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Framework for discovering significant spatial transmission pattern of crime occurrence
In this section, a framework for discovering significant spatial interaction pattern of crime is developed. As illustrated in Fig. 1, the proposed framework comprises the following three steps.
Figure 1

Overview of framework for discovering spatial transmission patterns of crime occurrence.

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The proposed method works on a collection of crime points with spatial and temporal information. Firstly, near repeat crime pairs are identified by specifying the spatio-temporal proximity. All near repeat crime pairs would form a network structure, making it difficult to discover the dominant patterns. Therefore, we simplify the network by overlaying with spatial girds and then aggregating it. Finally, some indicators are defined to measure the spatial interaction strength, and a spatial association pattern mining approach was developed. The whole framework is designed to discover the most probable spatial transmission routes and related high flow regions. Explanation for each step is further illustrated in following sections.
Construction of crime transmission network
This study aims to discover spatial interaction patterns from a collection of discrete points. Each point represents a location where crime incident happens. However, these crime incidents are not totally independent, but related with each other in spatial aspect. The typical phenomena demonstrating such interaction is the near repeat crime. The interaction between near repeat crime pairs can be represented as a “directed link”, and a directed network can well describe the spatial interaction of all crime incidents (denoted as “transmission network”).
The crime transmission network is composed of a node set V and an edge set E, which can be denoted as N = (V, E). Each node in V indicates a crime incident and each edge represents the spatio-temporal relation between two incidents. Because the influence of a crime only existed in a limited spatial and temporal range, spatio-temporal proximity should be defined to identify the near repeat crime. Specifically, given two crime incidents c1 and c2 occurring at timestamps tA and tB, their spatial distance and time difference are denoted as rAB and tAB, respectively. A directed edge eAB is added if the following conditions are satisfied:

$$left{ {begin{array}{*{20}l} {{0} le t_{B} – t_{A} le Delta t} hfill \ {r_{AB} le Delta s} hfill \ end{array} } right.$$
(1)

where Δs and Δt are two parameters to define the spatio-temporal proximity. In this manner, a crime transmission network can be constructed with the dual constraint of spatial and temporal proximity.
Spatial aggregation based on spatial grids
In the crime transmission network, each edge stands for an instance of near repeat crime pairs. As described above, crime transmission network indicates the “spatial interaction”. To explore the spatial interaction, the spatial analysis scale should be determined first. On the other hand, because “near repeat” pairs are judged by the spatio-temporal proximity, a single crime incident may be viewed as “close pair” with many other incidents, all the “close pairs” of crime incidents may form a complex structure (like a complex network), thus making it difficult to extract dominant patterns from such complex structure. As illustrated in Fig. 2, network nodes are usually clustered and network edges are usually intersected in an unregularly way. In situation of lots of nodes and edges, it is difficult to extract dominant spatial interaction patterns from the complex network.
Figure 2

Illustrative example of spatial aggregation of original network.

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To address the above issues, we then overlay the crime transmission network with spatial grids. The advantage of applying spatial grids lies in two aspects. First, the spatial interaction should be explored at a spatial scale. The analysis scale is closely related to spatial grid size. By setting different grid sizes, multiple scales analysis results can be achieved. Second, by overlaying spatial grids with the crime transmission network, each node and edge in the network can be associated with one or several spatial grids, then the crime network can be simplified greatly by spatial aggregation. As an example illustrated in Fig. 2, each circle in sub-figure (a) represents a crime incident, and crime pairs are connected by dashed lines. Obviously, it is not easy to identify the dominant spatial patterns. The complex network can be simplified by overlaying with spatial grids. The close crime pairs can be classified into two categories: “following in same grids” and “crossing different grids”, and those crossing different grids can be used to analyze spatial interaction between different regions. In sub-figure (d), each spatial region is represented as a square, and the numbers beside links represent number of close crime pairs crossing different regions (i.e. the by spatial aggregation). In this manner, the original crime transmission network has been simplified. It should be pointed out that the “spatial aggregation” does not discard any close crime pair. Those falling in a single grid can be used to measure strength of spatial interaction, which will be described in following section.
Discovery of significant spatial interaction patterns
From the above description, we can learn that the aggregated crime network is a directed network. Each node of network represents a spatial region (spatial grid) and edges indicates near repeat pairs crossing different grids. After the aggregated crime network is obtained, the spatial association rule mining technique can be applied to discover the spatial interactions patterns. The spatio-temporal association rule mining approach is a powerful tool for discovering the interdependence relation in both spatial and temporal domains. The existing research has proved that it can not only reveal a spatial dependence structure among various spatial features or spatial objects38,39 but also discover the dynamic interactions among different spatial regions37,40,41. For example, Verhein and Chawla describe spatial interaction patterns between different regions using spatio-temporal association rules37.
In this study, we also try to summarize the spatial interaction pattern by applying spatio-temporal association rules mining. To fulfil that, following definitions are first clarified.
Definition 1
Given two adjacent spatial grids (denoted as GA and GB) and two crime incidents (c1 and c2), if c1 falls in grid GA, c2 falls in GB, and their distance satisfies the spatio-temporal proximity constraint in Eq. (1), then the pair of c1 and c2 is called an instance of flow from GA to GB and denoted as: instance (GA → GB). The total number of instance (GA → GB) is called the out flow number of (GA) and denoted as outNum(GA). Correspondingly, total number of instance (GB → GA) is called the inflow number of (GA) and denoted as inNum(GA). In addition, the total number of close pair which totally falls in grid GA is denoted as statbleNum (GA).
Definition 2
The spatial region GA is termed as a source when out flow number outNum (GA) is higher than random assumption. Conversely, region is termed as sink if inflow number inNum (GA) is higher than random assumption. A thoroughfare is a region which meets both the source and sink requirements. Collectively, sources, sinks and thoroughfares are called high flow regions in which near repeat crime pairs can be frequently observed.
Definition 3
High flow regions and transmission routes together can describe spatial interaction pattern between different regions. For regions GA and GB, if the number of instance (GA → GB) is higher than random assumption, then it is called a significant transmission route from GA → GB, denoted as route (GA → GB), while GA is called antecedent and GB is consequent of the route.
Definition 4
Another two concepts are defined to evaluate the discovered spatial transmission routes. The spatial support of a transmission route r, denoted as Sup(r), is the sum of spatial areas referenced in the antecedent and consequent of the transmission route. The confidence of a transmission route r, denoted as Conf (r), is defined as the ratio of number of instance (GA → GB) to number of instances flowing out and falling in the antecedent grid. They can be represented formally as:

$$Supleft( r right) = arealeft( {G_{A} } right) + arealeft( {G_{B} } right)$$
(2)

$$confleft( r right) = frac{{sum {instance} ;left( {G_{A} to G_{B} } right)}}{{outNumleft( {G_{A} } right) + stableNum(G_{A} )}}$$
(3)

The first three definitions are used to discover the spatial interaction pattern, while the last one can be used to evaluate the discovered results. The definition of spatial support considers spatial semantic of discovered pattern (the size of spatial area) and confidence indicates the transmission possibility between antecedent and consequent regions. Both support and confidence indicators are commonly used in Apriori-like association rule mining approaches42, while these concepts have different meanings in this study.
Based on the above concepts, spatial interaction pattern can be discovered. In spatial association pattern mining process, thresholds for indicators measuring association strength should be determined in advance, e.g. outNum and inNum in this study. However, determination of the thresholds objectively is not easy. Thus, the discovered results are evaluated via the Monte Carlo (MC) testing. In another words, we aim to find out these patterns with their indicators significantly higher than that would be observed by chance. In the current study, MC methods are employed to generate N simulated spatial crime distributions with permutation of temporal information. For example, statistical significance of spatial transmission route r can be calculated as:

$$pleft( r right) = frac{{sum {left( {instance_num^{obs} left( r right) le instance_num^{ith_sim} left( r right)} right)} + 1}}{N + 1}$$
(4)

where (instance_num^{obs} left( r right)) represent the number of instance (r) calculated on real observed data, and (nstance_num^{ith_sim} left( r right)) represent the number calculated on a simulated spatial dataset. Then, given a significant level α (0.05 by default), if the p(r) value is less than the significance level, it can be treated as a significant pattern.
Study area and material description
To evaluate the effectiveness of the proposed approach, we aim to explore the spatial interaction pattern of a robbery in the city of Philadelphia, United States. Located in southeastern Pennsylvania, Philadelphia is an economic and cultural anchor of the greater Delaware Valley, with a population of 1,580,863 (based on 2017 census-estimated results). The crime occurrence in Philadelphia consistently ranks above the national average, which is a major concern for the government. The crime-related data can be freely accessed via the OpenDataPhilly website (https://www.opendataphilly.org/), which provides both crime datasets and basic geographic data. The geographic data include administrative division and road network. The crime incidents are recorded with detailed longitude, latitude and timestamps. In this study, we mainly focus on unarmed robbery during the period of January 1st, 2016, to June 30th, 2016. During this period, the total number of unarmed robberies was 1612. We selected robbery crime as a case study because robbery is frequently observed in the study regions and have a profound effect on the quality of life in urban neighborhood43. This study aims to find out: (1) whether robbery crime exhibits the near repeat phenomena? and (2) what kinds of spatial interaction patterns are embedded in the near repeat phenomena? The study region and distribution of robbery crime are showed in the Fig. 3.
Figure 3

Study region and distribution of robbery incidents.

Full size image More

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Plant material
Z. japonica used in this study was collected from Fangchenggang, Guangxi, China.
DNA extraction and primer design
Leaves of Z. japonica were used as materials to extract genomic DNA from young leaves that had grown well. A MiniBEST Plant Genomic DNA Extraction Kit (TaKaRa, 9768) was used to extract genomic DNA from the leaves of Z. japonica following the manufacturer’s instructions. Based on the full-length cDNA sequence of ZjFes1 obtained by RACE18, three identical and high annealing temperature specific primers (SP Primer) were designed, and four specifically designed degenerate primers, AP1, AP2, AP3 and AP4, were used for thermal asymmetric interlaced PCR (TAIL-PCR). Typically, at least one of these degenerate primers can react with specific primers by TAIL-PCR based on the difference of annealing temperature, and the flanking sequence of known sequence can be obtained by three nested PCR reactions. Because the length obtained in one experiment cannot meet the experimental requirements, we continue to acquire the flanking sequence according to the sequence information obtained in the first genome walking. Four genome walkings were conducted. Twelve SP Primers were designed. DNAMAN software was used to combine the four fragments described above into a consensus sequence by combining overlapping fragments. Specific primers were designed to amplify 2 kb sequences according to the results (Table 1), and the experimental results were verified.
Table 1 PCR primer sequences.
Full size table

Cloning and construction of the plant expression vector and sequence analysis of promoter
The full-length promoter sequence was amplified using high fidelity polymerase 2 × TransStart FastPfu PCR SuperMix (-dye) (TRANSGEN BIOTECH, AS221-01) using the DNA of Z. japonica as a template following the manufacturer’s instructions. The PCR products were detected using 1% gel electrophoresis. The results showed that the size of the bands was the same as that of the target fragments, and the PCR products were recovered using a MiniBEST Agarose Gel DNA Extraction Kit Ver. 4.0 (TaKaRa, 9762). The pCXGUS-P plasmid is a vector designed to detect the activity of plant promoters. The promoter activity is detected by the dyeing intensity of GUS. We used XcmI to digest the empty vector to obtain T vector. After recovery, the product was recombined with T vector, and then the recombinant vector was transformed into E. coli DH5α Competent Cells (TaKaRa, 9057) following the manufacturer’s instructions. The positive samples identified by PCR were verified by sequencing at the Guangzhou Sequencing Department of Invitrogen. The sequencing results were compared using DNAMAN software. The plasmid was extracted from the correct bacterial solution and designated pZjFes1::GUS. The sequence analysis of cis-acting elements that could possibly be found in the promoter was performed using the plant-CARE online prediction database (plant cis-acting regulatory element, https://bioinformatics.psb.ugent.be/webtools/plantcare/html/)20.
Agrobacterium-mediated genetic transformation of pZjFes1::GUS into Arabidopsis thaliana
The fusion vector pZjFes1::GUS was transformed into Agrobacterium Rhizobium strain GV3101 chemically competent cells (Biomed, BC304) using the freeze–thaw method following the manufacturer’s instructions. Transgenic plants of A. thaliana were obtained by floral dipping. Plants in nutrient soil were cultured to form a large number of immature flower clusters. The monoclone of A. tumefaciens GV3101 was selected and inoculated in liquid LB medium containing kanamycin and rifampicin (50 µg/mL). The monoclone was cultured overnight at 200 rpm and 28 °C. A volume of 2 mL bacterial solution was transferred to a 500 mL flask culture (containing 200 mL liquid LB with 50 µg/mL kanamycin and rifampicin added) and was cultured overnight at 200 rpm and 28 °C. The next day, the OD600 of Agrobacterium solution was 1.8–2.0. The solution was centrifuged at 5000 rpm for 15 min at 4 °C. The supernatant was discarded, and the precipitate of A. tumefaciens was resuspended in 1/2 volume (100 mL) osmotic medium (1/2 Murashige-Skoog, 5% sucrose, 0.5 g/L MES, 10 µg/mL 6-BA, 200 µl/L Silwet L-77, and 150 µM acetyleugenone, pH 5.7), resulting in an OD600 of approximately 1.6. The bacterial solution was adsorbed on the transformed plants using the floral dip method (5 min), wrapped with film to keep it fresh, and cultured overnight, followed by the removal of the film. The plants were cultured until the seeds were ripe, and they were harvested. A mixed disinfectant consisting of 70% ethanol and 30% bleaching water was used to soak the seeds for 3 min, suspend them continuously, and wash them three times with anhydrous ethanol. The dried seeds were evenly dispersed on the surface of solid screening medium containing hygromycin (25 µg/mL). After stratification at 4 °C for 2 days, the seeds were germinated in a light incubator and cultured for 2 weeks at 21 °C and 16 h light/8 h darkness. The development of seedlings and length of roots were used to determine whether they were transformants.
GUS dyeing and activity analysis
The expression of GUS reporter gene in Arabidopsis tissues was determined using a GUS staining kit (Solarbio, G3060) following the manufacturer’s instructions. The seedlings, leaves, flowers and siliques to be dyed were immersed in GUS dye solution and incubated overnight at 37 °C. The chlorophyll was removed with 75% ethanol until the background color disappeared completely. The results were documented by photography using a Canon 60d camera.
The material needed to determine Gus enzyme activity was frozen rapidly with liquid nitrogen, and then ground into powder by ball mill. The extraction buffer solution (50 mM NaH2PO4 (pH 7.0), 10 mM EDTA, 0.1% Triton X-100, 0.1 (w / v) sodium dodecyl sulfonate, 10 mM β-mercaptoethanol) were added to extract protein. After centrifugation at 4 °C, 12,000 r/min for 10 min, the supernatant was taken as protein extract. The protein concentration was determined by Bradford method. 4-MUG, the substrate of GUS reaction, was added and reacted at 37 °C for 30 min. Fluorescence measurement was carried out under the condition of 365 nm excitation light and 455 nm emission light. Three independent biological repeats were conducted. Finally, the GUS enzyme activity value was calculated according to the relative change of product in unit time. More

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Human gut microbiome data and preprocessing
The publicly available data that we re-analyzed here were generated by David et al.32 accessible on the European Nucleotide Archive (ENA) under the accession number ERP006059, and by Hsiao et al.31 on the NCBI Short Read Archive (SRA) under the accession number PRJEB6358. The downloaded reads were trimmed with V-xtractor version 2.146 a HMM scan based method of isolating variable regions from 16S rRNA sequences) to ensure the amplicon sequences could be aligned across consistent fractions of the 16S rRNA variable regions. Trimmed reads were then clustered into OTUs using usearch v9.2.6447 with a minimum cluster size of two. Representative sequences from each OTU were classified using mothur v1.36.148 and the RDP reference 16S rRNA sequences v1649.
Prochlorococcus data
Data from Malstrom et al.33 was obtained from the Biological and Chemical Oceanography Data Management Office (https://www.bco-dmo.org), accession number 3381.
Mapper
Conceptually, the Mapper algorithm accepts as input a matrix of distances or dissimilarities between data, and aims to represent the shape of the distribution of data points in high-dimensional phase space as an undirected graph. In this graph, vertices represent neighborhoods of phase space spanned by subsets of adjacent data points, and edges represent connectivity between neighborhoods. In brief, it does this by dividing the data into overlapping subsets that are similar according to the output of at least one filter function that assigns a scalar value to each data point, performing local clustering on each subset, and representing the result as an undirected graph, where each vertex represents a local cluster of data points, and edges between vertices represent at least one shared data point between clusters.
Distance matrix
We interpreted microbiome relative abundances to be probability distributions, and thus used the square root of the Jensen-Shannon divergence as a metric50. However, it is important to note that any other metric can be used in place of the Jensen-Shannon distance, such as the Aitchison distance51, calculated from centered10 or isometric12 log-transformed relative abundances.
Filter functions and binning
For the filter functions used by Mapper to bin data points, we performed principal coordinate analysis (PCoA, also known as classical multidimensional scaling) in two dimensions on the pairwise distance matrix, and used the ranked values of principal coordinates (PCo) 1 and 2 as the first and second filter values for Mapper, following Rizvi et al.28. PCo ranks are an appropriate filter for our purposes, as it assigns similar filter values to points that are relatively close together in the original phase space. We wish to note that while PCoA leads to loss of information, the following local clustering step is performed using subsets of distances from the original distance matrix, and is thus not affected. The data points were then binned by overlapping intervals of the two ranked principal coordinates. For hyperparameters specifying these bins and their overlaps, see Table 1.
Table 1 Hyperparameters used to generate the Mapper representation of each data set.
Full size table

Local clustering
The algorithm first performs hierarchical clustering from all pairwise distances between data points within a bin of filter values. Then, it creates a histogram of branch lengths using a predefined number of bins, and uses the first empty bin in the histogram as a cutoff value, separating the hierarchical tree into single-linkage clusters. The algorithm thus finds a separation of length scales within each neighborhood of phase space represented by a bin of the filter values. We used the default number of histogram bins, 10, for each data set (Table 1).
Creating the undirected Mapper graph
The final output is produced by representing each local cluster of data points as a vertex, and drawing an edge between each pair of vertices that share at least one data point. When plotting, the size of each vertex represents the number of data points therein. Layout and visualization of the Mapper graph may be performed with any graph layout algorithm; we used the Fruchterman-Reingold force-directed layout algorithm52. It is important to note that the visualized shape of the Mapper graph depends on the algorithm used, and may not be deterministic. When performing a Mapper analysis, one should rely on the connectivity of the graph rather than the overall shape.
Selection of hyperparameters
The Mapper algorithm is relatively new, and there are currently no standard protocols to optimize the values of the hyperparameters. For our purposes, it was important that the algorithm achieved a sufficiently high resolution in partitioning data, but also adequately represented connections between regions of phase space. We thus used the following heuristic to set the number of intervals and percent overlap for each data set.
1.
The largest vertex in the resultant Mapper graph should represent no more than ≈10% of the total number of data points in the set;

2.
the number of connected components representing only one data point should be minimized.

We acknowledge that a heuristic determination of appropriate hyperparameter values leaves much to be desired; as such, we recommend future in-depth theoretical explorations of how the Mapper output depends on the choice of hyperparameters.
Density estimation
We estimated the inverse density for each vertex by calculating the k-nearest neighbors (kNN) distance53 for each constituent data point i.
We first define the k-neighborhood N(k)i of a point i, to be the set of k nearest neighbors of i, choosing k equal to 10% of the number of samples in each data set, rounded to the nearest integer. Then the kNN distance of point i is defined as:

$${rm{kNN}}(i,k)=frac{{sum }_{jin N{(k)}_{i}}{d}_{ij}}{k}$$
(1)

where dij is the distance between points i and j.
For a vertex V representing n points, we define its inverse density as

$${D}_{{rm{inv}}}(V)=frac{{sum }_{iin V}{rm{kNN}}(i,k)}{{n}^{2}}$$
(2)

The n2 term in the denominator compensates for the differing sizes of vertices. Finally, we invert the inverse density to obtain the estimated density:

$$D(V)=frac{1}{{D}_{{rm{inv}}}}$$
(3)

State assignment
We then defined states as topological features of the density surrounding local maxima of D. We designated each vertex with higher D than its neighbors to be a local maximum of the potential. Connected vertices tied for maximum D were each assigned to be a local maximum. To approximate a gradient, we converted the undirected Mapper graph to a directed graph, with each edge pointing from the vertex with lower D to the one with higher D. For each non-maximum vertex, we found the graph distance dg to each local maximum constrained by edge direction. We defined the state Bx of a maximum Vx as the set of vertices V with uniquely shortest graph distance to Vx:

$$Vin {B}_{x},{rm{if}},{d}_{g}(V,{V}_{x}), More

NATURE PODCAST
14 October 2020

A high pressure experiment reveals the world’s first room-temperature superconductor, and a method to target ecosystem restoration.

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In this episode:
00:44 Room-temperature superconductivity
For decades, scientists have been searching for a material that superconducts at room temperature. This week, researchers show a material that appears to do so, but only under pressures close to those at the centre of the planet. Research Article: Snider et al.; News: First room-temperature superconductor puzzles physicists
08:26 Coronapod
The Coronapod team revisit mask-use. Does public use really control the virus? And how much evidence is enough to turn the tide on this ongoing debate? News Feature: Face masks: what the data say
19:37 Research Highlights
A new method provides 3D printed materials with some flexibility, and why an honest post to Facebook may do you some good. Research Highlight: A promising 3D-printing method gets flexible; Research Highlight: Why Facebook users might want to show their true colours
22:11 The best way to restore ecosystems
Restoring degraded or human-utilised landscapes could help fight climate change and protect biodiversity. However, there are multiple costs and benefits that need to be balanced. Researchers hope a newly developed algorithm will help harmonise these factors and show the best locations to target restoration. Research Article: Strassburg et al.; News and Views: Prioritizing where to restore Earth’s ecosystems
28:40 Briefing Chat
We discuss some highlights from the Nature Briefing. This time, a 44 year speed record for solving a maths problem is beaten… just, and an ancient set of tracks show a mysterious journey. Quanta: Computer Scientists Break Traveling Salesperson Record; The Conversation: Fossil footprints: the fascinating story behind the longest known prehistoric journey
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