Corn is a very important agricultural product, however, some pests may cause damage to the corn productivity such as

The fall armyworm is one of the pests that cause the greatest damage to the corn (

In general, the pest control is made through the chemical pesticides, which besides polluting the environment also cause the increasing of the resistance of the insect. For this reason control techniques for the insect are researched. For example ^{®}Viptera™ 3111 corn.

Statistical methods based on spatial point processes provide useful tools for points randomly distributed on the plane or in space in many fields of knowledge, such as ecology, biology, epidemiology, seismology, archaeology, astronomy, and geography. However, most of the analyses are made assuming isotropy of the process without an adequate confirmation (

The use of spatial point pattern analysis in agriculture was considered by

In this work we study the spreading of the fall armyworm along an agricultural experimental area at Paraná State, Brazil, comparing methods of anisotropy by means of the spatial point patterns. The main purpose is to examine its spatial distribution along the study area, investigating possible dominant direction of infestation of the fall armyworm, and if the pattern presents characteristics of clustering, regular or random spreading. This knowledge provides the farmer a more adequate management of his crop by considering the characteristics of the dispersion of the insect along the area in consortium with the techniques of precision agriculture. The lack of literature that incorporates spatial point processes and agriculture motivated this work.

The experiment was carried out in an agricultural commercial area of 27 ha located in the city of Cascavel, Paraná state, Brazil (approx. 24.95
^{o} S, 53.57
^{o} W and 650 m asl). Local soil is classified as Rhodic Hapludor (^{o}C. A representation of the location of the experimental area is presented in

For the data sampling, we considered as a possible event each plant of corn attacked with the fall armyworm in the experimental area during the crop year 2015/2016. For every observed infected plant we registered the geographical coordinates as an event of interest, which were obtained using a GEOEXPLORE 3 GPS positioning system receiver in a Universal Transverse Mercator (UTM) coordinate system.

The study area (

A spatial point pattern is a stochastic mechanism that generates a countable set of events
_{i} in the plane (

A spatial point process is considered to be stationary under translations if the distribution is unchanged when the origin of the index set is translated (

Consider

where E[.] is the expected value;

The estimation of the first-order property, or intensity of the point pattern, is usually performed through kernel functions (

where
_{i} (

The second-order properties of a point process can be observed using the covariance between events of two areas. The second-order intensity provides a way of describing the relationship between pairs of points and is defined as

where
_{2} (
_{2} (
_{2} (
_{2} (

The term λ

where
_{ij} is the distance between the

This function has shown its importance as a spatial dependency summary since it allows quantifying spatial dependence between different regions of the process, and is also known as reduced second-order analysis. It has been widely used in two-dimensional point processes because it enables the detection of the point pattern in different distance scales simultaneously and the observed point pattern can be compared to known stochastic process models for different point configurations (

The shape of
^{2}. For a cluster process the points have more neighbors than expected under CSR, and therefore the estimates for
^{2}. On the other hand, for a regular pattern, the estimates of
^{2 }(

A directional counterpart of Ripley

where λ is the intensity, and
_{u}

The function is obtained by counting how many pairs of events in the pattern have both their vector of difference angle less than

^{2}, be a function. The continuous directional wavelet transform for scale

where the over line denotes complex conjugate. There are many different directional wavelets
_{ a,b} (

with regard to identify the behavior of the process in different directions.

_{j }with
_{k},
_{i}_{j,}

Monte Carlo simulation determines the significance of the directional wavelet coefficients. By assuming that the null hypothesis is isotropy, an isotropic random point pattern should be expected to have the same value of the directional scalogram for all possible directions.

An alternative method to study the isotropic behavior of the data set is the histogram of the

To confirm the form of relationship of the point pattern we calculated the quadrat counting test (^{2} statistic based on quadrat counts where the study area
_{1},
_{2}, …,
_{q}) of equal area. The test counts the number of points that fall in each quadrat
_{j} = N(
_{j}) for
_{j }are independent and identically distributed Poisson random variables. The test statistic is given by

where
^{2} is the sample variance. If the distribution is Poisson, the variance is exactly the mean under CSR
^{2 }/
^{2 }/
^{2 }/

For the data analysis we used the software R version 3.3.3 (

The

By applying the kernel method of

A first study of anisotropy is performed by the histogram of the nearest neighbor angle, which shows a descriptive behavior of the isotropy of the point pattern.

With regard to correct the anisotropy observed in the histogram analysis, we rotated the data set by 45 degrees clockwise. The rotated point pattern is represented by

In the following, we calculated the directional
^{o}, the second most dominant direction was 45
^{o}. The other two directions (0
^{o} and 90
^{o}) showed almost the same behavior. Considering the rotated pattern shown in

We then performed the wavelet-based test by using the empirical logarithm of directional scalogram as described in _{i}),

In order to analyze the clustering feature of the data we applied the quadrat counting test, as defined by

The knowledge of the population spreading of

From the estimation of the intensity function of the fall armyworm data (

Our results also showed a significant anisotropy when considering the original data set. Both the

In order to avoid any border effect, we selected a central subset of the rotated pattern and we estimated the

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