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# Plotting Random Effects In R

3 odds ratios, mean difference and incidence rate ratio) for different types of data (e. Plotting separate slopes with geom_smooth() The geom_smooth() function in ggplot2 can plot fitted lines from models with a simple structure. In this post, I want to focus on the simplest of questions: How do I generate a random number? The answer depends on what kind of random number you want to generate. Random Integer Generator. glmer(fit, type = "fe", sort = TRUE) To summarize, you can plot random and fixed effects in the way as shown above. Random Effect Models The preceding discussion (and indeed, the entire course to this point) has been limited to fixed effects" models. In particular, I compare output from the lm() command with that from a call to lme(). Fixed and random factors can be nested or crossed with each other, depending on. Fonton N, Atindogbe G, Honkonnou N, and Dohou R. Single factors (~g) or crossed factors (~g1*g2) are. type = "std2" Forest-plot of standardized beta values, however, standardization is done by dividing by two sd (see 'Details'). It is also a R data object like a vector or data frame. That is, qqmath is great at plotting the intercepts from a hierarchical model with their errors around the point estimate. Function adonis evaluates terms sequentially. It internally calls via. Random forest is one of those algorithms which comes to the mind of every data scientist to apply on a given problem. effect, and summary. All packages except MIXOR can provide estimates of the random effects. But why?! Well, this will become clear if we understand what our interaction effect really means. 1 How do we describe populations? KEYWORDS: Blooms: Remember 2. None of the above. glmer(fit, type = "re. To obtain Type III SS, vary the order of variables in the model and rerun the analyses. , the fixed effects) and the population variation (i. The trials of intravenous magnesium after myocardial infarction provide an extreme example of the differences between fixed and random effects analyses that can arise in the presence of funnel plot asymmetry. Main Effects Residual Plots. 10 means that 10 percent of the variance in Y is predictable from X; an R 2 of 0. Let’s get started. Tutorial index. A normal probability plot of the effects is shown below. Model I and Model II anova. The syntax for including a random effect in a formula is shown below. Hundreds of charts are displayed in several sections, always with their reproducible code available. It covers a many of the most common techniques employed in such models, and relies heavily on the lme4 package. field (Intercept) 16. Nested plots with two plot sizes (12. The trace plot has a stationary pattern, which is what we would like to see. Plotting separate slopes with geom_smooth() The geom_smooth() function in ggplot2 can plot fitted lines from models with a simple structure. Alternative names: split-plot design; mixed two-factor within-subjects design; repeated measures analysis using a split-plot design; univariate mixed models approach with subject as a random effect. Note that each point on the plot corresponds to the odds ratio of each level of the fixed effect period relative to period=1. This is a basic introduction to some of the basic plotting commands. A video showing basic usage of the "lme" command (nlme library) in R. Pause During Mplus Analysis. Spatial Statistics using R-INLA and Gaussian Markov random ﬁelds DavidBolinandJohanLindstrom 1 Introduction In this lab we will look at an example of how to use the SPDE models in the. The dots in a scatter plot not only report the values of individual data points, but also patterns when the data are taken as a whole. Graphing change in SPSS The simplest way is to produce a scatter plot of the variable you are interested in over time, this is also called a profile or spaghetti plot. An example of blocking factor might be the gender of a patient (by blocking on gender), this is a source of variability controlled for, leading to greater accuracy. -You cannot make inferences to a larger experiment. Keywords: discrete choice models, random parameters, simulated maximum likelihood, R, individual-speci c. For the random intercept model, this thing that we're taking the covariance of, is just u j + e ij and we've actually written this here as r ij because, if you remember, in the variance components model, when we were calculating residuals we actually defined r ij to be just u j + e ij. So let's inspect our profile plots. qq-plot of random effects. And then this last point, the residual is positive. Figure 8 shows a diagnostic graph that contains a trace plot, a histogram and density plots for our MCMC sample, and a correlegram. This model is a three-level random intercepts model, which splits the variance between lecturers, students, and the residual variance. The summary effect and its conﬁdence interval are displayed at the bottom. Use an image as a free-writing exercise. n is of length > 1, random effects indicated by the values in sample. ## ## Random effects: ## Groups Name Variance Std. Not necessarily that is. That is, qqmath is great at plotting the intercepts from a hierarchical model with their errors around the point estimate. only parameter in the random part of the model. I illustrate this with an analysis of Bresnan et al. For details see here Epil. lmer and sjp. Select the data in which we want to plot the 3D chart. bmeta is a R package that provides a collection of functions for conducting meta-analyses and meta-regressions under a Bayesian context, using JAGS. qq") Probability curves of odds ratios. args = list ( family = "binomial" ), se = FALSE ) par ( mar = c ( 4 , 4 , 1 , 1 )) # Reduce some of the margins so that the plot fits better plot ( dat $mpg , dat$ vs ) curve ( predict ( logr_vm , data. JMP - AN INTRODUCTORY USER'S GUIDE by Susan J. population distribution b. The table result showed that the McFadden Pseudo R-squared value is 0. To produce a forest plot, we use the meta-analysis output we just created (e. With roots dating back to at least 1662 when John Graunt, a London merchant, published an extensive set of inferences based on mortality records, survival analysis is one of the oldest subfields of Statistics [1]. Weibull plot The fit of a Weibull distribution to data can be visually assessed using a Weibull plot. The first argument is the formula object describing both the fixed-effects and random effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. The R chart is used to evaluate the consistency of. Pareto plots, main effects and Interactions plots can be automatically displayed from the Data Display tool for study and investigation. plot_model() is a generic plot-function, which accepts many model-objects, like lm, glm, lme, lmerMod etc. RANDOM_WALK_2D_SIMULATION, a MATLAB program which simulates a random walk in a 2D region. Another diagnostic plot is the qq-plot for random effects. Model I and Model II anova. The workshop covers the new General Cross-Lagged Panel Model (GCLM) in Mplus. re requests the GLS random-effects (mixed) estimator. This is valid simple random sampling, because every part of the study area is equally likely to be sampled and the location of one line does not affect the location. A main‐effects plot clearly shows that depositional effects are the strongest, especially contrasting high versus either medium or low depositional areas along mMDS axis 1 (Figure 3a). I’m not super familiar with all that ggpubr can do, but I’m not sure it includes a good “interaction plot” function. • Caution if random effects return meaningfully different results from fixed effects. For example, in many experiments. + ( effect expression | groups ) The following are a few examples of specifying random effects. In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. In R, we’ll use the simple plot function to compare the model-predicted values to the observed ones. I found the combination of R/ggplot/maps package extremely flexible and powerful, and produce nice looking map based visualizations. In addition, it provides the weight for each study; the effect measure, method and the model used to perform the meta-analysis; the confidence intervals used; the effect estimate from each study, the overall effect estimate, and the statistical significance of the analysis. (PDF) Table S1. Cases or individuals can and do move into and out of the population. args = list ( family = "binomial" ), se = FALSE ) par ( mar = c ( 4 , 4 , 1 , 1 )) # Reduce some of the margins so that the plot fits better plot ( dat $mpg , dat$ vs ) curve ( predict ( logr_vm , data. extract() function from texreg package) as well as plot_model() function from the sjPlot package. This is a basic introduction to some of the basic plotting commands. ) The action “na. Use type = "re. It is assumed that you know how to enter data or read data files which is covered in the first chapter, and it is assumed that you are familiar with the different data types. when r is much smaller than 1 in magnitude. The random walk pattern shown in animation 2 indicates problems with the chain. Random Image. R Pubs by RStudio. Fixed effects model If the effect is the same in all. , random intercept / subject=block*year. A normal 3d surface plot in excel appears below, but we cannot read much from this chart as of now. For mixed effects models, only fixed effects are. This indicates that everyone has a different change rate. Scatter plots: This type of graph is used to assess model assumptions, such as constant variance and linearity, and to identify potential outliers. We apply five fertilizers, each of different quality, on five plots of land each of wheat. type = "std" Forest-plot of standardized coefficients. ) The action “na. The ggplot2 package is extremely flexible and repeating plots for groups is quite easy. The effects are instantaneous, they can be permanent or last for up to 24 hours depending on what is appropriate and/or funny. On the other hand, the log likelihood in the R output is obtained using truly Weibull density. Instant access to millions of Study Resources, Course Notes, Test Prep, 24/7 Homework Help, Tutors, and more. effects: Plot random effects of model in Bayesthresh: Bayesian thresholds mixed-effects models for categorical data rdrr. This form allows you to generate random integers. We have some repeated observations (Time) of a continuous measurement, namely the Recall rate of some words, and several explanatory variables, including random effects (Auditorium where the test took place; Subject name); and fixed effects, such as Education, Emotion (the emotional connotation of the word to remember), or $\small \text{mgs. observations independent of time. Discussion includes extensions into generalized mixed models and realms beyond. -You cannot make inferences to a larger experiment. So I present to you: a list of random potion effects. It comprises data, a model description, fitted coefficients, covariance parameters, design matrices, residuals, residual plots, and other diagnostic information for a linear mixed-effects model. 3D plotting 3d reasoning random effects. Plotting separate slopes with geom_smooth() The geom_smooth() function in ggplot2 can plot fitted lines from models with a simple structure. plot_model() is a generic plot-function, which accepts many model-objects, like lm, glm, lme, lmerMod etc. ggpubr is a fantastic resource for teaching applied biostats because it makes ggplot a bit easier for students. population d. ) The action “na. Processing is a flexible software sketchbook and a language for learning how to code within the context of the visual arts. Extreme weather increases the risk of large-scale crop failure. But why?! Well, this will become clear if we understand what our interaction effect really means. Here are two suggestions for how to use these images: 1. It helps to know that R has different functions to create an initial graph and to add to an existing graph. lmer and that of priming. 0, Shiny has built-in support for interacting with static plots generated by R’s base graphics functions, and those generated by ggplot2. the random effects slope of each cluster. Plot symbols and colours can be specified as vectors, to allow individual specification for each point. a two-sided linear formula object describing both the fixed-effects and random-effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. extract() function from texreg package) as well as plot_model() function from the sjPlot package. , a "trellis" object). list, print. There are some R packages that are made specifically for this purpose; see packages effectsand visreg, for example. pt = min(length(unique(pred. RANDOM_WALK_2D_SIMULATION, a MATLAB program which simulates a random walk in a 2D region. 37 m and a DBH less than 14 cm) trees, respectively. Scatter Plot; With a scatter plot a mark, usually a dot or small circle, represents a single data point. And then this last point, the residual is positive. • Partial plots and interpretation of effects. This is because the association is nonlinear. The package includes functions for computing various effect size or outcome measures (e. Psychological Methods, 2, 64-78. Main Effects Residual Plots. The lmfunction in R can handle factorial design with ﬁxed effects without taking the special experimental design or the random effects into account. A model for such a split-plot design is the following:. As a language for statistical analysis, R has a comprehensive library of functions for generating random numbers from various statistical distributions. The only reason that we are working with the data in this way is to provide an example of linear regression that does not use too many data points. The plotlines generated are not guaranteed to make sense but they do inspire writers by triggering a creative chain of thought. Because there are not random effects in this second model, the gls function in the nlme package is used to fit this model. The Spatial Patterns of Functional Groups and Successional Direction in a Coastal Dune Community. Bayesian random effects meta-analysis of trials with binary outcomes: methods for absolute risk difference and relative risk scales. The analysis based on a random-effects model is shown in Figure 2. For Example: If there were only one random effect per subject (e. 9919) [1] 0. Mixed Models and Random Effect Models. + ( effect expression | groups ) The following are a few examples of specifying random effects. The default is type = "fe", which means that fixed effects (model coefficients. If not, consider a random effects model. The Spatial Patterns of Functional Groups and Successional Direction in a Coastal Dune Community. To calculate the mixed effects limits of agreement, we analysed the paired differences of each device compared with the gold-standard using a mixed effects regression model, including participant as a random effect and activity as a fixed effect, using the nlme package in R software version 3. The dots should be plotted along the line. Students do NOT need to be knowledgeable and/or experienced with R software to successfully complete this course. • There is no built-in quantile plot in R, but it is relatively simple to produce one. Using R to Compute Effect Size Confidence Intervals. random variable). Conspiracy theory definition, a theory that rejects the standard explanation for an event and instead credits a covert group or organization with carrying out a secret plot: One popular conspiracy theory accuses environmentalists of sabotage in last year's mine collapse. Random effects probit and logit specifications are common when analyzing economic experiments. The lme function in thenlme(Pinheiro et al. Add something like + (1|subject) to the model for the random subject effect. plot = TRUE, then partial makes an internal call to plotPartial (with fewer plotting options) and returns the PDP in the form of a lattice plot (i. Note that each point on the plot corresponds to the odds ratio of each level of the fixed effect period relative to period=1. observations independent of time. The syntax for including a random effect in a formula is shown below. This model is a three-level random intercepts model, which splits the variance between lecturers, students, and the residual variance. The odds ratios is simply the exponentials of the regression coefficients. where z 1 and z 2 are Fisher transformations of r, and the two n i 's in the denominator represent the sample size for each study. They must be a representative or random sample. These will be either linear or generalized linear. plot_model() is a generic plot-function, which accepts many model-objects, like lm, glm, lme, lmerMod etc. 3 – Plotting Anomalies. I found the combination of R/ggplot/maps package extremely flexible and powerful, and produce nice looking map based visualizations. Each split plot. population growth c. If you use the ggplot2 code instead, it builds the legend for you automatically. In a random effects model, the values of the categorical independent variables represent a random sample from some population of values. (pdf file) Slides: Mixed Pattern-Mixture and Selection Models for Missing Data (pdf file). Scatter plots’ primary uses are to observe and show relationships between two numeric variables. Infix functions. For details see here Surg. Each whole plot was divided into four split plots, and b= 4 plant densities were randomly assigned to the split plots within each whole plot. Fabio Veronesi, data scientist at WRC plc. Alternative names: split-plot design; mixed two-factor within-subjects design; repeated measures analysis using a split-plot design; univariate mixed models approach with subject as a random effect. is there a significant variation due to the random effects) Test statistic: Chi-square (likelihood ratio test) H 0: µ 1 = µ 2 = … = µ t H 1: µ i ≠ µ j for some i, j in the set 1 … t H 0: σ g 2 = 0 H 1: σ g 2 > 0. Finally, a slight word of warning: our model assumed that the random. Yet, we do have choose an estimator for $$\tau^{2}$$. The plotlines generated are not guaranteed to make sense but they do inspire writers by triggering a creative chain of thought. The upper edge (hinge) of the box indicates the 75th percentile of the data set, and the lower hinge indicates the 25th percentile. population d. Supported model types include models fit with lm(), glm(), nls(), and mgcv::gam(). Change in size (grow shorter or taller). In R, I know how to do it. Feel free to suggest a chart or report a bug; any feedback is highly welcome. This, of course, is a very bad thing because it removes a lot of the variance and is misleading. For example, suppose the business school had 200. Now in the Insert Tab under the charts section click on the surface chart. These will be either linear or generalized linear. Question 8. These plotting functions have been implemented to easier. Because you’re likely to see the base R version, I’ll show you that version as well (just in case you need it). Model I and Model II anova. Following is a scatter plot of perfect residual distribution. The counts were registered over a 30 second period for a short-lived, man-made radioactive compound. These partial terms are often regarded as similar to random effects, but they are still fitted in the same way as other terms and strictly speaking they are fixed terms. Scatter plots: This type of graph is used to assess model assumptions, such as constant variance and linearity, and to identify potential outliers. This section is intended to supplement the lecture notes by implementing PPA techniques in the R programming environment. Plots involving these estimates can help to evaluate whether the random effects are plausibly normally distributed, whether there are extreme values, and whether predictors may have omitted nonlinear effects. lmer and sjp. That is, qqmath is great at plotting the intercepts from a hierarchical model with their errors around the point estimate. The results of the individual studies are shown grouped together according to their subgroup. Let us see how we can use the plm library in R to account for fixed and random effects. Even though the association is perfect, because you can predict Y exactly from X, the correlation coefficient r is exactly zero. Fixed effects model If the effect is the same in all. Each of the above offer different underlying engines and capabilities and therefore choice of package, will dependend on the nature of the data and the desired model. Processing. Because there are not random effects in this second model, the gls function in the nlme package is used to fit this model. McMurry Written specifically as material for CHANCE courses July 24, 1992 This guide is intended to help you begin to use JMP, a basic statistics package,. effects: Plot random effects of model in Bayesthresh: Bayesian thresholds mixed-effects models for categorical data rdrr. Look closely at. It is efficient at detecting relatively large shifts (typically plus or minus 1. Change in size (grow shorter or taller). For forecasting, o R2 matters (a lot!) o Omitted variable bias isn’t a problem! o We will not worry about interpreting coefficients in forecasting models o External validity is paramount: the model estimated using historical data must hold into the (near) future. The protagonist sets out to defeat something that threatens him/her or a group they belong to. The Spatial Patterns of Functional Groups and Successional Direction in a Coastal Dune Community. Marginal Effects (related vignette) type = "pred" Predicted values (marginal effects) for specific model terms. You can also include polynomial terms of the covariates. You will also learn about training and validation of random forest model along with details of parameters used in random forest R package. plot_model() allows to create various plot tyes, which can be defined via the type-argument. Partial dependence plot gives a graphical depiction of the marginaleffect of a variable on the class probability (classification) orresponse (regression). R has a built-in editor that makes it easy to submit commands selected in a script file to the command line. values <- seq(-4,4,. In this post I will explain how to interpret the random effects from linear mixed-effect models fitted with lmer (package lme4). Inference summary(m1) Linear mixed model fit by REML ['lmerMod'] Formula: Biomass ~ Temp + N + (1 + Temp | Site) Data: data REML criterion at convergence: 327. For now, we'll ignore the main effects-even if they're statistically significant. To calculate the mixed effects limits of agreement, we analysed the paired differences of each device compared with the gold-standard using a mixed effects regression model, including participant as a random effect and activity as a fixed effect, using the nlme package in R software version 3. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a nonlinear model is more appropriate. Welcome to the Python Graph Gallery. A First Course in Design and Analysis of Experiments Gary W. ) When both factors are fixed effects, as in this unit, you should look at both profile plots (see Problem 7. Of the remaining two parameters, one can be chosen to draw a family of graphs, while the fourth parameter is kept constant. The model can include main effect terms, crossed terms, and nested terms as defined by the factors and the covariates. n is of length > 1, random effects indicated by the values in sample. In a model with right-hand-side ~ A + B the effects of A are evaluated first, and the effects of B after removing the effects of A. We may try to relate the size of the effect to characteristics of the studies and their subjects, such as average age, proportion of females, intended dose of drug, or baseline risk. All packages except MIXOR can provide estimates of the random effects. frame ( mpg = x ), type = "response" ), add = TRUE ). There is a video tutorial link at the end of the post. (illustrated with R on Bresnan et al. " These words begin a report on a statistical study of the effects of logging in Borneo. Below is an example of a forest plot with three subgroups. This model is a three-level random intercepts model, which splits the variance between lecturers, students, and the residual variance. Discussion includes extensions into generalized mixed models and realms beyond. Rags to Riches. Scatter Plot; With a scatter plot a mark, usually a dot or small circle, represents a single data point. In Rangeland Ecology & Management. Next click on Add to specify the plot (see Figure 9-6) and then click Continue. Following is a scatter plot of perfect residual distribution. Therefore, there is significant individual difference in the growth rate (slope). Since 2001, Processing has promoted software literacy within the visual arts and visual literacy within technology. Another example is the amount of rainfall in a region at different months of the year. args = list ( family = "binomial" ), se = FALSE ) par ( mar = c ( 4 , 4 , 1 , 1 )) # Reduce some of the margins so that the plot fits better plot ( dat$ mpg , dat $vs ) curve ( predict ( logr_vm , data. Of the remaining two parameters, one can be chosen to draw a family of graphs, while the fourth parameter is kept constant. We will select the Bonferroni interval adjustment to control the. We try to group the samples based on two feature variables - age and bmi. , the -1 term in the Corr column under Random effects). Each example provides the R formula, a description of the model parameters, and the mean and variance of the true model which is estimated by the regression and observed values. Identification of correlational relationships are common with scatter plots. 02 Residual 2. It is efficient at detecting relatively large shifts (typically plus or minus 1. -You cannot make inferences to a larger experiment. Seasonal effects are apparent along mMDS axis 2 (from winter to spring to summer), while the contrast of rain versus dry was relatively much smaller (Figure 3a). Estimating fixed Effects & Predicting Random Effects For a mixed model, we observe y, X, and Z!, u, R, and G are generally unknown Two complementary estimation issues (i) Estimation of ! and u Estimation of fixed effects Prediction of random effects BLUE = Best Linear Unbiased Estimator BLUP = Best Linear Unbiased Predictor Recall V = ZGZ T + R. The upper left plot in the above figure shows the effect of the median income in a district on the median house price; we can clearly see a linear relationship among them. The term “split plot” derives from agriculture, where fields may be split into plots and subplots. A model for such a split-plot design is the following:. Partial dependence plot. type = "std2" Forest-plot of standardized beta values, however, standardization is done by dividing by two sd (see 'Details'). Random-effects terms are distinguished by vertical bars ("|") separating expressions for design matrices from grouping factors. The table result showed that the McFadden Pseudo R-squared value is 0. The random walk pattern shown in animation 2 indicates problems with the chain. (To reduce the scale of the y-axis, the largest two effects, X4: Direction and X5: Batch, are not shown on the plot. A protagonist is in some way misfortune, usually financially. Quantile Plots • Quantile plots directly display the quantiles of a set of values. None of the above. Fonton N, Atindogbe G, Honkonnou N, and Dohou R. More than 90% of Fortune 100 companies use Minitab Statistical Software, our flagship product, and more students worldwide have used Minitab to learn statistics than any other package. We have some repeated observations (Time) of a continuous measurement, namely the Recall rate of some words, and several explanatory variables, including random effects (Auditorium where the test took place; Subject name); and fixed effects, such as Education, Emotion (the emotional connotation of the word to remember), or$\small \text{mgs. This type of prediction incorporates the uncertainty for the population average (i. Mixed Models and Random Effect Models. Titanic: Getting Started With R - Part 5: Random Forests. I’ve ended up with a good pipeline to run and compare many ordinal regression models with random effects in a Bayesian way using the handy R formula interface in the brms package. In ANNALS OF FOREST SCIENCE. Mixed and Random Effect Model Reports and Options. This is because the association is nonlinear. glmer(fit, type = "fe", sort = TRUE) To summarize, you can plot random and fixed effects in the way as shown above. glmer(fit, type = "re. With either base R graphics or ggplot 2, the first step is to set up a vector of the values that the density functions will work with: t. According to Christopher Booker, there are seven types of story. 9919) [1] 0. Maybe I’m wrong. The following graph plots BCG treatment effect on the y axis by distance from the equator on the x axis, with an ab line from a meta-regression. Plot size for modeling the spatial structure of Sudanian woodland trees. dygraphs() is an R package that takes R input and outputs the JavaScript needed to display it in your browser, and as its made by RStudio they also made it compatible with Shiny. Most of the results might be off-the-wall but some are pure gold. io Find an R package R language docs Run R in your browser R Notebooks. Equation (11. Not necessarily that is. To obtain Type III SS, vary the order of variables in the model and rerun the analyses. 282, which indicates a decent model fit. Below the output window are two additional windows. list and plot. The McFadden Pseudo R-squared value is the commonly reported metric for binary logistic regression model fit. Going Further. Michael Zyphur has made available a free 3-day workshop held in July 2019 at the University of Melbourne. • There is no built-in quantile plot in R, but it is relatively simple to produce one. Rags to Riches. plot_model() allows to create various plot tyes, which can be defined via the type-argument. This means that for every 1% increase in biking to work, there is a correlated 0. The basics of random intercepts and slopes models, crossed vs. Yet, we do have choose an estimator for $$\tau^{2}$$. Random effects can be thought as being a special kind of interaction. Tutorial index. ) The action “na. , gender: male/female). population birth rate d. Titanic: Getting Started With R - Part 5: Random Forests. For the other one, the residual is negative one, so we would plot it right over here. To produce a forest plot, we use the meta-analysis output we just created (e. The program provides a complete set of numeric reports and plots to allow the investigation and presentation of the studies. Fonton N, Atindogbe G, Honkonnou N, and Dohou R. For example, suppose the business school had 200. RANDOM_WALK_2D_SIMULATION, a MATLAB program which simulates a random walk in a 2D region. Below the output window are two additional windows. 09 m) were used to measure large (both live trees with a DBH larger than 14 cm and dead trees with a height of 3. A First Course in Design and Analysis of Experiments Gary W. R supports two additional syntaxes for calling special types of functions: infix and replacement functions. The effects can either be harmful or helpful, but not lethal. Assuming the model fitted is saved in the mymodel object, one can get the random + fixed effects of a multilevel model in R as follows:. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. Random effects probit and logit specifications are common when analyzing economic experiments. An example for such a behavior is shown. Oehlert University of Minnesota. R language uses many functions to create, manipulate and plot the time series data. , random intercept / subject=block*year. Discussion includes extensions into generalized mixed models and realms beyond. after using runmed(x,7) we remove the outlier effect from trend so the random part will have the outlier effect –> raw data(has. Most functions in R are “prefix” operators: the name of the function comes before the arguments. To divide each block into three equal sized plots ( whole plots ), and each plot is assigned a variety of oat according to a randomized block design. The qqmath function makes great caterpillar plots of random effects using the output from the lmer package. For mixed effects models, only fixed effects are. Figure 8 shows a diagnostic graph that contains a trace plot, a histogram and density plots for our MCMC sample, and a correlegram. Pareto plots, main effects and Interactions plots can be automatically displayed from the Data Display tool for study and investigation. 8764 Number of obs: 100. Here are data on the number of tree species in 12 unlogged forest plots and 9 similar plots logged 8 years earlier:. 31 Beneficial effects on mortality, found in a meta-analysis of small studies,32 were subsequently contradicted when the very large ISIS-4. Variance components are found in the output under Random effects (the. This is a demonstration of using R in the context of hypothesis testing by means of Effect Size Confidence Intervals. This process is described in Baayen page 305, through the languageR function plot. The upper edge (hinge) of the box indicates the 75th percentile of the data set, and the lower hinge indicates the 25th percentile. The shrinkage amount is based on how much information is contained in a random effect groups. See full list on rcompanion. are covered. SAS calls this the G matrix and defines it for all subjects, rather than for individuals. Write about whatever it makes you think of. • Caution if random effects return meaningfully different results from fixed effects. So, what I am trying to do is to plot each of the 30 versions of b3, i. Repeated measures analysis with R Summary for experienced R users The lmer function from the lme4 package has a syntax like lm. If you use the ggplot2 code instead, it builds the legend for you automatically. The new independent variable improves the predictive power of the regression. So it is just like that. The easiest is to plot data by the various parameters using different plotting tools (color, shape, line type, facet), which is what you did with your example except for the random effect site. Supported model types include models fit with lm(), glm(), nls(), and mgcv::gam(). block effects (instead of removing it as in the intrablock analysis) in estimating the treatment effects to conduct the analysis of design. In conclusion, it is possible to meta-analyze data using a Microsoft Excel spreadsheet, using either fixed effect or random effects model. We read in the data and subtract the background count of 623. Read blog posts, and download and share JMP add-ins, scripts and sample data. The world's largest digital library. For now, we'll ignore the main effects-even if they're statistically significant. r i=1 X ij rc SST = a c j=1 a r i=1 (X ij-X)2 You compute the among-group variation, also called the sum of squares among groups (SSA), by summing the squared differences between the sample mean of each group, and the grand mean, weighted by the number of blocks, r. lmer and that of priming. The dots in a scatter plot not only report the values of individual data points, but also patterns when the data are taken as a whole. By selecting the appropriate parameters for the y and the x axis, one parameter (a, power (1b), effect size, or sam- ple size) can be plotted as a function of another parame- ter. The difference between homogeneity and heterogeneity therefore lies in the different approaches taken to calculate the pooled result. Free Mplus workshops - Dr. It is not necessary to specify it separately. Therefore, there is significant individual difference in the growth rate (slope). In a random effects model, the values of the categorical independent variables represent a random sample from some population of values. 68(8): 1315-1321. In this post I will explain how to interpret the random effects from linear mixed-effect models fitted with lmer (package lme4). 1) After the graphs are complete, you’ll put the infinity symbol on the legends to denote the df for the standard normal distribution. A group of individuals of the same species occupying a given area defines an: a. In Figure Figure1 1 we show the box plots of the sampled random effects in WinBUGS for the first 10 centers of the binary logistic random effects model applied to the IMPACT data. In Figure 9, the Q-Q plot of the predicted random slopes of model (1) Þt to the radon data was inserted into the lineup, while the lineup in Figure 10 included a Q-Q plot of the random slopes in model (1) where the random e! ects were simulated from a. Finally, a slight word of warning: our model assumed that the random. linear, non-linear). Inference summary(m1) Linear mixed model fit by REML ['lmerMod'] Formula: Biomass ~ Temp + N + (1 + Temp | Site) Data: data REML criterion at convergence: 327. In R, the likelihood ratio test is carried out with the anova function: The value listed under Chisq equals twice the diﬀer- ence between the log-likelihood (listed under logLik) for priming. • Some researchers believe that when there is evidence of heterogeneity, shouldnʻtʼcombine studies at all. In one recent project I needed to draw several maps and visualize different kinds of geographical data on it. ii) within-subjects factors, which have related categories also known as repeated measures (e. To calculate the mixed effects limits of agreement, we analysed the paired differences of each device compared with the gold-standard using a mixed effects regression model, including participant as a random effect and activity as a fixed effect, using the nlme package in R software version 3. In the past two years I’ve found myself doing lots of statistical analyses on ordinal response data from a (Likert-scale) dialectology questionnaire. The basics of random intercepts and slopes models, crossed vs. But why?! Well, this will become clear if we understand what our interaction effect really means. Welcome to the Python Graph Gallery. [12] The Weibull plot is a plot of the empirical cumulative distribution function F ^ ( x ) {\displaystyle {\widehat {F}}(x)} of data on special axes in a type of Q-Q plot. Seasonal effects are apparent along mMDS axis 2 (from winter to spring to summer), while the contrast of rain versus dry was relatively much smaller (Figure 3a). random-eﬀects parameters; and (4) the ability to ﬁt generalized linear mixed models (al-2 Linear Mixed Models with lme4 though in this paper we restrict ourselves to linear mixed models). Each split plot. In principle, a mixed-model formula may contain ar-bitrarily many random-eﬀects terms, but in practice the number of such terms is typically low. See full list on jaredknowles. New Mplus Technical Note: Random starting values and multistage optimization. In this plot, the scatter in X for a given value of Y is very small, so the association is strong. This is an introduction to mixed models in R. DISCLAIMER: Any opinions, findings, conclusions, or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of CIMSS, SECE, UW-Madison, NOAA, NESDIS, STAR, NASA, SPAWAR, or NRL-Monterey. RStudio is a set of integrated tools designed to help you be more productive with R. ## subject (Intercept) 0. Effects of the Shape Parameter, beta. These will be either linear or generalized linear. Looks good so far. Most of the results might be off-the-wall but some are pure gold. Priors can be defined for the residuals, the fixed effects, and the random effects. And so this thing that I have just created, where we're just seeing, for each x where we have a corresponding point, we plot the point above or below the line based on the residual. effects: Plot random effects of model in Bayesthresh: Bayesian thresholds mixed-effects models for categorical data rdrr. New Mplus Technical Note: Random starting values and multistage optimization. partialPlot {randomForest} R Documentation. Basic life-table methods, including techniques for dealing with censored data, were discovered before 1700 [2], and in the early eighteenth century, the old masters - de Moivre. The only reason that we are working with the data in this way is to provide an example of linear regression that does not use too many data points. Interpreting a Boxplot. The 'ggplot2' philosophy is to clearly separate data from the presentation. plot_model() is a generic plot-function, which accepts many model-objects, like lm, glm, lme, lmerMod etc. Plotting separate slopes with geom_smooth() The geom_smooth() function in ggplot2 can plot fitted lines from models with a simple structure. The trace plot has a stationary pattern, which is what we would like to see. Avoid the lmerTest package. The new independent variable improves the predictive power of the regression. Likelihood Ratio test (often termed as LR test) is a goodness of fit. It internally calls via. The protagonist sets out to defeat something that threatens him/her or a group they belong to. • Partial plots and interpretation of effects. If there are R random-effects terms, then the value of 'CovariancePattern' must be a string array or cell array of length R, where each element r of the array specifies the pattern of the covariance matrix of the random-effects vector associated with the rth random-effects term. glmer(fit, type = "fe", sort = TRUE) To summarize, you can plot random and fixed effects in the way as shown above. Similarly to AFT models, when we consider a more complex model that includes interactions and nonlinear terms, it is more useful to communicate the results of the model using an effects plot. The default is type = "fe", which means that fixed effects (model coefficients) are plotted. The following graph plots BCG treatment effect on the y axis by distance from the equator on the x axis, with an ab line from a meta-regression. Random effects can be thought as being a special kind of interaction. Partial Dependence Plots: also referred to as PD plots, shows the minor effect of a feature(s) on the model’s predictions Stat Trekking: Let’s take a look at the data…. The upcoming version of my sjPlot package will contain two new functions to plot fitted lmer and glmer. In particular, I compare output from the lm() command with that from a call to lme(). Point pattern analysis in R. In Figure 9, the Q-Q plot of the predicted random slopes of model (1) Þt to the radon data was inserted into the lineup, while the lineup in Figure 10 included a Q-Q plot of the random slopes in model (1) where the random e! ects were simulated from a. An example of the lmer and qqmath functions are below using the built-in data in the lme4 package called Dyestuff. Arguments formula. Below the output window are two additional windows. The Mixed ANOVA is used to compare the means of groups cross-classified by two different types of factor variables, including: i) between-subjects factors, which have independent categories (e. , gender: male/female). Add something like + (1|subject) to the model for the random subject effect. The random effects in the model can be tested by comparing the model to a model fitted with just the fixed effects and excluding the random effects. qq-plot of random effects. Main Effects Residual Plots. Has to be put in "" (e. Here, µis a grand mean, αh is an eﬀect for the hth level of the whole plot factor (e. , random intercept / subject=block*year. A mathematical line'' has no thickness, so it's invisible; but when we plot circular dots at each point of an infinitely thin line, we get a visible line that has constant thickness. Select the data in which we want to plot the 3D chart. population. The package includes a command to produce funnel plots to assess small study effects, and L'Abbe plots to examine whether the assumption of a common odds ratio, risk ratio or risk difference is reasonable. Association of physical activity with BMI, waist circumference, body fat percentage, risk of obesity, and risk of overweight in a random effects meta-analysis of up to 218,166. For more informations on these models you can browse through the couple of posts that I made on this topic (like here, here or here). So we've written that here because it takes less space. This, of course, is a very bad thing because it removes a lot of the variance and is misleading. Basic life-table methods, including techniques for dealing with censored data, were discovered before 1700 [2], and in the early eighteenth century, the old masters - de Moivre. If there are two random effects, such as block and year, both affects must appear in the same random statement i. LN(1+r) ≈ r. To get p-values, use the car package. The effects are instantaneous, they can be permanent or last for up to 24 hours depending on what is appropriate and/or funny. The subject effect is, in a sense, "factored out" of the random effects. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. In the past week, colleagues of mine and me started using the lme4-package to compute multi level models. Forest plot of the association of the FTO rs9939609 SNP with physical activity in a random effects meta-analysis of 19,268 children and adolescents. A video showing basic usage of the "lme" command (nlme library) in R. term: name of a polynomial term in fit as string. Pareto plots, main effects and Interactions plots can be automatically displayed from the Data Display tool for study and investigation. These partial terms are often regarded as similar to random effects, but they are still fitted in the same way as other terms and strictly speaking they are fixed terms. 84536 Random effects: Groups Name Variance Std. The default is type = "fe", which means that fixed effects (model coefficients) are plotted. Let us study the effect of fertilizers on yield of wheat. The subject effect is, in a sense, "factored out" of the random effects. Plotting Estimates (Fixed Effects) of Regression Models Daniel Lüdecke 2020-05-23. You can have more effects for the plotting and the dendrogram with the R packages ggplot2 and dendextend respectively, but I will leave them out of the scope of this article. Thanks for visiting our lab's tools and applications page, implemented within the Galaxy web application and workflow framework. If asked, the effect function will compute effects for terms that. This is the same plot as is used as an example in the User Manual. The calculation of P-values for complex models with random effects and multiple experimental unit sizes is not a trivial matte. Students do NOT need to be knowledgeable and/or experienced with R software to successfully complete this course. It includes a console, syntax-highlighting editor that supports direct code execution, and a variety of robust tools for plotting, viewing history, debugging and managing your workspace. While R’s traditional graphics offers a nice set of plots, some of them require a lot of work. The first argument is the formula object describing both the fixed-effects and random effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. Sam, the function is plotting based on the model object, not the data itself, that is why aes_string and the model parameters are in there. To show how this works, we will study the decompose( ) and STL( ) functions in the R language. Rows in the dot-plot are determined by the form argument (if not missing) or by the row names of the random effects (coefficients). Different values of the shape parameter can have marked effects on the behavior of the distribution. But if I’m not, here is a simple function to create a gg_interaction plot. For the center population plot, we are going to use posterior predicted means for a new (as yet unobserved) participant. Immediately we have a special case of a general model Y = fixed parameters + random effects where the only fixed parameter is. observations independent of time. The results of the individual studies are shown grouped together according to their subgroup. Infix functions. Mixed Models and Random Effect Models. Psychological Methods, 2, 64-78. 2% decrease in the incidence of heart disease. R uses recycling of vectors in this situation to determine the attributes for each point, i. If you use the ggplot2 code instead, it builds the legend for you automatically. Cases or individuals can and do move into and out of the population. In this example, the estimate of variance of random effects location x genotype (LC:CLT), year x genotype (YR:LC) and year (YR) is zero. Minitab is the leading provider of software and services for quality improvement and statistics education. The program RANDOM_WALK_2D_PLOT plots the trajectories of one or more random walks. Random sampling definition, a method of selecting a sample (random sample ) from a statistical population in such a way that every possible sample that could be selected has a predetermined probability of being selected. 19) shows the computa-tion for the among-group variation. OpenGL is the industry's most widely used and supported 2D and 3D graphics application programming interface (API), incorporating a broad set of rendering, texture mapping, special effects, and other powerful visualization functions. This document describes how to plot marginal effects of various regression models, using the plot_model() function. Random Effect Models The preceding discussion (and indeed, the entire course to this point) has been limited to fixed effects" models. But if I’m not, here is a simple function to create a gg_interaction plot. A model with uncorrelated random e ects The data plots gave little indication of a systematic relationship between a subject’s random e ect for slope and his/her random e ect for the intercept. Look closely at. Using R to Compute Effect Size Confidence Intervals. Supported model types include models fit with lm(), glm(), nls(), and mgcv::gam(). Plot the estimates of random effects with confidence intervals plot. In this plot, the scatter in X for a given value of Y is very small, so the association is strong. Our first mixed model. Introduction to R Overview. (Missing values in R appear in the data frame as NA. The formula and data together determine a numerical representation of the model from. Amongst all the packages that deal with linear mixed models in R (see lmm, ASReml, MCMCglmm, glmmADMB,…), lme4 by Bates, Maechler and Bolker, and nlme by Pinheiro and Bates are probably the most commonly used -in the frequentist arena-, with their respective main functions lmer. # scatter plot of expense vs csat plot (sts. Figure 9-6 Specifying plot. It is instructive to review completely randomized design (CRD) and randomized complete block. If there were two random effects per subject, e. mle requests the maximum-likelihood random-effects estimator. Using R to Compute Effect Size Confidence Intervals. The basics of random intercepts and slopes models, crossed vs. code Seeds: random effects logistic regression. Sam, the function is plotting based on the model object, not the data itself, that is why aes_string and the model parameters are in there. You can also create infix functions where the function name comes in between its arguments, like + or -. Here are the estimators implemented in meta , which we can choose using the method. ,2016) package handles the mixed effect model, and in this function, the user can specify the factors with a random effect. Welcome the R graph gallery, a collection of charts made with the R programming language. Main Effects Residual Plots. Each whole plot is divided into 4 plots ( split-plots) and the four levels of manure are randomly assigned to the 4 split-plots. 45609 for the first entry, which corresponds to the first point. (PDF) Table S1. Rags to Riches. Random Forests for Regression and Classification. This can be used to get a look at what what observations may be stressing the model. Amongst all the packages that deal with linear mixed models in R (see lmm, ASReml, MCMCglmm, glmmADMB,…), lme4 by Bates, Maechler and Bolker, and nlme by Pinheiro and Bates are probably the most commonly used -in the frequentist arena-, with their respective main functions lmer. This chapter describes how to compute and. A mathematical line'' has no thickness, so it's invisible; but when we plot circular dots at each point of an infinitely thin line, we get a visible line that has constant thickness. nested models, etc. Weibull plot The fit of a Weibull distribution to data can be visually assessed using a Weibull plot. The mechanisms involved are complex and intertwined, hence undermining the identification of simple adaptation levers to help improve the resilience of agricultural production. These plotting functions have been implemented to easier interprete odds ratios, especially for continuous covariates, by plotting the probabilities of predictors. You will also learn about training and validation of random forest model along with details of parameters used in random forest R package. interactions. Write about whatever it makes you think of. Hundreds of charts are displayed in several sections, always with their reproducible code available. and it is often called the random (or stochastic) part of the model. In this post I will demonstrate in R how to draw correlated random variables from any distribution The idea is simple. population distribution b. glmer(fit, type = "fe", sort = TRUE) To summarize, you can plot random and fixed effects in the way as shown above. If the p-value is significant (for example <0. A video showing basic usage of the "lme" command (nlme library) in R. F),Cigarettes) #resid() calls for the residuals of the model, Cigarettes was our initial outcome variables - we're plotting the residuals vs observered. The main advantage of nlme relative to lme4 is a user interface for ﬁtting models with structure in the residuals (var-. (2005)’s dative data (the version supplied with the languageR library). In ADS, the DOE tool comes with full supporting plots that enable designers to determine simultaneously the individual and interactive effects of many factors that could affect the output results in any design. Fixed and random effects models Random effects model Less powerful because P values are larger and confidence intervals are wider. In above code, the plot_summs(poisson.


text:
Transcendence of numbers with a low complexity expansion.(English)Zbl 0895.11029

The (block-) complexity of a sequence on a finite alphabet is the function $$n\to p(n)$$, where $$p(n)$$ counts the number of (different) blocks of length $$n$$ occurring in the sequence. A small step towards the conjecture that any algebraic irrational number is normal in base $$k$$ (hence contains all possible blocks in its base $$k$$ expansion) would be to prove that, if too many blocks are missing (i.e., $$p(n)$$ is small), then the number is either rational or transcendental. Let us call $$C$$ the set of sequences on a given alphabet $$\{0,1,\dots, k-1\}$$ having low complexity, and such that the corresponding real numbers in base $$k$$ are either rational or transcendental. For automatic sequences, we have $$p(n)= O(n)$$ and results of Loxton and van der Poorten seem to indicate that these sequences are in $$C$$.
In the paper under review the authors translate in combinatorial terms Ridout’s theorem; they obtain a combinatorial criterion of transcendence that implies: Sturmian sequences (i.e., sequences of complexity $$n+k-1$$ on the alphabet $$\{0,1,\dots, k-1\}$$) are in $$C$$ (this was known only for subclasses); Arnoux-Rauzy sequences are in $$C$$ (these sequences have complexity $$2n+1)$$; fixed points with overlaps of primitive morphisms are in $$C$$. This last result has been extended in the binary case by Luca Q. Zamboni (misspelled in the paper under review) and the reviewer to any fixed point of a binary morphism primitive or of constant length and non-trivial [J.-P. Allouche and L. Q. Zamboni, J. Number Theory 69, No. 1, 119–124 (1998; Zbl 0918.11016)].

MSC:

 11J81 Transcendence (general theory) 11B85 Automata sequences

Zbl 0918.11016
Full Text:

References:

 [1] Adams, W.W.; Davison, J.L., A remarkable class of continued fractions, Proc. amer. math. soc., 65, 194-198, (1977) · Zbl 0366.10027 [2] P. Alessandri, 1996, Codages de rotations et basses complexités, University Aix-Marseille 2 [3] J.-P. Allouche, L. Q. Zambon, Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive substitutions [4] P. Arnoux, S. Ferenczi, P. Hubert, 1997, Trajectories of rotations, Institut de Mathématiques de Luminy, Marseille · Zbl 0921.11033 [5] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité 2n, Bull. soc. math. France, 119, 199-215, (1991) · Zbl 0789.28011 [6] Berstel, J.; Séébold, P., A characterization of overlap-free morphisms, Disc. appl. math., 46, 275-281, (1993) · Zbl 0824.68093 [7] Bullett, S.; Sentenac, P., Ordered orbits of the shift, square roots, and the Devil’s staircase, Math. proc. camb. phil. soc., 115, 451-481, (1994) · Zbl 0823.58012 [8] Cobham, A., Uniform tag sequences, Math. systems theory, 6, 164-192, (1972) · Zbl 0253.02029 [9] Davison, J.L., A series and its associated continued fraction, Proc. amer. math. soc., 63, 29-32, (1977) · Zbl 0326.10030 [10] Dekking, M., Transcendance du nombre de thue – morse, C. R. acad. sci. Paris Sér. A-B, 285, A157-A160, (1977) [11] Ehrenfeucht, A.; Lee, K.P.; Rozenberg, G., Subword complexities of various classes of deterministic developmental languages without interactions, Theoret. comput. sci., 1, 59-75, (1975) · Zbl 0316.68043 [12] Ferenczi, S., LES transformations de Chacon: combinatoire, structure géométrique, liens avec LES systèmes de complexité 2n, Bull. soc. math. France, 123, 271-292, (1995) · Zbl 0855.28008 [13] Hedlund, G.A.; Morse, M., Symbolic dynamics II: Sturmian trajectories, Amer. J. math., 62, 1-42, (1940) · JFM 66.0188.03 [14] Komatsu, T., A certain power series and the inhomogeneous continued fraction expansions, J. number theory, 59, 291-312, (1996) · Zbl 0872.11033 [15] Loxton, J.H.; van der Poorten, A.J., Arithmetic properties of automata: regular sequences, J. reine angew. math., 392, 57-69, (1988) · Zbl 0656.10033 [16] K. Mahler, 1961, Lectures on diophantine approximations. Part 1.g, University of Notre Dame · Zbl 0158.29903 [17] Nishioka, K.; Tamura, J.-i.; Shiokawa, I., Arithmetical properties of a certain power series, J. number theory, 42, 61-87, (1992) · Zbl 0770.11039 [18] Ostrowski, A., Bemerkungen zur theorie der diophantischen approximationen I, II, Abh. math. sem. Hamburg, 1, (1922) · JFM 48.0185.01 [19] Pansiot, J.-J., Complexité des facteurs des mots infinis enegendrés par morphismes itérés, ICALP ’84, Lecture notes in computer science, 172, (1984), Springer-Verlag New York/Berlin, p. 380-389 · Zbl 0554.68053 [20] Queffélec, M., Substitution dynamical systems – spectral analysis, Lecture notes in math., 1294, (1987), Springer-Verlag New York/Berlin [21] Tamura, J.-i., A class of transcendental numbers having explicitg, Acta. arith., 71, 301-329, (1995) · Zbl 0828.11036 [22] Thue, A., Über die gegenseite lage gleiche teile gewisser zeichenreihen, Norske vi. selsk. skr. mat. nat. kl., 1, 1-67, (1912) [23] Zhu, Y., The algebraic independence of certain transcendental continued fractions, Acta. math. sin., 2, 127-134, (1991) · Zbl 0736.11035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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Spring Deals - hours only!Up to 70% off on all courses and bundles.-Close
Recap
Using sets to compute unique elements
Set operations
7. Set operations – review
Simulating machines
Summary

## Instruction

Great! Let's have a quick review of basic set operations in Python before we learn more about them:

1. To find elements that are both in set A and set B (in other words, to find the intersection of sets A and B), you can use the .intersection() method or an & operator:
A.intersection(B)
A & B

2. To find elements that are either in set A or set B (in other words, to find the union of sets A and B), you can use the .union() method or a | operator:
A.union(B)
A | B

3. To find elements that are in set A but not in set B (in other words, to find the difference of sets A and B), you can use the .difference() method or a - operator:
A.difference(B)
A - B

4. To find elements that are either in set A or set B but not in both sets (in other words, to find the symmetric difference of sets A and B), you can use the .symmetric_difference() method or an ^ operator:
A.symmetric_difference(B)
A ^ B

5. You can also check if set A is a subset of set B. Use the .issubset() method or a <= operator:
A.issubset(B)
A <= B

6. Similarly, to check if set A is a superset of set B, you can use the .issuperset() method or an >= operator:
A.issuperset(B)
A >= B

Note: set A is a superset of set B if and only if set B is a subset of set A.

## Exercise

Write a function named check_words(word1, word2) that accepts two strings and checks if we can create one of these strings using the letters from the other string (using each letter as many times as we need to). The function should return True if we can create word1 from the letters in word2 or if we can create word2 from the letters in word1. Otherwise, it should return False.

### Stuck? Here's a hint!

For each string, create a set of letters in that string. Then, check if the first set of letters is a subset or superset of the other string.


text:
# Functions and vectors

In this introduction to functions in C++, we have so far seen examples of functions that return values of the basic types of the language, like int and double. But, could a function return a vector? In addition, the functions used in previous examples only have arguments of the basic types of the language. But, could a vector be an argument of a function? For instance, one may want to write a function that displays the list of values stored in a given vector.

## Vectors as return values

Yes, functions in C++ can return a value of type std::vector. Let us see an example of how this can be used in practice.

Recall again the example about the course grades given in the beginning of the motivation section for functions. A vector is used to count the number of grades that fall in each of the $11$ clusters (or intervals). Slot with index $0$ counts the number of grades in the cluster $[0,9]$, slot with index $1$ counts the number of grades in the cluster $[10,19]$, $\cdots$, slot with index $10$ counts the number of grades in the last cluster $[100,100]$.

A useful function in this context is one that reads the grades of a class of students, at same time counts the number of grades that fall in each cluster, and then returns a vector with the $11$ cluster counters.

// Read a list of grades
// Return a vector of 11 cluster counters

Since the program should handle the grades of two different classes of students, two vectors are used, named clustersA and clustersB. These vectors can then be initialized by calling function read_grades().

int main() {
std::cout << "Enter grades for  class A:\n";

std::cout << "Enter grades for  class B:\n";

// ...
}

Function read_grades can be defined as follows.

// Read a list of grades
// Return a vector of 11 cluster counters
std::vector<int> C(11, 0); //create a vector of 11 cluster counters initialized to zero

}

return C;
}

The for-loop above could also be written as a while-loop.

int grade = 0;

int cluster = grade / 10;

++C[cluster];
}

Note that function read_grades is called twice in the program’s main in order to read the grades of both student classes. This illustrates an advantage of using functions: code reusability. By using a function, the programmer was kept away from re-writing the same block of code in different places of the program, what would have made the code lengthier and more complicated.

## Vectors as function arguments

Functions can also have arguments which are vectors.

Continuing with the same example of the course grades, another useful function is one that can display the counters stored in a vector. Thus, one could declare this function as follows. Note that the function has an argument which is a vector of integer values.

// Display the cluster counters in vector C
void display_cluster_counters(std::vector<int> C);

The main can then call this function for variables clustersA and clustersB (lines $12$ and $16$ below).

The function display_cluster_counters can be defined as follows. The $11$th cluster is handled separately (lines $6$ and $7$) from the other ten clusters because the $11$th cluster corresponds to an interval with a single value $[100,100]$.

Note that call-by-value is used to pass a vector to function display_cluster_counters. One can motivate this decision by the fact that the function should not modify the actual arguments, clustersA and clustersB (not even accidentally by a programming bug in the function). Logically speaking, all the function should do is to walk through the list of values in the vector passed to it and display them, but the vector should not be modified in any way. So, call by-by-value gives us this guarantee, since a copy of the vector clustersA (clustersB) is made when the function is called in display_cluster_counters(clustersA) (display_cluster_counters(clustersB)). However, the problem is that the vector clustersA (or clustersB) is a variable with too many bytes to be copied into the function. In many systems, an int occupies $4$ bytes. Thus, the vector clustersA (or clustersB) occupies $11 \times 4 = 44$ bytes, at least. The bottom line is that copying vectors is not efficient and unnecessary copying of vectors should be avoided.

To tackle this problem one should use call by const reference, instead.

Similar to call-by-reference, the function display_cluster_counters receives the address of the vector passed to it and, therefore, it has access to the actual arguments defined in the main, clustersA and clustersB. Thus, there is no copying of the vector passed to the function. In addition, in C++, it is possible to state that an argument passed to the function should be handled as a constant within the function. In the case of the example above, this is expressed by adding const to the function’s argument type (const std::vector<int>&) which implies that the vector referred by C cannot be modified by the function. Any such attempts would result in a compiler error. In this way, it’s possible to combine the efficiency of call-by-reference with the safety of call-by-value for function arguments that occupy many bytes of memory and should not be modified by the function.

We suggest that you make the following experiment: add a line of code to the function display_cluster_counters which attempts to modify vector C, like line of code $17$ in the function below. Note then the compilation error.

You can find below the complete program for the course grades example introduced in motivation section for functions and discussed also in the previous sections.

#include <iostream>
#include <iomanip>
#include <vector>

// Return a vector of 11 cluster counters

// Display the cluster counters in C
void display_cluster_counters(const std::vector<int>& C);

/* ***************************** */

int main() {
std::cout << "Enter grades for  class A:\n";

std::cout << "Enter grades for  class B:\n";

// Display cluster counters, for class A
std::cout << "\nClass A\n";
display_cluster_counters(clustersA);

// Display cluster counters, for class B
std::cout << "\nClass B\n";
display_cluster_counters(clustersB);
}

/* ***************************** */

// Return a vector of 11 cluster counters
std::vector<int> C(11, 0); //create a vector of 11 cluster counters initialized to zero

}

return C;
}

// Display the cluster counters in C
void display_cluster_counters(const std::vector<int>& C) {
int counter = 0;

for (int c : C) {
if (counter == 10) {
std::cout << "[" << 100 << "," << 100 << "]: ";
} else {
std::cout << "[" << counter * 10 << "," << (counter + 1) * 10 - 1 << "]: ";
}

std::cout << c << "\n";

++counter;
}
}

### Good programming practices

In the section Passing arguments to functions, we presented some guidelines for which method to use when passing arguments to functions. We now complete that information with one more case: call by const reference.

Call-by-value Call-by-reference Call by
const
reference
Type T is cheap to copy. Thus, T occupies few bytes (e.g. int, double, bool). Function modifies the value stored in the variable referred by x. Type T is expensive to copy. Thus, T occupies many bytes. But, x cannot be modified in f.

Good programming practice: use call by const reference for a vector passed to a function f, if f should not modify the vector passed to it.

When using call-by-reference (or call by const reference) to pass a vector as argument to a function f, the memory address of the vector is sent to f. A memory address is a non-negative integer usually ocuppying not more than $8$ bytes, while a vector often occupies many more bytes of memory.

Recall that C++-programmers assume that if a function uses call-by-reference to pass an argument x then the function can, and it will, modify x.

We also stress that call by const reference shouldn’t be used for function arguments that occupy few bytes of memory. The reason is that call-by-value results in code that executes faster than call by const reference, in the case of variables that require a few bytes of memory.

## Remove an item from a vector

Programs that store a list of values in a vector may need, at some point, to remove a given value from the list. Thus, it’s useful to create a function to remove a given value x from a given vector V.

Recall that it’s only possible to remove the last element stored in a vector (this operation is called std::pop_back()). Removing an element which is not the last element in the vector is not directly supported by the language and, therefore, requires some extra coding to accomplish this task. We present two functions based on two different algorithms.

### Algorithm $1$

The first function is based on the following simple algorithm:

1. Copy all elements of V to a new vector, with exception of the elements equal to x. Let us call this new vector aux (from auxliary).
2. Return the newly created vector aux.
// Return a vector with all elements in V except those equal to x
std::vector<int> remove(const std::vector<int>& V, int x) {
std::vector<int> aux;  // empty vector

for (int e : V) {
if (e != x) {
aux.push_back(e); // add e in the end of aux
}
}

return aux;
}

As you can see, the function above considers vectors storing values of type int, though the code could be easily generalized for other types. In C++ is also possible to write generic functions which are not dependent on the type of elements stored in the vector. These are called function templates and will be introduced in a more advanced C++ programming course.

Note that call by const reference is used to pass a vector V to the function, since the function does not modify the vector passed to it, while call-by-value is used for argument x. In addition, if there are multiple occurrences of x in V then none of them will be copied to the new vector aux. As an exercise, we suggest that you modify the function above so that it copies all elements in V to a new vector, with exception for the first element in V equal to x.

You can find below a main that calls function remove.

int main() {
std::vector<int> list = {2, 4, 5, 8, 10, 12, 23, 16, 10, 300};

int value = 0;

std::cout << "Enter an integer:\n";
std::cin >> value;

// Remove all occurrences of value from the list
list = remove(list, value);

// Display the list
std::cout << "List = ";

for (int e : list) {
std::cout << e << " ";
}
}

### Algorithm $2$

Our second function is based on a slighly more complicated algorithm with three steps.

1. Search for the first occurrence of x in vector V.
2. Shift all elements after (the first occurrence of) x one slot to the left.
3. Remove the last element of vector V.

Let us illustrate this algorithm with a small example. Consider the vector V given below and that one wants to remove 3.

V[0] V[1] V[2] V[3]
2 3 6 8

Step $1$: The first occurence of 3 in V is found in slot with index $1$.

Step $2$: All elements in V stored after slot $1$ are shifted to the left, i.e. V[2] is copied into V[1] and V[3] is copied into V[2].

V[0] V[1] V[2] V[3]
2 6 8 8

Step $3$: The last element in the vector is now removed (V.pop_back() can be used here).

V[0] V[1] V[2]
2 6 8

Next, we translate this second algorithm into C++-code. Since the first step of the algorithm requires a search, we define first a function to make a linear search for a value in a vector (i.e. to search for the value to be removed). Note that the call by const reference is used to pass a vector to the search function, since a vector usually occupies too many bytes of memory to be copied and a search function should not modify the values stored in the vector.

constexpr int kNot_found = -1;

// Return the index of the first occurrence of value in V
// Return kNot_found, if value does not occur in V
int linear_search(const std::vector<int>& V, int value) {
int counter = 0; // to count slots visited

for (int e : V) {
if (e == value) {
return counter;
} else {
++counter;
}
}

return kNot_found;
}

We can now define the function remove based on the second algorithm presented above. Note that call-by-reference is used to pass a vector to this function because this version of function remove modifies the vector passed to it.

void remove(std::vector<int>& V, int x) {
// Step 1: find first occurrenc of x in V
int found = linear_search(V, x);

if (found == kNot_found) // end the function, if x does not exist in V
return;

// Step 2: shift all values after x to the left
for (int i = found + 1; i < V.size(); ++i) {
V[i - 1] = V[i];
}

// Step 3: remove last element of V
V.pop_back();
}

You can find below a main that calls this function remove.

int main() {
// Declare a vector and initialize it
std::vector<int> list = {2, 4, 5, 8, 10, 12, 23, 16, 10, 300};

int value = 0;

std::cout << "Enter an integer:\n";
std::cin >> value;

remove(list, value);

// Display the list
std::cout << "List = ";

for (int e : list) {
std::cout << e << " ";
}
}

Let us now compare both algorithms given above for removing an element from a vector.

• Algorithm $1$ does not modify the vector V passed to it, while algorithm $2$ modifies V. Thus, call by const reference is used to pass the vector to the first function remove, but call-by-reference is used to pass the vector to the second function.
• Algorithm $1$ creates a new vector and, consequently, requires more memory space than Algorithm $2$.
• Algorithm $1$ can be used to remove all values equal to x from a list stored in a vector. Algorithm $2$ removes only the first occurrence of x.

Algorithm $1$ can be easily modified to remove only the first occurence of x. An interesting exercise that we suggest that you try is to modify the function remove based on the $2$nd algorithm so that it removes all occurrences of x from the vector.


text:
Still alive…

but not teaching math at Yale, for all of you JoCo fans. Goal for the summer, in addition to passing comps, is blogging about math. I have some ideas, but we’ll see how they pan out.

So, umm, yea…

I still exist, despite not posting for almost a year and a half. Go. Me.

While I still intend to (some day) continue with quandle posts, for the time being I’ll be switching to talking about the work I’m doing this summer (and probably will talk a bit about most of the work I did last summer, which is very tangentially related.)

Kludging an MP3 Bookmark

Suppose you, a run of the mill Linux user, just downloaded a very long mp3 – for example a 10 hour mp3 of Cory Doctorow reading The Hacker Crackdown. Naturally you’re not going listen to this in one sitting; but, you cannot find a media player that implements bookmarking. What do you do?

Posted in Computer Science, Off Topic | 2 Comments

Well-Definededness

In Algebra, we just finished chapter 10 of Gallian. One of the big theorems in the chapter is the First Isomorphism Theorem – and since I’m a lazy typist I’ll call it FIT from now on. (Isomorphism theorems 2 and 3 are left as exercises.) Skipping over what a group is, what a homomorphism is, what the kernel of said homomorphism is, and what normal subgroups and factors groups are (see, FIT talks about all sorts of cool stuff!), FIT says:

Suppose $\theta :G \longrightarrow G^\prime$ is a homomorphism. Then $G/\text{ker}\theta \cong G^\prime\theta$; moreover, the map $\theta : G/\text{ker}\theta \longrightarrow G^\prime\theta$ by $g\text{ker}\theta \mapsto g\theta$ is an isomorphism.

One of the points of the proof, since $\theta$ is defined on the cosets of $G/\text{ker}\theta$ by specifying the image of a representative, is whether or not the map $\theta$ is even well defined for some $g\text{ker}\theta \in G/\text{ker}\theta.$ In particular, will taking different representatives of the coset $g\text{ker}\theta$ always have the same image? It turns out that the map is indeed well defined – it’s a rather straightforward argument – but this just brings up another question:

If a map is well defined, does it have the property of well-definededness?

Yay, English!

EDIT: Fixed my accidental interspersing of $\varphi$ for $\theta.$

Quandles

Since I have to begin somewhere, I might as well start with something I find interesting. Back in the summer of 2005, I participated in an REU; our research group (one of four or so) studied objects called quandles. I’ll go into the specifics of what we were trying to do in a later post — today I just want explain what a quandle is.

This being my first mathy post, it’s probably obvious that quandles are a pet topic of mine. And no, I don’t like them just because they may have the best name in a technical field this side of quarks (future nomenclaturists take note – both begin with q.) Joking aside quandles fascinate me because, as we will learn in a moment, quandles are nonassociative. We’ll see, in later posts, that this makes quandles behave quite differently than objects like groups and vector spaces that we may be familiar with.

I’ll talk in more depth about the motivations for studying quandles at a later point; suffice it to say that quandles arise in a very natural way in the context of knot theory. For those of you familiar with knot theory, with the proper setup quandles encode the Reidermeister moves. Quandles also have a significant connection with the fundamental group of the knot complement, if I ever get to algebraic geometry I’ll talk about this connection.[1]

So what exactly are quandles? They are an algebraic system like groups. If you don’t know what a group is, I’ll get there eventually. In defining a quandle, we start with a set Q and a closed binary operation $\vartriangleright : Q \times Q \rightarrow Q$. We then impose the following axioms:

$\forall a,b,c \in Q$ the following holds:

1. $a \vartriangleright a = a$;
2. $\exists x \in Q$ such that $a \vartriangleright x = b$; and lastly,
3. $(a \vartriangleright b) \vartriangleright c = (a \vartriangleright c) \vartriangleright (b \vartriangleright c)$.

For those of you familiar with more mainstream things like group theory, these axioms probably seem strange. We’ll start with axiom 1. This is like saying that $\forall x \in \mathbb{Z}: x + x = x!$ This is certainly true for $0: 0+0 = 0$. But for every other integer, it’s absured. The reason is that in things like groups we can prove:

Theorem:

Let $(G, \cdot)$ be a group and let $x \in G$. Then $x \cdot x = x \implies x = \text{id}_G.$

I’m going to withhold the proof of this until the groups section, but if you’re familiar with groups, this is a very straightforward thing to prove.

Put group theoretically: in groups only the the identity element has order 1; in quandles every element has order 1. This axiom will give us some quaint strangeness. If you’ve no idea what the order of an element is – patience grasshopper (alternatively, hop over to John Armstrong’s excellent The Unapologetic Mathematician which, in addition to inspiring this blog, has a great write up of group theory from a while back.)

Axiom 1 causes some quaint strangeness like the fact that, give quandles $Q, P$ then $\forall p \in P: Q \cong Q \oplus \{p\}.$ Axiom 3 causes strangeness that gives me headaches. It’s also what makes quandles so interesting – axiom 3 is when quandles become nonassociative. Compare the following:

1. $(a \cdot b) \cdot c = a \cdot (b \cdot c)$;

2. $(a \vartriangleright b) \vartriangleright c = (a \vartriangleright c) \vartriangleright (b \vartriangleright c)$.

The first is familiar, it says something is associative – for example (3 + 4) + 5 = 3 + (4 + 5). The second is weird. It says that in quandles, our operation distributes over itself! That’s like saying (3 + 4) + 5 = (3 + 5) + (4 + 5). Ultimately, this is what drew me to quandles – the fact that such a weird structure would crop up very naturally in a number of places.

As always, if you spot any mistakes or would like references on any of the above, just ask. And sorry for the many forward references.

Up next: Examples of Quandles – The trivial quandle on N elements, the Tait-3 quandles, conjugation quandles, and more…

PS. As a side note, today I checked out Baby Rudin 2nd ed. from the science library today. Time permitting (which is a huge if this semester), and in spite of the fact I’m taking Advanced Calc right now, I’m planning on working my way through is – proving his theorems before reading his proofs, doing the problems, and just generally putting some mathematical hair in my beard. (For those of you that don’t know, Rudin’s Principles of Mathematical Analysis is held in quite high esteem by the many people who learned analysis from its dense but rich pages.) Expect updates. Funny, no?

[1] Some history: As far as I know, this connection (specifically, that there’s a presentation of the fundamental group of the knot complement with only conjugation relations – groups under conjugation form quandles as we’ll see later) was first established by David Joyce. While Joyce coined the name quandle and – to my knowledge – first established their topological usage, quandles were considered at least as early as the 1950′s by John Conway (who called them wracks and didn’t impose axiom 3.)

Posted in Quandles, Rudin | 4 Comments

Introduction

So I suppose I should start with the obligatory “Who the hell are you and why should I read this?” post. My name’s Geoff, I’m currently a junior year (I think… long story that I may or may not tell) undergraduate in Theoretical Math at the University of Akron.

I study math because I legitimately enjoy the subject, hard as that may be to believe if you’re not familiar with upper level mathematics. I also study computer science because I like things like programming, operating systems, and databases. Yes, I run Linux. Beyond that… we’ll see where it goes.

Posted in Computer Science, Math, Off Topic | 1 Comment


text:
### Author Topic: Classification of PDEs (Chapter Problems 1) clarification  (Read 1340 times)

#### Tristan Fraser

• Full Member
• Posts: 30
• Karma: 11
##### Classification of PDEs (Chapter Problems 1) clarification
« on: January 07, 2018, 08:08:17 PM »

From the textbook, we defined ( I might be paraphrasing)

Quote
"Equations of the form $Lu = f(X)$ where Lu is a linear partial differential expression with respect to an unknown function u is called a Linear Equation. The equation is homogenous if $f = 0$ and inhomogenous otherwise
and used the example of $Lu := a_{11}u_{xx} + 2a_{12}u_{xy} +a_{22}u_{yy} + a_{1}u_{x} + a_{2}u_{y} + au = f(x)$ (i) . We also know it is Non linear if it cannot be expressed in the form (i).

This leads me to my question as Problem 1 asks us to classify equations (1-10), if they are linear homogenous, inhomogenous or , non linear
and for two of them:
$u_{t} + uu_{x} = 0$ (2) and $u_{t} + uu_{x} + u = 0$ (4).

Paraphrasing the text:
Quote
If $Lu = f(x)$ , and both the coefficients and right hand side expression depend on lower order derivatives, then the equation is Quasilinear

So does this mean that eqns (2) and (4) are Quasilinear then? Is that wholly seperate from the options of Linear Homogenous, Linear Inhomogenous and Nonlinear, or are Quasilinear and Semilinear sub-catagories of those?

« Last Edit: January 08, 2018, 03:26:16 PM by Victor Ivrii »

#### Victor Ivrii

• Elder Member
• Posts: 2553
• Karma: 0
##### Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #1 on: January 07, 2018, 11:08:26 PM »
Quasinear are subclass of Nonlinear , and Semilinear are subclass of Quasilinear

It will not be an error to call $u_{xx}+u_{yy}=u^2$ nonlinear or quasilinear, but it is in fact semilinear (linear expression, plus non-linear expression, depending only on lower order terms).

It will not be an error to call $u_{x}+u^2u_{y}=0$ nonlinear, but it is in fact quasilinear.

These problems ask for the most precise classification

#### Jingxuan Zhang

• Elder Member
• Posts: 106
• Karma: 20
##### Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #2 on: January 08, 2018, 01:36:02 PM »
The term $uu_{x}$ determines $(2),(4)$ to be both Quasilinear.
« Last Edit: January 08, 2018, 03:25:09 PM by Victor Ivrii »

#### Zhuoyan Zou

• Newbie
• Posts: 1
• Karma: 0
##### Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #3 on: January 11, 2018, 06:46:47 PM »
I have some more questions regarding classification; from the homework questions:

(1) ut+ux-u2=0
(2) ut2-ux2-1=0
(3) xux+yuy+zuz=0

My answers are semi-linear for all of them since the coefficients are not dependent on u, and they are not linear operator. Please help me if I am wrong. I am not quite sure about the definition of semi-linear.
Quote
Equations of the type (11) where the right-hand expression f depend on the lower-order derivatives are called semilinear.
What does it exactly mean the right-hand expression f? For example, for (1), if f=u2, then it's not semi-linear.

#### Victor Ivrii

• Elder Member
• Posts: 2553
• Karma: 0
##### Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #4 on: January 11, 2018, 07:12:19 PM »
Quote
What does it exactly mean the right-hand expression?
Quote
It is only for semi-linear. That you can move some terms containing lower order derivatives only (including function) to the right, so on the left only linear part remains.

#### Jaisen Kuhle

• Full Member
• Posts: 20
• Karma: 2
##### Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #5 on: January 12, 2018, 02:19:54 PM »
I have some more questions regarding classification; from the homework questions:

(1) ut+ux-u2=0
(2) ut2-ux2-1=0
(3) xux+yuy+zuz=0

My answers are semi-linear for all of them since the coefficients are not dependent on u, and they are not linear operator. Please help me if I am wrong. I am not quite sure about the definition of semi-linear.
Quote
Equations of the type (11) where the right-hand expression f depend on the lower-order derivatives are called semilinear.
What does it exactly mean the right-hand expression f? For example, for (1), if f=u2, then it's not semi-linear.

(1) ut+ux-u2=0

Semi-Linear because of u2

(2) ut2-ux2-1=0

Fully Non-Linear because of ut2, so not quasi or semi linear.

3) xux+yuy+zuz=0

Linear Homogenous with variable coefficients.


text:
# Solving special exponential integral

1. Feb 2, 2014

### simone8888

Has anybody got an idea how to solve this integral. I tried integration by parts, and that made the things even more complicated, substitutions as well. I used Mathematica to Solve that problem. Here is the integral:

https://scontent-b-ams.xx.fbcdn.net/hphotos-ash3/t1/1010082_10202689859479307_236334784_n.jpg

Or in Mathematica, it would be:

Integrate[E^(-((x^2 b c^2)/(x^2 + b)))/(x^2 + b)^2,x]

2. Feb 2, 2014

### D H

Staff Emeritus
http://www.wolframalpha.com/input/?i=Integrate[E^(-((x^2+b+c^2)/(x^2+++b)))/(x^2+++b)^2,x]
When Mathematica says "no result found in terms of standard mathematical functions" the odds are pretty good that the function to be integrated doesn't have a nice closed form solution.

Even relatively simple integrals such as $\int \exp(-x^2)\,dx$ and ##\int \exp(x)/x\,dx don't have closed form solutions in the elementary functions. However, these two integrals raise their heads so often that special functions have been invented to represent these integrals. Just by appearance, yours almost certainly is not integrable in the elementary functions, and because it's rather esoteric, I doubt there is an existing special function that can represent it, either.

There are lots of numerical techniques to solve such integrals when expressed as a definite integral. The generic name for such numeric integration is quadrature.


text:
## Conceptual Physics (12th Edition)

Published by Addison-Wesley

# Chapter 17 - Think and Solve: 38

#### Answer

320 calories. It is less.

#### Work Step by Step

Changing 1 gram of $-273^{\circ}C$ ice to $0^{\circ}C$ ice requires (273)(0.5) = 140 calories. Melting that gram of $0^{\circ}C$ ice to $0^{\circ}C$ of water requires 80 calories. Raising the temperature of 1 gram of $0^{\circ}C$ water to $100^{\circ}C$ boiling water requires 100 calories. The subtotal is 320 calories. This is to be compared with the energy needed to change 1 gram of $100^{\circ}C$ water to steam at the same temperature, which is 540 calories. The latter process, perhaps surprisingly, takes more energy than it took to bring the water all the way from absolute zero to $100^{\circ}C$!

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# www.campusplace.co.in

One stop blog for all competitive examinations

Directions( from questions 1 to 5) An automobile company produces five models. The following are the details of units of cars manufact...

# Data Interpretation Solved Questions for Bank Tests - Data Interpretation on Tables for CAT, GRE, GMAT

Directions( from questions 1 to 5)
An automobile company produces five models. The following are the details of units of cars manufactured in the six years mentioned. Study the table and answer the questions given.
 Model 2011 2012 2013 2014 2015 2016 Alpha Beta Gamma Omega Theta 2200 2500 2100 1400  2800 2500 2800 2400 1800 3200 1980 2750 2600 1250  3650 2800 2900 2250 1485 1950 1800 3200 2680 1680 2450 3200 1885 2356 1800 1800

1. For which of the following models, the productions in 2014 showed the highest change over that in 2013?
a) Alpha   b) Omega and Alpha       c) Beta and Omega   d) Beta only  e)None of these
Explanation:
Alpha = 2800 – 1980 =820
Beta = 2750- 2900 =-250
Gamma = 2600 – 2250= 350
Omega = 1250 – 1485 = -235
Theta = 1950-3650 = - 1700
Change may be increase or decrease. So model Theta has highest change of production in 2014 over 2013.

2.  In which year does the production in model Omega showed the highest percentage increase over the previous year?
a) 2012                   b) 2013         c) 2014                   d) 2015    e)2015
Explanation :
% Increase in 2012 = $\frac{1800-1400}{1400}$ x 100 % =28.57%

% Increase in 2014= $\frac{1485-1250}{1250}$ x 100% =18.8%

% Increase in 2015 =$\frac{1680-1485}{1485}$ x 100% =13.13%

% increase in 2016 =  $\frac{1800-1680}{1680}$ x 100% = 7.14%

The highest percentage increase in the production of model Omega is in 2012 over the previous year

3. The production of which model increase continuously from 2013 to 2016?
a) Alpha only           b) Beta and Gamma           c) Both a and b
d)Omega                 e) None of the above
Explanation :
Model Omega only has continuous  increase in its production from 2013 to 2016.

4. In which year did the productions of Theta fall for the first time?
a)2014                 b) 2015         c) 2016         d) 2012         e)2013
Explanation :
For the first time, in 2014, production is decreased to 1950 from 3650.

5. In 2011, which of the following models contributed approximately 25% of the total productions of all the five models?
a) Alpha                  b) Mango       c) Omega                d) Gamma     e) Theta
Explanation :
Total production of all five models in 2011 = 2200+2500+2100+1400+2800 =11000

% contribution  of Alpha model in 2011 = $\frac{2200}{11000}$ x 100% = 20%

% contribution  of Beta model in 2011 = $\frac{2500}{11000}$ x 100% = 22.7%

% contribution  of Gamma model in 2011 = $\frac{2100}{11000}$ x 100% = 19.09%

% contribution  of Omega model in 2011 = $\frac{1400}{11000}$ x 100% = 12.72%

% contribution  of Theta model in 2011 = $\frac{2800}{11000}$ x 100% = 25.45%

In 2011 , Theta model contributed approximately 25% of the total productions of all the five models .

Directions (Questions from 1 to 5): Trends in expenditure in lacs of various departments of a company for different years.
 2008-09 2009-10 2010-11 2011-12 2012-13 2013-14 Marketing Finance Production HR Other Dept 3120 1200 13860 2100 250 1480 1500 23670 2520 375 1680 2400 15450 3500 400 2000 2800 17250 4500 450 2480 3280 18250 4250 137 1816 4000 10539 5100 600

1. The average marketing expenditure from 2008-09 to 2013-14?
a) 2196                   b) 2076         c) 4015                   d) 2167      e)2096
Explanation:
Total expenditure of marketing department in 6 years
=  3120 + 1480 + 1680 +2000+ 2480 +1816 =12576

Average expenditure =$\frac{12576}{6}$ 12576/6 = 2096

2. In which year, the percentage increase in the expenditure of Finance department with respect to previous year is maximum?
a) 2009-10              b) 2010-11               c)2011-12               d) 2012-13   e)2013-14
Explanation :
% increase in 2009-10 over previous year=$\frac{1500-1200}{1200}$ x 100% = 25%

% increase in 20010-11 over previous year=$\frac{2400-1500}{1500}$ x 100% = 60%

% increase in 2009-10 over previous year=$\frac{2800-2400}{2400}$ x 100% = 16.66%

% increase in 2009-10 over previous year=$\frac{3280-2800}{2800}$ x 100% = 17.14%

% increase in 2009-10 over previous year=$\frac{4000-3280}{3280}$ x 100% = 21.95%

In year 2010-11 , % increase in the expenditure of finance department over previous year is maximum

3. The difference in the total expenditure of all departments between the years 2012-13 and 2013-14?
a) 6328                   b) 4387         c) 6154                   d) 6342         e)None of these
Explanation :
Total expenditure in the year 2012-13 = 2480 + 3280 + 18250 + 4250 +137 =28397
Total expenditure in the year 2013-14 = 1816 + 4000+ 10539 + 5100+ 600 =22055
The required difference is => 28397 – 22055 =6342

4. The percentage increase in expenditure is highest for which department in the year 2011-12  over the previous year 2010-11?
a) Marketing            b) Finance     c) Production            d)HR   e) Other depts.
Explanation:
% increase in expenditure in Marketing department = $\frac{1680-1480}{1480}$1680-1480/1480 x 100% =13.51%

% increase in expenditure in Finance department = $\frac{2800-2400}{2400}$ x 100% =16.66%

% increase in expenditure in Production department =$\frac{17250-15450}{15450}$ x 100% =11.65%

% increase in expenditure in Marketing department =$\frac{4250-4000}{4000}$ x 100% =6.25%

There is not increase in other departments from 2010-11 to 2011-12.
From the year 2010-11 to 2011-12 , the highest percentage increase in expenditure is in Finance department.
5. What is the approximate percentage of Production department’s expenditure in the total expenditure of all departments for the year 2012-13?
a) 64                      b) 38             c) 42                      d) 18            e) None of these
Therefore, required percentage = $\frac{18250}{28397}$ x 100% = 64


text:
This study was conducted to determine the biological effects of inhaled <tex-math>${}^{238}{\rm PuO}{}_{2}$</tex-math> over the life spans of 144 beagle dogs. The dogs inhaled one of two sizes of monodisperse aerosols of <tex-math>${}^{238}{\rm PuO}{}_{2}$</tex-math> to achieve graded levels of initial lung burden (ILB). The aerosols also contained <tex-math>${}^{169}{\rm Yb}$</tex-math> to provide a γ-ray-emitting label for the <tex-math>${}^{238}{\rm Pu}$</tex-math> inhaled by each dog. Excreta were collected periodically over each dog's life span to estimate plutonium excretion; at death, the tissues were analyzed radiochemically for plutonium activity. The tissue content and the amount of plutonium excreted were used to estimate the ILB. These data for each dog were used in a dosimetry model to estimate tissue doses. The lung, skeleton and liver received the highest α-particle doses, ranging from 0.16-68 Gy for the lung, 0.08-8.7 Gy for the skeleton and 0.18-19 Gy for the liver. At death, all dogs were necropsied, and all organs and lesions were sampled and examined by histopathology. Findings of non-neoplastic changes included neutropenia and lymphopenia that developed in a dose-related fashion soon after inhalation exposure. These effects persisted for up to 5 years in some animals, but no other health effects could be related to the blood changes observed. Radiation pneumonitis was observed among the dogs with the highest ILBs. Deaths from radiation pneumonitis occurred from 1.5 to 5.4 years after exposure. Tumors of the lung, skeleton and liver occurred beginning at about 3 years after exposure. Bone tumors found in 93 dogs were the most common cause of death. Lung tumors found in 46 dogs were the second most common cause of death. Liver tumors, which were found in 20 dogs but were the cause of death in only 2 dogs, occurred later than tumors in bone and lung. Tumors in these three organs often occurred in the same animal and were competing causes of death. These findings in dogs suggest that similar dose-related biological effects could be expected in humans accidentally exposed to <tex-math>${}^{238}{\rm PuO}{}_{2}$</tex-math>.

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