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We are introducing a new class of regressions called Taylor Regression and we are using it to create a unified method that can estimate a root, an extreme and an inflection point when we have discrete functional data. The method works with good accuracy for both simple and noisy data, while it is not necessary to use equidistant abscissae.

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... n} we want to find the roots, extremes and inflections for it. A general method for computing all above has been presented in [1], which uses an optimization procedure after excessive regressions for a properly defined Taylor Polynomial. We want to avoid regressions due to the many problems related with them (matrix inversions, multicollinearity) and focus on the geometric properties. ...

... But if we just have only a set of discrete (x i , y i ) points there is not any known method for root or extreme finding without usage of either interpolation (for not noisy) or regression modeling (for noisy curves). A mixed strategy called 'Taylor Regression' can be found at [1] which can find roots, extrema and inflections for all kind of discrete data sets, but even that strategy uses extensive regressions with all known issues related to needed matrix inversions. ...

By classifying planar curves under their symmetry around extreme points we present the 'tulip' and 'bell' methods for extreme computation. If curve is not symmetric, then we develop the 'integration' method for estimating extremes, which uses BESE or BEDE methods for inflection finding. After observing that absolute value of a function creates a set of local edge minima in the vicinity of roots, we use 'integration' method for estimating roots. All methods are based on the trapezoidal rule for numerical integration which gives consistent estimations for surfaces and consequently makes introduced estimators statistically consistent. New methods can constitute the core of a new discipline called 'Noisy Numerical Analysis'.

... The relationships between the estimated genetic gains with selection as a function of the strategies ranked by B V were analyzed graphically. In the curve obtained, the inflection point was found using the Taylor regression procedure (Christopoulos 2017). From the inflection point, the strategies were separated and characterized in conservative and non-conservative. ...

For continuous genetic gains over time, a balance between genetic gain and maintaining the genetic base must be a constant concern of forest breeders. This study aims at determining the best thinning strategies for a population of Eucalyptus dunnii, by incorporating the effects of environmental heterogeneity and competition in the analysis, as well as the best growth trait regarding precision and accuracy. The population studied consisted of 160 open-pollinated families. The survival and growth (height, HT; diameter at breast height, DBH; and volume, VOL) were evaluated 4 years after planting. The growth rate data were analyzed and compared by four mixed models. Selection and thinning strategies were simulated by varying the number of families, individuals within families, and selected individuals, considering the estimated genetic gains and the effective size. The species showed good survival (89.7%) and productive performance (mean annual increment = 42 m³ ha⁻¹ y⁻¹). The Spatial+Competition Model provided the best fit for DBH and VOL. The strategies that allow a balance between improvement (genetic gains) and genetic conservation (effective size) consist of keeping 36 to 50% of the individuals in the test (370 to 510 trees ha⁻¹), by reducing more intensively the number of individuals from the worst-performing families. The selection of 100 individuals with a restriction of at most one individual per family generates the largest number of effective size (Ne), with more than double the Ne obtained without restricting the individuals per family, with a small drop in genetic gain.

... Accuracy (objective performance) was treated as a nonlinear function of the SOA and the point of the fastest increase in accuracy, corresponding to the inflection point is the access threshold to conscious processing, characterizing the transition between unconscious (i.e., subliminal and/or preconscious processing in the existence of the lack of awareness) and conscious processing (Del Cul, Baillet, & Dehaene, 2007). To obtain access threshold to conscious processing (inflection point) for both each person and each group, we used nonlinear regression to estimate the objective performance as a function of SOA using Taylor regression estimator which is available in the RootsExtremaInflections package of R (Christopoulos, 2014) and can work for discrete and noisy values. In this method, Taylor polynomial of a smooth function f around a point ρ is used: ...

Although multiple sclerosis (MS) is frequently accompanied by visuo‐cognitive impairment, especially functional brain mechanisms underlying this impairment are still not well understood. Consequently, we used a functional MRI (fMRI) backward masking task to study visual information processing stratifying unconscious and conscious in MS. Specifically, 30 persons with MS (pwMS) and 34 healthy controls (HC) were shown target stimuli followed by a mask presented 8–150 ms later and had to compare the target to a reference stimulus. Retinal integrity (via optical coherence tomography), optic tract integrity (visual evoked potential; VEP) and whole brain structural connectivity (probabilistic tractography) were assessed as complementary structural brain integrity markers. On a psychophysical level, pwMS reached conscious access later than HC (50 vs. 16 ms, p < .001). The delay increased with disease duration (p < .001, β = .37) and disability (p < .001, β = .24), but did not correlate with conscious information processing speed (Symbol digit modality test, β = .07, p = .817). No association was found for VEP and retinal integrity markers. Moreover, pwMS were characterized by decreased brain activation during unconscious processing compared with HC. No group differences were found during conscious processing. Finally, a complementary structural brain integrity analysis showed that a reduced fractional anisotropy in corpus callosum and an impaired connection between right insula and primary visual areas was related to delayed conscious access in pwMS. Our study revealed slowed conscious access to visual stimulus material in MS and a complex pattern of functional and structural alterations coupled to unconscious processing of/delayed conscious access to visual stimulus material in MS.

... Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publications, https://www.researchgate.net/publication/ 261562841 Examples #Load data: data(xydat) # #Extract x and y variables: x=xydat$x;y=xydat$y # #Find root, plot results, print Taylor coefficients and rho estimation: b<-rootxi(x,y,1,length(x),5,5,plots=TRUE);b$an;b$froot; # #Find extreme, plot results, print Taylor coefficients and rho estimation: c<-extremexi(x,y,1,length(x),5,5,plots=TRUE);c$an;c$fextr; # #Find inflection point, plot results, print Taylor coefficients and rho estimation: d<-inflexi(x,y,1,length(x),5,5,plots=TRUE);d$an;d$finfl; # Create a relative big data set... f=function(x){3*cos(x-5)};xa=0.;xb=9; ...

Implementation of the Taylor Regression Estimator method described in Christopoulos (2014, https://www.researchgate.net/publication/261562841) for finding the root, extreme or inflection point of a curve, when we only have a set of probably noisy xy points for it. The method uses a suitable polynomial regression in order to find the coefficients of the relevant Taylor polynomial for the function that has generated our data. Installation: install.packages('RootsExtremaInflections')

... We used model predictions and fitting linear models to fit curve (Legendre and Legendre 1998). We have calculated inflection points by using a proper Taylor regression procedure (Demetris 2014) in Package RootsExtremaInflections of R (Online Resource 2). ...

This paper aims at detecting the relationships between phytogeographical patterns of genera of Chinese endemic seed plants and latitude or climatic factors. The landmass of China was divided into four latitudinal zones, each of c. 8°. Based on a total of 1664 indigenous genera of Chinese endemic seed plants which were grouped into fifteen geographical elements, belonging to three major categories (cosmopolitan, tropical and temperate) and which were absent or present in 28 provinces in China, we analyzed the phytogeographical patterns of genera of Chinese endemic seed plants and detected the relationships between them and main climatic factors. Our results showed that the proportion of tropical genera decreases with the increase in latitude; the proportion of temperate genera increases with the increase in latitude; and the proportion of cosmopolitan genera increased gradually increased gradually with latitude. There are a slow decrease in the proportion of tropical genera and slow increase in the proportion of temperate genera at latitudes 35°–40°. Alternatively, the tropical genera and the temperate genera have the same proportion at latitude c. 25°. These changes and issues about the different genera also appeared in main climate factors. In general, the genera present in a more northerly flora are a subset of the genera present in a more southerly flora. In summary, the large-scale patterns of phytogeography of endemic flora in China are strongly related to latitude, which covary with several climatic variables such as temperature.

... Demetris T. Christopoulos (2014), Roots, extrema and inflection points by using a proper Taylor regression procedure, ResearchGate publications, https://www.researchgate.net/publication/ 261562841 ...

Implementation of the Taylor Regression Estimator method described in Christopoulos (2014,<https://www.researchgate.net/publication/261562841>) for finding the root, extreme or inflection point of a curve, when we only have a set of probably noisy xy points for it. The method uses a suitable polynomial regression in order to find the coefficients of the relevant Taylor polynomial for the function that has generated our data.

We are introducing two methods for revealing the true inflection point of
data that contains or not error. The starting point is a set of geometrical
properties that follow the existence of an inflection point p for a smooth
function. These properties connect the concept of convexity/concavity before
and after p respectively with three chords defined properly. Finally a set of
experiments is presented for the class of sigmoid curves and for the third
order polynomials.

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.