Violent movement of crop stems can lead to failure under high winds. Known as lodging, this phenomenon is particularly detrimental to cool-season cereals such as oat, barley, and wheat; contributing to yield and economic losses. Phenotyping the movement of cereal crops in real-time could aid in the breeding and selecting of lodging resistant cereals. Since no methods exist to quantify dynamic, real time plant responses in an agricultural setting, we devised a video analysis protocol to quantify mean frequency and amplitude of plant movement for a 360° field of view camera system.We present both the image analysis method for identifying predefined regions of a 2D field design as they appear on 360° field of view video, as well as a signal processing pipeline to quantify movement from time varying color signals from plot canopies within these predefined field regions. We detected significant differences in the natural frequency and amplitude of plant movement from video of 16 cereal cultivars planted in a randomized complete block design on five different windy days. Natural frequencies quantified by this method averaged 1.37 Hz, while over 2.5-fold differences in amplitude within similar frequency ranges were detected across the 16 cereal cultivars.This method is sensitive enough to systematically differentiate small frequency and amplitude differences in cultivar movement, and shows promise for investigating the physiological basis for differences in cereal movement and lodging resistance. The relative accuracy of the plot demarcation protocol suggests it could be used for other high-throughput phenotyping applications that require both high image resolution and a large field of view.The online version of this article (10.1186/s13007-019-0437-5) contains supplementary material, which is available to authorized users. Prior to the failure (lodging) of plant stems under wind stress, an entire plant experiences movement. This movement occurs in plants of all sizes and taxa; it is dependent on plant structural traits and wind conditions, which together govern the failure velocity at which a plant stem will fail in the wind , 2. Incrωn) at which a plant stem moves is governed by the ratio of stem stiffness to mass per unit length (mp) of the stem [Plant movement has been studied theoretically from the scale of individual plants to entire canopies , 2. At athe stem :1\documemp constant, will increase the natural frequency of the plant oscillations (movement). If the intermittent gusts across a cereal canopy encounter plants whose ωn matches that of the gusts, the movement of the plant will not dissipate and will likely be amplified. If the vibration is sustained long enough to cross a failure threshold, the plants will lodge. However, additional physical parameters and the wind conditions all interact in reality to make plant movement a function of various physical properties at any given time point. These can include the structural (ζs) and aerodynamic (ζa) damping coefficients of the plant, stem acceleration (θ), and the wind moment . These values can be used to estimate the ωn at a given moment by the equation [From Eq. equation :2\documeAn increase in plant natural frequency should increase the wind velocity required to induce failure and result in greater lodging resistance, because stem oscillations would dissipate after being excited by gusts that occur at a lesser frequency . RealistEmpirical studies on plant-wind interaction employ a variety of technologies to quantify movement in plant systems primarily developing or validating theoretical models on a small scale. A large portion of experimental research on plant movement has focused on understanding tree sway to minimize timber losses from storms . The earωn) and the amount of movement (amplitude) at that frequency as plant phenotypes could enable genetic studies and breeding based on a crops’s real-time response to field wind stress. Implementing a method to quantify phenotypes of plant movement at high throughput in experimental designs such as the randomized complete block (RCB) pose another use for relating physiology, movement, and lodging: to better optimize physical trait combinations that are desirable in theory but in practice will require high statistical power to validate under variable field environments. While no systematic approaches exist for quantifying plant movement for purposes of improving crop lodging resistance from video, many studies have devised methods to quantify crop lodging from image and spectral data. Recent lodging phenotyping efforts include analyzing lodging in the visible spectrum from unmanned aerial vehicle (UAV) images in wheat, rice, and maize [Treating plant natural frequency  of movement and amplitude of single, homogenous rows of cereal cultivars without mechanical interference. Our method accomplishes this primary objective by (1) identifying plots from a predefined field design to systematize data analysis, (2) employing signal processing techniques to analyze plant movement as changes in the RGB color space, and (3) demonstrating the method’s capabilities by quantifying plant movement in the field for 16 cereal varieties planted in a randomized complete block (RCB) design.In this paper, we develop novel video analyses and demonstrate that they can be used to quantify plant movement as a phenotype in the field from a stationary, 360° field of view camera system for purposes of understanding cereal crop lodging resistance under field wind stress. The context of this method focuses on individual plant rows as opposed to canopy dynamics of uniform fields, with the idea that this method could be generalized to an applied breeding scheme. The method presented quantifies the natural frequency   Figs. c and nots) Figs. d. Regions) Figs. d. Along Θoff) calculation and cropping with the automatic Θoff calculation and cropping functions generated by the plotfinder script. The small Θoff differences (averaging between 0.66° and 2.42°) between automatic and manual estimation propagated larger transformation errors for outer replicates compared to the inner replicates adjacent to the camera track , high amplitude color changes detectable due to changes in sunlight exposure from passing clouds during a single video, which substantially altered the red color values of the plots relative to the red panel. This is evidenced by the shared low frequency, high amplitude wave movements across all plots shown  for the bandpassed peak distribution  and high ωn, low amplitude movement (‘Linkert’ Wheat).With the plotfinder method enabling the systematic cropping and color quantification of 929 cmωn) obtained from these distributions, modeled as a response in Eq. ωn  within a 0.2 Hz frequency interval (bin) between 1.1 and 1.3 Hz. The amplitude within this frequency bin encompassed many mean natural frequencies of the cultivars in this study varied significantly among cultivars and planting dates across data from the five video dates. When modeled as a response in Eq. nt Table . Near-unnt Table . Finallynt Table .Table 8AMeasurements of cereal crop lodging, while increasingly possible at high throughput through UAVs and image analysis, cannot capture the dynamic response that a cereal crop experiences during high winds or storms. Treating movement as a response variable could improve a breeder’s understanding of cereal wind resistance by relating cereal physiology to repeated measures of plant movement under different wind conditions and final lodging outcomes. There are obvious challenges to phenotyping plant movement on a large scale agricultural experiment using mechanical methods. Manual vibration and timing tests would prove laborious on the scale of common field experimental designs, while mechanical sensors on tens or hundreds of replicates would present data synchronization challenges for large field experiments. Thus a video analysis method using a colorimetric, signal processing approach to quantify movement offers a non-invasive measure of this dynamic phenotype that is better scaled to agricultural experiments. The desired capabilities of a scalable phenotyping method for plant movement are twofold: the method must be systematic enough to eliminate sources of human or equipment measurement errors under the intermittent winds that excite plant movement, while providing enough data of replicated varieties to elucidate trends in movement that require increased statistical power to discern in the field. The pairing of novel hemispherical image analysis of agricultural plots with the analysis of plant canopy movement in a systematic way enables differentiation of crop movement at the cultivar level. Though the analysis of plant movement from video data predates digital video, we consider our method generalizable to available camera technology and common field experimental designs for measuring movement as a plant phenotype for breeding or other biological purposes .Demarcating individual crop rows from 360 video is novel with respect to high throughput image analysis of agricultural research plots, which is often performed using UAV captured images. Demarcating plots from a 360 image negates the need for stitching images captured by UAV mounted cameras, though at the cost of a smaller field coverage on a per image basis compared to multiple images stitched into one from a UAV . In meth2 in our case), even if the magnitude of the absolute amount of movement is less due to lighting conditions in a given video. Thus, a researcher employing this method over an entire growing season to investigate biological questions would be advised to employ a repeated measures ANOVA account for variation due to windspeed and lighting conditions among sampling dates in a final model of plant movement comparing varieties or crops. The RCB field design we used allowed us to statistically test the results from the video and signal processing pipelines by comparing relevant experimental effects and their interactions at seperate video dates. The presence of a significant effect of replicate position (inner vs outer) for natural frequency indicates that blocking the experiment with camera distance, as was done here, is critical in minimizing the effect of distance from the camera on mean natural frequency. Such effects on the variation in mean natural frequency are likely manifestations of errors in angle calculation identified in lower correlations between automatic and hand plot demarcation, indicating that there is more error in quantifying color changes at greater distance from the camera. This primarily stems from errors in calculating Θoff, resulting in a shift of the 929 cm2 region to another plot and capturing a color signal that is not representative of the intended plot canopy. While the plots captured in the hemispherical still frames occur at varying latitudes (viewing angles) in the images that are in theory affected by different lighting conditions, the predominant lack of a significant position effect on plant movement where the total amount of color change over time is directly quantified suggests that this issue is not as critical as improper angle calculation for accurate measurement of plant movement. That said, care should be taken to account for variation in lighting conditions at different viewing angles within hemispherical images for other applications if specific indices  are desired.The plot demarcation and plant movement quantification methods are well integrated via the use of the panels that enable normalization of color values in RGB video frames and identification of plots within the RCB field design. From this normalization and subsequent bandpass filtering of raw waveforms, we were able to compare parameters of plant movement across videos where different lighting conditions were present. Bandpass filtering also allowed for the removal of higher frequency (> 5 Hz) color changes due to camera vibrations induced at sufficiently high windspeeds on the camera system, leaving a signal of plant movement for analysis. The ability to treat parameters of plant movement such as natural frequency and amplitude as phenotypes that can be analyzed in common experimental designs such as the RCB represents a rigorous analytical framework for further testing models of plant wind interactions in the field. In the test of the plot demarcation and signal processing pipelines, we analyzed the RCB separately at five different dates. Lighting and wind conditions are expected to vary over different dates, and changes in lighting conditions (due to sun elevation or cloud cover) at the time of imaging will affect the absolute amplitude measurements as they are quantified based on the area under the bandpassed waveforms. However, a variety prone to larger amplitude movements will still be detected relative to those with lower amplitude movements when measured over a constant area of the plant canopy  values estimated by our method fall within ωn values either theoretically or empirically determined for cereal crops. Theoretical natural frequencies of a generic cereal crop range between 1.10 and 5.31 Hz, based on dimensionless parameters while empirical data on in-field wheat stems (cultivar ‘Mercia’) using a manual free vibration resulted in an average natural frequency of 0.91 Hz [ωn values calculated for the cultivars in our study were higher on average (1.37 Hz) than those reported for different cultivars of the same cereal crops, barley cultivars in our study generally had lower resonant frequencies than the wheat cultivars, a trend also apparent in [ωn values for cultivars and planting dates, which is to be expected as the wind moment increases thereby reducing the natural frequency by the square root of the increase in wind moment, all else equal [The cultivar mean natural frequency  are not suitable. When applied over a growing season, this method should amend itself to discerning trends in plant movement over time. These could include the physiological relationships between physiology, movement, and lodging, or even thigmotropic responses of plant tissues to plant movement . In the We designed and constructed an automatable camera track system to collect hemispherical video of single row research plots . BrieflyThe 360fly 4k hemispherical video camera (referred to as 360 camera) recorded plant movement in the field. The 360 camera contained an 8 Elements Glass Ultra Fisheye Lens with an aperture of F2.5, effective focal length of 0.88 mm, and a minimum focus distance of 30 cm. The horizontal field of view was 360°, while the vertical field of view was 240°. Brightness was set to full brightness, while the aperture and contrast settings were set in the middle values for each. Videos were recorded at 24 frames per second with a per frame image size of 2880 × 2880 pixels. The 360fly ios app (V 2.0.0) was used to maintain consistent settings and initiate camera recordings while the camera was mounted on the track system, and the 360fly Director (V 0.10.4.0) software for Microsoft Windows 10 was used to download the videos from the camera and export them into.mp4 format.The camera system imaged a randomized complete block (RCB) design. The design contained four cultivars of four different cereal grains Table  for a toCultivar_Crop_PlantingDateRep (i.e. Gopher_oat_4A).Prior to identifying plot regions within the 360 video and subsequent signal processing, we defined the field coordinate system and plot naming conventions. This allowed us to generalize the matlab scripts in the pipeline across different parts of the field RCB as indicated by the unique panels. The lodging RCB was numbered in a serpentine pattern beginning in the northeastern corner of the field, moving to the west, and then back to the east in the next replicate at each panel  so that the date information could be obtained throughout the video analysis and signal processing pipelines. We used the following file naming convention for videos: FLYmm_dd_yy_pn.mp4, with n being the panel number imaged in that video.This portion of the methods section details the demarcation of agricultural plots from images taken using the 360 camera along with researcher defined field coordinates and plot names. The text below explains the processes coded in Additional file The panel, identified by the red square in the northwest corner of each panel, provides a reference upon which all subsequent transformations of the field design are based. Masking the red square in the northwest corner of the panel provides a reference point for all measurements to conduct this transformation. The red square is masked by applying a threshold  to the 360 image in RGB color space . Depending on ambient lighting conditions, these threshold values may need to be changed for individual videos to completely identify the red square. To ensure that the masked object using the threshold is the red square on the panel, square structuring elements of 4 pixels in size helped define the right angle edges of the red square, while all masked objects whose areas were < 5000 pixels (the area of the true red square is approximately 6000 pixels in the center of the image) were then unmasked, leaving only a masked image of the red square. While not experienced in this experiment, sunlight conditions that create an excessively saturated image might necessitate altering the minimum red square size or ensuring that the red square is adjacent to black panels instead of white to better define the location of the red square. With the red square masked from the original image, the left-most corner of the square serves as the reference point for all subsequent measurement  is calculated using the relationshipEstimating the direction and angle (in degrees) of rotation of the panel within the image allows the field design to be rotated to the correct orientation in the image. The column index of the leftmost corner of the red square is used to determine which direction the panel is facing in relative terms within the 360 image, as the camera axes might not be in line with the field axes when the camera is installed on the track system. The direction of rotation is determined by obtaining the RGB color values of the 40 pixels above and below the leftmost corner column Fig. b. A meanWhile in Fig. With the points A,B,C representing the same triangulation (in row indices for AB and column indices for BC) of the panel offset angle in the image. The leftmost corner of the red square will not always represent the same point on the physical panel Fig. c, d. ThuThe researcher assigns the panel in frame a number that corresponds to those in the field coordinates or plot names files so that the correct versions of these files are used in subsequent transformations. The panel is identified based on user input at line 14 of the plotfinder script  coordinates are extracted from the file into an n × 2 matrix, which is then transposed to a 2 × n matrix. All points in the 2 × n matrix are rotated clockwise around the origin by multiplying with a 2 × 2 rotation matrix. In the case of a northeastern rotation of the panel  for a given coordinate n is calculated by taking the inverse sine of the ratio of plant height to the distance from the camera height to the nth point on the field:Following rotation of the field coordinates in relation to the camera axes, the rotated field coordinates are transformed from the Cartesian  to spherical  coordinates. This is accomplished by two separate triangulations in three dimensional space Fig. . The latφn ranging from 0° to 90°. The longitude angle (Θn) for a given coordinate is calculated by taking the inverse 4-quadrant tangent of the ratio of y to x distance to the rotated field coordinate:With the value of Θn ranging from − 180° to 180°. The nth plot coordinate is then represented in a new n × 2 matrix containing the latitude and longitude values for each coordinate in the rotated hemispherical image. These values are stored in plot order corresponding to that given in the plot coordinates and plot names files.With the value of Dn) in an image is thus a quadratic function of a point’s latitude Once plot coordinates are transformed from Cartesian to spherical coordinates, the latitude for a given point can then be represented in terms of the distance from the center of the image as a percentage of total pixels. This assumes that the image was taken directly above the panel so that the image center (pole) and the panel center are in alignment. The relationship between degrees latitude and number of pixels is non-linear: as latitudinal coordinates approach the equator (0°), fewer pixels are required to represent the change in latitude. Given the 240° field of view captured by the 360Fly 4K camera, the edge of the hemisphere does not represent the equator. Instead, the radius from the image center to the equator is assumed to be located at the 1440th row for frames taken at a 2880 × 2880 pixel resolution. Due to the field of view and other unknowns regarding the lens optics, we empirically determined the relationship between degrees latitude and the percentage of pixels in the image along the radius at the 1440th row. The percent distance  along the x and y axes within the image. The percentage of the image in terms of rows (Rn) is calculated asAs the Cn) is calculated asAnd the percentage of the image in terms of columns , they represent the percent distance in rows and columns to a given point from the field design on the hemispherical image.The resulting Rn and Cn values calculated above are expressed in percent distance from the image center without the correct sign indicating where the points fall with respect to the image center. Longitude angles for the nth point are used to assign the appropriate sign to the Rn and Cn values. For rows, if the longitude angle for the nth point is between 0 and 180°, Rn will be negative while if the longitude angle is between 0 and − 180°, Rn will be positive. For columns, if the longitude angle for the nth point is between 90 and − 90°, Cn will be positive while if the longitude angle is between 90 and 180° or − 180 and − 90°, Cn will be negative. The correction matrix will have dimensions of n × 2, with the first column containing the row distance as a percentage of image size, the second column containing the column distance as a percentage of image size, and n rows for the number of points in the field design.Though the row and column indices are expressed as percentages of the radius of the hemispherical image, the calculations in section 1.5 do not assign the correct sign to these indices to indicate their location in regards to the image center. The The correction matrix indicating the correct sign for the row, column index percentages is applied, with the index percentages converted to actual pixel indices based on the image size, and then added to the image center row, column index to obtain the final transformation of the field coordinates to their location in the hemispherical image. The correction matrix for each coordinate is multiplied element-wise by the matrix of row and column distance percentagesRn and Cn values whose signs reflect the correct orientation with respect to the image center. The Rn and Cn distance percentages are then multiplied element-wise by the image size (2880 pixels) express these values as image pixels. Since these values now represent the number of pixels to the nth point along the image y and x axes from the center of the image, half the value of the image size (1440 pixels) is added to every element within this matrix. The resulting array contains the indexed pixel value for the row and column in the image where a plot coordinate is located.This resulting matrix contains the nth row of each of the four arrays containing row and column pixels are used to demarcate a polygon cropping region on the original image with the plot name appended to the filename according to the nth row of the plot names file. The script will produce the same number of images as the number of rows in the input plot coordinates and plot names files. Additionally, a cropping function is generated by the script that can demarcate the plot region across frames in a video. Examples of the resulting cropped plots in different rotation conditions and region of interest sizes are shown  represents a unique plot in the field design that corresponds with the order in the original plot coordinates and plot names files. The own Fig. a–d. The Θoff and obtain the rotated coordinates for each corner of the demarcated 929 cm2 area. We then manually calculated the angle of the red square by measuring the distance manually using the line tool in Powerpoint (Office 365), which gives the height and width that the line travels in the image and used those values to calculate Θoff manually  and time location (in frames) of each local maxima of the waveform color signal with at least a 0.5% red change prominence. Next, we calculated the distance (in frames) between peaks, skipping the first peak. A visualization of the peak identification process is presented in Fig. This frequency and area data was run through an R script  of the peak distribution for each row in a video date, as well as the total (absolute) area between peaks in the 1.1–1.3 Hz frequency band. Each response variable was modeled using the following linear model:Plant row movement was modeled using the components of a randomized complete block design  nested within the ith cultivar, ß4 the cultivar by planting date interaction effect, and ß5 the cultivar by jth position nested within the ith planting date interaction effect. The ANOVA for estimating the significance of each effect was conducted using the expected mean squares and F values presented in the table below (Table With ow Table .Table 11The F values were tested according to the relevant degrees of freedom for each mean square component in the ratio using the probability density function for an F distribution in R. For all of these linear effects, an LSD test with a false discovery rate correction in the R package agricolae was used to quantify the mean value for each level within the response variable and assess significant differences in either mean frequency or total area at alpha = 0.05 . The ANOAdditional file 1. Plotfinder_published. Matlab script to demarcate user-defined regions of a field from hemispherical videoAdditional file 2. FLY07_12_17_p2. MP4 video file from 360Fly 4k camera, taken 12 Jul 2017 at planting date 2. Used as sample video for published scriptsAdditional file 3. Fieldmap_readme. Field design at the panel in planting date 2. Corresponds to analyzable plots in Additional file Additional file 4. FieldPlotnames7. Field plot names file for 25 Apr 2017 planting dateAdditional file 5. FieldPlotnames5. Field plot names file for 5 May 2017 planting dateAdditional file 6. FieldPlotnames3. Field plot names file for 15 May 2017 planting dateAdditional file 7. FieldPlotnames1. Field plot names file for 25 May 2017 planting dateAdditional file 8. FieldCoordinates7. Field Coordinates file for 25 Apr 2017 planting dateAdditional file 9. FieldCoordinates5. Field Coordinates file for 5 May 2017 planting dateAdditional file 10. FieldCoordinates3. Field Coordinates file for 15 May 2017 planting dateAdditional file 11. FieldCoordinates1. Field Coordinates file for 25 May 2017 planting dateAdditional file 12. FLY07_12_17_p2. Matlab script to obtain normalized waveforms from sample video (Additional file Additional file 13. BandpassParse_p2. Matlab script to filter and parse waveforms from Additional file Additional file 14. Stationary_organization_published. R script to organize the frequency, amplitude, and plot data from Additional file Additional file 15. Stationary_analysis_published. R script to analyze the data from Additional file


An established statistical mechanical theory of amorphous polymer deformation has been incorporated as a plastic mechanism into a constitutive model and applied to a range of polymer mechanical deformations. The temperature and rate dependence of the tensile yield of PVC, as reported in early studies, has been modeled to high levels of accuracy. Tensile experiments on PET reported here are analyzed similarly and good accuracy is also achieved. The frequently observed increase in the gradient of the plot of yield stress against logarithm of strain rate is an inherent feature of the constitutive model. The form of temperature dependence of the yield that is predicted by the model is found to give an accurate representation. The constitutive model is developed in two-dimensional form and implemented as a user-defined subroutine in the finite element package ABAQUS. This analysis is applied to the tensile experiments on PET, in some of which strain is localized in the form of shear bands and necks. These deformations are modeled with partial success, though adiabatic heating of the instability causes inaccuracies for this isothermal implementation of the model. The plastic mechanism has advantages over the Eyring process, is equally tractable, and presents no particular difficulties in implementation with finite elements. Solid polymers are mechanically nonlinear, time-dependent and capable of attaining large deformations, especially in processing regimes. Progress has been made in recent years in understanding their stress-strain behavior, and in implementing complex constitutive equations that reflect experimental observations. A viscoplastic approach is customarily adopted. At the heart of such a model there is a plastic mechanism, in which the stress must be nonlinearly dependent on strain rate to give a realistic representation for polymers.The plastic mechanism most frequently used in polymer modeling is the Eyring process  in E in E5\doentclass1pt{minima Spencer , a constHere we outline a two-dimensional plane stress approach. This was implemented within finite element analyses using the package ABAQUS, with the material model incorporated via a ‘UMAT’ user-defined subroutine. The kinematics are similar to those used previously for high temperature stretching of polypropylene  on 5 mg specimens of polymer. A TA Instruments Discovery DSC was used, programmed with a mean temperature ramp of 3 K/min from 273 K (0 °C) to 573 K (300 °C). In order to decouple the reversible heat flow measurements from non-reversible thermal events a modulation amplitude of After drying the granules in a vacuum oven overnight, sheets nominally 0.9 mm thick were compression molded between platens heated to 553 K (280 °C) at a pressure of up to 4 MPa. This method of manufacture is established as one that produces no significant skin-core effects (Kiraly and Ronkay Tensile specimens of Tensile testing was carried out using an Instron 5568 testing machine combined with an environmental chamber. Images of the specimens were taken through the window of the environmental chamber using a PixeLINK model PL-D722MU-T video camera operating at capture rates up to 75 frames/s. After testing, the images were analyzed using the software Fiji, which is based on ImageJ (Schindelin et al. The adiabatic heating of polymers, which results from the conversion of mechanical energy into heat when they are extended, is a well-established phenomenon (Liu and Harrison In all cases the deformations in the specimen gauge lengths were initially uniform. Depending on the temperature and strain rate, uniform stretching continued until the end of the test, or was followed by shear banding or necking. Strains were measured using image analysis, and this enabled the rates of strain for the uniform stages of deformation to be calculated. These were compared with the nominal rates based on machine speed and on the assumption that all deformation occurs in a specimen gauge length of 30 mm. The measured rates were found to be significantly lower at higher temperatures and low speeds. This is a result of deformation outside the gauge length. At 323 K, the measured rates of strain do not decrease significantly with speed. At higher temperatures, the discrepancy broadly increases with temperature and decreases with testing speed, as shown in Fig. At 323 K, instabilities were observed at all testing speeds. In Fig. At 333 K, the behavior is similar to that at 323 K except that the transition from necking to banding occurs between 18 and 54 mm/min, as shown in Figs. At 341 K, strains are uniform at the lowest two speeds 0.54 and 1.8 mm/min (nominal strain rates Figure mentclasspt{minimaYield stresses can be readily identified. At all temperatures, we observed maxima in engineering stress that we equate with yield stresses. Some examples of stress-strain curves at an intermediate speed are shown in Fig. Using the same techniques and software as for the PVC yield data in Sect. entclass1pt{minimaentclass1pt{minimaAt the temperatures 341 K and 346 K, there is a clear increase in gradient with strain rate, and the model fits reflect this. At the lower temperatures the increasing slope is less obvious, corresponding to the higher values of We analyze the temperature dependence of the parameter entclass1pt{minimaentclass1pt{minimaIn Fig. This the temperature change can be compared with that available from the strain energy. For a peak engineering stress e et al. .The utility of any material model will be limited if it is difficult to implement via the finite element method. In this section we explore this aspect of the model of Fig. The mesh for the ASTM specimen is shown in Fig. First, the predictions of strain rate at uniform strains will be evaluated. As shown in Fig. We now turn attention to the deformation field after yield. The possibilities here are that the specimen remains homogeneous or develops an instability in the form of a neck or shear band. As noted previously (Sweeney et al. For parameters corresponding to 333 K, the model performance is similar to that at 323 K, with bands predicted rather than necks and the delay between yielding and unstable deformation greater in the model than in the experiments. For both these temperatures Fig. For the higher temperatures 341 and 346 K simulations, deformations are uniform at the lowest speeds, but bands begin to appear at higher speeds. This is shown in Figs. The onset of an instability in strain is controlled by the strain rate sensitivity (Ward and Sweeney Rate-dependent tensile yielding has been modeled to good accuracy by a constitutive model that incorporates a single plastic mechanism as defined by Chen and Schweizer , 2011 anThe model has been implemented as the core of a finite element analysis using a user-defined subroutine and material parameters derived from yield data. Finite element models of the tensile specimens have been created on this basis to give predictions of the strains. The predictions of strain rate compared well with those derived from the strain measurements when the state of strain in the gauge length was uniform. Predictions of non-uniform strain fields were less successful. There are three principal reasons for the discrepancies. First, the model is based on the assumption of isothermal conditions, whereas our observations show the specimens to be highly anisothermal once the strains have become non-uniform. Secondly, the model includes no provision for strain hardening. Finally, the values of the model parameters were derived from yield behavior at uniform strains. The first two are controlling factors for strain localization, which is in many cases observed after yield. To give accurate predictions of instabilities, a constitutive model would need to be thermally coupled; this is independent of whether the core plastic mechanism is that of Chen and Schweizer or some other, such as Eyring’s. The addition of a parallel elastic mechanism could give the required strain hardening. Thus, a more complex model could be developed to give more accuracy. We can conclude that the Chen–Schweizer process can be implemented with no special difficulties and is capable of acting as the core of a constitutive model that returns the principal features of the observed behavior of PET.The PET yield stresses show dependence on rate of strain that changes with temperature, so that at higher temperatures the slope of the plot of yield stress against the logarithm of rate displays a distinctly increasing slope as rate increases. At lower temperatures the slope is almost constant, corresponding to an Eyring-like behavior. Varying the parameter The present three-dimensional model has been created by causing the Chen–Schweizer mechanism to operate via a flow rule, and adding an elastic mechanism. These appear to be the minimum requirements for a viable model. There is no pressure-dependence of yield. This is a measurable effect in polymers that is customarily incorporated into Eyring-based models using a pressure activation volume term (Ward and Sweeney


This article proposes a general mixture item response theory (IRT) framework that allows for classes of persons to differ with respect to the type of processes underlying the item responses. Through the use of mixture models, nonnested IRT models with different structures can be estimated for different classes, and class membership can be estimated for each person in the sample. If researchers are able to provide competing measurement models, this mixture IRT framework may help them deal with some violations of measurement invariance. To illustrate this approach, we consider a two-class mixture model, where a person’s responses to Likert-scale items containing a neutral middle category are either modeled using a generalized partial credit model, or through an IRTree model. In the first model, the middle category (“neither agree nor disagree”) is taken to be qualitatively similar to the other categories, and is taken to provide information about the person’s endorsement. In the second model, the middle category is taken to be qualitatively different and to reflect a nonresponse choice, which is modeled using an additional latent variable that captures a person’s willingness to respond. The mixture model is studied using simulation studies and is applied to an empirical example.The online version of this article (10.3758/s13428-017-0997-0) contains supplementary material, which is available to authorized users. IRT measurement invariance (MI), in order to emphasize that we are considering the assumption that a single IRT model is appropriate for all persons in the sample.There are a large number of different item response theory (IRT) models available in the literature  explain the observed response patterns. The different measurement models do not need to be nested, nor do they have to be special cases of a more general measurement model. The approach proposed in this paper makes it possible to obtain information for all persons about the attributes intended to be measured, even if there are important qualitative differences across classes in the cognitive processes that relate these attributes to the responses. The approach requires the researcher to formulate competing measurement models that may hold for an unknown subsection of the population, after which a mixture model can be estimated that includes these different measurement models. In this framework, class membership and IRT person and item parameters can be estimated concurrently.The structure of the remainder of the paper is the following. “In many applications in the social sciences, attributes are measured using Likert scales Likert, . These sacross persons in the way persons interpret and make use of these categories. This complicates the analysis of the response data using polytomous IRT methods, because it implies that different measurement models are appropriate for different persons.It is important to emphasize that while the coded responses result in numerical item scores, the response categories are qualitative in nature. Thus, it is not necessarily the case that the differences between an item score of 1  and 2  in terms of the severity of the position are the same as the difference between an item score of 2 and 3 . Furthermore, due to the qualitative nature of the categories, there are also likely to be differences Of particular relevance in this context is the middle category that is often present in Likert items, for example formulated as “neither agree nor disagree” or “neutral”. While including such a middle category gives respondents the possibility to communicate a neutral position towards the presented statement, respondents differ in their interpretation and use of this neutral category: Some respondents select the neutral category to indicate that their position falls somewhere in between the two adjacent categories , but others treat the neutral category as a nonresponse option that indicates that they do not have (or do not want to communicate) an opinion regarding the statement that is presented  but qualitatively similar . Thus, the item scores are considered to be ordinal, and it may be appropriate to model these using polytomous IRT models. Let m to indicate that m categories are modeled. In the gPCM-m, item scores are modeled throughp on dimension di that item i is designed to capture, αi is the slope of item i, and δi = δi1,…,δim is a vector of thresholds of the m categories, with δi1 = 0 for identification. For each response the gPCM-m can be represented as a decision tree with one node and m possible outcomes .Several polytomous IRT models exist that could be used to analyze Likert-type data, which differ in their specification of the IRF Andrich, . BecauseD dimensions, such that for each person p the set of latent variables 𝜃p = {𝜃p1,…,𝜃pD} is of interest. Let us by d = {d1,…,dK} denote a design vector specifying to which dimension each item belongs, where di = d indicates that item i belongs to dimension d. Since each item only captures one dimension, the item scores can be modeled through Eq. D = 1, the subscript di can be dropped and Eq. It is rather common for questionnaires to consist of multiple Likert scales each designed to measure a single latent trait. Suppose a test is intended to measure Using a gPCM (or a mixture of gPCMs) may be appropriate if respondents consider the categories to differ only quantitatively, meaning that they take each category to reflect a particular degree of endorsement and hence that we can treat the item scores as ordinal. However, this would not be defensible if respondents interpret the middle response category as being qualitatively different from the other categories, for example by approaching it as a noninformative response option. In that case, standard polytomous IRT models are not appropriate for modeling the item scores, because a respondent’s choosing the middle category cannot be explained by an appeal to the dimension that the item is supposed to capture.Instead, it may be possible to model the response process through the use of IRTree models  or not (Xpi∗ = 0 ). For the second node, we recode Xpi into an ordinal outcome variable Xpi∗∗ = 1,…,m − 1, such thatTo apply the IRTree model, we need to re-code the item score 𝜃p0 denotes the person parameter of person p, αiS and δiS denote the slope and the location parameter of item i, and the subscript S refers to “S kipping” giving an informative response. Here, 𝜃p0 is a latent variable that captures a respondent’s tendency to skip items by choosing the nonresponse option, with low values of 𝜃p0 indicating a relatively strong tendency to provide informative answers . It can be emphasized here that the model in Eq. 𝜃p0), but items also differ in the extent to which people are inclined to provide noninformative responses to it . This is in line with the idea that item content and formulation can play an important role in determining the extent to which nonresponse occurs, but that the probability of observing a noninformative response will also depend on person characteristics such as response style.Both nodes of the IRTree can be modeled using common IRT models. In line with Jeon and De Boeck m − 1) remaining categories  is modeled throughαiT and δiT denote the slope and the threshold parameters of item i, and subscript T refers to the “IRT ree” in order to differentiate these parameters from those in the gPCM-m.The second node of the IRTree captures the selection of a response category relevant for the attribute of interest, for which we propose to use the gPCM. Thus, a respondent’s choice for one of the  or on the item-side  that play a role in the second node of the IRTree model are the same as those present in the gPCM-m  × (D + 1) covariance matrix Σ. The mean is constrained to 0 for identification, because in IRT models only the difference (𝜃 − δ) is identified and not the parameters themselves , a log-normal prior distribution (to ensure these parameters to be positive) with a mean of 0 and variance of 4 is used. Using a relatively large variance compared to the range of values that the logs of slope parameters normally take on ensures that the prior is relatively uninformative and that the posterior will be dominated by the data , respectively; δS is a vector of δis of all the items; Z is a vector Zps of all persons. To sample from the posterior distribution in Eq. The model can be estimated by sampling from the joint posterior distribution of the model parameters:Z when a finite number of iterations are used for the burn-in period. For example, if at the start of the algorithm none of the persons who belong to the IRTree are assigned to the IRTree class, then the IRTree item parameters will not be sampled optimally and it is possible that the class will initially become empty, because the non-optimized IRTree model will not fit the data of these persons better than the gPCM-m. To avoid ending up with a chain stuck in such a local maximum , we recommend the use of multiple chains , number of items , number of dimensions that the test is intended to measure , and proportion of persons belonging to the IRTree class . For the simulation study a full factorial 3 × 2 × 2 × 3 design was used. Five-point Likert scales were considered with persons either belonging to the gPCM-5 class or to the IRTree class with the 2PL in the first node and the gPCM-4 in the second node. In each condition, 50 replicated data sets were generated using Eq. Four design factors were considered: sample size  and the remaining Zps were set to one . All 𝜃 s were sampled independently from For each condition the item and the person parameters were generated in the same way. For the first αiT and αiR were sampled from a bivariate normal distribution with means equal to 0, variances equal to 0.25 and correlation of .5. Here a moderate correlation was chosen, as it accommodates the fact that having less categories available to provide an informative response may alter the discriminative properties of the item to some degree, while still being relatively similar under both models. The threshold parameters were sampled through αiS ∼LogNorm and αiS was chosen to match the fact that items were not designed to measure a tendency to avoid giving informative responses.Because the process of giving an informative response is assumed to be relatively similar across the two classes, the item parameters of the gPCM-5 and gPCM-4 were set to be correlated. The logs of p was considered to be correctly classified if true class membership was equal to the estimate of Zp. For P = .25 and P = .5 several outcome variables were considered. Pall is the proportion of overall correctly classified persons, while PTree and PPCM are the proportions of correctly classified persons among those whose true class membership is IRTree and gPCM-5, respectively. Pcert is the proportion of persons assigned to the correct class with high certainty . For these four outcome measures, the average and standard deviation across the 50 replications were considered. For P = 0, the only outcome measure was the proportion of replications in which the IRTree class was estimated to be empty .Under each condition the accuracy of the classification of the persons in the two classes was investigated. A person P = 0 were very similar across all conditions, and are for that reason not displayed in Table P = 0, the IRTree class is consistently estimated to be empty: Across all replications in all conditions, it only happened once that the IRTree class did not become empty , and in that one replication only two persons out of 2000 were assigned to the IRTree class. Thus, there does not appear to be a risk of overfitting, meaning that empty classes are consistently identified as such.The results of the simulation study are displayed in Table Pall) exceeded .80 and in many cases .90  when K = 20 and P = .25. Here, little information is available per person (because K = 20) and for N = 500 there is also little information available for estimating the IRTree item parameters (because only 125 persons belong to that class). Using the mixture model in this challenging condition may not be ideal, as the IRTree class was estimated to be empty in about 25% of replications, and the average PTree was low and its variance was high. For K = 20 and P = .25, with larger N the issue of the empty IRTree class disappeared, the average PTree improved, and its variance decreased.As can be observed in Table K from 20 to 40. The overall proportion of misclassified persons  is more than halved by this increase in test length, with Pall exceeding .90 in all conditions , the relative size of the two classes did not appear to affect the overall proportion of correct classifications. However, the class proportions did have a strong impact on the classification accuracy for persons belonging to the IRTree. For P = .25, relatively few persons belong to the IRTree class. As will be further discussed in the next section, this complicates the estimation of the item parameters for that class. Additionally, because a hierarchical prior was used, the procedure’s posterior probability of a person belonging to a class depends on the estimated proportion of persons belonging to that class  rather than the location and threshold parameters, because the former were considered in the estimation procedure  provides a better insight into the degree to which the item location is correctly recovered by the procedure. On the person side, we investigated how well 𝜃1 was recovered, and compared this with the recovery of 𝜃1 under a nonmixture gPCM-5.To assess whether using the mixture model may improve measurement, the accuracy and precision of the estimates of both dure see  as theirdure see . Hence, P = 0,.25,.5). A single set of item parameters and continuous person parameters was generated . The average absolute bias in these parameters of the gPCM-4 appears to be approximately halved when the number of persons belonging to that class increases from 250 to 500 (P = .5). The only exception seems to be the intercept in the first node of the IRTree (βiS), where bias is low in both conditions.The item parameter recovery results are presented in Table βikT) appears to be the most difficult to recover, especially when P = .25. The variance of the IRTree item parameter estimates is more than halved when class size is doubled (P = .5). Similarly, the variance of the estimates of the gPCM-5 parameters is also lowest when the gPCM-5 class is largest (P = 0) and gets worse when P > 0.With respect to the variance of the estimates, the item category intercept of the IRTree , item recovery of all relevant parameters appears to be adequate.As the bias in the parameter estimates is small compared to their variance, the patterns observed for the MSE largely match those that were found for the variance. Thus, the MSE results indicate that of all parameters considered 𝜃1 are displayed in Fig. 𝜃1 observed under both the gPCM-5 and the mixture model. The MSEs for these two models are displayed in Fig. The results for the recovery of P = 0 . Deviations from the pattern observed for P = 0 can be taken to indicate lack of robustness of the model inferences for P > 0, that is, that estimates of 𝜃1 differ from those that would have been obtained if all persons had belonged to the gPCM-5 group.The results of the gPCM-5 and the mixture model are practically identical when P = .25, the average absolute bias and the average MSE was slightly lower for the mixture model  than for the gPCM-5 model , while the average variance was similar (0.045 for both models). As can be seen in Fig. 𝜃1 of persons belonging to the IRTree class is highly overestimated at the lower end and underestimated on the higher end of the 𝜃1-scale, resulting in a relatively large average absolute bias and MSE for this group . For persons with high or low true values of 𝜃1 and belonging to the IRTree class the MSEs were much lower when the mixture model was used  was observed as in the condition with P = 0 , which may be due to the lower number of informative responses available per person.When using the nonmixture gPCM-5, the parameters of the persons whose true class is the gPCM-5 were only slightly biased (0.03), and while there is still shrinkage to the mean for this group, the observed effect was smaller than for P = .5, the gPCM-5 shows a larger average absolute bias (0.12) and MSE (0.09) than the mixture model , while the average variance was similar (0.05 for both models). As can be seen in Fig. 𝜃1 being highly overestimated on the lower end and underestimated at the higher end of the 𝜃1-scale (absolute bias of 0.19). Figure For  bias 0.1 and MSE 𝜃1 for persons whose true membership is gPCM-5 were only slightly biased under the gPCM-5 (absolute bias of 0.04), but the direction of the bias is different compared to P = 0: For P = .5 there is underestimation on the lower end and overestimation on the higher end of the scale. Thus, instead of the shrinkage towards the mean observed for P = 0, the estimates are slightly inflated under the gPCM-5 when P = .5 for persons belonging to the gPCM-5 class, indicating that when using the nonmixture model the estimation of 𝜃1 is also not robust for persons for whom a gPCM-5 model would in fact be appropriate.The estimates of P = 0  to that observed when see Fig. f and d. see Fig. f, comparsee Fig. c.The mixture model was applied to data on the ‘Experiences in Close Relationships’ (ECR) questionnaire developed by Brennan, Clark, and Shaver . The queThe mixture model was estimated using the Gibbs Sampler with 10 chains with 10000 iterations each (including 5000 iterations of burn-in). With respect to the number of iterations, we decided to stay on the safe side compared to the simulation study by taking both a longer burn-in and using more iterations for the post-burn-in, because computational time is less of an issue when only one data set needs to be analyzed.pD, defined as the difference between the deviance averaged across iterations and the deviance computed for the parameter estimates. In hierarchical models pD is typically smaller than the number of parameters present in the model, because the contribution of a parameter to pD depends on the ratio of the information about the parameter in the likelihood to its posterior precision . Both models were estimated using the same estimation procedure as for the mixture model, but where for all persons class membership was fixed to that of the model that was considered. The relative fit of the three models was compared using the deviance information criterion (DIC), which was used because it adequately takes the complexity of hierarchical models into account  was higher than that of the gPCM-5, but lower than that of the IRTree model, where 𝜃0 is estimated for every person. The mixture model has a lower pD than the nonmixture IRTree model due to the fact that in the former 𝜃0s do not contribute (or hardly contribute) to pD for the persons who are classified in the gPCM-5 class with high certainty, because for these persons 𝜃p0 is effectively sampled from the prior and is not informed by the data. The fit of the mixture model , the items do reasonably well in differentiating persons on this tendency to produce noninformative responses (𝜃0) for persons belonging to the IRTree class, and—as indicated by the interquartile range of the αiSs – the items also differ in the extent to which they capture this tendency. Furthermore, the item location parameter δiS showed notable variance across items (standard deviation of 0.76), and ranged from 0.82 to 3.94 (mean of 2.13). This indicates that there was a substantial difference between the items in terms of how likely persons were to provide a noninformative response. While investigating whether certain item properties can be linked to higher or lower values of δiS is beyond the scope of this example, such study might be of interest to test constructors who likely want to avoid designing items that evoke noninformative responses.Consideration of the parameter estimates obtained using the mixture model yielded substantively relevant results that would have been unavailable if only a gPCM-5 would have been used. For the mixture model the median of the estimated 𝜃0 and the ‘avoidance’ dimension of the ECR is estimated to be .27 . This indicates that persons who show a high degree of avoidance in close relationships also display a stronger tendency to avoid giving informative responses, at least on this questionnaire about close relationships. This can be seen as providing some indication that 𝜃0 might capture a substantively relevant dimension that can be related to other relevant person attributes, such as the avoidance tendency that the scale was designed to measure. These findings invite further research into the nature of the tendency to produce noninformative responses as captured by 𝜃0 and its relation to other traits.Interestingly, the correlation between This manuscript considered the application of the proposed general mixture IRT framework to address between-person differences in how the middle response category on Likert-scale items is interpreted and used. For this, a mixture of a gPCM and an IRTree model was used, where the IRTree model assumes a person-specific ‘information-avoidance tendency’ to influence the usage of the middle response category. By using a mixture of two structurally different measurement models, the model can accommodate the possibility that persons show qualitative differences in their usage of this category and take this into account for the measurement of the attribute(s) of interest.It may be noted that our approach to modeling the usage of the middle category in Likert-scale items is distinct from but related to other approaches that consider response styles and nonresponse choice  as well as items (the extent to which the item evokes noninformative responses) that may be of interest to researchers or test constructors.In this article we proposed a general mixture IRT framework that allows researchers to use a mixture of structurally different measurement models. The approach was illustrated in the context of a two-class mixture model for Likert data, but it can readily be applied using any set of measurement models for which Bayesian estimation procedures are available and for which the concurrent estimation of these different models is tractable. Usage of these mixture models may lead to improved recovery of the relevant person parameters  as well as improved understanding of the response processes that are involved .While this manuscript has considered a particular application of the general mixture IRT framework, it is to be expected that the framework can be relevant in a variety of other contexts. Whenever different response processes are expected to play a role for different persons, it may be relevant to consider using a mixture of structurally different measurement models. That is, assuming the same measurement model  to adequately capture qualitatively different response processes may not be realistic, and a mixture of different models may do more justice to the actual underlying processes and improve measurement. For example, in the context of educational measurement researchers often have to deal with the fact that items on educational tests can be solved using different strategies, some of which may only be known to a subgroup. Likewise, students may differ in their willingness to guess on multiple choice items, or may differ in the way they guess . The framework may also be useful for dealing with the effects of confounding factors such as dyslexia or test anxiety, which may only play a role for part of the sample. Examples such as these are likely to be present in many other fields as well.It may be noted that for the successful application of the framework the measurement models should result in differential predictions for the expected response patterns, resulting in differences in the likelihood for individual response patterns. That is, for the different measurement models to be separable, they should be empirically nonequivalent. The larger the differences in prediction are, the more easily persons are assigned to the right class and the more can be gained from using a mixture of measurement models instead of a single model. While the simulation results obtained for the specific mixture model that was considered here were encouraging, more research is needed to provide a complete picture of the general conditions under which the procedure performs well.As a recommendation, we suggested to consider measurement models linked through the inclusion of a shared set of latent variables. While assuming this weak form of MI is not necessary for the mixture model to be estimable, it has strong appeal from the measurement point of view, as it entails that the same attribute is measured in each class. Whether assuming this weak form of MI is reasonable will need to be assessed in the context of the application at hand, which should be tested empirically  alternative would be to consider person-by-subscale mixtures, where class membership is taken to be fixed within a subset of the items but class switching across subscales is allowed.(R 43.7 KB)(C 10.9 KB)(R 7.03 KB)(R 3.46 KB)Below is the link to the electronic supplementary material.


A and G denote the arithmetic and geometric means. Sharp bounds for the identric mean in terms of For  The study of inequalities involving means has become very popular in recent years because of their applications in various kinds of areas of mathematics. Finding sharp bounds for inequalities is an important task in order to have more accurate results in the aforementioned areas.a and b, we recall that the arithmetic mean Let us fix some notation in order to describe our results. For distinct positive real numbers Inequalities relating means in two variables have attracted and continue to attract the attention of mathematicians. Many articles studying the properties of means of two variables have been published, and there is a large body of mathematical literature about comparing pairs of means. The interested reader may consult –7, 9–11 a and b, if and only if For example, Alzer and Qui considered in  the folla and b, we haveThis was later complemented by Trif  who prova and b if and only if a and b if and only if Similarly, it is proved in  that thesults of  and 12]\documentIn this paper we continue the search for nontrivial bounds for the identric mean by studying a new family of two parameter means of two variables. The article is organized as follows. In Sect. a and b byMotivated by the works  and 13]13], we cthors in , 13 compThe aim of this work is to produce a general result comparing the identric mean to members of the family sults of  and 13]\documenta and b and given a and b, we haveIndeed, in Corollary u et al.  and Wangu et al. .The following lemmas pave the way to the main theorem. In Lemma Forandwe consider the real functiondefined onbyThe necessary and sufficient condition to haveforis thatThe necessary and sufficient condition to haveforis thats and positive on We consider only the case ∘If ∘If First, It follows that in this case unique zero ∘If ∘If Second, This shows that the necessary and sufficient condition for Noting that This achieves the proof of Lemma Next we introduce the set Considerand letForandwe haveAlso, for the identric mean, we haveIndeed, (a) follows from the simple fact thatThe next corollary is an immediate consequence of part (a) of Lemma For distinct positive real numbersaandb, and for giventhe functionis continuous and increasing on the intervalCombining (a) and (b) from Lemma a and b if and only ifIn this section we prove our main result which states that, for Further, this result is used to obtain in Corollary Letsbe a real number such thatand define the setsThenFirst note thatWhen mentclass2pt{minimThe necessary and sufficient condition onp, qfromto havefor every distinct positive numberaandbis thatSimilarly, when mentclass2pt{minimThe necessary and sufficient condition onp, qfromto havefor every distinct positive numberaandbis thatIn the next corollary, the lower bound is an inequality due to Seiffert , it appeFor every positive numberaandb, we haves tend to +∞, we obtainIndeed, for In fact, because of the “limit argument” in the proof of Corollary The necessary and sufficient condition onp, qfromto haveis thatandIn this work, we have considered a new two-parameter family of means, and we have compared them to the identric mean giving sharp upper and lower bounds.


All the existing intrinsic dimension estimators are not reliable whenever the dataset is locally undersampled, and this is at the core of the so called curse of dimensionality. Here we introduce a new intrinsic dimension estimator that leverages on simple properties of the tangent space of a manifold and extends the usual correlation integral estimator to alleviate the extreme undersampling problem. Based on this insight, we explore a multiscale generalization of the algorithm that is capable of (i) identifying multiple dimensionalities in a dataset, and (ii) providing accurate estimates of the intrinsic dimension of extremely curved manifolds. We test the method on manifolds generated from global transformations of high-contrast images, relevant for invariant object recognition and considered a challenge for state-of-the-art intrinsic dimension estimators.Identifying the minimal number of parameters needed to describe a dataset is a challenging problem known in the literature as  Manifold learning and dimensional reduction3 are the main techniques employed to perform this task. Several of these approaches work under the reasonable assumption that the points (or samples) of a dataset, represented as vectors of real numbers lying in a space of large embedding dimension D, actually belong to a manifold intrinsic dimension (ID) d is much lower than D. The problem of providing accurate estimates for this number has been recognized multiple times and in different contexts: in psychometry by Shepard5, in computer science by Trunk6, Fukunaga and Olsen7, in physics by Grassberger, Procaccia9, and Takens10. More recently, Intrinsic Dimension Estimation (IDE) has been reconsidered with the advent of big data analysis, artificial intelligence and demanding molecular dynamics simulations, and several estimators to measure the intrinsic dimension have been proposed22.Processing, analyzing and extracting information from high dimensional data is at the core of the modern research in machine learning and pattern recognition. One of the main challenges in this field is to decompose and compress, without losing information, the redundant representations of complex data that are produced across diverse scientific disciplines, including computer vision, signal processing and bioinformatics. curse of dimensionality23. When a manifold has large ID . The ID is then estimated by looking for jumps in the magnitude of the sorted eigenvalues of CX  and a multiscale version (mPCA) of the algorithm are used24. In the former one evaluates the covariance matrix on the whole dataset X, whereas in the latter one performs the spectral analysis on local subsets X of X, obtained by selecting one particular point x0 and including in the local covariance matrix only those points that lie inside a cutoff radius rc, which is then varied.Algorithms for intrinsic dimension estimation can be roughly classified in two groupsrc needs to be small enough, implying that the sampling of the manifold must be dense enough to guarantee that a sufficient number of samples lies inside the subsets X. Another technical issue that makes mPCA difficult to employ for IDE is the fact that the amplitude of the “jump” in the magnitude of the sorted eigenvalues depends on the data, and the choice of a threshold size is somewhat arbitrary.The main limitation of the global PCA is that it can only detect the correct ID of linearly embedded manifolds  methods. Their common ancestor is the correlation dimension (CorrDim) introduced by Grassberger and Procaccia9 to measure the fractal dimension of strange attractors in the context of dynamical systems. This estimator is based on the observation that the density of neighbours  with a given cutoff distance rr r → 0 and therefore one can extract the ID by measuring the slope of the linear part of ρ as a function of r in log-log scale, since the relation The other group of estimators belongs to the so-called 18 and is only partially mitigated by more recent and advanced generalizations of CorrDim based on k-nearest-neighbors distances12. The reason why all these algorithms systematically fail for 23 that the accurate estimation of the ID requires a number of samples N which grows exponentially in the intrinsic dimension d. As a consequence, one observes a systematic undersampling of the small radius region of the density of neighbors ρX(r), as shown for the CorrDim estimator in the bottom left panel of Fig. CorrDim is very effective for the estimation of low IDs  and considers this quantity at finite radius r. We show in the following that this method overcomes the extreme undersampling issue caused by the curse of dimensionality and displays a remarkable robustness to non-uniform sampling and noise. Based on the intriguing features reported above, we propose a multiscale generalization of the estimator, capable of providing the correct ID for datasets extracted from highly curved or multidimensional manifolds.In order to try and break the curse of dimensionality, additional information about the probability distribution of the data must be assumed. The main ingredient of our spell is the assumption that data are locally isotropic. This suggests to consider the average correlation integral for hyperspheres d-dimensional disks linearly embedded. The average correlation integral for the boundary of this manifold, which is the (d − 1)-dimensional sphere of radius rs, can be analytically evaluated as The tangent space approximation suggests that a special role in the IDE problem is played by uniformly sampled 2F1 is the -hypergeometric function, Ωd is the d-dimensional solid angle and full correlation integral, to stress that we work away from the small radius limit employed for CorrDim.where d of linearly embedded spherical datasets, by performing a non-linear regression of the empirical density of neighbours using the FCI in  of the dataset uniformly drawn from {0, 1}d, linearly embedded in D = 60 dimensions , we provide accurate estimates even in the regime N < d with a relative average error that decays quickly with the number of samples N ; as a matter of fact, for N = 100 the error is already below one percent. The method is also particularly robust to Gaussian noise, showing a sharp crossover between the regime where the ID is correctly retrieved to a phase where the noise covers the signal.Our estimator is robust to extreme undersampling and to noise, as highlighted in the center and right panels of Fig. global, thus it is expected to fail on manifolds with high intrinsic curvature. As in the case of the global PCA24, we can overcome this issue by providing a suitable multiscale generalization of the FCI estimator.So far, we have applied and verified the performance of our algorithm only on linearly embedded manifolds. More in general, we have verified that the FCI estimator correctly identifies the ID even in the case of simple non-linear polynomial embeddings or of slightly curved manifolds. However, the method presented above is x0 in the dataset and a cutoff radius rc. We then apply the FCI estimation method to the set of points whose distance from x0 is less then rc. In this way, we obtain a local ID estimate dx0(rc) that depends on the cutoff radius. Varying rc and x0, we obtain a family of curves that describes the local ID of the dataset at different scales , chosen from regions with very different curvatures. When the curvature is large, the height of the plateau identifies an overestimated ID, which drifts towards the embedding dimension as expected. On the other hand, when the curvature is small, the correct ID is identified. This observation suggests that the best estimator for the multiscale analysis could be the minimum ID identified in the plateau region. We use this heuristic in the tests below, but we leave a more detailed investigation of the multiscale method for future work.First, we look at a paradigmatic curved manifold studied in the literature, the Swiss roll N = 1000 + 1000), representing two intersecting hypercubes with different IDs. Two plateaux at d = 20 and d = 30 are clearly visible and allow to infer the multidimensionality of the dataset . In the top center panel of Fig. N = 2500), first introduced in13 and considered a challenging dataset for ID estimation for its high intrinsic curvature. Although a thorough comparison of the performance of our algorithm with state-of-art ID estimators is out of the scope of our investigation , we observe that our prediction Now we move to multidimensional datasets and non-trivially curved manifolds, which are considered challenges for state-of-the-art ID estimators. In the bottom center panel of Fig. at the same time– with the removal of the well known underestimation issue common to all geometric methods, provides two major advantages of our algorithm over other traditional schemes used for ID estimation and suggests to test and validate it on the manifolds of global transfomations  generated by high contrast images, relevant in the context of invariant object recognition. These manifolds often feature high local curvature, since even infinitesimal transformations produce almost orthogonal tangent spaces28.This excellent accuracy on highly curved manifolds, combined –d = 5 and d = 15 respectively). It is worth noticing that whereas single points are affected by the high curvature of the manifold (resulting in higher ID estimations), the minimum of the plateaux works again nicely as the estimator of the correct ID.As a more demanding test for our multiscale FCI estimator, we report a preliminary investigation of a manifold  that belongs to this class, where we can keep under control the intrinsic dimensionality  eigenvalues of the correlation matrix should go much faster to zero than the remaining relevant ones when the cutoff radius is reduced. In practice, we are not able to identify a clear signature of this phenomenon and we can only establish loose bounds on the ID  as proposed inN < d).In this manuscript we introduced the FCI estimator for the ID of spherically sampled, linearly embedded datasets, showing that it is robust to noise and non-idealities and more importantly that it works effectively in the extreme undersampled regime .We leave open to future investigations the analysis of high dimensional manifolds of high contrast images taken from the Machine Learning literature, as well as the possibility of combining our estimator with state-of-the-art techniques for dimensional reduction and manifold learning, or, even more ambitiously, to elaborate on it in order to propose a novel more effective toolbox for these tasks.30 by the Machine Learning community as a tool to understand how deep neural networks transform and compress information in their hidden layers. Here the authors observe that the range of IDs of many training sets (such as Fashion-MNIST and CIFAR-10) processed through the hidden layers of a DNN is between 10 and 100. This is the typical regime where our ID estimator overcomes standard methods, so that it would be interesting to use it to reproduce these analysis.Beyond these applications, ID estimation has been very recently usedrs, i.e.α and β are spherical coordinates ∀i = 2…d and α1 ∈  and the same for βi), d-dimensional spherical volume element and x(⋅) is the function that converts spherical coordinates into (d + 1)-dimensional Euclidean coordinates on the sphere of radius rs. d-dimensional solid angle. The integral can be evaluated by choosing the spherical coordinates β such that their azimuth axis is in the direction of α, so thatHere we derive the average correlation integral Eq.  for a daα and in β1…βd−1 are trivial, givingThe integrals in The empirical full correlation integral is easily computed in Mathematica 12 using the following one-liner:Module] – 1)/Length[dists]\}]]d and rs as free parameters. The empirical full correlation integral was preprocessed before the application of FindFit by extracting a RandomSample of The non-linear fit for the FCI estimator was performed using the default function FindFit, with D − d zeros to its argument, and then rotates it in In this section we briefly describe the datasets used in the presented numerical simulations. In the following, a linear embedding is the map d, linearly embedded.d, linearly embedded.π]d, embedded with the mapd, embedded with the mapn blobs. n = 1 images generation is described in the following section. n > 1 images are generated by summing n different n = 1 images.l × l bitmap the following value vi,j:The high-contrast images shown in Fig. Morover, any pixel of value less then 0.01 was manually set to 0 to increase the contrast of the image.


Publisher’s correction to: Cognitive Research: Principles and Implications, volume 4Cognitive Research: Principles and Implications. Several articles were published in volume 4 with a duplicate citation number.An error occurred during the publication of a number of articles in In this correction article the old and new citation metadata are published in Table The original articles have been updated. The publisher apologizes for the inconvenience caused to our authors and readers.


Diffraction gratings are important optical components and are used in many areas of optics such as in spectroscopy. A diffraction grating is a periodic structure that splits and diffracts the impinging light beam into several beams travelling in different directions. The diffracted beams from a grating are commonly called diffraction orders. The directions of the diffraction orders depend on the grating period and the wavelength of the impinging light beam so that a grating can be used as a dispersive element. In the diffraction of a plane wave from a conventional grating, the intensities of diffracted beams decrease with increasing order of diffraction. Here, we introduce a new type of grating where in the diffraction of a plane wave, the intensity of a given higher order diffracted beam can be higher than the intensity of the lower orders. We construct these gratings by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating that has a sinusoidal profile and we call them azimuthally-modified linear phase gratings (AMLPGs). In this work, in addition to introducing AMLPGs, we present the generation of varied radial carpet beams over different diffraction orders of an AMLPG with controlled intensity sharing among the generated beams. A radial carpet beam is generated in the diffraction of a plane wave from a radial phase grating. We show that for a given value of the phase amplitude over the host linear phase grating, one of the diffraction orders is predominant and by increasing the value of the phase amplitude, the intensity sharing changes in favor of the higher orders. The theory of the work and experimental results are presented. In comparison with the diffraction of a plane wave from radial phase gratings, the use of AMLPGs provides high contrast diffraction patterns and presents varied radial carpet beams over the different diffraction orders of the host linear phase grating. The resulting patterns over different diffraction orders are specified and their differences are determined. The diffraction grating introduced with controlled intensity sharing among different diffraction orders might find wide applications in many areas of optics such as optical switches. We show that AMLPG-based radial carpet beams can be engineered in which they acquire sheet-like spokes. This feature nominates them for potential applications in light sheet microscopy. In addition, a detailed analysis of the multiplication of the diffraction pattern of an AMLPG by the 2D structure of a spatial light modulator is presented. The presented theory is confirmed by respective experiments. In addition, the intensity pattern over a plane includes the propagation axis and the grating vector  that is called the Talbot carpet2. In the far-field regime, the diffraction orders appear as the impulses of the Fourier transform of the grating’s transmission function.A conventional optical diffraction grating, say linear grating, is a periodic structure in the Cartesian coordinates. In the diffraction of a plane wave from a linear grating, the beam is split and diffracted into several beams travelling in different directions. These diffracted beams are known as diffraction orders. For the conventional gratings, the intensities of diffraction orders decrease with increasing the number of order of diffraction. In the near-field regime, superposition of diffraction orders form self-images and sub-images of the grating’s structure at certain propagation distances. This effect is known as Talbot effect3, interferometry6, lithography8, strain and stress analysis9, displacement measurement10, optical alignment technique, color printing11, and 3D displays13. In recent two to three decades amplitude and phase gratings with topological defects have found serious applications in generating vortex beams and/or characterizing such beams21. The use of phase hologram gratings instead of the conventional one has some advantages25. For instance, a computer generated hologram phase grating provides the implemention of a desired phase map on the incident wave in a controlled way26.The linear gratings have numerous applications in optics and other areas of sciences and technologies such as in astronomical spectroscopy, moiré fringe technique including moiré deflectometry and moiré topography27. The conventional Talbot carpet is formed on the longitudinal plane when a grating is illuminated by a plane beam29. It has also been shown that the plane boundaries between the optical regimes including the geometric shadow and near-field and far-field diffraction regimes have acquired curvature27. The diffraction of a vortex beam from an amplitude radial grating presents a simple way for measuring the topological charge alteration30. That study also presents a Poisson-Arago spot-like effect on the optical axis when an amplitude grating is illuminated by a vortex beam in which the number of grating spokes is equal to the topological charge of the incident beam. More precisely, a needle-like optical beam forms along the optical axis in far-field diffraction. The diffraction of a plane wave from a radial phase grating was also theoretically and experimentally investigated. It was shown that the diffraction of a plane wave from radial phase gratings is a simple way to generate radial carpet beams that are nondiffracting, accelerating, and shape preserving31. These beams have unprecedented patterns that are shape-invariant during propagation. The patterns can be easily turned into tunable 2D optical lattices. We also proposed a new solution for the paraxial Helmholtz equation and introduced combined half-integer Bessel-like beams32. It was shown that a huge family of beams can be produced in diffraction from designed radial structures and not necessarily from radial gratings. In that work spatially asymmetric, radial carpet, petallike, and twisted-intensity ringlike vortex beams were introduced through detailed theory and suitable practical approaches were proposed to generate each of them.Another specific type of grating is radial grating. A radial grating has a periodic structure in the azimuthal direction in polar coordinates. Unlike conventional gratings, there is no diffraction order in diffraction from a radial grating. However, the diffraction of a plane or Gaussian wave from radial gratings shows other interesting features. On illuminating an amplitude radial grating by a spatially coherent light beam the Talbot carpet is formed on transverse planes31 the use of AMLPGs provides a set of high contrast and varied radial carpet patterns over the different diffraction orders of the host linear phase grating. Since an SLM has a two dimensional periodic structure it multiples the diffraction pattern of the AMLPG. A detailed formulation is also presented and the theoretical predictions are verified by experiments.In this work, we integrate the features of linear and radial gratings to introduce a new grating that, in addition to the properties of linear and radial gratings, has additional properties. We construct this type of grating by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating that has a sinusoidal profile and we call them azimuthally-modified linear phase gratings (AMLPGs). In the diffraction of a plane wave from an AMLPG, we observe different diffraction orders in which the pattern of each order is very similar  to the diffraction pattern of a radial grating. Unlike the conventional gratings, here the intensity of a given higher order diffracted beam can be higher than the intensities of the lower orders. We show that the intensity sharing among different diffraction orders of an AMLPG can be adjusted by the value of the phase amplitude of the host linear grating. This kind of grating might find wide applications in many areas of optics such as optical switches. The diffraction of a plane wave from AMLPGs is formulated and using a spatial light modulator (SLM) the respective experimental works are presented. In comparing with the diffraction of a plane wave from radial phase gratingsIt is worth noting that, as the transmittance of an AMLPG is not the product of the transmittances of the radial and linear gratings, the diffraction pattern from an AMLPG is not the convolution of the diffraction patterns of radial and linear gratings. However, the transmittance of an AMLPG imposed on an SLM is the product of an ideal AMLPG and the two-dimensional periodic structure of the SLM, therefore the resulting diffraction pattern is the convolution of the diffraction pattern of the ideal AMLPG and the two dimensional impulses of the SLM.Here, we formulate the diffraction of a plane wave from an AMLPG. When the phase structure of a linear phase grating with a sinusoidal profile hosting an additional radial phase grating, it can be considered as “azimuthally-modified linear phase grating (AMLPG)”. Some AMLPGs with almost sinusoidal phase profiles are shown in Fig. γ, l, d, and We introduce an AMLPG with the following transmission function:33. Therefore the diffraction pattern from an AMLPG is not the convolution of the diffraction patterns of the radial and linear gratings. Below we formulate the diffraction of a plane wave from an AMLPG.As is apparent from Eq. 34Js is the s–th Bessel function of the first kind, the grating’s transmission function, Eq. Using the Jacobi-Anger identitymentclass2pt{minimλ is the wavelength of the light beam, By illuminating this phase structure with a plane wave, the complex amplitude of the light field after the structure is given byBy substituting Eq. Equation Now using Eqs As By substituting Eq. 35:Now using the following integral identitys shows the number of the diffraction order of the AMLPG, and the corresponding light field isHere This is the main result, showing the complex amplitude of the diffracted beam from an AMLPG and indicates that all the diffraction patterns forming over the individual diffraction orders are different.It is worth noting that unlike the case of diffraction from the product of a given function and a linear grating in which the spectrum of the function is multiplied over different diffraction orders of the grating, here such behavior does not occur. The reason is that the structure of an AMLPG is not separable into a linear grating structure and another definite function.Equation 31, here again we see that each of individual diffraction patterns has a radial form with a considerable structural complexity. All of the patterns are shape invariant under propagation. Therefore we consider each of them a “radial carpet beam”. In one of the next subsections we will show that for a specific value of the phase amplitude of an AMLPG having a given value of l, one of the individual diffraction patterns will get the same intensity distribution of the radial carpet beam produced directly in the diffraction of a plane wave from a radial phase grating with the same spokes number, l.Similar to the diffraction of a plane wave from a radial phase gratings are similar in form but have a relative rotation with respect to each other.We show that the diffraction patterns generated over a pair of diffraction orders ±s, substituting s with −s in Eq. For a given positive value of Now we use Eq. Now by comparing Eqs s are similar except there is a relative rotation with a value of This means that the resulting patterns over a pair of diffraction orders ±γ. Figure γ at a distance J0, J1, and J2 their first zeros appear, respectively, and 5.5201 leads to the second zero of J0. As is apparent, the visibility of the patterns and intensity sharing among different diffraction orders depend to the value of γ.For For For For For Here we show that the intensity of an incident beam on an AMLPG can be divided among different diffraction orders with desired proportions. For an AMLPG, unlike the conventional gratings, the intensity share of the higher diffraction orders may be larger than the intensity on the lower diffraction orders, when given values are chosen for the phase amplitude of the grating, γ the energy of the incident light transfers to the higher diffraction orders. As is apparent, at larger radii, all individual diffraction patterns have the same number of spokes equal to the number of spokes of the AMLPG, but at smaller radii, the fine structure of the patterns strongly depends on the value of s. Also, it is seen that for larger values of s, the spoke patterns become needle-like and therefore the path of each needle-like pattern under propagation becomes a light sheet. These features might find some application in light sheet microscopy.This means that by increasing the value of γ.Now, we estimate the diffraction coefficients of an AMLPG by calculating the percentage of the incident power flows among different diffraction orders. This is done by calculating the ratios of the mean values of the intensities over the diffracted patterns to the mean value of the incident beam intensity. The mean value of the intensity over each of the individual patterns, separated by the dashed lines in the first row of Fig. Ps to the power of the incident beam PiIi and Is are the intensity values over the incident beam and the diffracted beam of order s, respectively. Their mean values over the corresponding areas are given by Using Eq. mentclass2pt{minimFor For For For For γ correspond to the first and second zeros of J0.For the values of J1 and the value of J2.For the values of The following results in Table 31 with the diffraction pattern of an AMLPG. The diffracted light field in the diffraction of a plane wave from a radial phase grating having a sinusoidal transmission function can be written as31Here we compare the diffraction pattern of a plane wave diffracted directly from a radial phase grating31, and the s–th diffracted order of an AMLPG are similar when the phase amplitude and spokes number of the AMLPG are equal to the spokes number of the radial phase grating.Comparing Eqs s–th diffraction order has the maximum value of intensity between all the diffraction orders. From Figs γ, the higher diffraction orders get a larger fraction of the energy of the incident beam.In Fig. entclass1pt{minimaγ, this feature can be used in optical switching.Since the value of intensity at different diffraction orders can be adjusted with the value of the grating phase amplitude, Here we investigate the replication of the spectrum of an AMLPG by the SLM structure under diffraction. First we consider the diffraction of a plane wave from an SLM when a secondary structure is not embedded in it.fx and fy show two one dimensional (1D) linear gratings placed together at right angles. The Fourier transform of 36δ is the delta function, Bm and Bn are the Fourier coefficients, fx and fy, respectively, and Assume that the structure of the SLM is a rectilinear 2D grating with the following transmission:Now suppose that the structure of an AMLPG with a transmission function of The diffracted light field distribution or equally the spectrum of the AMLPG is replicated by the impulses of the SLM. The replicated spectrum by the z is given byThis means that the spectrum of an AMLPG is replicated by each of the diffraction impulses of the SLM. Using Eqs s shows the order of diffraction of the AMLPG imposed on the SLM.The patterns of various radial carpet beams over different diffraction orders of the AMLPG with controlled intensity sharing among the generated beams are replicated over the SLM diffraction orders. Each of the generated radial carpet beams is given byγ, one can control the intensity sharing between different diffracted beams  to provide the desired AMLPGs. Maximum amplitude of the phase modulation was limited to mentclass2pt{minimAn AMLPG was imposed on the SLM and the collimated wavefront of the laser beam propagated through it. At different distances from the SLM, the diffracted patterns were recorded with a camera (Nikon D7200). We recorded the diffraction patterns by two methods. In some parts of the experiments, the diffraction patterns were directly formed over the active area of the camera by removing the imaging lens of the camera. Also in some of the experiments, the desired diffraction patterns were formed on a diffuser and then their patterns were imaged by the imaging lens of the camera on its active area. In the latter case, a magnification in the size of the images appeared. The active image area of the camera was 23.4 mm × 15.6 mm.x) and vertical (y) directions corresponding to the SLM’s main structure.Figure x direction and a period of 0.11 mm is imposed on it. The transmittance of a linear phase grating with a sinusoidal profile is given by Eq. cos(lθ) = 0. The sets of numbers indicate the orders of the diffraction from the SLM structure and the linear phase grating, Figure x direction (see Eq. s, are the same. For two given individual diffraction patterns with orders In Fig. mentclass2pt{miniml is odd, the intensity patterns of Figure In Fig. In this work, we introduced a new kind of phase grating with controlled intensity among different diffraction orders. We constructed an AMLPG by adding an azimuthal periodic dependency to the argument of the transmission function of a linear phase grating having a sinusoidal profile. Generation of diverse radial carpet beams was investigated over different diffraction orders of an AMLPG with controlled predominant diffraction order. A detailed theoretical analysis was reported and its experimental verification was presented by generating diverse radial carpet beams with controlled shift of intensity in the illumination of an AMLPG with a spatially coherent light beam. We specified diverse radial carpet beams produced over different diffraction orders of the host linear phase grating. It was shown that all the diffraction patterns are different and only the pairs of positive and negative orders with the same order numbers are similar except that there is a relative rotation between them. The diffraction grating introduced with controlled intensity sharing among different diffraction orders might find wide applications in many areas of optics such as optical switches. Also, the radial carpet beams produced over different diffraction orders might find applications in light sheet microscopy.