Parametric and nonparametric methods have been developed for purposes of predicting phenotypes. These methods are based on retrospective analyses of empirical data consisting of genotypic and phenotypic scores. Recent reports have indicated that parametric methods are unable to predict phenotypes of traits with known epistatic genetic architectures. Herein, we review parametric methods including least squares regression, ridge regression, Bayesian ridge regression, least absolute shrinkage and selection operator (LASSO), Bayesian LASSO, best linear unbiased prediction (BLUP), Bayes A, Bayes B, Bayes C, and Bayes Cπ. We also review nonparametric methods including Nadaraya-Watson estimator, reproducing kernel Hilbert space, support vector machine regression, and neural networks. We assess the relative merits of these 14 methods in terms of accuracy and mean squared error (MSE) using simulated genetic architectures consisting of completely additive or two-way epistatic interactions in an F2 population derived from crosses of inbred lines. Each simulated genetic architecture explained either 30% or 70% of the phenotypic variability. The greatest impact on estimates of accuracy and MSE was due to genetic architecture. Parametric methods were unable to predict phenotypic values when the underlying genetic architecture was based entirely on epistasis. Parametric methods were slightly better than nonparametric methods for additive genetic architectures. Distinctions among parametric methods for additive genetic architectures were incremental. Heritability, i.e., proportion of phenotypic variability, had the second greatest impact on estimates of accuracy and MSE.
Complex quantitative traits are measured on a continuous scale and are controlled by a network of many genes, by the environment, and by genetic by environment interactions. Most traits of economical interest in agriculture (e.g., grain yield) are measured on continuous scales, i.e., they are quantitative. Understanding the complexity of these traits and accounting for the effects that are contributed by these genes and their interactions is not trivial.
The gene-by-gene interaction or epistasis is an important research topic in quantitative genetics. Epistasis can be modeled in different ways (Cordell 2002). Physiological epistasis is the difference in the phenotype when the genotype at a locus is influenced by the genotype at another locus or loci. Fisher (1918) defined epistasis as the deviation of the genotypic value from the contribution of the sum of additive effects at all functional loci in the genome. Fisher's definition of epistasis is also known as statistical epistasis and has been used to quantify deviations from independence (Wilson 2004). Epistasis has an important role in accounting for the genetic variation for quantitative traits, and excluding it from the prediction equations for simplicity can result in poor predictions of genetic gain (Cooper et al. 2002).
Most simulation studies of genomic selection (GS) methods (Meuwissen et al. 2001) have considered genetic architectures in which the number and relative magnitudes of quantitative trait loci (QTL) have varied. To our knowledge, no studies of GS methods have considered epistatic genetic architectures, although Gianola et al. (2006) predicted nonparametric methods would be better-suited for epistatic genetic architectures. Although theoretic models predict a significant role for epistasis in speciation (Dobzhansky 1937; Mayr 1942), adaptation (Lewontin 1974; Wade 2000), and canalization (Waddington 1949; Rice 1998), there is little empirical evidence from biometric studies of significant epistatic contributions to genetic variability. Biometric approaches, however, average across epistatic genotypic values at individual loci and contribute primarily to additive genetic variance (Cockerham 1954; Cheverud and Routman 1995). With development of low-cost high-throughput marker technologies, it has become possible to estimate epistatic interactions based on genotypic values for all possible pairwise genotypes in genome-wide association studies, although searches for higher-order interactions are still limited by experimental and computational resources (Moore and Williams 2009). These studies are beginning to reveal that epistasis is not the exception, but rather the most prevalent form of genetic architecture for quantitative traits (Flint and Mackay 2009; Huang et al. 2012). Nonetheless, it was hypothesized that GS should provide accurate predictions because epistatic gene action will be translated primarily into additive genetic variance (Crow 2010). Thus, for purposes of this study, we decided to evaluate GS methods for an extreme case of epistasis with 10 pairs of loci, each consisting of two alleles at equal frequencies and modeled using the principle of orthogonality (Goodnight 2000).
The development of DNA markers in the 1980s was an important step in the process of identifying DNA segments that are statistically associated with quantitative traits, i.e., QTL mapping and for marker-assisted selection (MAS). In MAS, markers and phenotypic information are used to guide indirect selection of a trait of interest (Fernando and Grossman 1989). This approach is considered an improved and more efficient method for selection in plant breeding relative to phenotype pedigree–based approaches (Mohan et al. 1997). Extensive resources have been devoted to develop QTL mapping methodology as a component of MAS (Young 1996; Melchinger et al. 1998). Marker-assisted backcrossing (MABC) is one of the simplest examples of MAS. In MABC, genomic regions defined by markers closely linked to QTL are identified. These genomic regions are then introgressed into the elite lines through backcrossing (Bernardo 2010). In MABC, a plant with a desired gene, called a donor parent, is crossed with an elite or breeding line, called a recurrent parent. The goal is to introgress the desired gene into the genome of the recurrent parent (Visscher et al. 1996). Developing varieties can also involve accumulating multiple desired genes into a recurrent parent. The marker-assisted process for alleles at multiple loci is called gene pyramiding. MAS is widely used in gene pyramiding because the use of molecular markers gives the advantage of selecting the desired plants without extensive phenotyping. With traditional phenotyping, it is often impossible to distinguish among plants with all desirable alleles and the plants with some of the desirable alleles (Huang et al. 1997).
MAS has been shown to be efficient and effective for traits that are associated with one or a few major genes with large effect but does not perform as well when it is used for selection of polygenic traits (Bernardo 2008). QTL detection also results in some false-negative and false-positive rates, and further QTL mapping does not guarantee that estimates of genetic effects are correct (Beavis 1994). Also, for MAS to be useful, the interaction between the QTL and the genetic background has to be minimal, so the QTL has the same effect in different genetic backgrounds (Bernardo 2010 p. 223). The genetic background of an organism refers to all of its alleles at all loci that can interact with the locus where the QTL is located (Yoshiki and Moriwaki 2006).
The parametric models and statistical methods introduced for QTL mapping and MAS do not address genetic improvement for quantitative traits that are influenced by a large number of genes with small effects. Some of the statistical challenges arising in MAS include the specification of threshold for multiple testing, the "large p, small n" problem [which refers to the situation when the number of predictors, p (marker data points) greatly exceeds the number of individuals, n, that have been evaluated in the study], difficulty of interpretation of effects due to collinearity among the explanatory/predictor variables, model assumptions that cannot be satisfied, and nonadditivity among genetic effects.
With advanced molecular techniques that provide dense marker maps, it is possible to overcome some shortcomings of MAS. Meuwissen et al. (2001) proposed predicting the genotypic value for individuals using all marker information simultaneously. Their proposed method and the subsequent derivative methods have been referred to as GS. They modeled the associations between the markers and a phenotype focusing on calculating a breeding value for an individual (which can be calculated as the sum of the average effect of the alleles for the individual's genotype) instead of identifying significant marker–trait associations. In their approach they estimated the effect of each QTL and then used the sum of all estimates to calculate a genotypic value for the individual.
In GS, individuals with both phenotypic and marker information (called the training set) are used to model the association between the phenotype and the genotype. The model is used to predict the phenotypic value of individuals for which only the marker information is available (called the validation set or testing set). In GS, all available markers are included in the model, not just those above a significant threshold, thus eliminating the problem of multiple testing.
Effort is underway to find ways to model epistasis, the gene-by-gene interaction. In the presence of epistasis, the effect of one locus changes the effect of another locus on the phenotype. Usually several loci are involved, which means that multiway interactions may need to be modeled. Because the volume of marker data points available is huge, the number of epistatic interactions can be overwhelming and computationally intractable to estimate with parametric methods (Moore and Williams 2009).
More recently, Gianola et al. (2006) stated that parametric approaches to GS have several drawbacks. The parametric model assumptions do not always hold (e.g., normality, linearity, independent explanatory variables), which suggests the use of nonparametric methods. Also, the convenient partitioning of genetic variance into additive, dominance, additive × additive, additive × dominance, etc., only holds under conditions of linkage equilibrium, random mating of male and female parents, no inbreeding, no assortative mating, no (natural or artificial) selection, and no genotyping errors. In breeding programs, these conditions are all violated. Gianola et al. (2006) proposed nonparametric and semi-parametric methods to model the relationship between the phenotype and the markers that are available within the GS framework. Gianola et al. (2006) proposed nonparametric methods capable of accounting for complex epistatic models without explicitly modeling them.
Herein, we review some existing statistical methods used in GS. First, we discuss the parametric methods in more detail. Then, we focus on the nonparametric and semi-parametric methods. Among parametric methods, we review linear least squares regression, penalized ridge regression, Bayes ridge regression, least absolute shrinkage and selection operator (LASSO), and Bayes LASSO methods, best linear unbiased prediction (BLUP), and some Bayesian alternatives used in GS (Bayes A, Bayes B, Bayes C, and Bayes Cπ). We also explain the nonparametric kernel regression using the Nadaraya-Watson estimator (NWE) and the semi-parametric reproducing kernel Hilbert space (RKHS) regression. Finally, we describe support vector machine (SVM) regression and neural networks (NN) applications to GS. de los Campos et al. (2013) give an overview of some of the parametric methods used in GS, and Gianola et al. (2010) provide information about some of the nonparametric models used in GS. Heslot et al. (2012) compared some parametric and nonparametric GS methods. However, they did not consider epistatic genetic architectures in their simulated data. Daetwyler et al. (2010) discussed the impact of the genetic architecture in GS, but they defined genetic architecture by the effective population size and the number of QTL.
Here, we use simulated data to compare the performance of the parametric models with the nonparametric procedures for predicting the genetic value for individuals in a F2 and a backcross (BC) populations. We simulate the F2 and the BC populations with low and high heritabilities and compare the two extreme genetic architectures. One architecture had only additive genetic effects from alleles at 30 loci, and the other had only two-way epistatic genetic effects among 30 loci. The performance of the methods is illustrated by comparing the accuracy of prediction, which we defined by the correlation between the true phenotypic value and the predicted phenotypic value and the mean squared error (MSE). We demonstrate the advantage of some nonparametric methods for the epistatic genetic architecture. Because the results for the F2 and BC populations were similar, we only illustrate the F2 population in this article. In the Supporting Information, we provide accuracy and MSE values for a simulated BC population with low and high heritabilities and with two extreme genetic architectures.
. For more details about linear models, the reader can refer to Linear Models in Statistics (Schaalje and Rencher 2000) or Linear Models with R (Faraway 2006). The elements of the design matrix X depend on the number of different alleles present. For example, individuals having marker genotypes AA, Aa, aa have elements coded as −1, 0, and 1 in Xij, respectively.
One obvious problem with linear regression is that usually the number of markers (explanatory variables) available is much greater than the number of individuals with phenotypic information (response variables), which means that p is much greater than n, and it is impossible to perform the estimation. Using a subset of the markers can be an alternative (using a variable selection method like the forward, backward, or stepwise selection procedure) (George 2000), but it can still perform poorly if the relative ratio of the number of markers and the number of individuals is large or has multicollinearity, e.g., linkage disequilibrium (LD) exists among the markers.
, where q is the number of QTL. With this approach, they eliminated the problem of having more predictor (explanatory/independent) variables than regressands (response/dependent variables), but it does not fully take advantage of the whole marker information because only markers with a significant effect are included in the final model. To overcome some of the drawbacks of the linear regression approach, other methods for GS have been introduced.
is the vector of least squares estimates of regression coefficients.
represents additive genetic variance expressed among individuals and nk is the number of marker loci (Habier et al. 2007).
It can be shown that ridge regression is a special case of the BLUP (Ruppert et al. 2003), which we demonstrate after introducing BLUP. Thus, the mixed linear model can be implemented. Within the mixed model context, the restricted maximum likelihood (REML) estimation is a good choice for finding a reasonable value for the penalty parameter and estimating the variance components (Henderson 1988). Piepho (2009) discusses some models that feature the ridge regression in terms of mixed models and uses REML for variance and penalty parameter estimation in GS.
The BLUP theory and the mixed model formulation were first discussed by Henderson (1949), and they were influential for selection purposes in animal breeding (Henderson 1959). BLUP is a statistical procedure, and it is useful in situations when the data available are unbalanced (for example, in different locations the number of individuals is not the same), and it can accommodate family information (Bernardo 2010). Since Henderson's first work in BLUP, the theory has been widely expanded (Henderson 1959, 1963, 1975a, 1975b; Harville 1976). Since the 1990s, BLUP has been used not only in animal breeding applications (Henderson 1984) but also in plant breeding (Bernardo 1994).
. In addition within the unbiased linear predictors, it is the best in the sense of minimizing the MSE. BLUE is similar to BLUP in that it is a linear function of the data y, it is unbiased among the linear estimators, and it is best in the sense that it minimizes the MSE.
where R = Var(e) and G = Var(m). The solution to the MME is the BLUE of β and the BLUP of m. Henderson's derivation assumes that m and e are normally distributed and maximizes the joint likelihood of (y, m) over the unknowns β and m. Maximizing the likelihood implies an optimization criterion of (y − Xβ − Zm)′R−1(y − Xβ − Zm) + m′G−1m, and it can be viewed as the "ridge regression formulation" of the BLUP (Ruppert et al. 2003).
. This constraint shrinks some of the marker effects and sets some of them to zero. One of the major differences between LASSO and ridge regression is that in LASSO as we increase the penalty, more marker effects will shrink to zero and in ridge regression all parameters will be reduced but still remain nonzero.
. Repeat steps 2 and 3.
The algorithm described above is computationally intensive. Efron et al. (2004) proposed a new model selection algorithm called least angle regression (LARS) that can be used in combination with LASSO estimation. LARS is similar to the traditional forward selection method. It starts with all the coefficients (marker effects) at zero. First, the marker that has the highest correlation with the phenotypic values is added into the model. The next marker added has to have a correlation with the residual that is at least as large. The third marker entered into the model is equiangular with the first two markers already in the model. At each iteration, a new marker is added, and the algorithm is accomplished in p iterations where p is the number of the available markers. However, for LASSO, the LARS procedure is modified. Because the LASSO has a constraint, the LARS procedure has to apply a restriction, so this model selection method is more closely related to the stepwise selection method. For a detailed description of LARS and the LARS–LASSO relationship, the reader can refer to Efron et al. (2004).
One other important question is how to find the upper bound of the sum of the absolute value of the marker effects, s. Finding the best value for s can be viewed as the selection of the size of the best subset of markers. Usai et al. (2009) used the cross-validation approach of Kohavi (1995) with random subsampling replication. In every replication, the data are randomly divided into a training set and a validation set. The training set is used to estimate the marker effects using the LARS algorithm for the LASSO method. The estimated marker effects were used to calculate the genomic breeding values (GEBV) for the individuals in the validation set, and then the correlation coefficients between the GEBV and the true phenotypic value were reported. The LARS iterations were carried forward until the maximum correlation was reached.
Inferences about model parameters are based on the posterior distribution. By Bayes' Theorem, the posterior is obtained by combining the prior distribution and the likelihood function. For detailed information about Bayesian methods, the reader can refer to Kruschke (2010) or Gelman et al. (2003).
where nj is the number of haplotype effects at marker position j.
The Bayes B approach seems more realistic for GS than Bayes A. The only difference between the two methods is the prior for the variance components. Bayes B assumes that not all markers contribute to the genetic variation. It has a prior density on the variance that is a mixture. It has a high probability mass at σmj = 0 and an inverted chi-square distribution when σmj > 0. It can be summarized as σmj = 0 with prob= π and σmj ∼ χ−2(ν, S) with prob=(1 − π).
is the data that are corrected for the mean and for all genetic effects except mj.
Using simulated data, it was shown that the Bayesian methods perform better in terms of prediction accuracy than the linear least squares regression, the ridge regression, and the BLUP method (Meuwissen et al. 2001; Habier et al. 2009, 2010). However, as Gianola et al. (2009) pointed out, the choice of the degrees of freedom and the scale parameters of the scaled inverse chi-square distribution can influence the outcome. Improved Bayesian methods were developed by Habier et al. (2011) to deal with the weakness of Bayes A and Bayes B. Bayes C uses a common variance for all SNPs, and for Bayes D the scale parameter of the scaled inverse chi-square distribution is estimated instead of specified by the user. Bayes Cπ and Bayes Dπ (Habier et al. 2011) are the modification of Bayes C and Bayes D where the probability of having a zero effect SNP π is estimated.
. In the LASSO setting, λ controls the penalty for minimizing the MSE.
In this section, we review some of the nonparametric estimation methods that have been proposed for the case where the form of the relationship between a response variable and a set of predictors is unknown. A popular approach, at least in terms of usage, is based on the kernel method proposed by Silverman (1986) in the context of density estimation. In that context, the goal is to estimate the unknown density using a smooth curve (Schucany 2004). The kernel method is the most commonly used nonparametric estimation procedure.
. The window width provides information about the range of observations that are included (Sheather 2004). Figure 1 shows how the kernel density estimation changes with different bandwidth values. Using simulated data from a mixture of two normal distributions, the second, third, and fourth panels show how the estimation changes with the change of the bandwidth value.
The influence of the bandwidth in kernel density estimation. From left to right, the first plot shows simulated data from a mixture of two normal distributions. The second, third, and fourth plots show the Gaussian kernel density estimates using bandwidth values h = 0.1, h = 2, and h = 10.
When h = 0.1, the data have strong influence on the density estimate, resulting in little bias and large variability among estimates. It is called an undersmoothed estimate. As we increase the bandwidth value, the estimates become smoother. When h = 10, the spread is too big and even the bimodal feature of the data disappears, which implies that the estimate is oversmoothed. Setting the bandwidth too large results in a large bias with little variance.
(8)i = 1, 2, …, n where yi phenotypic measurement on individual i, xi is a p × 1 vector of dummy SNP covariates observed on individual i, g(.) is some unknown function relating genotypes to phenotypes, g(xi) = E(yi|xi), and ei is a random residual effect for individual i where ei ∼ (0, σ2) and is independent of xi.
The estimator is just a weighted sum of the observations yi, i = 1…n and is called the NWE.
The CV estimate of h is the value of h that minimizes CV(h).
is the additive genetic variance and A is the additive relationship matrix.
, and then form semi-parametric draws of the total genetic value.
where K(.,.) is the reproducing kernel.
SVM was proposed by Vapnik and discussed by Cortes and Vapnik (1995). SVM is a supervised learning technique that was originally developed as a classifier. A training data set is used to develop a maximum margin classifier that produces the largest possible separation between two classes of observations. In the linearly separable case, if observations (xi) ∈ Rp, then the separator is a hyper-plane in Rp−1.
Because fitting a regression model essentially consists of finding an optimal projection of the observations on a lower-dimensional hyper-plane, the idea can be used to estimate the unknown regression function subject to restrictions. The reader can refer to Hastie et al. (2009), Steinwart and Christmann (2008), and Christianini and Shawe-Taylor (2000) for a review of SVM. SVM regression was adopted by Maenhout et al. (2007) and Long et al. (2011) for GS in plant breeding. A nice feature of SVM regression in plant breeding applications is that the relationship between the marker genotypes and the phenotypes can be modeled with a linear or nonlinear mapping function that takes samples from a predictor space to an abstract, multidimensional feature space (Hastie et al. 2009).
. L(.) denotes the loss function that measures the quality of the estimation. The regularization parameter λ quantifies the trade-off between the sparsity and the complexity of the model. Increasing λ implies a higher penalty on the error. The norm ‖w‖ of vector w is inversely associated with model complexity; by choosing w to minimize ‖w‖, we reduce model complexity.
There are many loss functions used for SVM regression. Some of the popular loss function choices include the squared loss, absolute loss, and the ε-insensitive loss. Here, we present these loss function formulations.
1. The squared loss function has the form L(y − f(x)) = (y − f(x))2. It scales the loss quadratically by the size of the error. Using this loss function indicates that outliers are also weighted quadratically, which requires the user to deal with the outliers before the regression analysis.
2. The absolute loss function has the form L(y − f(x)) = |y − f(x)|. The absolute loss function scales the loss linearly by the size of the error eliminating the difficulty of using data sets with outliers.
where ε determines the number of support vectors used in the regression function. By definition (Vapnik 1995; Vapnik and Vashist 2009), a support vector is a vector xi that satisfies the equation yi(wxi + b) = 1. Increasing ε implies that fewer support vectors are used in the fitting. The ε-insensitive loss function ignores the errors in the regression that have size less than ε. When the error is greater than ε, the loss is |y − f(x)| − ε.
Figure 2 illustrates the absolute loss, squared loss, and ε-insensitive loss functions as a function of the error y − f(x).
Loss functions used for SVM regression. The first panel shows the absolute loss function. The second panel is the square loss function, and the last panel is the ε-insensitive loss function.
In the remainder, we focus on the ε-insensitive loss function, which needs a more robust representation to account for the noise in the data.
ξ2i ≥ f(xi) − yi − ε, where i = 1, …, n.
(Nocedal and Wright 1999). The solution depends on the training data through the inner product 〈xi, xj〉, which is a linear function of the observations.
3. The Laplace radial basis function k(x, z) = exp(−σ‖x − z‖).
. The choice of the kernel function and of the tuning parameters λ, ε, and σ are not straightforward. Because optimizing SVMs is not the focus of this article, we refer the reader to Cherkassky and Ma (2004).
NNs represent a nonparametric prediction procedure that captures additivity and epistasis by being able to model linear and complex nonlinear functions. The original idea of NN came from the theory of how neurons in the human brain work and interact, and how the brain conducts computations. In the NN, every unit is analogous to a brain neuron and the connections between them are analogous to synapses (Hastie et al. 2009). The first introduction of NNs in the context of brain architecture was presented by Bain (1873) and James (1890). McCulloch and Pitts (1943) developed a mathematical model for NNs.
The basic layout of the NN is a two-stage network with three types of layers: an input layer; a hidden layer; and an output layer. This model is called the feed-forward NN and is illustrated in Figure 3.
A three-layer feed-forward neural network with K input layer units, L hidden layer units, and M output layer units.
(12)where K is the number of units in the input layer, Ik is the kth input, β0 ε RM is the intercept (bias terms), M is the number of output layer units, L is the number of hidden layer units, βl (l = 1, 2, …, L) are the output layer weights connecting the lth hidden layer unit to the output layer units, σ is the activation function modeling the connection between the hidden layer and the output layer, and wl ε RK and bl ε R are the unknown learning parameters of the hidden layer unit l (l = 1, 2, …, L) connecting the kth neuron in the input layer to them (Romero and Alquézar 2012).
In GS, typically Ik represents a vector of predictors (marker genotypes or other information) collected on individual k (k = 1, 2, …, K), where K is the number of individuals in the analysis. The activation function σ is typically chosen to be the sigmoid (logistic) or the Gaussian radial basis function.
Gianola et al. (2011) implemented NNs for GS using two real data sets. Other examples are shown in work by Lampinen and Vehtari (2001) and Titterington (2004).
For the purpose of illustrating the parametric and nonparametric prediction approaches, simulated data were created by the R (R Development Core Team 2008) package QTL Bayesian interval mapping ("qtlbim") (Yandell et al. 2012). R can be downloaded from http://www.r-project.org, the qtlbim package can be accessed by library(qtlbim) in R, and the description of the package can be found at http://cran.r-project.org/web/packages/qtlbim/qtlbim.pdf. The reader can refer to Yandell et al. (2007) for detailed information about the qtlbim package. There are other publications as well where the qtlbim package is used to implement statistical methods. Some examples include Yi et al. (2007), Yi and Shriner (2008), and Piao et al. (2009). For comparing methods, we used a simulated F2 population with specifications listed in Table 1.
We simulated four sets of phenotypic and genotypic information for a F2 and a BC population. The results for the BC population can be found in the supporting information section. For each set we created 20 replicates, which yielded to a total of 80 phenotypic and 80 genotypic data sets. Within each replicate we created 25 different training–testing data sets. Half of the data sets assume only additive effects and half assume only epistatic effects without any additive effects. We only evaluated the two extreme genetic architectures. Finally, for each genetic architecture, we generated data with two different narrow sense heritabilities. The low heritability was determined to be 0.30, and the high heritability was 0.70. For each of the simulated combinations of population, genetic architecture, and heritability, the data contain phenotypic information for 1000 individuals and genotypic information for 2000 biallelic markers (the possible values coded as "A" and "H") for each individual. Out of the 1000 individuals, 800 were chosen randomly to be in the training set to fit the model, and 200 individuals were in the testing set. We predicted the phenotype for the individuals in the testing set. The qtlbim package uses Cockerham's model as the underlying genetic model. The simulated genome has 10 chromosomes, each having a specified length. The 2000 markers were distributed throughout the genome in such a way that each chromosome had 200 markers and the markers were equally spaced over the chromosomes. We assumed no missing genotypic values and no missing phenotypic values. The phenotypic values are normally distributed.
For the additive model, we placed two QTL on each of the 10 chromosomes with either positive or negative additive effect. For the additive model we assumed no epistatic interaction.
For the epistatic model, we only considered two-way interactions between the QTL. The interacting QTL were at the same genomic location as the QTL for the additive model, and only neighboring QTL were associated with each other, resulting in 10 two-way epistatic interactions, with each having either positive or negative epistatic effect on the phenotype. For the epistatic model, we assumed that the QTL contributed no additive effect. The phenotypic values were drawn from a normal distribution and are based on the P = G + E model. Figure 4 shows the histograms of the simulated phenotypic values for the four population–genetic architecture–heritability combinations.
The histograms of the simulated phenotypic values. The histograms represent the distribution of the phenotypic values for the F2 population.
To compare the performance of the methods, we used cross-validation, where we divided the data into training sets and testing sets. The training sets were used to fit the models, and the testing sets were used to determine the performance of the particular method. The performance of the methods was calculated by the accuracy of prediction and the MSE. We define accuracy of prediction as the correlation between the true phenotypic values and the predicted phenotypic values. We evaluated parametric methods including parametric least squares regression, ridge regression, Bayesian ridge regression, BLUP, LASSO, Bayesian LASSO, Bayes A, Bayes B, Bayes C, and Bayes Cπ. We also evaluated nonparametric methods, including NWE, RKHS method, SVM, and NN. To implement the parametric and nonparametric methods, the statistical software R and software written in C++ provided by the Animal Science Department at Iowa State University were used. Specifications of the parameters and inputs for each method are described below.
Because the number of markers exceed the number of the individuals in all simulated data sets, the idea of Meuwissen (2001) was adopted, and we first performed simple regression by coding each marker genotype as −1, 1. After fitting the 2000 simple linear models, one for each marker, we chose 300 of the markers with the most significant p-values. Then, these 300 markers were included into a final model, and we simultaneously used them to fit a linear model. To perform the linear regression, the lm function was used that can be found in the stats package (R Development Core Team 2008) in R. Finally, the prediction for the testing set was performed by using the marker data for the testing set and the output for the estimated marker effects provided by the lm function.
, where h2 is the narrow sense heritability and Var(X) is the sum of the 2000 marker variances. For all of the scenarios, we used h2 = 0.4.
is assigned to have a scaled inverse χ2 distribution with degrees of freedom dfBR = 5 and scale parameter SBR = 0.01. The residual variance σE has a scaled inverse χ2 distribution with degrees of freedom dfE = 4 and scale parameter SE = 1. The BLR function implements the Gibbs sampler, and the number of iterations is specified to be 20,000. We used 2000 iterations for the burn-in period without any thinning. To fit the Bayesian ridge regression, we used all available phenotypic and genotypic data.
To implement BLUP, we used the mixed.solve function in R that can be found in the rrBLUP package (Endelman 2011). The available marker data were used as the design matrix for the random marker effects, and there was no fixed effect specified. The prediction was performed using the marker data for the testing set and the output for the predicted marker effects provided by the mixed.solve function.
To predict phenotypic values in the testing set using the LASSO method, we used the glmnet function of the glmnet package (Friedman et al. 2010) in R. For the initial parameter values, the default setting was applied. The prediction was performed with the tuning parameter, λ, that minimized the average cross-validation error (cvm).
To fit the Bayesian LASSO method, the function BLR of the BLR package (de los Campos and Rodriguez 2010) in R was used. The regularization parameter, λ, is specified to be random and has a Gamma prior distribution with shape parameter α1 = 0.53 and rate parameter α2 = 0.00005. The residual variance σE has a scaled inverse χ2 distribution with degrees of freedom dfE = 4 and scale parameter SE = (df − 2)(1 − h2)Var(y), where we specify that df = 4, h2 = 0.5, and Var(y) is the phenotypic variance. The Gibbs sampler was applied with 20,000 iterations, and 2000 iterations were in the burn-in period. The chain was not thinned.
To implement the Bayes A, Bayes B, Bayes C, and the Bayes Cπ models, software called Gensel (version 2.12) was used. GenSel was written by Fernando and Garrick (2008) in C++ and is used for GS in animal breeding populations. The software is not available for the public. However, it is available to Iowa State University research collaborators working on GS.
In GenSel Bayes A, Bayes B, Bayes C, and Bayes Cπ have been implemented. We used the settings for the four methods that are listed in Table 2.
To use the NWE for predicting the phenotypic values in the testing set, first we formed the cross-validation criteria and we evaluated it on a grid of values. We examined the cross-validation criteria between 1 and 1000 and chose the value to be the bandwidth, h, that minimized the criteria. Table 3 shows the bandwidth values that minimized each of the four data combinations for the NWE prediction. The code for calculating the optimal bandwidth value and for the prediction was written in R.
The RKHS regression was based on methods and algorithms described by de los Campos et al. (2010b) and the R implementation was developed by Gustavo de los Campos et al. (2010b). To specify the RKHS regression, we chose the Gaussian reproducing kernel with the Euclidean distance for all of the eight combinations of genetic architectures, heritabilities, and population types. We fitted the model using three arbitrarily chosen bandwidth values. We performed the prediction for the testing set with each of the bandwidth values, and we averaged the three values of accuracy of selection and the three MSE values.
To implement the SVM regression, we used the ksvm function of the kernlab package (Karatzoglou et al. 2004) in R. For the ksvm function we used epsilon-regression as the type and the radial basis (Gaussian) kernel as the kernel function. After fitting the model, the predict function was used to perform the prediction of the phenotypic values for the testing set. For the other input parameters, the default values were used.
We implemented the NN model using the brnn function of the brnn package (Rodriguez and Gianola 2013) in R. This function fits a two-layer NN. We first map the input information into some basis function. Then, the inputs of the NN model are the marker-derived principal components. We specified the number of neurons to be three and the number of epochs to train to be 30 in the model. The other parameters were left at the default setting. For a detailed description of the application of the NN using the R package brnn, the reader can refer to Pérez-Rodiguez and Gianola (2013) and Pérez et al. (2013).
We compared 10 parametric and four nonparametric statistical GS methods. Comparisons were based on predicted accuracies of a simulated F2 progeny derived from crosses of inbred lines where genotypic variability was responsible for either 30% or 70% of the phenotypic variability. The underlying genetic architectures responsible for the genotypic variability consisted of 20 independently segregating biallelic loci that contributed equally either in an additive manner to a quantitative phenotype or through additive by additive epistatic interactions among 10 pairs of loci. Each GS method was applied to 20 sets of simulated progeny with 25 replicates for each of the four combinations of genetic architecture and heritability, which yielded 500 total replicates for each combination. Training sets were used to develop a model, and the model was used to predict phenotypes in the testing sets. Training sets consisted of simulated phenotypes and 2000 marker genotypes for 800 random progeny while the testing sets associated with the training sets consisted of the same information for 200 progeny derived from the same cross. The accuracy of prediction was determined by calculating the correlation between the predicted phenotypic values for the 200 individuals in the testing set with the simulated phenotypic values for the same 200 individuals. The MSE values were determined by calculating the sum of the squared differences between the 200 predicted phenotypic values in the testing set and the 200 simulated phenotypic values, and then dividing the sum by 200.
Table 4 and Table 5 report the average prediction accuracies and SE (sampling variabilities) of the 10 parametric and four nonparametric methods applied to the 500 replicates of the four combinations of genetic architecture and heritability. Table 6 and Table 7 report the average MSE values and SE of the MSE values of the 14 methods applied to the 500 replicates of the four combinations of genetic architecture and heritability. Figure 5, Figure 6, Figure 7, and Figure 8 each contain 14 boxplots of accuracy of prediction values for the 14 different methods. The boxplots show the distribution of the accuracy of prediction values for the 500 runs. Figure 9, Figure 10, Figure 11, and Figure 12 each contain 14 boxplots of MSE values for the 14 different methods. In each figure, the first 10 boxplots are for the parametric methods, and the last four (shaded) are for the nonparametric methods. These boxplots show the distribution of the MSE values for the 500 runs. The first plot of Figure 13 shows the ratio of the accuracy averaged over the parametric methods (excluding the least squares method because it is an outlier) and the accuracy averaged over the nonparametric methods, and the second plot of Figure 13 shows the ratio of the MSE averaged over the parametric methods (excluding the least squares method) and the MSE averaged over the nonparametric methods. The left sides of the plots show the ratios for the additive genetic architecture, and the right sides of the plots show the ratios for the epistatic genetic architecture. These summary plots clearly show the advantage of using nonparametric methods when epistasis is present. In both heritability scenarios, the parametric-to-nonparametric accuracy ratio is lower for the epistatic genetic architecture than for the additive genetic architecture. The parametric-to-nonparametric MSE ratio is higher for the epistatic genetic architecture than for the additive genetic architecture.
The boxplots of accuracy of prediction for the F2 population with additive genetic architecture and heritability of 0.70. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of accuracy of prediction for the F2 population with epistatic genetic architecture and heritability of 0.70. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of accuracy of prediction for the F2 population with additive genetic architecture and heritability of 0.30. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of accuracy of prediction for the F2 population with epistatic genetic architecture and heritability of 0.30. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of mean squared error for the F2 population with additive genetic architecture and heritability of 0.70. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of mean squared error for the F2 population with epistatic genetic architecture and heritability of 0.70. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of mean squared error for the F2 population with additive genetic architecture and heritability of 0.30. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
The boxplots of mean squared error for the F2 population with epistatic genetic architecture and heritability of 0.30. The first 10 boxplots correspond to the parametric methods, and the last four (gray) boxplots correspond to the nonparametric methods.
Plots of the parametric to nonparametric accuracy and MSE ratios. The left sides of the plots show the additive cases, and the right sides of the plots show the epistatic cases.
Genetic architecture responsible for the genetic contribution to the phenotypes had the greatest impact on differences of accurate predictions among the GS methods. If the genetic architecture for the trait is due to additive-by-additive epistasis among 10 pairs of independently segregating loci, then parametric GS methods are unable to predict the phenotypes in the testing sets (shown in Figure 6, Figure 8, Figure 10, and Figure 12). In contrast, nonparametric methods, particularly the NWE, the RKHS, and SVM, provided predictions that are reasonably accurate, especially for traits with higher heritabilities (shown in Figure 6 and Figure 8). Our results are consistent with the statement by Gianola (2006) that nonparametric methods should be able to better predict phenotypes that are based on genetic architectures consisting of epistatic interactions. If the underlying genetic architecture is additive, then parametric GS methods are slightly better than the nonparametric methods for both levels of heritability and types of segregating progeny. Both the accuracy of prediction and the MSE results suggest the same about the models in terms of predictive performance. When additive effects are present, the least squares regression performs the worst among the parametric methods, and the NWE performs the worst among the nonparametric methods (shown in Figure 5, Figure 7, Figure 9, and Figure 11). When epistasis is present, the nonparametric NWE, the RKHS, and the SVM perform significantly better than the parametric methods (shown in Figure 6, Figure 8, Figure 10, and Figure 12). Among the parametric methods, the least squares regression has the highest accuracy of prediction values when epistasis is present. However, least squares has the highest MSE values among the parametric methods as well when epistasis is present. It suggests that the least squares method estimates the QTL effects from both loci involved in the epistasis more accurately than the other parametric methods in the F2 population. The parametric methods other than the least squares are shrinking the QTL effects too much. Among the nonparametric methods, the NN showed poor predictive ability when epistasis is present. We know that NN is prone to over-fitting (Lawrence et al. 1997; Smith 1996), which would affect prediction ability. Most of the results are consistent with the fact that parametric approaches assume that the explanatory variables fitted in the model are independent. When we only simulate additive effects, but not epistasis, the markers are assumed to be independent. In this case, we satisfy the parametric model assumption of having independent explanatory variables, so the parametric models have a larger predictive power than the nonparametric models. However, when we simulate epistasis, the markers are dependent, which violates the parametric model assumption. Nonparametric models can handle epistatic models without explicitly modeling the interactions.
Recently, the inability of parametric GS methods to predict has been observed in experimental data. Parametric GS methods were unable to predict chill coma recovery, a quantitatively measured adaptive trait in Drosophila (Trudy F. C. MacKay, personal communication). Two-dimensional scans of the whole genome had previously revealed that the genetic architecture of this trait is composed primarily of interactions involving many loci. Thus, the simulated architectures used in our study are reasonable for many quantitative traits.
The clear distinctions of estimated accuracies and MSE values between parametric and nonparametric methods when underlying genetic architecture is epistatic suggest that data analyses consisting of a combination of parametric and nonparametric GS methods could be used as a diagnostic to reveal the prevalent genetic architecture of the trait. It is likely that the true underlying genetic architecture consists of mixtures of additive and epistatic genetic effects, so the inferential limits of applying pairs of GS methods to data from samples of breeding populations as a diagnostic needs further investigation. However, the first step is to look at the extremes in terms of genetic architecture.
Our results also suggest that if the goal of the research is to accurately predict the genotypic value of an individual, particularly for purposes of selection, and if the underlying genetic architecture of the traits are not known, then it is best to use the nonparametric NWE, the RKHS, or the SVM. Unfortunately, these methods do not provide interpretable inferences about relative weighting that is being applied to various regions of the genome, i.e., inferences about specific allelic contributions to the trait are limited. If the goal is genetic improvement and the underlying genetic architecture is known to be additive, then parametric GS methods will provide better predictions for selection. It has previously been hypothesized (Xu et al. 2011) that if all specific desirable alleles are known, then gene stacking (genome construction) based on optimization approaches will be more effective and efficient than GS approaches. Thus, in the interest of both immediate and long-term genetic improvement goals, a combination of data analyses consisting of parametric, nonparametric GS methods as well as genetic mapping (Guo et al. 2013) should be applied to data derived from plant breeding populations.
Although heritability did not affect the ability to distinguish among GS methods, it did affect estimated accuracies. When heritability is high and genetic architecture is additive, predictions are more accurate than for low heritability. When genetic architecture is based on epistasis and the trait exhibits low heritability, predictions are not very accurate for almost all GS methods. Even when the heritability is 0.70, the highest mean for prediction accuracy is 0.35, which indicates that further improvement of the models is necessary. Also, further research is needed to determine the affects of more complex plant breeding population structures. Typically, plant breeding population structures consist of inbred progeny derived from multiple crosses involving related and unrelated inbreds (Guo and Beavis 2011; Guo et al. 2013). Thus, GS needs to accurately predict phenotypes not only among subsets of progeny from related families within generations but also among generations of related and unrelated families.
In practice, plant breeders do not know the genetic architecture responsible for quantitative traits and the dynamics of selection for genetic improvement will tend to favor alleles that contribute to additive components. Genetic improvement is affected not only by the underlying genetic architecture but also by additional types of unpredictable genetic contributions including intra-locus dominance and genotype by environment interactions. Herein, we have demonstrated the superior ability of the nonparametric NWE, the RKHS, and the SVM methods to accurately predict phenotypes for additive by additive inter-locus interactions. We hypothesize that nonparametric GS methods also will enable more accurate predictions of individual genotypic value for traits that are affected by dominance and genotype by environment interactions.
We thank Rohan Fernando (Animal Science Department, Iowa State University) and Gustavo de los Campos (Biostatistics Department, University of Alabama) for their valuable input and help with the software development for this project. Also, we gratefully acknowledge the help of Dorian Garrick (Animal Science Department, Iowa State University) for providing access to the GenSel software.
Supporting information is available online at http://www.g3journal.org/lookup/suppl/doi:10.1534/g3.114.010298/-/DC1.
Copyright © 2014 Howard et al.
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text:
Q: Google maps dynamic center point PHP I'm trying to find out a way which i can have a dynamic center point, rather than have it static, Is there a way i can do this via the bounds method ?
For now i am getting the dynamic location but when the map loads, it is showing the static location, This new google.maps.LatLng(37.775196, -122.419204);
Which i want to point my dynamic location when the map loads.

<?php

  $res = $con->query("SELECT mrk_lat , mrk_lng FROM `marker_info` ORDER BY mrk_id DESC
                      LIMIT 1 ");

  $mrk_cnt = 0;
  while ($obj = $res->fetch_object()){
    $lat[$mrk_cnt] = $obj->mrk_lat;  // save the lattitude
    $lng[$mrk_cnt] = $obj->mrk_lng;  // save the longitude
    $mrk_cnt++;                      // increment the marker counter
  }
  $res->close();
?>

A: Just put your PHP variables containing the location using the echo shorthand:
new google.maps.LatLng(<?=$lat[$mrk_cnt]?>, <?=$lng[$mrk_cnt]?>)

Or in your code at the end:
echo "var point$lcnt = new google.maps.LatLng(" . $lat[$lcnt] . "," . $lng[$lcnt] . ");\n";

redpajama_set_name: RedPajamaStackExchange


I went for a swim… and I wanted to drown.
Thinking that life is just a faded bloom.
My world was dark by my own device.
Pushing the envelope and paying the price.
Choosing a path that kept me alone.
There is a way to cut myself free.
I can turnabout. I can discover me.
There is an easier and softer way.
I can start anew and fresh each day.
I can fill the hours and give myself joy.
A gift of bliss that I never employ.
All I need is faith; all I need is hope.


text:
Q&A with Kathleen Freeman
How many years have you worked in the AEC industry? 14 Years (not including two 6-month internships)
Imagine you on your 1st day of work at your very first job out of college, what advice would you give yourself? If you don't like how you're being treated, speak up and let it be known. Stay away from office politics and gossip.
What has been your biggest professional achievement throughout the duration of your career? Passing the Professional Engineering Exam.
Have you encountered any hurdles in your professional life that you believe were due to being a woman? I can't think of any.
What would you tell a 5-year-old girl to get them excited about being an engineer? You can be or do whatever you want to do. There are no barriers just because you are a female.
How has your career path evolved since you started at 3J ? 1) I am now the direct supervisor of another person. 2) I am able to work from home three days a week.
Where do you see yourself professionally in 10 years? Retired!
Is there anything that makes 3J different for women than other firms? I've worked in just two companies since finishing school. Both companies have treated women well and with respect.
If you could have any super power, what would it be? Get by on 5 hours of sleep a night.
Kathleen leads 3J's Water Resources and Flood Plain Management services. She has been at 3J since 2011, working extensively in the region for public agencies, including the City of Independence, Parkrose School District, Tualatin Hills Parks & Recreation District and Washington County. Kathleen designs low impact stormwater management systems, preparation of watershed analysis including capacity analysis, modeling of channels and floodplain/floodways, and downstream analyses.
© 2023 3J CONSULTING

redpajama_set_name: RedPajamaCommonCrawl


2 "Tell the Israelites to turn back and encamp near Pi Hahiroth, between Migdol and the sea. They are to encamp by the sea, directly opposite Baal Zephon.
4 And I will harden Pharaoh's heart, and he will pursue them. But I will gain glory for myself through Pharaoh and all his army, and the Egyptians will know that I am the LORD." So the Israelites did this.
5 When the king of Egypt was told that the people had fled, Pharaoh and his officials changed their minds about them and said, "What have we done? We have let the Israelites go and have lost their services!"
9 The Egyptians--all Pharaoh's horses and chariots, horsemen and troops--pursued the Israelites and overtook them as they camped by the sea near Pi Hahiroth, opposite Baal Zephon.
11 They said to Moses, "Was it because there were no graves in Egypt that you brought us to the desert to die? What have you done to us by bringing us out of Egypt?
12 Didn't we say to you in Egypt, 'Leave us alone; let us serve the Egyptians'? It would have been better for us to serve the Egyptians than to die in the desert!"
13 Moses answered the people, "Do not be afraid. Stand firm and you will see the deliverance the LORD will bring you today. The Egyptians you see today you will never see again.
14 The LORD will fight for you; you need only to be still."
15 Then the LORD said to Moses, "Why are you crying out to me? Tell the Israelites to move on.
18 The Egyptians will know that I am the LORD when I gain glory through Pharaoh, his chariots and his horsemen."
23 The Egyptians pursued them, and all Pharaoh's horses and chariots and horsemen followed them into the sea.
25 He made the wheels of their chariots come off so that they had difficulty driving. And the Egyptians said, "Let's get away from the Israelites! The LORD is fighting for them against Egypt."
28 The water flowed back and covered the chariots and horsemen--the entire army of Pharaoh that had followed the Israelites into the sea. Not one of them survived.
31 And when the Israelites saw the great power the LORD displayed against the Egyptians, the people feared the LORD and put their trust in him and in Moses his servant.


text:
"""Test the tempered SMC steps and routine"""
import functools
import itertools
from typing import List

import chex
import jax
import jax.numpy as jnp
import jax.scipy.stats as stats
import numpy as np
from absl.testing import absltest, parameterized

import blackjax
import blackjax.smc.resampling as resampling
import blackjax.smc.solver as solver
from blackjax import adaptive_tempered_smc, tempered_smc
from blackjax.smc.tempered import TemperedSMCState

def inference_loop(kernel, rng_key, initial_state):
    def cond(carry):
        _, state, *_ = carry
        return state.lmbda < 1

    def body(carry):
        i, state, op_key, curr_loglikelihood = carry
        op_key, subkey = jax.random.split(op_key, 2)
        state, info = kernel(subkey, state)
        return i + 1, state, op_key, curr_loglikelihood + info.log_likelihood_increment

    total_iter, final_state, _, log_likelihood = jax.lax.while_loop(
        cond, body, (0, initial_state, rng_key, 0.0)
    )

    return total_iter, final_state, log_likelihood

class TemperedSMCTest(chex.TestCase):
    """Test posterior mean estimate."""

    def setUp(self):
        super().setUp()
        self.key = jax.random.PRNGKey(42)

    def logprob_fn(self, log_scale, coefs, preds, x):
        """Linear regression"""
        scale = jnp.exp(log_scale)
        y = jnp.dot(x, coefs)
        logpdf = stats.norm.logpdf(preds, y, scale)
        return jnp.sum(logpdf)

    @chex.all_variants(without_jit=False, with_pmap=False)
    @parameterized.parameters(itertools.product([100, 5000], [True, False]))
    def test_adaptive_tempered_smc(self, N, use_log):
        x_data = np.random.normal(0, 1, size=(1000, 1))
        y_data = 3 * x_data + np.random.normal(size=x_data.shape)
        observations = {"x": x_data, "preds": y_data}

        def prior(x):
            return (
                stats.expon.logpdf(jnp.exp(x[0]), 0, 1) + x[0] + stats.norm.logpdf(x[1])
            )

        conditioned_logprob = lambda x: self.logprob_fn(*x, **observations)

        log_scale_init = np.log(np.random.exponential(1, N))
        coeffs_init = 3 + 2 * np.random.randn(N)
        smc_state_init = [log_scale_init, coeffs_init]

        iterates = []
        results = []  # type: List[TemperedSMCState]

        hmc_parameters = {
            "step_size": 10e-2,
            "inverse_mass_matrix": jnp.eye(2),
            "num_integration_steps": 50,
        }

        for target_ess in [0.5, 0.75]:
            tempering = adaptive_tempered_smc(
                prior,
                conditioned_logprob,
                blackjax.hmc,
                hmc_parameters,
                resampling.systematic,
                target_ess,
                solver.dichotomy,
                use_log,
                5,
            )
            init_state = tempering.init(smc_state_init)

            n_iter, result, log_likelihood = self.variant(
                functools.partial(inference_loop, tempering.step)
            )(self.key, init_state)
            iterates.append(n_iter)
            results.append(result)

            np.testing.assert_allclose(
                np.mean(np.exp(result.particles[0])), 1.0, rtol=1e-1
            )
            np.testing.assert_allclose(np.mean(result.particles[1]), 3.0, rtol=1e-1)

        assert iterates[1] >= iterates[0]

    @chex.all_variants(without_jit=False, with_pmap=False)
    @parameterized.parameters(itertools.product([100, 1000], [10, 100]))
    def test_fixed_schedule_tempered_smc(self, N, n_schedule):
        x_data = np.random.normal(0, 1, size=(1000, 1))
        y_data = 3 * x_data + np.random.normal(size=x_data.shape)
        observations = {"x": x_data, "preds": y_data}

        prior = lambda x: stats.norm.logpdf(x[0]) + stats.norm.logpdf(x[1])
        conditionned_logprob = lambda x: self.logprob_fn(*x, **observations)

        log_scale_init = np.random.randn(N)
        coeffs_init = np.random.randn(N)
        smc_state_init = [log_scale_init, coeffs_init]

        lambda_schedule = np.logspace(-5, 0, n_schedule)
        hmc_parameters = {
            "step_size": 10e-2,
            "inverse_mass_matrix": jnp.eye(2),
            "num_integration_steps": 50,
        }

        tempering = tempered_smc(
            prior,
            conditionned_logprob,
            blackjax.hmc,
            hmc_parameters,
            resampling.systematic,
            10,
        )
        init_state = tempering.init(smc_state_init)

        kernel = self.variant(tempering.step)

        def body_fn(carry, lmbda):
            rng_key, state = carry
            _, rng_key = jax.random.split(rng_key)
            new_state, info = kernel(rng_key, state, lmbda)
            return (rng_key, new_state), (new_state, info)

        (_, result), _ = jax.lax.scan(body_fn, (self.key, init_state), lambda_schedule)
        np.testing.assert_allclose(np.mean(np.exp(result.particles[0])), 1.0, rtol=1e-1)
        np.testing.assert_allclose(np.mean(result.particles[1]), 3.0, rtol=1e-1)

def normal_logprob_fn(x, chol_cov):
    """minus log-density of a centered multivariate normal distribution"""
    dim = chol_cov.shape[0]
    y = jax.scipy.linalg.solve_triangular(chol_cov, x, lower=True)
    normalizing_constant = (
        np.sum(np.log(np.abs(np.diag(chol_cov)))) + dim * np.log(2 * np.pi) / 2.0
    )
    norm_y = jnp.sum(y * y, -1)
    return -(0.5 * norm_y + normalizing_constant)

class NormalizingConstantTest(chex.TestCase):
    """Test normalizing constant estimate."""

    @chex.all_variants(without_jit=False, with_pmap=False)
    @parameterized.parameters(itertools.product([500, 1_000], [2, 10]))
    def test_normalizing_constant(self, N, dim):
        rng_key = jax.random.PRNGKey(2356)
        rng_key, cov_key = jax.random.split(rng_key, 2)
        chol_cov = jax.random.uniform(cov_key, shape=(dim, dim))
        iu = np.triu_indices(dim, 1)
        chol_cov = chol_cov.at[iu].set(0.0)
        cov = chol_cov @ chol_cov.T
        conditionned_logprob = lambda x: normal_logprob_fn(x, chol_cov)

        prior = lambda x: stats.multivariate_normal.logpdf(
            x, jnp.zeros((dim,)), jnp.eye(dim)
        )

        rng_key, init_key = jax.random.split(rng_key, 2)
        x_init = jax.random.normal(init_key, shape=(N, dim))

        hmc_parameters = {
            "step_size": 10e-2,
            "inverse_mass_matrix": jnp.eye(dim),
            "num_integration_steps": 50,
        }

        tempering = adaptive_tempered_smc(
            prior,
            conditionned_logprob,
            blackjax.hmc,
            hmc_parameters,
            resampling.systematic,
            0.9,
            solver.dichotomy,
            True,
            10,
        )
        tempered_smc_state_init = tempering.init(x_init)
        n_iter, result, log_likelihood = self.variant(
            functools.partial(inference_loop, tempering.step)
        )(rng_key, tempered_smc_state_init)

        expected_log_likelihood = -0.5 * np.linalg.slogdet(np.eye(dim) + cov)[
            1
        ] - dim / 2 * np.log(2 * np.pi)

        np.testing.assert_allclose(log_likelihood, expected_log_likelihood, rtol=1e-1)

if __name__ == "__main__":
    absltest.main()

redpajama_set_name: RedPajamaGithub


text:
ZIP code 55084 Health Insurance Charts
This section of charts shows ZIP code 55084 Minnesota healthcare data based on the most recent 2019 data from the Census Bureau which was released in December of 2020 and tracks healthcare in the United States. Figure 1 shows the percentage of people who acquired some form of health coverage insurance during open enrollment. 55084 depicts it has a With Coverage of 96.1% which is in the center range of other zip codes in the metro area. The zip code with the highest health insurance coverage in the area is 55045 which depicts an insured of 97.0% ( about the same size). The next chart (Figure 2) shows the change in the percent of people covered from 2018 to 2019 (latest year available) in the percentage of people in the greater ZIP code 55084 area who had some form of health insurance coverage. ZIP code 55084 shows it has a Change in Coverage of 2.9% which is the third most change in insured of all other zip codes in the greater ZIP code 55084 region. The zip code with the highest change in health insurance coverage in the area is 54824 which depicts a change in insured of 3.7% (26.3% larger).
In Figure 3 the relative proportions of different types of health insurance coverage benefits are shown (broken down by major categories of public versus private.) Not public insurance includes affordable care act/Obamacare, Medicare, and Medicaid. ZIP code 55084 has the percentage of private health insurance in the mid range of other zip codes in the greater region at 39.2% of the total.
The next two charts show the percentage of men and women with coverage. First, Figure 6 shows men and it has a With Healthcare Coverage-Men of 95% which is in the middle range of other zip codes in the local area. The zip code with the highest men with health care insurance coverage in the area is 55012 which depicts an insurance coverage of 96% ( only slightly larger). In the next chart women are shown for the percentage that are covered by health insurance. Figure 7 shows it has a With Healthcare Coverage-Women of 98% which is the third most of all other zip codes in the greater region. The zip code with the highest women with health care insurance coverage in the area is 54006 which depicts an insurance coverage of 99% ( only slightly larger).
The next two charts show the percentage of people who do not have health care insurance. Both charts show the data broken out by household income with Figure 8 showing the percent covered by income group. ZIP code 55084 depicts it has the hightest change in the percentage of people not covered with insurance of 11.5% for the $25k to $50k group. Figure 9 shows the change in the percentage of people who have no insurance from 2018 to 2019. ZIP code 55084 shows it has the hightest change in the percentage of people not covered with insurance of 16.7% for the $50k to $75k-change classification. The next chart shows the percentage of children (17 years of age and younger) who do not have health insurance. In Figure 11, it has a Children without Health Insurance of 1.5% which is the second smallest as measured by children without health insurance of all the other zip codes in the surrounding region. The zip code with the highest children without health insurance in the area is 55074 which depicts a children without insurance of 9.1% ( substantially bigger).
ZIP code 55084 Area Hospital Info Charts
Figure 14 shows a listing of hospitals in the area which provide a variety of health services/treatment including: primary care, urgent care, hospice, ICU, orthopedic, family medicine, clinic, home health care and other health coverage to the community. Data from Medicare and Medicaid Services (CMS) and not the Minnesota Department of Health. Some of the Area Hospital Info are: St Croix Regional Medical Ctr, Ladd Memorial Hospital, Fairview Lakes Medical Center, Amery Regional Medical Ctr, and Burnett Medical Center. In Figure 15, the number of patients discharged from each hospital is shown. Looking at Discharges for Area Hospital Info we find that Cambridge Medical Center ranks the largest with a value of 1,443 discharges. The next largest values are for: Fairview Lakes Medical Center (1,424), St Croix Regional Medical Ctr (0), Ladd Memorial Hospital (0), and Amery Regional Medical Ctr (0). The difference between the highest value (Cambridge Medical Center) and the next highest (Fairview Lakes Medical Center) is that the discharges is about only slightly larger. The next Figure 16 shows the average level of illness of an average patient treated at each hospital using the Medicare Case Mix Index (CMI). This indicator is considered an important healthcare industry metric for a hospital and shows the average severity level of procedures performed by a physician at the location. Looking at CMI for Area Hospital Info we find that Cambridge Medical Center ranks the largest with a value of 1.08 cmi. The next largest values are for: Fairview Lakes Medical Center (1.07), St Croix Regional Medical Ctr (0.00), Ladd Memorial Hospital (0.00), and Amery Regional Medical Ctr (0.00). The difference between the highest value (Cambridge Medical Center) and the next highest (Fairview Lakes Medical Center) is that the cmi is about about the same size.
ZIP code 55084 Patient Experience Charts
The next set of charts are based on a national survey of patient experience in hospitals and covers a range of caregiver staff (doctor, primary care, medical assistant, behavioral health, pain management, and other provider.) These metrics provide overall and important quality of care metrics for the hospitals in the community and how comfortable your child or loved ones will be at the facility. Figure 17 shows the percent of patients surveyed who said their hospital room and bathroom were always clean. Looking at Room and bathroom Always clean for Patient Experience we find that Amery Regional Medical Ctr ranks the largest with a value of 84.0% room and bathroom always clean. The next largest values are for: Burnett Medical Center (84.0%), Fairview Lakes Medical Center (80.0%), St Croix Regional Medical Ctr (78.0%), and Westfields Hospital And Clinic (78.0%). The difference between the highest value (Amery Regional Medical Ctr) and the next highest (Burnett Medical Center) is that the room and bathroom always clean is about about the same size.
Figure 18 shows the percent of patient residents surveyed who said the area around the hospital room was always quiet. Looking at Always quiet at night for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 72.0% always quiet at night. The next largest values are for: Amery Regional Medical Ctr (66.0%), Westfields Hospital And Clinic (65.0%), Cambridge Medical Center (65.0%), and Fairview Lakes Medical Center (63.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Amery Regional Medical Ctr) is that the always quiet at night is about 9.1% larger.
In Figure 19, the percent of patients who said that they always received help when they need it. Looking at Always received help when needed for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 82.0% always received help when needed. The next largest values are for: Cambridge Medical Center (79.0%), Burnett Medical Center (78.0%), Fairview Lakes Medical Center (75.0%), and Amery Regional Medical Ctr (75.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Cambridge Medical Center) is that the always received help when needed is about only about 3.8% larger.
Figure 20 provides a pain management analysis using the percent of patient residents who reported that their pain was always well controlled for all the hospitals in the community. Looking at Pain Always Well Controlled for Patient Experience we find that Burnett Medical Center ranks the largest with a value of 76.0% pain always well controlled. The next largest values are for: Ladd Memorial Hospital (73.0%), Cambridge Medical Center (73.0%), Amery Regional Medical Ctr (72.0%), and St Croix Regional Medical Ctr (69.0%). The difference between the highest value (Burnett Medical Center) and the next highest (Ladd Memorial Hospital) is that the pain always well controlled is about only about 4.1% larger.
The next chart, Figure 21, is the percent of patients who said that the nurses always communicated well with them. Looking at Nurses Always Communicated Well for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 85.0% nurses always communicated well. The next largest values are for: Burnett Medical Center (82.0%), Cambridge Medical Center (81.0%), Amery Regional Medical Ctr (79.0%), and St Croix Regional Medical Ctr (78.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Burnett Medical Center) is that the nurses always communicated well is about only about 3.7% larger.
Figure 22, shows the percent of patients who said that the doctors always communicated well with them. Looking at Doctors Always Communicated Well for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 85.0% doctors always communicated well. The next largest values are for: Burnett Medical Center (85.0%), Cambridge Medical Center (84.0%), Westfields Hospital And Clinic (80.0%), and St Croix Regional Medical Ctr (79.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Burnett Medical Center) is that the doctors always communicated well is about about the same size.
In Figure 23, each hospital in the community is rated based on a scale from 1 to 10 (where 10 is the best) and provides a high level metric of favorable wellness from treatment. Looking at Hospital Rating for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 83.0% hospital rating. The next largest values are for: Cambridge Medical Center (75.0%), Westfields Hospital And Clinic (72.0%), Burnett Medical Center (71.0%), and Fairview Lakes Medical Center (70.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Cambridge Medical Center) is that the hospital rating is about 10.7% larger.
The final chart (Figure 24) provides comparative data for ZIP code 55084 community hospitals looking at overall patient experience. This section specifically shows the percent of patients who would definitely recommend this adult specialty hospital to other people in the community. Looking at Would Recommend Hospital for Patient Experience we find that Ladd Memorial Hospital ranks the largest with a value of 85.0% would recommend hospital. The next largest values are for: Burnett Medical Center (72.0%), Cambridge Medical Center (72.0%), St Croix Regional Medical Ctr (70.0%), and Westfields Hospital And Clinic (70.0%). The difference between the highest value (Ladd Memorial Hospital) and the next highest (Burnett Medical Center) is that the would recommend hospital is about 18.1% larger.
Zip code 55084, Minnesota Healthcare Data
Area Hospital Info
Figure 1: Percent of People with Health Insurance
Figure 2: Change in People with Health Coverage (2018 to 2019)
Figure 3: Type of Health Insurance
Figure 4: Amount of Health Insurance Coverage
Figure 5: Detailed Types of Health Insurance Coverage in 55084, MN
Figure 6: Men with Health Insurance Coverage
Figure 7: Women with Health Insurance Coverage
Figure 8: Percent of People with No Insurance by Income Group
Figure 9: Change in Percent of People with No Insurance from 2018 to 2019
Figure 10: People without Health Insurance Coverage by Race
Figure 11: Children without Health Insurance
Figure 12: Advertisement
Figure 13: Insurance Coverage Broken Down by Age and Detailed Type
Figure 14: Hospitals in the 55084, MN Area and Key Information
Figure 15: Patients Discharged
Figure 16: Level of Illness Measured by Case Mix Index
Figure 17: Patients Room and Bathroom Always clean
Figure 18: Patients Room Always Quiet at Night
Figure 19: Patients Always Received Help When Needed
Figure 20: Patients Pain Always Well Controlled
Figure 21: Nurses Always Communicated Well
Figure 22: Doctors Always Communicated Well
Figure 23: Overall Hospital Rating of 9 or 10 by Patients
Figure 24: Would Definitely Recommend Hospital
Cities marked with an asterisk ("*") should resemble a city or town but do not have their own government (i.e. Mayor, City Council, etc.) These places should be recognizable by the local community but their boundaries have no legal status. Technically these include both Census Designated Places (CDP) and Census County Divisions (CCD) which are defined by the Census Bureau along with local authorities. (For more information, see: Census Designated Place or "CDP") and Census County Division "CCD".)
For comparison purposes, the US national average and the state average value are provided. Additionally, the "Combined Statistical Area" or CSA is shown that is closest to the city, county, or zip code shown. A CSA is a large grouping of adjacent metropolitan areas that identified by the Census Bureau based on social and economic ties. (See: Combined Statistical Area)
Data sources - Mouse over icon in upper right corner of each chart for information.
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text:
Q: Error en condicional Hola necesito crear una cola utilizando vectores en Java y no comprendo por qué la condición no funciona cuando mi vector esta lleno.
import java.util.Vector;

public class ColaVector {
    private Vector<Integer> cola = new Vector<Integer> (3);
    public int elementosEncolados = 0;

    public void insertarAlFinal(int elemento) {

        if (this.elementosEncolados <= this.cola.size()) {

        this.cola.add(elemento);
        this.elementosEncolados++;

        } else {
            System.out.println("No se puede insertar. Cola llena");
    }

    public int primerValor() {
       return cola.firstElement();
    }

}

Aquí está la clase main desde donde llamo a todas las clases mostradas en el cuadro anterior y como se puede ver mi vector de 4 elementos esta lleno y al imprimir por pantalla el primer valor debería mostrar también que está lleno.
public class Main {

public static void main(String[] args) {

    ColaVector c1 = new ColaVector();

    c1.insertarAlFinal(1);
    c1.insertarAlFinal(2);
    c1.insertarAlFinal(3);
    c1.insertarAlFinal(4);
    c1.insertarAlFinal(5);
    c1.insertarAlFinal(6);
    c1.insertarAlFinal(7);

    System.out.println( c1.primerValor() );

    }

}

A: La condicion si se aplica pero usted piensa que hace una cosa, cuando hace otra:
public void insertarAlFinal(int elemento) {

        System.out.println("" + this.elementosEncolados);
        System.out.println("" + this.cola.size());

        if (this.elementosEncolados <= this.cola.size()) {

        this.cola.add(elemento);
        this.elementosEncolados++;

        }

Lo anterior tendria una salida mas o menos asi:
0
0
1
1
2
2
3
3

Puede ver que tanto elementos como size crecen al mismo tiempo, ahora sin hacer muchos cambios y viendo como inicializa el vector usted podria usar lo siguiente:
public void insertarAlFinal(int elemento) {

        if (this.elementosEncolados + 1 < this.cola.capacity()) {

        this.cola.add(elemento);
        this.elementosEncolados++;

        } else {
            System.out.println("No se puede insertar. Cola llena");
    }
    }

Cambiamos this.cola.size() por this.cola.capacity() y <= por this.elementosEncolados + 1 < aunque esto ultimo es solo para que su codigo
pueda hacer lo que busca usted lo puede cambiar y ajustar mejor. pues pienso que su principal error era que quizas estaba confundido con size() y capacity()
Lo anterior tendria una salida asi:
0
3
1
3
2
3
No se puede insertar. Cola llena

Por cierto le falta un } en el metodo insertarAlFinal

*

*size() devuelve el numero de componentes el el vector en este caso.

*capacity() devuelve la capacidad actual del vector.

Vector class

A: Primeramente, debes usar capacity() para obtener tu capacidad inicial en tu vector, luego hacer algunas adecuaciones para que tu if valide correctamente los 3 elementos que deben entrar en la cola :
Nota: La clase Vector se indica una capacidad inicial y como segundo argumento el tamaño que irá incrementando:
Cuando creamos un vector u objeto de la clase Vector, podemos especificar su dimensión inicial, y cuanto crecerá si rebasamos dicha dimensión.

Vector vector=new Vector(20, 5);

package com.stack.over.flow.ejemplos;

import java.util.NoSuchElementException;
import java.util.Vector;

public class ColaVector {

    private Vector<Integer> cola = new Vector<Integer>(3);
    public int elementosEncolados = 0;
    static int capacidadCola = 0;

    public void insertarAlFinal(int elemento) {
        try {

            if (this.elementosEncolados < capacidadCola) {

                this.cola.add(elemento);
                this.elementosEncolados++;

            } else {
                System.out.println("No se puede insertar. Cola llena");
            }
        } catch (NoSuchElementException e) {
            e.printStackTrace();

        } catch (Exception e2) {
            e2.printStackTrace();

        }

    }

    public int primerValor() {
        return cola.firstElement();
    }

    public static void getCapacidadInit() {
        ColaVector obj = new ColaVector();
        capacidadCola = obj.cola.capacity();
        System.out.println("capacidad es : " + capacidadCola);
    }

    public static void main(String[] args) {
        getCapacidadInit();
        ColaVector c1 = new ColaVector();

        c1.insertarAlFinal(1);
        c1.insertarAlFinal(2);
        c1.insertarAlFinal(3);
        c1.insertarAlFinal(4);
        c1.insertarAlFinal(5);
        c1.insertarAlFinal(6);
        c1.insertarAlFinal(7);

        System.out.println(c1.primerValor());

    }

}


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My eight-year-old daughter has become quite the Disney expert and when she saw your question, her initial answer: take photos!
Disney allows its Guests to take personal belongings such as cameras on its attractions. Just note that some attractions have a "no flash photography" rule. For thrill attractions, there are typically pockets in the seat backs in which you can put your camera and other items.
If you're not comfortable bringing your camera on to an attraction, I would suggest leaving it your Resort room or renting a locker in the Theme Park.


Incorporating a business or forming an LLC is a relatively quick and painless process. However, while it may seem straightforward, there are some common mistakes that business owners make that can have a significant impact on their business as discussed in this blog post.
Many questions invariably arise throughout the process of incorporating or forming an LLC for your business. By far, one of the most common questions is…where? And more often than not, the question is framed as, "Should I incorporate in Delaware or Nevada?" These two states are hot choices for incorporation; and for good reason. However, as a general rule of thumb, if your corporation or LLC will have less than five shareholders or members (a condition which applies to the bulk of small businesses), it's best to incorporate or form an LLC in the state where your business has a physical presence. In other words, unless your business has a physical office in Delaware or Nevada, it's going to be much easier and less expensive in the long run to incorporate or form an LLC in your home state.


text:
Very Sad News On Christmas Day
25. December 2016 · Comments Off on Very Sad News On Christmas Day	· Categories: Losses
I had read of this last night, but hadn't seen enough news to post. Now it's in print and we know it's true. Very, very sad.
Russian military plane carrying 92 people, including dozens of Red Army Choir singers, dancers and orchestra members, crashed into the Black Sea on its way to Syria on Sunday, killing everyone on board, Russian authorities said.
The Russian Defence Ministry said one of its TU-154 Tupolev planes had disappeared from radar screens at 0525 MSK (9.25 p.m. ET), two minutes after taking off from Sochi in southern Russia, where it had stopped to refuel from Moscow, on its way to Syria.
Major-General Igor Konashenkov, a ministry spokesman, told reporters that nobody had survived.
"The area of the crash site has been established. No survivors have been spotted," he said. An unnamed ministry source told Russian news agencies no life rafts had been found, while another source told the Interfax agency that the plane had not sent an SOS signal.
In televised comments, President Vladimir Putin, speaking in St Petersburg, declared Dec. 26 a national day of mourning.
The jet, a Soviet-era Tupolev plane built in 1983, had been carrying 84 passengers and eight crew members.
At least 60 were members of the Alexandrov Ensemble, better known internationally as the Red Army Choir, and were being flown out to Russia's Hmeymim air base in Syria to entertain troops in the run-up to the New Year.