Disease Progression via Single Sketches.

for the VNIR ranges, single sketches of the barley diseases based on mean reflectance archetypes are shown in Fig. 5 (for the sake of clarity, the figure omits results for SWIR ranges, which are provided in supporting information). Each sketch consists of parts (line segments) which encode major stages of the dynamics using similar weights. Thus, the shorter a part, the higher the impact of the corresponding period.

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Figure 5. Single sketches of hyperspectral dynamics of plant diseases for visible-near infrared (VNIR) wavelengths.

Each sketch consists of parts encoding major states during pathogenesis of the plant disease with similar weights. Thus, the shorter a part, the higher the impact of the corresponding period. (Best viewed in color)

https://doi.org/10.1371/journal.pone.0116902.g005

Major states of the dynamics appeared to differ between diseases: Net blotch diseased leaves exhibited a steady development in the first 7 days. From 7 dai to 12 dai significant reflectance changes appeared in the VNIR and SWIR ranges which corresponds to biological processes during pathogenesis. First symptoms are chlorosis and small necroses 4 dai to 7 dai, reflected in changes in the VNIR. Severe structural and biochemical changes of the leaf tissue occur 7 dai to 12 dai, and a considerable water loss can be deduced from a period with high impact in the sketch of the SWIR data. The impact of rust on barley leaves was comparatively minor and more consistent over time. The biotrophic pathogen aims to feed from living host cells to produce new rust spores. Time points with crucial processes appear in the VNIR between 5 dai to 7 dai where first chloroses appeared and changed into small brownish rust pustules. Starting 10 dai, the rust spores ruptured the epidermis and massive amounts of spores were released from each pustule; this relevant step during rust pathogenesis was clearly marked in the sketches (Fig. 5). The dynamics of rust pathogenesis in the SWIR were minor as the effect depends on the biotrophic lifestyle of P. hordei. The characteristic powdery mildew pustules on the leaf surface constantly grew during the first 9 days of pathogenesis. Notable changes in pustule size could be recorded; however, the color of powdery mildew colonies stayed whitish and the surrounding barley tissues seemed to remain intact to the greatest possible extent. From 9 dai onwards, the color of the powdery mildew mycelium changed from whitish over gray to brownish. This process can be traced in the sketches of hyperspectral dynamics of powdery mildew (Fig. 5). Barley tissue became necrotic beginning 12 dai, and, in turn, SWIR reflectance increased noticeably. The sketches based on disease archetypal signatures highlight relevant steps during disease progression. This characteristic information is hidden when using mean signatures without archetypal selection.

Metro Maps of Disease Dynamics.

So called metro maps provide an innovative and intuitive way to illustrate spectral dynamics and changes during pathogenesis (Fig. 6) as they can be seen as a concise structured set of salient information. Metro maps of disease development therefore provide a metric of the interaction between each pathogen and the host tissue as well as a simply structured summary of the spectral information during disease development. The connectivity of a map conveys the underlying biophysical and biochemical processes during disease development as well as similarities between the three diseases at different developmental stages.

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Figure 6. Collective disease progression via Metro Maps of hyperspectral dynamics of diseased plants for visible-near infrared (VNIR) (top) and short-wave infrared (SWIR) wavelengths (bottom).

Each disease track from hyperspectral images exhibits a specific route in the metro map, the direction and the dynamic steps are in correspondence to biophysical and biochemical processes during disease development. The beginning of all routes is at the same time point/train station (day of inoculation, gray circle). (Best viewed in color)

https://doi.org/10.1371/journal.pone.0116902.g006

Each disease track exhibits a specific route in the metro map. The start of all routes is set at the same time point/train station (day of inoculation, gray circle). From the first days after inoculation the appearance of powdery mildew on barley is different to that of other diseases; the relatively intact tissue is covered by white mycelium colonies producing an increasing amount of conidia with a dominant influence on the VNIR spectrum (Fig. 6, top). Net blotch and rust share the occurrence of chloroses in early stages of disease development. However, net blotch causes early necrotic tissue damage and followed a different path until 10 dai; here, the routes of rust and net blotch disease are interlaced in the VNIR, as the epidermis of rust diseased barley leaves is ruptured and thousands of brownish rust spores are released. The path of healthy barley plants has a different general direction than the other treatments in the VNIR and SWIR metro map, because only natural senescence processes influence the spectral course of healthy barley plants (Fig. 6). SWIR spectral reflectance is mainly influenced by water content. During rust pathogenesis only tiny chlorotic spots with rust uredina appear on the tissue; necrosis and loss of water occurred only at later stages and were not present in our experiments. Thus, the deviation of the rust track from the one of healthy plants is minor (Fig. 6, bottom). A major impact on SWIR reflectance by powdery mildew and net blotch was observed and described by archetypal signatures. Metro maps are an easily interpreted graphical image of this effect. The course of powdery mildew and net blotch is summarized in a similar manner until day 9, afterwards the destructive characteristic of net blotch affects SWIR reflectance to a greater extent than powdery mildew which causes rather local necrosis and tissue damage.

Discussion of Methodology.

The present paper provides a novel and efficient data analysis cascade for plant phenotyping data recorded from plants diseased with different pathogens. The developed framework aims at an automatic information extraction from hyperspectral images and at an intuitive visualization of results; a particular focus is on potential practical impact and the potential for implementation in automated phenotyping platforms.

The benefits of the presented sketching approaches, summarized in Fig. 2, are manifold: Metro maps are natural tools for passengers to navigate in large urban areas [26]. Because people easily read and understand such maps, the metro map metaphor currently emerges as an easy-to-understand technique for visualization of abstract interconnected “trains of thoughts” [27] and as a tool to automatically construct structured summaries of information in scientific texts or news articles [28]. An additional favorable property is that metro maps illustrate relations between case lines which, in our study, are the dynamics of healthy and diseased barley leaves. Single sketches as well as metro maps are expressed in terms of disease archetypal signatures which carry meaningful information for plant biologists. They conform to plant physiological knowledge, explicitly illustrate the interaction between plants and pathogens, and provide an interpretable summary of disease progressions.

Disease archetypal signatures can be discovered at massive scale by using a recent linear time matrix factorization approaches, which re-parameterizes each hyperspectral image of a disease (a data matrix) in terms of a distribution on the simplex spanned by a few extreme signatures (a small number of pixels in the hyperspectral images). This facilitates high-throughput phenotyping and avoids running the risk of loosing information when selecting typical signatures manually at some diseased spots only. Since extreme data points (approximately) form a simplex, there are natural candidates to describe their statistical distribution, namely simplex distributions such as the Dirichlet distribution. In contrast to other “data cloud” embedding methods, see e.g. [29], we therefore do not make any assumptions as to the true generating distribution of each input data matrix, which is good practice since these are typically unknown. In contrast to histogram based embedding approaches such as, for instance, proposed by Sakurai et al.[30], probabilistic inference can be performed to quantify the “impact” of extremes in a dispersion model or to determine the “distance” between the data matrices in an information theoretic manner, cf. Kersting et al.[14]. Generally, distributions pave the way to statistical data analysis methods such as regression techniques, similarity metrics, low-rank embeddings, and advanced visualization techniques.

To summarize, linking disease symptomatology with disease archetypal signatures and metro maps of the diseased plants clearly shows that questions (MQ) and (SQ) can be answered affirmatively.

Plant Material and Pathogens.

A data set of hyperspectral images recorded from barley plants formed the basis for further analysis in this research. Barley plants (Hordeum vulgare cv. Leibniz, KWS Lochow, Bergen-Wohlde, Germany) grown in a controlled greenhouse environment were used for hyperspectral measurements after reaching growth stage (GS) 32 [31]. For each pathogen treatment, 9 plants were inoculated with Pyrenophora teres (causing net blotch disease), Puccinia hordei (causing leaf rust of barley) and Blumeria graminis hordei (causing powdery mildew), respectively. A control group was kept non-inoculated. Conidia of P. teres were harvested from diseased barley leaves, sampled from fields during spring and incubated in a moist chamber for 24 h at room temperature. Pyrenophora teres was inoculated by spaying a spore suspension (1×10⁴ conidia ml⁻¹) onto leaves using a hand sprayer. Subsequently, plants were placed in transparent plastic boxes to realize 100% relative humidity (RH) at 23/20^∘C for 24 h. Uredospores of P. hordei for inoculation were obtained of from diseased leaves and were stored at 4^∘C. Suspension of P. hordei (4×10⁴ uredospores ml⁻¹) were sprayed onto leaves before placing the barley plants in plastic boxes and incubating them for 24 h at 23/20^∘C and 100% RH. After 24 h the plants were removed from the plastic boxes. For inoculation of B. graminis hordei, conidia from sporulating colonies were distributed over barley leaves in a ventilator chamber. After incubation, all plants were kept in the greenhouse at 21/18^∘C and 60% RH, whereas B. graminis hordei inoculated plants were kept in an extra chamber to avoid cross-contamination of the other treatments. Disease progress of the foliar pathogens Pyrenophora teres, Puccinia hordei, and Blumeria graminis hordei were observed for 2 weeks after inoculation. For each treatment 9 leaves (each belonging to a different plant, respectively) were assessed at each measuring day.

Hyperspectral Imaging and Data Representation.

Hyperspectral images of barley leaves were recorded with two line scanning spectrographs 4, 6, 8, 10, 12 and 14 days after inoculation (dai). The ImSpector V10E (Spectral Imaging Ltd., Oulu, Finland) covers the VIS and the NIR ranges from 400 to 1000 nm, with a spectral resolution of up to 2.8 nm and a spatial resolution of 0.12 mm per pixel, resulting in 210 hyperspectral bands. A SWIR-camera (Spectral Imaging Ltd., Oulu, Finland) was used to record hyperspectral images in the SWIR ranges from 1000 to 2500 nm, with a spectral resolution of 5.8 nm and a spatial resolution of 0.4 mm per pixel, resulting in 226 bands. Constant illumination was provided by six ASD-Pro-Lamps (Analytical Spectral Devices Inc., Boulder, USA). The hyperspectral cameras and the illumination system were installed on a motorized line scanner (Spectral Imaging Ltd., Oulu, Finland) to obtain a second spatial dimension. The camera settings and the control of the motorized line scanner were adapted using the SpectralCube software (Spectral Imaging Ltd., Oulu, Finland). The hyperspectral images were recorded in a dark chamber in order to realize constant and reproducible illumination and measurement conditions. For a detailed description of the measuring setup, see Mahlein et al. [12]. Normalization and smoothing of raw hyperspectral images were performed using the software ENVI 4.6 + IDL 7.0 (EXELIS Visual Information Solutions, Boulder, USA). Reflectance was calculated relative to a white reference bar and to dark current measurement. Then the Savitzky-Golay filter [32] was applied to remove noise and to smooth the spectral information of the hyperspectral images. Furthermore the background of the images was masked applying band thresholds of R551 nm < 0.045, R667 nm > 0.085 and R798 nm < 0.8 in the VNIR, and of R1124 nm < 0.1 in the SWIR (R = reflectance at wavelengths indicated).

For subsequent analysis, each hyperspectral image was represented as dense Λ×N matrix, where N denotes the number of pixels and Λ the number of spectral bands (see Fig. 2 (top) for an abstract illustration). By stacking all data matrices recorded during pathogenesis, we obtained a data matrix with approximately 10 millions columns or about 2 billion matrix entries (encoding the reflected energy at different spectral bands) for the VNIR and about 200 million entries for the SWIR data set. Furthermore, each hyperspectral data set contains temporal information (day after inoculation) and one of the four labels “healthy”, “rust”, “powdery mildew” and “net blotch” which were used throughout the analysis in this work.

Step 1: Interpretable Matrix Factorization.

In the first step of our framework, we employ interpretable matrix factorization at massive scale. Matrix factorization methods allow for embedding high dimensional data in lower dimensional spaces and can therefore mitigate effects due to noise, uncover latent relations, or facilitate further processing and ultimately help finding patterns in the data set distribution. More precisely, they factorize a matrix $X\in {ℝ}^{m\times n}$ into two matrices (1) where the matrix of basis vectors $W\in {ℝ}^{m\times c}$ and the coefficient matrix $H\in {ℝ}^{c\times n}$ containing the low dimensional coordinates, are typically determined by minimizing a cost function, such as the squared Frobenius norm. A widely used technique for the factorization in Eq. (1) is Principal Component Analysis (PCA) [34] which is known to retain as much of the variation present in the data as possible. It is effective in terms of data compression, noise and redundancy reduction and typically projects the data points into a lower dimensional space spanned by the top eigenvectors of the covariance matrix. While these basis vectors are optimal in a statistical sense, PCA has been criticized for being less faithful to the nature of the data at hand. For instance, data mining practitioners often tend to assign a “physical” meaning to the resulting factors. Such reification must be based on an intimate knowledge of the application domain and can often not be justified from mathematics. This also holds for other techniques such as Non-Negative Matrix Factorization due to [35] and K-means clustering which implicitly performs matrix factorization. More importantly, classical approaches pose the difficulty of characterizing sophisticated patterns of data point distributions in a unified parametric and interpretable form. This is generally intractable [36].

An alternative are interpretable matrix factorizations methods, which compute low rank approximations from selected columns of a data matrix [37]. They are increasingly popular in the data mining community [33, 38–43] since they preserve properties such as sparseness or non-negativity and were successfully applied in many important applications, e.g. fraud detection, fMRI segmentation, drought detection in plants or collaborative filtering. Here we consider interpretable data driven methods in the context of convexity constrained matrix factorization that can be written as (2) where ‖⋅‖_F denote the Forbenius norm, the values in matrix H have non-negativity constraints and the matrix G is restricted to be binary. Consequently, the basis vectors W are real data points, i.e. columns of the data matrix X that represent actual observations. Therefore, they have naturally a biological meaning and can easily be interpreted by a domain expert. As pointed out by Cutler and Breiman [44], basis vectors in convexity constrained matrix factorization correspond to the most extreme, rather than to average columns. Moreover, it was shown that a good subset of columns maximize their volume [45,46]. Therefore, we determine c columns from the input data matrix X as basis vectors W, such that the volume of the simplex spanned by the columns of W, is maximized. However, the maximum-volume criterion is provably NP-hard [45]. An approximation called Simplex Volume Maximization (SiVM), was introduced by Thurau et al. (2012) [33] and empirically proven to be feasible and reliable. The authors presented an efficient and linear time greedy approach that iteratively determines basis vectors using the notion of distance geometry. An example of how SiVM works is illustrated in Fig. 7 on a synthetic data set consisting of three Gaussians components. For the first data points to be selected, we simply take the two points which are most likely furthest away from each other. Pairwise distances computed in one iteration can be reused in later iterations so that, for retrieving c columns, we need to compute distances from the last selected column to all other data points exactly c + 1 times. As c is constant, we have an overall effort of $\mathcal{O}(n)$ since the coefficients in H can computed in a single pass over the data set solving a constrained quadratic program [47] of fixed dimensions per datapoint. We refer to [33] for more details.

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Figure 7. Example in which way Simplex Volume Maximization iteratively determines basis vectors for interpretable matrix factorization.

https://doi.org/10.1371/journal.pone.0116902.g007

We make use of SiVM in (Step 1) of our approach in order to determine the most extreme columns (hyperspectral signatures) and to use them as basis vectors W in Eq. (2). For the plant data, however, before computing the decomposition, we stacked all matrices of hyperspectral images. On the resulting matrix we first selected c = 25 extreme signatures per data set (VNIR and SWIR) that form the matrix of basis vectors W. This allows us to capture global dependencies. Since the basis vectors correspond to actual data points, they can be identified as signatures of a diseased, dry or healthy leaf. Given the matrix W, we then computed the coefficients H, i.e. the coordinates of all signatures in the space spanned by the extreme data points, in a single pass over the entire data set. Experiments were run on a standard Intel Quad-Core CPU with 3.06 GHz and 8 GB main memory. For the larger VNIR data, it took about 80 minutes on only a single core to determine the c basis vectors; for the smaller SWIR data set, it took less than 1 minute. The computation of the reconstruction matrix H for all images took less than 2 hours by using all cores. However, computing the factorization was the most time consuming part of the cascade, whereas performing the remaining steps was a matter of minutes in each case.

Steps 2–3: Archetypal Bayes Factors.

One of our goals is to find the archetypal hyperspectral signatures for a disease. Up to now, hyperspectral signatures are represented by means of convex combinations H of extreme signatures W discovered by stacking the hyperspectral signatures of all plants and diseases. Therefore, we next address the question of how to turn these “global” reconstructions into disease-specific reconstructions? That is, how to make use of the available disease label of each hyperspecral image?

Step 2: From Reconstructions to Distributions on the Simplex: The resulting reconstructions H are proportions that sum to one and describe the relative contribution of each of the c columns in W to a hyperspectral pixel. From a geometric point of view, the columns h₁, …,h_n of H can be considered as data points residing in a simplex spanned by W so that there are natural parametric distributions for h_i on the simplex. Probably the best known one is the Dirichlet distribution which is parametrized by a vector $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_c)$ and where Γ denotes the gamma function, see Fig. 2 (Step 2) for an illustration. Dirichlets naturally enforce convexity constraints on the reconstructions, 0 ≤ h_ij ≤ 1 and ${\displaystyle {\sum }_{j=1}^c{h}_{ij}}=1$, so that changing one h_ij impacts all other h_ik. To estimate the parameters α from the reconstructions H_i, we follow a maximum-likelihood approach [48]. The simplex distribution induced by the reconstructions H_i per class/disease i results without having made any priori assumption as to the underlying distribution next to the distance measure used within SiVM.

Step 3: Using Labels to Select Disease Archetypes: Having distributions at hand opens the door to statistical machine learning. In particular, we make use of the label information. Given the Dirichlet distributions with parameters α_k for a disease k and α_h for the healthy plant, the task is to decide which is the best distribution describing signatures S_k within a diseased plant k. To this end, we apply Bayes factor [49, 50]: $BF(S)=\frac{P(S\mid {\mathscr{H}}_i)}{P(S\mid {\mathscr{H}}_j)}$. Here, two distributions ${\mathscr{H}}_i$ and ${\mathscr{H}}_j$ are compared by forming the posterior odds. Note that we assume that the prior over the models is uniform, i.e. $P(\mathscr{H})$ is a constant. Since our models are distributions of classes learned on the simplex we are dealing with the simplest case where Bayes factors coincide with the likelihood ratio [49]. Specifically, we advocate the use of log-likelihood ratio together with Dirichlets: (3)

The interpretation of LLR_ij w.r.t. class i is as follows: if LLR_ij ⩽ log(1) there is no difference to j but if LLR_ij > log(1) there are a differences, and, the larger the LLR_ij, the more pronounced the differences. Fig. 2 (Step 3) illustrates this using three overlapping Gaussians; the darker data points for Gaussian 2 denote a higher difference, i.e. a higher min_j LLR_2j value for j = 1,3. Using the LLR, we can select highly differential archetypal signatures for each disease. Examples of diseased spots and their differences to a healthy plant, which were selected in (Step 3) of our approach are shown in Fig. 4. The more yellow or red a pixel in this figure is, the more it differs.

Step 4: Dirichlet Aggregation Regression and Mean Archetypal Signatures.

Given sequences of disease distributions, we are interested in capturing their dynamics over time. To capture the dynamics of a single disease, we make use of Dirichlet-Aggregation Regression (DAR) [15]. DAR allows for modeling and predicting the distribution over disease archetypal signatures at any time. Specifically, DAR assumes a Bayesian perspective and supposes that the reconstruction $h_t^k$, computed for a single pixel in hyperspectral image of a plant k at time t, was generated from a hidden Dirichlet distribution parameterized by the variable ${\alpha }_t^k={[{\alpha }_{t1}^k,\dots ,{\alpha }_{tc}^k]}^T$, where c denotes the reconstruction dimension. To do so, it puts a Gaussian process [51] prior on the Dirichlet distributions induced on the simplex spanned by the extremes. For more details we refer to [15].

DAR is realized in (Step 4) of our approach (cf. Fig. 2). In order to sketch the hyperspectral dynamics of diseased plants, we first predict Dirichlets ${\alpha }_t^k$ using the GP for T intermediate time points and compute the Dirichlet mean archetypal signature ${\varsigma }_t^k=[{\varsigma }_{t1}^k\dots ,{\varsigma }_{t\Lambda }^k]$ for a plant k at time t as (4) where $E[{\alpha }_{ti}^k]={\alpha }_{ti}^k/({\displaystyle {\sum }_j{\alpha }_{tj}^k})$ and $W\in {ℝ}^{\text{Λ}\times c}$ contains c extreme signatures selected across all plants and time steps and λ denotes the current dimension (wavelength in hyperspectral signature). Examples of mean archetypal signatures for different diseased plants are shown in Fig. 3. As discussed above, they allow us to automatically sketch the hyperspectral dynamics of plant diseases, as illustrated in Fig. 5 and Fig. 6. Next, we discuss how to draw such interpretable summaries.

Step 5: Mapping Hyperspectral Dynamics of Diseased Plants.

Given the K sequences of automatically assessed mean disease archetypal signatures for different stages of diseases progression (where K is the number of diseases), we are left with summarizing their dynamics, highlighting interesting time points, and describing the behavior of disease symptoms. We distinguish between multiple and single sketches, which we formally define as

Definition 1 (Summary) A sketch $\mathbb{S}$ is a directed connected graph G = (V,E), where V is a set of nodes and E is a set of edges, and Π a set of D paths in G. Each edge e ∈ E belongs to at least one path Π and each node v_i ∈ V consists of at most D incoming and D outgoing edges.

Here, D denotes the number of different paths, say, diseases. A single sketch follows from this definition by setting D = 1 and thus consists of a sketch summarizing interesting time points of one particular diseased plant.

Single Sketches: To create a single sketch we are looking for a segmentation of ordered objects in B equally weighted bins which preserves the original ordering of the objects. Given a matrix $X\in {ℝ}^{K\times N}$ where the columns denote the hyperspectral signatures representing different stages of diseases progression, we can achieve a segmentation into B bins as follows: First, we compute the distances of consecutive spectra (columns) using their Euclidean distance and compute the average bin size as (5) Then, we fill the B−1 bins successively with the objects according to the bin size δ. The last bin is filled with the remaining objects. Drawing a single sketch is now simple. Each node denotes the begin (resp. end) of a period, and the length of the edge e_b between two consecutive nodes v_b and v_b+1 is set relatively to the length of the period covered by the objects in bin b. Thus, the shorter a part, the higher the impact of the corresponding period. This is illustrated in Fig. 5 for several diseased plants. Each sketch highlights the interesting periods of a disease, where a small edge denotes a period of high impact (change in hyperspectral signature).

The limitation of a single sketch, however, is that it only represents the progression of a single diseased plant. In order to uncover (dis-)similarities in progression and behavior of several diseased plants, we thus consider multiple sketches.

Multiple Sketches via Metro Maps of Diseased Plants: As an addition to the above sketches, we provide an abstract map showing spectral dynamics and changes during the pathogenesis. Here, we first embed hyperspectral images of plants, represented by means archetypal signatures, in a low dimensional Euclidean space. classical approaches for doing so are multidimensional scaling (MDS) [52] and IsoMap [53]. IsoMap first creates a graph by connecting each object to l of its neighbors and then considers path lengths in the graph for embedding using multidimensional scaling. Nevertheless, when analyzing such embeddings, it is easy to focus on a narrow aspect such as how close two plants are at a specific point in time and to lose track of the big picture. As small difference in distances are not of great importance overall, we abstract the embedding further into a multiple sketch, which is motivated by the idea of metro maps, the schematic diagrams of public transport networks. To draw a metro map of diseased plants, we used the approach described by Nöllenburg and Wolff [26] which takes as input the two dimensional embedding computed by, e.g. by MDS or IsoMap, and turns into into a metro map by solving a linear program. We refer to [26] for more details.

A modified version of IsoMap that makes use of temporal information was already used for the creation of simplex traces [14]. Unlike in classical MDS (which is employed by IsoMap), we here seek a solution for the minimization problem using any gradient-based optimization method, e.g. via quasi Newton methods [54] or via the SMACOF algorithm for Stress functions [52] which is based on iterative majorization. In particular, we seek a configuration of points representing the plants such that the distances between these points matches their similarity as close as possible (6) where δ_ij denotes the dissimilarity between plant i and j, X denotes a point configuration, the weights in a_ij indicate if the value δ_ij is missing (a_ij = 0, imposing no restrictions on the configuration in X) or are known (a_ij = 1), and d_ij(X) is the Euclidean distance used in Eq. (5). Thus, matrix A can indicate whether two matrices belong to the same time slice or plant/sequence but at consecutive days. Additionally, we may fix some of the points to predefined coordinates. That is —(i) since all plants were healthy on the first day, we fix the coordinates of the first measurements, and (ii) since we have observations over time, the x-coordinates are set to 1,2,3, …,t— we find an embedding by adjusting only the unknown points $\widehat{x}\in X$ such that the Stress function is minimized.

To see that this two-step approach is able to sketch dynamics in general, consider the two moving Gaussians shown in Fig. 8 (left). Both Gaussians start in the upper left corner, then, the green one moves in L-shape to the bottom right corner, and the red one moves first to the bottom right corner and then to the bottom left corner. Each Gaussian was evaluated at 16 positions sampling 100 data points. To obtain an embedding, we took the mean coordinates of each Gaussian. The matrix A was defined as described above by preserving the temporal information (see e.g. Fig. 8 right). The resulting embedding is shown in Fig. 8 (middle), where the black dots were fixed on both dimensions whereas for the remaining green and red dots only the x-axis coordinates were fixed (x_i = t_i, where t is the temporal information) and y-coordinates were unknown (learned by MDS). This way, the behavior of both Gaussians is captured while respecting their similarities. To compute a sketch of this embedding, we first connect two elements i and j if d_ij(X) < ε, for a small threshold ε > 0. The resulting metro map is illustrated in Fig. 2 (Step 5) and provides an abstract summary describing the behavior of two moving Gaussians.

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Figure 8. Example for multiple sketches on a synthetic data set.

The data set was generated by two moving Gaussians (green and red) where the brightness encodes time (the darker, the later) (left). The corresponding embedding using MDS (middle), the black dots were fixed on both dimensions whereas for the green and red dots only the x-axis coordinates were fixed (x_i = t_i, where t is the temporal information) and y-coordinates were unknown (learned by MDS). (right) The connection graph defining the weight matrix W between different states of N sequences over time, e.g. the moving Gaussians: w_ij = 1 if the two elements i and j belong to the same time slice (column) or to the same entity (raw or the same color), say moving Gaussian, but at consecutive time points.

https://doi.org/10.1371/journal.pone.0116902.g008