedia Britannica. Retrieved 2018-10-16.
2. Born and Wolf (1959). Principles of Optics . New York, NY: Pergamon Press INC. p. 37.
3. R. Paschotta, article on chromatic dispersion Archived 2015-06-29 at the Wayback Machine in the Encyclopedia of Laser Physics and Technology Archived 2015-08-13 at the Wayback Machine , accessed on 2014-09-08
4. Carl R. Nave, page on Dispersion Archived 2014-09-24 at the Wayback Machine in HyperPhysics Archived 2007-10-28 at the Wayback Machine , Department of Physics and Astronomy, Georgia State University, accessed on 2014-09-08
5. Hecht, Eugene (2002). Optics. Addison-Wesley. p. 101. ISBN   0-321-18878-0.
6. "Refraction". RP Photonics Encyclopedia. RP Photonics Consulting GmbH, Dr. Rüdiger Paschotta. Retrieved 2018-10-23. It results from the boundary conditions which the incoming and the transmitted wave need to fulfill at the boundary between the two media. Essentially, the tangential components of the wave vectors need to be identical, as otherwise the phase difference between the waves at the boundary would be position-dependent, and the wavefronts could not be continuous. As the magnitude of the wave vector depends on the refractive index of the medium, the said condition can in general only be fulfilled with different propagation directions.
7. Dill, Lawrence M. (1977). "Refraction and the spitting behavior of the archerfish (Toxotes chatareus)". Behavioral Ecology and Sociobiology. 2 (2): 169–184. doi:10.1007/BF00361900. JSTOR   4599128. S2CID   14111919.
8. "The effect of heat haze on image quality". Nikon. 2016-07-10. Retrieved 2018-11-04.
9. "Refraction". eyeglossary.net. Archived from the original on 2006-05-26. Retrieved 2006-05-23.
10. "Shoaling, Refraction, and Diffraction of Waves". University of Delaware Center for Applied Coastal Research. Archived from the original on 2009-04-14. Retrieved 2009-07-23.
11. Navy Supplement to the DOD Dictionary of Military and Associated Terms (PDF). Department Of The Navy. August 2006. NTRP 1-02.
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13. Hogan, C. Michael (1973). "Analysis of highway noise". Water, Air, & Soil Pollution. 2 (3): 387–392. Bibcode:1973WASP....2..387H. doi:10.1007/BF00159677. S2CID   109914430.