text: To evaluate the HOMR Model, we followed the procedure outlined in Vergouwe et al (2016) and estimated four logistic regression models. The area under the receiver operating characteristic (ROC) curve (C-statistic) is often cited; it was developed for a different purpose, however, and may not address the clinically relevant questions. There are two main measures for assessing performance of a predictive model: Discrimination and Calibration. model: a logical value indicating whether the estimated area under the curve should be displayed. First time it happens. In its current usage, ROC curves are a nice way to see how any predictive model can distinguish between the true positives and negatives. Plots the ROC Curve rocplot: ROC Plot for a logistic regression model in Deducer: A Data Analysis GUI for R rdrr. Although SVM produces better ROC values for higher thresholds, logistic regression is usually better at distinguishing the bad radar returns from the good ones. In this analysis, the power of the model's predicted values to discriminate between positive and negative cases is quantified by the Area under the ROC curve (AUC). 5 is no better than random guessing. The objective of this case is to get you understand logistic regression (binary classification) and some important ideas such as cross validation, ROC curve, cut-off probability. The ROC curve does this by plotting sensitivity,. This feature is not available right now. The ROC curve for naive Bayes is generally lower than the other two ROC curves, which indicates worse in-sample performance than the other two classifier methods. scikit-learn makes it super easy to calculate ROC Curves. I By the Bayes rule: Gˆ(x) = argmax k Pr(G = k |X = x). Plot ROC curve and lift chart in R « Heuristic Andrew: "If this programming is too much for you, try rattle (a GUI interface to R for data mining) or Weka (a machine learning suite). Is there any. A one-column table with the area(s) under the ROC curve(s) Views ROC Curves ROC curves Best Friends (Incoming) Scorer (27 %) Decision Tree Predictor (14 %) Joiner (10 %) Logistic Regression Predictor (4 %) X-Aggregator (4 %) Random Forest Predictor (3 %) Streamable Deprecated; Naive Bayes Predictor (3 %) Streamable Deprecated; SVM Predictor (2. Logistic Regression Variable Selection Methods. To construct the ROC curves for comparing the performances of different classification models, we will first need to build the classification models and save the estimated probability of classifying each case into the ‘positive’ category. This means that the Red curve is better. And, probabilities always lie between 0 and 1. To compute the points in an ROC curve, we could evaluate a logistic regression model many times with different classification thresholds, but this would be inefficient. In other words, the observations should not come from repeated: measurements or matched data. Open the menu in the upper right corner of the interactive view, and select “Chart Subtitle”. Our study can contribute to the choice of the best approach in CR prediction for small businesses. Logistic regression allows us to estimate the probability of a categorical response based on one or more predictor variables ( X ). scikit-learn makes it super easy to calculate ROC Curves. Logistic Regression Logistic Regression Preserve linear classiﬁcation boundaries. If you don't want to draw the ROC curve (because it takes time), just comment out plot line. You can use the ROC Curve procedure to plot probabilities saved with the Logistic Regression procedure. auc The value of the area under the ROC curve. The logistic curve is displayed with prediction bands overlaying the curve. Calculating an ROC Curve in Python. Predicting Frank Harrel writes Is Medicine Mesmerized by Machine Learning? Some time ago I wrote about predictive modeling and the differences between what the ROC curve may tell us and how well a model 'calibarates. specificity (complement) •Originally designed to grade radar detection methods for German planes •Decades later, their usefulness in classification problems was realized •But the name stuck. 2 eliminates the need for the output data set creation in order to obtain and plot the fitted logistic curve and ROC curve. 2 by using the PLOTS=ROC option on the PROC LOGISTIC line. Using cross validation and AUC-ROC for a logistic regression model in sklearn. The ROC curve is constructed by plotting sensitivity on the Y-axis and the quantity (1-specificity) on the X-axis; thus, the optimum DL is the point closest to the upper left-hand corner of the graph , ie, where the area under the curve approaches 1. Artificial neural networks are commonly thought to be used just for classification because of the relationship to logistic regression: neural networks typically use a logistic activation function and output values from 0 to 1 like logistic regression. Download Open Datasets on 1000s of Projects + Share Projects on One Platform. Many statisticians and SAS programmers who are familiar with logistic regression have seen receiver operator characteristic (ROC) curves. 5 to predict the labels. 'lroc' graphs the ROC curve of a logistic regression model. Setting up Example 1 Modeling ROC and cutoff point AUC References This is part 1 of a two part article on logistic regression. In the end, we will consider two di erent classi ers (i. Now, suppose we decide to make the definition of hypothyroidism less stringent and now consider patients with T4 values of 7 or less to be hypothyroid. Ensure that you are logged in and have the required permissions to access the test. The ideal way to do this is to fit the model and test the model on independent data sets. I would like to know how can I draw a ROC plot with R. Code in this case is built upon lecture slides and sample code of Shaonan Tian and Shaobo Li. • PROC LOGISTIC in version 8 contains a CLASS statement, meaning that this is now the procedure of choice for logistic regression in SAS. ROC CURVE INTERPRETATION The chance diagonal is a line joining (0, 0) and (1, 1) dividing the curve into two equal parts. If this is not possible, we can use cross validation. producing ROC curves. Is it right way to use values predicted by logistic regression with markers considered as. Request the "Coordinate points of the ROC Curve. Hi Mary,=0A=0AMay be the answer is to consider separate logistic regression= models instead of a single multinomial model and then use the option in pr=. Logistic Regression and ROC Curves Using JMP Exercises Important note about ordering of the outcome levels For all the exercises the outcome is Significant Disease which is coded as 1=yes and 0=no, and is a nominal variable. , positive or negative) classification problems such as Logistic Regression. The objective of the dataset is to diagnostically. > which derives the ROC curve from a logistic regression, SPSS does so > entirely from the Graph menu. When you do a simple logistic regression with a binary outcome, there is a platform option to request a ROC curve for that analysis. Logistic Regression using Excel uses a method called a logistic function to do its job. • An additional beneﬁt of PROC LOGISTIC is that it contains options speciﬁc to logistic regression, such as goodness-of-ﬁt tests and ROC curves. Logistic Regression is likely the most commonly used algorithm for solving all classification problems. Tree induction and logistic regression are two standard, off-the-shelf methods for building models for classification. The C-statistic (sometimes called the “concordance” statistic or C-index) is a measure of goodness of fit for binary outcomes in a logistic regression model. Discrimination: A model has good discrimination if the distribution of risk scores for cases and controls separate out. Explaining ROC curves and PR curves in depth is a very ambitious take which is unlikely to be achieved in one single post. Therefore, in logistic regression, larger values of covariates that have positive coe cients will tend to increase the probability that Y = 1. Logistic Regression Model or simply the logit model is a popular classification algorithm used when the Y variable is a binary categorical variable. And, probabilities always lie between 0 and 1. Both anterior and inferior refer to sites on the heart where damage occurs. roc_curve (y_true, y_score, pos_label=None, sample_weight=None, drop_intermediate=True) [source] ¶ Compute Receiver operating characteristic (ROC) Note: this implementation is restricted to the binary classification task. Lastly we're going to add a constant term for our Logistic Regression. ROC curve example with logistic regression for binary classifcation in R. In practice, most of the classification models have an AUC between 0. Calculating an ROC Curve in Python. Receiver Operating Characteristic (ROC) Curves The « Receiver Operating Characteristic » function (ROC function) is a measure of the performance of a binary classifier. • Assessing Goodness to Fit for Logistic Regression • Assessing Discriminatory Performance of a Binary Logistic Model: ROC Curves. Look at the ROC Curve for the other "Evaluate Model" visualization, we can see that the Logistic Regression model is slightly more accurate than the Support Vector Machine. Logistic Regression 1)Fits a linear relationship between the variables 2)Transforms the linear relationship to an estimate function of ROC Curve Receiver. • Usually a concave shaped curve connecting (0, 0) and (1, 1). ROC curves can be used to evaluate how well these methods perform. 2 by using the PLOTS=ROC option on the PROC LOGISTIC line. For nominal variables, by default JMP will make the lowest ordered category (either in numerical or alphabetical order) the. For logistic regression you can create a 2 × 2 classification table of predicted values from your model for your response if $$\hat{y}=0$$ or 1 versus the true value of y = 0 or 1. Logistic Regression will estimate binary (Cox (1970)) and multinomial (Anderson (1972)) logistic models. Sensitivity = probability of predicting a non-event, given that the. It is also one of the first methods people get their hands dirty on. a disease or condition) had a higher risk score than a patient who had not experienced the event. We replicate their analysis here. An higher AUC is good. will be stored as tables. If that DV was coded 0 and 1, LOGISTIC REGRESSION predicts the 1 value and 1 will be the 'Value of state variable' in the ROC curve dialog. Logistic regression allows us to estimate the probability of a categorical response based on one or more predictor variables ( X ). use the Linear Regression procedure. For this example, I'm going to make a synthetic dataset and then build a logistic regression model using scikit-learn. Printer-friendly version; Navigation. Logistic Regression, ROC curve & AUC updragon 2018년 8월 4일. 5, because the curve follows the diagonal. This value should be the shoulder at the top left of the ROC (receiver operating characteristic curve). It’s a graph of sensitivity vs. logistic regression, and bagging applied to both methods. In doing so, you'll make use of the. The following example shows how to train binomial and multinomial logistic regression models for binary classification with elastic net. To evaluate the HOMR Model, we followed the procedure outlined in Vergouwe et al (2016) and estimated four logistic regression models. Here the ROC curve for the response scores from the logistic regression model is calculated with the widely used pROC package and plotted as a yellow line. The html output contains the regular stuffs, along with the ROC curve for the training data as well as the ROC curve of the validation data. It is possible to do this using the logistic linear predictors and the roccomp. It should be lower than 1. ## Cross validation Q = np. Toward the end, we will build a logistic regression model using sklearn in Python. 2 The Hosmer-Lemeshow Tests page 150 Table 5. 'lroc' graphs the ROC curve of a logistic regression model. Example of Receiver Operating Characteristic (ROC) metric to evaluate classifier output quality. Instructions: This web page calculates a receiver operating characteristic (ROC) curve from data pasted into the input data field below. Our study can contribute to the choice of the best approach in CR prediction for small businesses. The advantages of logistic regression models include the comparatively easy implementation, the availability in all standard statistical software packages, and short computation times. Basically the code works and it gives the accuracy of the predictive model at a level of 91% but for some reason the AUC score is 0. If the motivation for the logistic regression analysis is prediction it is important to assess the predictive performance of the model unbiasedly. The blue "curve" is the predicted probabilities given by the fitted logistic regression. ROC Analysis. This feature is not available right now. ROC curve example with logistic regression for binary classifcation in R. In order to do this, a model needs to not only correctly predict a positive as a positive, but also a negative as a negative. Many statisticians and SAS programmers who are familiar with logistic regression have seen receiver operator characteristic (ROC) curves. 5, while AUC for a perfect classifier is equal to 1. model: a logical value indicating whether the estimated area under the curve should be displayed. 3 - Receiver Operating Characteristic Curve (ROC) up. Logistic Regression 1)Fits a linear relationship between the variables 2)Transforms the linear relationship to an estimate function of ROC Curve Receiver. You can also think of logistic regression as a special case of linear regression when. In my previous post, I showed how to run a linear regression model with medical data. 1-specificity. Multi-label Classification Metrics This pages shows an example of applying logistic regression for a9a binary classification task. Which I will be explaining later, in this post I will target mostly on how logistic regression works. Therefore, this post will serve as an opening for following posts by introducing how to plot ROC and PR curves. 20 provided a sensitivity of 65% and a specificity of 96%. The first portion of the analysis from Comparing Logistic Regression Models is shown here in Figure 1. The ROC curve allows for easy visual comparison among a number of models, in this case, the Naive Bayes model appears to be the best performing of the models applied to date (much of it overlaps with the Logistic Model, however there is a bit of the red dotted line appearing above the Logistic model. Logistic Regression will estimate binary (Cox (1970)) and multinomial (Anderson (1972)) logistic models. a graphic means of assessing the ability of a screening test to discriminate between healthy and diseased people. Key words: hemoglobin A1c (HbA1c), diabetes, predict, receiver operating characteristic curve (ROC curve), logistic regression analysis In diabetes screening with hemoglobin A1c in lieu of plasma glucose, the optimum cut-off point for predicting the incidence of diabetes mellitus in the four-year period was examined. A probability value of 0. Simonoff , Claudia Perlich , Claudia Perlich , Foster Provost , Foster Provost , Jeffrey S. Logistic Regression is one of the regression model algorithms that can be used to predict the binary outcome like TRUE or FALSE based on input variables (predictors). 10 Video 6: ROC Curves 3. For this example, I'm going to make a synthetic dataset and then build a logistic regression model using scikit-learn. Example 1: Create the ROC curve for Example 1 of Comparing Logistic Regression Models. Ensure that you are logged in and have the required permissions to access the test. The most common type of neural networks, a feed-forward back-ward propagation neural network, was selected for this study. ROC Analysis. To avoid such pitfalls, it is necessary to understand ROC curves and PR curves thoroughly. 4 You can use the ROC Curve procedure to plot probabilities saved with the 4 IBM SPSS Regression 22. What do you really want to do?. For logistic regression you can create a 2 × 2 classification table of predicted values from your model for your response if $$\hat{y}=0$$ or 1 versus the true value of y = 0 or 1. 931 for the logistic regression model. The ROC curve is plotted using the sensitivity and specificity for all possible cut-offs, i. 2 by using the PLOTS=ROC option on the PROC LOGISTIC line. Read more in the User Guide. Concept Description. Titanic Data Set and the Logistic Regression Model. 1-specificity. In order to optimize the predictive model, the six potential predictors were further evaluated using multivariate logistic regression model. Maybe you’ve avoided logistic regression before because it’s seemed quite complex or overwhelming… or simply because it wasn’t a required part of your previous statistics coursework. Radiotherapy is a standard treatment option for breast cancer, but it may lead to significant late morbidity, including radiation heart damage. Which I will be explaining later, in this post I will target mostly on how logistic regression works. Discrimination: A model has good discrimination if the distribution of risk scores for cases and controls separate out. ROC, or Receiver Operator Characteristic, is used to examine the performance of a diagnostic test over a range of decision levels (medical decision points). See the section for the ROC Curve chart for more information on this measure. In this paper we present a method that simultaneously accomplishes both tasks. 5 which is basically the worst possible score because it means that the model is completely random. Briefly, an empirical distribution was obtained for the area under curve (AUC) derived from the ROC analysis and the determination coefficient (R2) derived from the logistic regression analysis, respectively, by randomly reallocating all of the patients into two groups (improvers and non-improvers) and re-computing the AUC and R2 based on the. ROC curves typically feature true positive rate on the Y axis, and false positive rate on the X axis. By comparing the ROC curves with the area under the curve, or AUC, it captures the extent to which the curve is up in the Northwest corner. How do I create ROC curve for combined biomarkers in SPSS? I plan to use marker combination in my thesis. In order to optimize the predictive model, the six potential predictors were further evaluated using multivariate logistic regression model. auc The value of the area under the ROC curve. The area under the curve is 0. The predictors can be continuous, categorical or a mix of both. Plots the ROC Curve rocplot: ROC Plot for a logistic regression model in Deducer: A Data Analysis GUI for R rdrr. (View the complete code for this example. Example 1 : Create the ROC curve for Example 1 of Comparing Logistic Regression Models. This is the logistic regression curve we have received which is basically the ROC curve. In the ROC Curve procedure, use the predicted probability variable as the test variable and the dependent variable from the logistic regression as the state variable with 1 as the State value. That is, $\hat{p}(x) = \hat{P}(Y = 1 \mid { X = x})$ The solid vertical black line represents the decision boundary , the balance that obtains a predicted probability of 0. In the window. Plots: ROC curve. Here the ROC curve for the response scores from the logistic regression model is calculated with the widely used pROC package and plotted as a yellow line. • An additional beneﬁt of PROC LOGISTIC is that it contains options speciﬁc to logistic regression, such as goodness-of-ﬁt tests and ROC curves. A random classifier has an area under the curve of 0. To analyze your data, use the following steps to fill out the data form on this page. Code in this case is built upon lecture slides and sample code of Shaonan Tian and Shaobo Li. The AUC corresponds to the probability such that a positive event has a higher probability given to it by the model than a negative event. This model will use labels with values in the set {0, 1} and will try to predict a continuous value that is as close as possible to 0 or 1. Explaining ROC curves and PR curves in depth is a very ambitious take which is unlikely to be achieved in one single post. FULL TEXT Abstract: Penalized regression methods offer an attractive alternative to single marker testing in genetic association analysis. 20 provided a sensitivity of 65% and a specificity of 96%. We then cover the area under curve (AUC) of the ROC curve as a measure of the predictive power of the model and the apply that to both training and validation datasets and compare against each. Or copy & paste this link into an email or IM:. In such cases, you should use the ROC (Receiver Operating Curve) - which is a plot of %True positves against % False positives. Logistic regression is similar to linear regression, with the only difference being the y data, which should contain integer values indicating the class relative to the observation. Multi-label Classification Metrics This pages shows an example of applying logistic regression for a9a binary classification task. Although PROC LOGISTIC creates many tables, I've used the ODS SELECT statement to suppress all output except for the ROC curve. Receiver Operating Characteristic (ROC) Curves The « Receiver Operating Characteristic » function (ROC function) is a measure of the performance of a binary classifier. This page briefly describes methods to evaluate risk prediction models using ROC curves. Hi: In my opinion, we should look at all three. So how well do we do across all cut-offs? ROC (Receiver Operating Characteristic) Curves: "Coin Flip" Rate Summarize an ROC curve by the area under the curve (AUC): 0. In my sample the occurence is 16. SARKAR 3 [1,2,3]Laboratory of Applied and Computational Statistics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, MALAYSIA E-mail: [email protected] We then cover the area under curve (AUC) of the ROC curve as a measure of the predictive power of the model and the apply that to both training and validation datasets and compare against each. See the section for the ROC Curve chart for more information on this measure. Performance of Logistic Regression Model. The estimate of the area under the ROC curve can be computed either nonparametrically or parametrically using a binegative exponential model. scikit-learn makes it super easy to calculate ROC Curves. Logistic regression interview questions – Set 1; Logistic regression interview questions – Set 2; Logistic Regression Concepts (Brushing up) Evaluation of Logistic regression models using some of the following techniques: Deviance calculations (Null and Residual Deviance) R-squared (McFadden, Cox and Snell, Likelihood ratio) ROC curve; AUC. 56 and the specificity is 0. Briefly, an empirical distribution was obtained for the area under curve (AUC) derived from the ROC analysis and the determination coefficient (R2) derived from the logistic regression analysis, respectively, by randomly reallocating all of the patients into two groups (improvers and non-improvers) and re-computing the AUC and R2 based on the. R” is included into the archive which is distributed with this tutorial. Also try practice problems to test & improve your skill level. In the ROC Curve procedure, use the predicted probability variable as the test variable and the dependent variable from the logistic regression as the state variable with 1 as the State value. 2 Modeling the Expert: An Introduction to Logistic Regression » 3. Probit regression can used to solve binary classification problems, just like logistic regression. r logistic-regression roc auc this question asked Dec 20 '14 at 20:34 rk567 94 1 12. Performance of Logistic Regression Model. diseased vs. Therefore, the logistic regression model performs better than the decision tree model in predicting the gender value of the people in the dataset. , all the probability scores. com Wan Nor Arifin, 2015. Logistic Regression is likely the most commonly used algorithm for solving all classification problems. Prediction and adjusted means The prediction option allows you to calculate values of the outcome (as response proportion) using your fitted logistic model coefficients with a specified set of values for the predictors (X1…p). In the logistic regression the constant (b 0) moves the curve. In general, I like STATA better for > ROC, but SPSS is nice by allowing one to plot several ROC curves on one > plot. This feature is not available right now. This vary based on the business problem. 931 for the logistic regression model. Logistic curve Note that from this curve we see some important characteristics of logistic regression: I The logistic curve is increasing. (Partial) area under the curve (AUC) can be compared with statistical tests based on U-statistics or bootstrap. I When z>0, then g 1(z) >1=2; when z<0, then g 1(z) <1=2. Logistic regression CART Regression Classi cation Example Estimation Partitioning Model complexity Pruning Surrogates Random Forests Torben Tvedebrink [email protected] The logistic curve is displayed with prediction bands overlaying the curve. 0 with larger values indicative of better fit. I have separated the graphs for Validation and Training data. Do Over-Sampling, use the over-sampled data to model 2. For an ideal model, AUC=1 and for a random model, AUC = 0. Output: ROC -AUC Curve: ROC stands for Receiver Operating Characteristic and AUC stands for Area Under Curve. Logistic Regression employs the logit model as explained in Logit / Probit / Gompit (see 7. Not only is there the lost revenue as a result of the customer cancelling, but this can also cause difficulty in coordinating bookings and adjusting revenue management practices. 2) enables users to specify nested logistic regression models, estimate their ROC curves and compare them using the AUC test. The various outputs like parameter estimate, concordance-discordance, classification table etc. In the situation where you have imbalanced classes, it is often more useful to report AUC for a precision-recall curve. A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied. Maybe you’ve avoided logistic regression before because it’s seemed quite complex or overwhelming… or simply because it wasn’t a required part of your previous statistics coursework. In this post, I will show how to conduct a logistic regression model. But even if could do that, when a i run a regression (xtlogit) with some particular group of independent variables, there is always the message "backed up" after some iterations. This is why logistic regression makes use of the sigmoid function. Tools for visualizing, smoothing and comparing receiver operating characteristic (ROC curves). Generally, one of the two levels of the response is considered the level of interest. Read more in the User Guide. Receiver operating characteristic (ROC) curve and logistic regression analyses were also used to evaluate the diagnostic value of single and multiple markers of lung cancer. com Wan Nor Arifin, 2015. A novel feature of RegressItLogistic is that nearly all of the tables and charts in the logistic regression output have interactive outputs that are controlled with spinners. false positive rate The false positive rate (or equivalently the complement of sensitivity) of the classifier at the evaluated thresholds. Conﬁdence intervals can be computed for (p)AUC or ROC curves. The area under the receiver operating characteristic (ROC) curve (C-statistic) is often cited; it was developed for a different purpose, however, and may not address the clinically relevant questions. Logistic Regression, ROC curve & AUC updragon 2018년 8월 4일. pyplot as plt rs = np. The goal of this exercise is to walk through a logistic regression analysis. Change the threshold to selected value, rather than using default 0. The Area Under the ROC Curve is another popular summary statistic for binary classification. After fitting a binary logistic regression model with a set of independent variables, the predictive performance of this set of variables - as assessed by the area under the curve (AUC) from a ROC curve - must be estimated for a sample (the 'test' sample) that is independent of the sample used to predict the dependent variable (the 'training' sample). See the section for the ROC Curve chart for more information on this measure. The receiver operating characteristic (ROC) curve is the plot that displays the full picture of trade-off between the sensitivity (true positive rate) and (1- specificity) (false positive rate) across a series of cut-off points. Logistic Regression is likely the most commonly used algorithm for solving all classification problems. predict_proba (X_test). The ideal classifier always passes through this point (TPR=1, FPR=0), and this ROC curve is a characteristic curve for such a classifier. Concept: Sensitivity and Specificity - Using the ROC Curve to Measure Concept Description.  Models: multiple logistic regression, prognostic score models, artificial neural networks  Statistics: c-index (equivalent to area under the ROC curve) • Validation of models on independent dataset: 3/99 - 12/99. pancakeapps. So how well do we do across all cut-offs? ROC (Receiver Operating Characteristic) Curves: "Coin Flip" Rate Summarize an ROC curve by the area under the curve (AUC): 0. predict_proba() method and become familiar with its functionality. a logistic regression and evaluate the model using an ROC curve. In other words, we can say: The response value must be positive. [email protected] receiver operating characteristic (ROC) curve. More than 800 people took this test. Of 13 clinical factors, six potential predictors were selected from 434 patients in the training cohort, based on nonzero coefficients in the LASSO logistic regression model. Concept: Sensitivity and Specificity - Using the ROC Curve to Measure Concept Description. my / wnarifin. Evaluating the model: Overview. Titanic Data Set and the Logistic Regression Model. How can I compare areas under ROC curves if I want to adjust the model for additional covariates? I guess, "rocfit" does not allow the inclusion of covariates; and after multiple logistic regression you will get of course only 1 ROC curve. In order to achieve this goal, logistic regression and survival analysis methods are applied to a large dataset of mortgage portfolios recorded by one of the national banks. The data and logistic regression model can be plotted with ggplot2 or base graphics, although the plots are probably less informative than those with a continuous variable. t-tests, the ROC curve and the Area under the ROC curve (AROC). This means that the top left corner of the plot is the “ideal” point - a false positive rate of. Logistic regression requires the observations to be independent of each: other. This value should be the shoulder at the top left of the ROC (receiver operating characteristic curve). R” is included into the archive which is distributed with this tutorial. I would highly recommend the book "Analyzing Receiver Operating Characteristic Curves with SAS: by Gonen. Plots the ROC Curve rocplot: ROC Plot for a logistic regression model in Deducer: A Data Analysis GUI for R rdrr. For example, if we want to examine the association between the vomiting and both age 1 *gender , ===,,,,,. (View the complete code for this example. ROC Analysis. logistic regression, and bagging applied to both methods. The purpose of this tutorial-based lecture is to show the usefulness of performing a receiver operating characteristic (ROC) curve analysis. scikit-learn makes it super easy to calculate ROC Curves. For each movie that the user has watched, we can encode the ids and as a string (e. my / wnarifin. To compute the points in an ROC curve, we could evaluate a logistic regression model many times with different classification thresholds, but this would be inefficient. 9999 would be too good to be true and will indicate overfitting. ROC Curve Plot for a No Skill Classifier and a Logistic Regression Model for an Imbalanced Dataset We can also repeat the test of the same model on the same dataset and calculate a precision-recall curve and statistics instead. Chi-Square Goodness Of Fit Tests and Deviance In linear regression, residuals can be de ned as y i y^ i where y. Applied Logistic Regression, Second Edition by Hosmer and Lemeshow Chapter 5: Assessing the fit of the model | SAS Textbook Examples 5. In this module, we will discuss the use of logistic regression, what logistic regression is, the confusion matrix, and the ROC curve. Hi All, I'd like to run a ROC curve analysis and was wondering if there are any statistical assumptions for this test. The random forest increases the GINI index further by 4 per cent and the AUROC by 2 per cent. • Area under ROC curve = measure of predictive power of the model. 0), SAS (version 9. Crafting the Features and Logistic Regression. The Receiver Operator Curve (ROC) is a graphical plot that illustrates the diagnostic ability of a binary classifier system, in our case the logistic regression, as its discrimination threshold is varied. This is a simplified tutorial with example codes in R. Receiver Operating Characteristic(ROC) curve is a plot of the true positive rate against the false positive rate. Logistic regression allows us to estimate the probability of a categorical response based on one or more predictor variables ( X ). Logistic Regression, ROC curve & AUC updragon 2018년 8월 4일. I am trying to perform Logistic regression on the sample data set. This means that the top left corner of the plot is the "ideal" point - a false positive rate of. Stepwise Logistic Regression and Predicted Values Logistic Modeling with Categorical Predictors Ordinal Logistic Regression Nominal Response Data: Generalized Logits Model Stratified Sampling Logistic Regression Diagnostics ROC Curve, Customized Odds Ratios, Goodness-of-Fit Statistics, R-Square, and Confidence Limits Comparing Receiver Operating Characteristic Curves Goodness-of-Fit Tests and. In a regression classification for a two-class problem using a probability algorithm, you will capture the probability threshold changes in an ROC curve. It shows the tradeoff between sensitivity and specificity. See the second picture below for an example of the spinners that control the cutoff value in the classification tables and the ROC chart. Basically the code works and it gives the accuracy of the predictive model at a level of 91% but for some reason the AUC score is 0. The first included the HOMR linear predictor, with its coefficient set equal to 1, and intercept set to zero (the original HOMR model). The ROC curve is plotted using the sensitivity and specificity for all possible cut-offs, i. This feature requires SPSS® Statistics Standard Edition or the Regression Option. The predictors can be continuous, categorical or a mix of both. The lecture will explain the background terminology associated with a ROC curve analysis, show the necessary SAS software coding to run a ROC curve analysis, and finally help interpret the analysis output in order to make informed research decisions. In the window. use the Linear Regression procedure. Receiver Operating Characteristic (ROC) Curves The « Receiver Operating Characteristic » function (ROC function) is a measure of the performance of a binary classifier. We saw the same spirit on the test we designed to assess people on Logistic Regression. It is basically a TPR versus FPR (true positive rate versus false-positive rate) curve for all the threshold values ranging from 0 to 1.


text:
American Institute of Mathematical Sciences

June  2019, 8(2): 315-342. doi: 10.3934/eect.2019017

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

 Technische Universität Berlin, Institut für Mathematik, Straẞe des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author

Received  May 2018 Revised  August 2018 Published  March 2019

Fund Project: The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 901 "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" (projects A2, A8).

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert–Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.

Citation: Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
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 [1] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [2] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [3] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [4] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [5] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [6] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [7] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [8] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [9] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [11] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [12] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [13] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122 [14] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [15] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [16] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [17] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [18] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [19] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [20] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

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# Basic Integral Formulas

### Diﬀerentiation Formulas Integration Formulas

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Diﬀerentiation Formulas d dx k = 0 (1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) g(x ...

### The Indefinite Integral and Basic Rules of Integration

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INTEGRATION OF TRIGONOMETRIC INTEGRALS . Recall the definitions of the trigonometric functions. ... We will assume knowledge of the following well-known, basic indefinite integral formulas : , where is a constant , where is a constant Most of the following problems are average. A few are challenging.

### Lists of integrals - Wikipedia

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Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives

### Integration Techniques - CliffsNotes

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Another integration technique to consider in evaluating indefinite integrals that do not fit the basic formulas is integration by parts. You may consider this method when the integrand is a single transcendental function or a product of an algebraic function and a transcendental function. The basic formula for integration by parts is.

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### Basic Formulas - John Abbott College

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Math 201-203-RE - Calculus II Basic Integration Formulas Page 6 of 6 Example 9: Indeﬁnite integral with conditions The problem gives the second derivative of f′′(x) with 2 given conditions. To solve for f(x), ﬁnd the integral twice and use the conditions to solve 2 constants of integration.

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Also find Mathematics coaching class for various competitive exams and classes. INTEGRATION FORMULAE - Math Formulas - Mathematics Formulas - Basic Math Formulas Javascript is …

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Basic Integration Formulas and the Substitution Rule 1The second fundamental theorem of integral calculus Recall fromthe last lecture the second fundamental theorem ofintegral calculus. Theorem Let f(x) be a continuous function on the interval [a,b]. Let F(x) be any

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x is the variable of integration. The anti-derivatives of basic functions are known to us. The integrals of these functions can be obtained readily. In the upcoming discussion let us discuss few important formulae and their applications in determining the integral value of other functions. Integration Formulas

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Start studying Basic Integration Formulas. Learn vocabulary, terms, and more with flashcards, games, and other study tools.

## Basic Integral Formulas

### Calculus II - Integration Techniques

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Jun 06, 2018 · Integrals Involving Trig Functions – In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.

### BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL …

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### Table of Basic Integrals1

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### Basic Integral Formulas | eMathZone

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22) $$\int {{{\csc }^2}xdx = – \cot x + c}$$ 23) $$\int {\sec x\tan xdx = \sec x + c}$$ 24) $$\int {\csc x\cot xdx = – \csc x + c}$$ 25) \int {\sinh xdx ...

### DERIVATIVES AND INTEGRALS - Fort Bend Tutoring

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### Integration Formula - Science HQ

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Integration Formula. Integration. Integration is the operation of calculating the area between the curve of a function and the x-axis. Integral also includes antiderivative and primitive. Integration works by transforming a function into another function respectively.. Some of the important integration formula s are listed below:-. See also: integration formulas

## Basic Integral Formula Sheet

### Basic Calculus: Rules & Formulas - Video & Lesson ...

In basic calculus, we learn rules and formulas for differentiation, which is the method by which we calculate the derivative of a function, and integration, which is the process by which we ...

### What is Calculus? List of Basic Calculus Formulas & Equations

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Jun 09, 2018 · You can calculate the derivative of a function, integration, anti-derivates etc. with the help of basic calculus formulas. Parts of Calculus. As discussed earlier, calculus is the study of instantaneous changes over tiny intervals of time. It has two major parts – One is Differential Calculus and the other is Integral Calculus.

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### Integral Calculus - Elaine Cheong

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Integral Calculus. Indefinite Integrals. Definition: A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ, . F'(x) = ƒ(x) ƒ(x) dx = F(x) + C, where C is a constant.Basic Integration Formulas. General and Logarithmic Integrals


text:
## Abstract

Many pulsars are formed with a binary companion from which they can accrete matter. Torque exerted by accreting matter can cause the pulsar spin to increase or decrease, and over long times, an equilibrium spin rate is achieved. Application of accretion theory to these systems provides a probe of the pulsar magnetic field. We compare the large number of recent torque measurements of accreting pulsars with a high-mass companion to the standard model for how accretion affects the pulsar spin period. We find that many long spin period (P ≳ 100 s) pulsars must possess either extremely weak (B < 1010 G) or extremely strong (B > 1014 G) magnetic fields. We argue that the strong-field solution is more compelling, in which case these pulsars are near spin equilibrium. Our results provide evidence for a fundamental link between pulsars with the slowest spin periods and strong magnetic fields around high-mass companions and pulsars with the fastest spin periods and weak fields around low-mass companions. The strong magnetic fields also connect our pulsars to magnetars and strong-field isolated radio/X-ray pulsars. The strong field and old age of our sources suggest their magnetic field penetrates into the superconducting core of the neutron star.

## INTRODUCTION

The vast majority of the more than 2000 known neutron stars are observed in the radio as rotation-powered pulsars, which lose rotational energy and spin-down due to their electromagnetic radiation. Measurement of the rate of energy loss, or spin-down rate, allows one to infer the pulsar magnetic field B (Gunn & Ostriker 1969; Spitkovsky 2006). However, the spin-down torque due to electromagnetic radiation ($$\sim -10^{30}{ erg }B_{12}^2P^{-3}$$, where B12 = B/1012 G and P is the spin period) is relatively weak compared to torque due to matter accretion by the neutron star from a close binary companion. In this latter case, interactions between the pulsar magnetic field and matter from the companion star produces a torque ($$\sim \pm 10^{34}{ erg }\skew4\dot{M}_{-9}^{6/7}B_{12}^{2/7}$$, where $$\skew4\dot{M}_{-9}=\skew4\dot{M}/10^{-9}{ {M}_{\odot }\,yr^{-1}}$$ and $$\skew4\dot{M}$$ is the mass accretion rate) which can spin the pulsar up or down, depending on whether matter is accreted or ejected. By measuring the rate of change in spin period $$\skew4\dot{P}$$ (which is related to torque N by $$N=-2\pi I\skew4\dot{P}/P^2$$, where I is the stellar moment of inertia), one can obtain insights into the physics that governs the accretion process, including important properties such as the role of the magnetic field. However, observing the long-term $$\skew4\dot{P}$$ in accretion-powered pulsars is very difficult due to short-term fluctuations, with approximately 20 sources detected (see Klus et al. 2013a, and references therein).

Our study is based on the work of Klus et al. (2013b), who report a detailed analysis of Rossi X-ray Timing Explorer observations of high-mass X-ray binaries (HMXBs) in the Small Magellanic Cloud, specifically pulsar systems with an OBe main-sequence companion; these sources are designated Small Magellanic Cloud X-ray pulsars (SXPs; Coe et al. 2005). The data set spans 13 years and contains 42 sources for which the spin period P and count rate as a function of time have been measured. Spin period time derivatives $$\skew4\dot{P}$$ are calculated from the former, and X-ray luminosities L are determined from the latter and can be related to mass accretion rate by $$\skew4\dot{M}=LR/GM$$, where M and R are the neutron star mass and radius, respectively. From their binary parameters, Klus et al. (2013b) find that matter accretes on to the neutron star in each SXP system via an accretion disc (cf. wind). Different methods are then used to determine the magnetic field of each pulsar.

Here, we extend the work of Klus et al. (2013b). In Section 2, we examine torque models, especially that of the standard disc accretion model of Ghosh & Lamb (1979b) and Kluźniak & Rappaport (2007), and demonstrate a simple method for matching the detailed result of Ghosh & Lamb (1979b). In Section 3, we show that the standard model can explain fast-spinning, weak magnetic field pulsars in low-mass X-ray binaries (LMXBs) and slow-spinning, strong-field pulsars in HMXBs, and we consider the broader context of neutron star magnetic fields in light of our findings. We summarize in Section 4.

## STANDARD disc ACCRETION MODEL AND EQUILIBRIUM SPIN PULSARS

Fig. 1 shows the spin period time derivative $$\skew4\dot{P}$$ as a function of spin period P for known pulsars. For isolated sources, i.e. those not in a binary system, pulsars with the highest $$\skew4\dot{P}$$ values are magnetars, i.e. neutron stars that predominantly have magnetic fields B ≳ 1014 G and can exhibit a variety of high-energy emission (Woods & Thompson 2006; Mereghetti 2008). The vast majority of sources are normal rotation-powered radio pulsars whose spin-down rate (i.e. $$\skew4\dot{P}>0$$) is measured very accurately from coherent timing analysis (Lyne & Graham-Smith 1998). The pulsar magnetic field is then estimated by assuming that the electromagnetic energy radiated produces a torque (after using the conversion: torque $$=-2\pi I\skew4\dot{P}/P^2$$)

(1)
$$\skew4\dot{P}\approx 9.8\times 10^{-16}{ s s^{-1} }R_6^{6}I_{45}^{-1} B_{12}^2P^{-1},$$
where R6 = R/10 km and I45 = I/1045 g cm2. While the exact nature of the mechanism that causes radio emission is not known for certain, there is general agreement that there exists a ‘death line'below which observable emission ceases (Ruderman & Sutherland 1975; Bhattacharya et al. 1992). An example death line is shown in Fig. 1.

Figure 1.

Pulsar spin period P versus spin period time derivative $$\skew4\dot{P}$$. Open squares are pulsar values taken from the ATNF Pulsar Catalogue (Manchester et al. 2005), and solid squares denote magnetars. Open and closed circles denote (accreting) sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Esposito et al. 2013; Klus et al. 2013a and references therein; Klus et al. 2013b). Triangles denote (accreting) LMXBs (Patruno 2010; Haskell & Patruno 2011; Riggio et al. 2011). The dashed lines indicate spin-down age ($$=P/2\skew4\dot{P}$$) and inferred magnetic field [$$=3.2\times 10^{19}{ G }(P\skew4\dot{P})^{1/2}$$]. The dotted line indicates the (theoretically uncertain) death line for pulsar radio emission; note that the death line shown here is calculated using equation (1), and therefore does not apply to accreting pulsars (cf. Fig. 4). The solid line indicates the minimum P and maximum $$\skew4\dot{P}$$ that a pulsar can possess as a result of matter accretion from a binary companion.

Figure 1.

Pulsar spin period P versus spin period time derivative $$\skew4\dot{P}$$. Open squares are pulsar values taken from the ATNF Pulsar Catalogue (Manchester et al. 2005), and solid squares denote magnetars. Open and closed circles denote (accreting) sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Esposito et al. 2013; Klus et al. 2013a and references therein; Klus et al. 2013b). Triangles denote (accreting) LMXBs (Patruno 2010; Haskell & Patruno 2011; Riggio et al. 2011). The dashed lines indicate spin-down age ($$=P/2\skew4\dot{P}$$) and inferred magnetic field [$$=3.2\times 10^{19}{ G }(P\skew4\dot{P})^{1/2}$$]. The dotted line indicates the (theoretically uncertain) death line for pulsar radio emission; note that the death line shown here is calculated using equation (1), and therefore does not apply to accreting pulsars (cf. Fig. 4). The solid line indicates the minimum P and maximum $$\skew4\dot{P}$$ that a pulsar can possess as a result of matter accretion from a binary companion.

In contrast to radio pulsars, the period derivative of accreting neutron stars in an X-ray binary is determined by measuring and finding the difference between the spin periods at different epochs (see, e.g. Townsend et al. 2011). We see from Fig. 1 that $$\skew4\dot{P}$$ for accreting pulsars with P ≳ 1 s, such as the SXPs, is much larger than that of most radio pulsars and that $$\skew4\dot{P}$$ for SXPs is comparable to other previously known long spin period sources. All these pulsars possess a binary companion (some companions are low-mass stars and others are high-mass main-sequence or supergiant stars) from which the neutron stars are accreting. Because accretion is thought to suppress radio emission (Bhattacharya & van den Heuvel 1991; Archibald et al. 2009) and the torque from accretion is much stronger than that of electromagnetic spin-down (see Section 1), the magnetic field of accreting pulsars in an LMXB or HMXB cannot be estimated using equation (1). To determine their magnetic field, one can use the standard disc accretion model of Ghosh & Lamb (1979b) (see also Ghosh & Lamb 1979a; we also examine the model of Kluźniak & Rappaport 2007, see below). This model is based on detailed calculations of the interaction between a rotating pulsar magnetosphere and an accretion disc surrounding the pulsar. The predicted torque yields

(2)
\begin{eqnarray} \skew4\dot{P} &=& -4.3\times 10^{-5}\,\mathrm{s\,yr}^{-1}\ M_{1.4}^{-3/7}R_6^{6/7}I_{45}^{-1} \times n(\omega _{\mathrm{s}})\mu _{30}^{2/7}\left(PL_{37}^{3/7}\right)^2, \nonumber \\ \end{eqnarray}
where M1.4 = M/1.4 M, μ30 = μ/1030 G cm3, μ (= BR3) is the magnetic moment of the neutron star, and L37 = L/1037 erg s−1, and is shown in Fig. 2 for three values of the magnetic field. Hereafter, we ignore mass and radius dependencies since they can only vary by a factor of about 2, while the magnetic field can vary by several orders of magnitude. The dimensionless torque ns) accounts for coupling between the magnetic field and the disc plasma (Ghosh & Lamb 1979b; see also Wang 1995) and depends on the fastness parameter ωs [≡ Ω/ΩK(rm)], which is given by
(3)
$$\omega _{\mathrm{s}}= 3.3\,\xi ^{3/2}M_{1.4}^{-2/7}R_6^{-3/7}\mu _{30}^{6/7} \left(PL_{37}^{3/7}\right)^{-1}.$$
Ω = 2π/P is the pulsar spin frequency, ΩK(rm) is the Kepler orbital frequency at radius rm [=ξrA = ξ(GMμ4/2R2L2)1/7], where the energy density of accreting matter transitions from being kinetically to magnetically dominated, and ξ ≈ 1 (see, e.g. Wang 1996). The sign of the dimensionless torque ns) is determined by whether the centrifugal force due to stellar rotation ejects matter and spins down the star (‘fast rotator’ regime with ωs > 1) or accretes matter and spins up the star (‘slow rotator’ regime with ωs < 1) (see, e.g. Wang 1995). It is important to note that a long spin period (P ≫ 1s) pulsar can still be classified as a fast rotator since the fastness parameter ωs depends on the strength of the pulsar magnetic field.

Figure 2.

Rate of change of spin period $$\skew4\dot{P}$$ versus PL3/7. Open and closed circles denote sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Esposito et al. 2013; Klus et al. 2013a, and references therein; Klus et al. 2013b). The lines indicate the theoretical prediction from the standard disc accretion model of Ghosh & Lamb (1979b) with different values of magnetic field (see equation 2).

Figure 2.

Rate of change of spin period $$\skew4\dot{P}$$ versus PL3/7. Open and closed circles denote sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Esposito et al. 2013; Klus et al. 2013a, and references therein; Klus et al. 2013b). The lines indicate the theoretical prediction from the standard disc accretion model of Ghosh & Lamb (1979b) with different values of magnetic field (see equation 2).

A good approximation to equation (2) can be derived very simply. Matter accreting at the magnetosphere transition rm has specific angular momentum

(4)
$$l_{\mathrm{acc}}=\pm r_{\mathrm{m}}^2\Omega _{\mathrm{K}},$$
while matter rotating with the magnetosphere at rm has specific angular momentum
(5)
$$l_{\mathrm{m}}=r_{\mathrm{m}}^2\Omega .$$
Assuming prograde rotation between the accretion disc and neutron star (i.e. lacc > 0), the total torque on the neutron star is then
(6)
$$N_{\mathrm{tot}}= N_{\mathrm{acc}}+N_{\mathrm{m}}= \skew4\dot{M}\Delta l = \skew4\dot{M}r_{\mathrm{m}}^2\Omega _{\mathrm{K}}\left(1-\omega _{\mathrm{s}}\right),$$
which gives
(7)
\begin{eqnarray} \skew4\dot{P}&=& -7.1\times 10^{-5}\,\mathrm{s\, yr^{-1}}\ \xi ^{1/2}M_{1.4}^{-3/7}R_6^{6/7}I_{45}^{-1} \nonumber \\ && \times\, \left(1-\omega _{\mathrm{s}}\right)\mu _{30}^{2/7}\left(PL_{37}^{3/7}\right)^2. \end{eqnarray}
Fig. 3 shows that equation (7) agrees with equation (2) to within ≈15 per cent over a large range in PL3/7, except when ωs approaches and exceeds unity. Similar accuracy is obtained as the magnetic field is varied over many orders of magnitude.

Figure 3.

Top panel: rate of change of spin period $$\skew4\dot{P}$$ versus PL3/7 for B = 1012 G. The solid line labelled NGL indicates the theoretical prediction of Ghosh & Lamb (1979b) (see equation 2). The short–long-dashed line labelled Ntot indicates the analytic approximation given by equation (7). Bottom panel: relative difference between the exact and approximate values of $$\skew4\dot{P}$$ for different values of ξ (see equation 3).

Figure 3.

Top panel: rate of change of spin period $$\skew4\dot{P}$$ versus PL3/7 for B = 1012 G. The solid line labelled NGL indicates the theoretical prediction of Ghosh & Lamb (1979b) (see equation 2). The short–long-dashed line labelled Ntot indicates the analytic approximation given by equation (7). Bottom panel: relative difference between the exact and approximate values of $$\skew4\dot{P}$$ for different values of ξ (see equation 3).

The linear dependence of $$\log \skew4\dot{P}$$ at relatively high values of log PL3/7 is simply the result of the standard spin-up torque $$\skew4\dot{M}(GMr_{\mathrm{m}})^{1/2}$$ due to disc accretion (for prograde rotation, or spin-down torque for retrograde rotation) and is the first term Nacc in equation (6) (Pringle & Rees 1972; Rappaport & Joss 1977). In the slow rotator regime (ωs ≪ 1; see equation 3), the rate of change of spin period $$\skew4\dot{P}$$ is simply given by Nacc, and the magnetic field is $$B\propto \skew4\dot{P}^{7/2}$$. The second term in equation (6) is the spin-down torque due to mass accretion on to a rotating object. These two torques and their total (recall that $$N=-2\pi I\skew4\dot{P}/P^2$$) are shown in Fig. 3. We note that an equation similar to equation (6) is given in Shakura et al. (2012) but with different coefficients for Nacc and Nm.

The sharp decrease and then increase in $$\skew4\dot{P}$$ for B = 1012 G (and 2 × 1014 G in Fig. 2) is due to the change of sign from $$\skew4\dot{P}<0$$ (spin-up) to $$\skew4\dot{P}>0$$ (spin-down). This marks the fast rotator regime near spin equilibrium (ωs ≈ 1) (Davidson & Ostriker 1973; Alpar et al. 1982; Wang 1987; Kluźniak & Rappaport 2007): after sustained epochs of spin-up and spin-down, a steady state will be achieved where the net torque on the pulsar is negligible ($$\skew4\dot{P}\sim 0$$). The resulting equilibrium spin pulsar (ESP) spins near the period

(8)
$$P_{\mathrm{eq}}= 23{ s }\xi ^{3/2}M_{1.4}^{-2/7}R_6^{15/7}L_{37}^{-3/7} \left(B/10^{13}{ G}\right)^{6/7},$$
which is obtained by setting ωs = 1 in equation (3). Note that for spin equilibrium at ωs ∼ 0.35 (Ghosh & Lamb 1979b), the coefficient in equation (8) becomes 67 s, and the inferred magnetic field is lower by a factor of (0.35)7/6 ∼ 0.3.

## COMPARISON TO SXP OBSERVATIONS

Previously only ∼20 X-ray pulsars have had large torques (i.e. $$\skew4\dot{P}$$) measured (see, e.g. Klus et al. 2013a). Fig. 2 shows these, as well as the 42 SXPs from Klus et al. (2013b). Using the observed values of P, $$\skew4\dot{P}$$ and L for each SXP, Klus et al. (2013b) find that the standard disc accretion model prediction for $$\skew4\dot{P}$$ with magnetic fields in the range B ≈ 106-1015 G can fit all SXPs (the model of Kluźniak & Rappaport yields magnetic fields ∼ 10-20 per cent weaker). In particular, fits to each SXP can yield two possible solutions, either a weak field (<1010 G) or strong field (>1012 G). The weak field solution arises from the match between the observed $$\skew4\dot{P}$$ and the torque Nacc (cf. Ntot; see equation 6) if the pulsar is in the slow rotator regime (see also Fig. 3). On the other hand, if the SXP is in the fast rotator regime near spin equilibrium, then the strong pulsar field is simply determined from equation (8), and the observed $$\skew4\dot{P}$$ is due to small fluctuations around torque balance (NaccNm). Both solutions are remarkable and unexpected, based on previous observations of other types of pulsars (see Fig. 4). In the following, we argue for the strong magnetic field solution. We note that, prior to this work, only four pulsars have been found to possibly be near spin equilibrium (Patruno 2010; Haskell & Patruno 2011; Riggio et al. 2011), although many pulsars in LMXBs are assumed to be in this regime (White & Zhang 1997; Patruno, Haskell & D'Angelo 2012). The strong field solution we find (see also Klus et al. 2013b) suggests that a vast majority of the 42 SXPs are near spin equilibrium and thus are ESPs.

Figure 4.

Pulsar magnetic field B versus spin period P. Open squares are pulsars from the ATNF Pulsar Catalogue (Manchester et al. 2005), and solid squares denote magnetars. Open and closed circles denote sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Ghosh & Lamb 1979b; Baykal, Stark & Swank 2002; Cui & Smith 2004; Ikhsanov 2012; Reig, Torrejon & Blay 2012; Klus et al. 2013a,b). Triangles denote LMXBs (Patruno et al. 2012). Stars denote HMXBs whose field is determined from an electron cyclotron spectral line (see http://www.sternwarte.uni-erlangen.de/wiki/doku.php?id=cyclo:start). The short dashed lines indicate spin-down age ($$=P/2\skew4\dot{P}$$), the long dashed line indicates the QED field BQED = 4.414 × 1013 G, and the dotted line indicates the (theoretically uncertain) death line for pulsar radio emission. The solid lines bounding the shaded region indicate the B and P (see equation 8) that a pulsar can possess as a result of accretion from a binary companion at the mass accretion rate observed for SXPs/ESPs (red circles).

Figure 4.

Pulsar magnetic field B versus spin period P. Open squares are pulsars from the ATNF Pulsar Catalogue (Manchester et al. 2005), and solid squares denote magnetars. Open and closed circles denote sources that have $$\skew4\dot{P}<0$$ and $$\skew4\dot{P}>0$$, respectively (Ghosh & Lamb 1979b; Baykal, Stark & Swank 2002; Cui & Smith 2004; Ikhsanov 2012; Reig, Torrejon & Blay 2012; Klus et al. 2013a,b). Triangles denote LMXBs (Patruno et al. 2012). Stars denote HMXBs whose field is determined from an electron cyclotron spectral line (see http://www.sternwarte.uni-erlangen.de/wiki/doku.php?id=cyclo:start). The short dashed lines indicate spin-down age ($$=P/2\skew4\dot{P}$$), the long dashed line indicates the QED field BQED = 4.414 × 1013 G, and the dotted line indicates the (theoretically uncertain) death line for pulsar radio emission. The solid lines bounding the shaded region indicate the B and P (see equation 8) that a pulsar can possess as a result of accretion from a binary companion at the mass accretion rate observed for SXPs/ESPs (red circles).

### Magnetic fields: weak or strong?

For many SXPs, we find that their observed P, $$\skew4\dot{P}$$ and L can be fitted with the standard accretion model (equation 2) and one of two possible magnetic fields: a relatively weak field (<1010 G) or strong field (>1012 G). While either solution is valid, the strong field is more likely from primarily age/time-scale arguments. Most neutron stars are born with B > 1012 G, which can be inferred from the lines of magnetic field and spin-down age shown in Fig. 1, as well as from population synthesis studies (Faucher-Giguère & Kaspi 2006; Popov et al. 2010). Therefore, if SXPs have weak magnetic fields, their initial field either decayed to the current level or has been buried below the surface. The global magnetic field in neutron stars decays rather slowly and on the time-scale ∼106 yr (Haensel, Urpin & Yakovlev 1990; Goldreich & Reisenegger 1992; Glampedakis, Jones & Samuelsson 2011), while the maximum age of an OBe star is ∼107 yr. Thus, there is insufficient time for the field to decay the many orders of magnitude required. Burial of the field by accretion (Romani 1990) is unlikely in so many SXPs because of the very short time over which a large amount of accretion must take place (Chevalier 1989; Geppert, Page & Zannias 1999). On the other hand, if SXPs have strong magnetic fields, then many are currently near spin equilibrium, which they can easily achieve given their short spin-evolution time-scale ($$P/|\skew4\dot{P}|\sim 10^3{ yr}$$) compared to their age. Finally, spin equilibrium also allows spin-down ($$\skew4\dot{P}>0$$) without retrograde rotation.

### Connection to LMXBs and millisecond pulsars

The evolutionary scenario that leads to the formation of fast-spinning (P ≲ 10 ms, hence millisecond) pulsars is briefly described here (see, e.g. Alpar et al. 1982; Bhattacharya & van den Heuvel 1991; Tauris 2012). A millisecond pulsar begins as a standard pulsar in a wide binary that has spun down past the death line. This pulsar can be brought back to radio activity by accreting matter from its (low-mass) companion star, either when the orbit shrinks due to gravitational radiation or the companion evolves off the main sequence. Accretion of matter spins up the pulsar, thus recycling it back from beyond death, although it is only seen as a millisecond radio pulsar after accretion ceases. However some pulsars are observed in the X-rays which are emitted during the accretion process, and these are the LMXBs (Bildsten et al. 1997). Thus, LMXBs are predecessors of millisecond radio pulsars. In this scenario, there is a lower limit to the spin period that accretion can produce. This minimum period is obtained by considering a pulsar that has reached spin equilibrium through accretion at the maximum accretion rate or luminosity ∼1038 erg s−1, above which radiation pressure prevents further accretion. Equation (8) then yields the magnetic field of the pulsar, and these fields turn out to be low for LMXBs (B ∼ 108–109 G). After accretion ceases, the pulsar spins down by the standard electromagnetic energy loss. The resulting maximum spin-up line is shown in Fig. 1. We see that almost all observed pulsars which are likely to be recycled are below this line and above the death line.

Fig. 4 shows the magnetic field and spin period of known pulsars. The magnetic field of rotation-powered pulsars are obtained from their P and $$\skew4\dot{P}$$ and using equation (1). Only four LMXBs have a measured $$\skew4\dot{P}$$ or upper limit on $$\skew4\dot{P}$$, and spin equilibrium is assumed in order to calculate their magnetic field. Many LMXBs have large distance errors which affects the determination of their luminosity. Thus, their inferred magnetic fields are very uncertain, and this uncertainty limits our ability to understand the physics of accretion using LMXBs.

The SXPs bridge this gap in knowledge. The weak magnetic field solution to these pulsars would place them in the extreme bottom right of Fig. 4, in a region far away from all previously known sources, and thus is not shown. Instead, we show the strong magnetic field solution, which is determined from their measured spin period and X-ray luminosity and equation (8), since SXPs are near spin equilibrium for this solution. The shaded region in Fig. 4 is the spin equilibrium region for the range of luminosities [L = (0.2 − 4) × 1037 erg s−1] spanned by our sources. Evidently SXPs (as ESPs) are governed by the same accretion physics as that used to explain the recycling of LMXBs into millisecond pulsars even though the physics regime spans many orders of magnitude in magnetic field. While LMXBs, with their low magnetic fields (B ∼ 108-109 G), produce pulsars with millisecond spin periods, SXPs, with their high magnetic fields (B ∼ 1012-1015 G), produce pulsars with long spin periods (P ≳ 10 s). An important difference between the LMXBs and SXPs is that the latter are now known to be close to spin equilibrium, as well as having a more accurate distance, and thus their magnetic fields are much better determined.

### Connection to magnetars and high-B pulsars

The strong magnetic field of SXPs spans a wide range (B ≈ 1012-1015 G), similar to the range for rotation-powered pulsars. There are also <20 Galactic HMXBs for which electron cyclotron spectral lines have been detected; however, there is uncertainty in where these lines are generated and likely selection effects (see Klus et al. 2013b, for more details). Of particular note are those 24 of 42 SXPs with B > BQED = me2c3/eℏ = 4.414 × 1013 G, where BQED is the critical quantum electrodynamics (QED) magnetic field (and 13 SXPs have B > 1014 G). There are about two dozen previously known neutron stars (∼1 per cent of the total population) with these fields. They are composed of magnetars, which are strong X-ray/gamma-ray sources and can undergo transient outbursts both of which are powered by their strong magnetic fields (Woods & Thompson 2006; Mereghetti 2008), and high-B pulsars, which behave similar to the bulk of the radio pulsars (Ng & Kaspi 2011). The difference between these two groups could be due to the strength of internal toroidal fields (Pons & Perna 2011). Magnetar-like behaviour has not been seen in SXPs, which indicates that SXPs with B > BQED are in the latter group and possess weak toroidal fields. This may suggest that formation of high-B pulsars is more efficient than that of magnetars and there exists many more of the former than latter.

The relative number of SXPs with superstrong fields (B > BQED) is much higher than in the normal pulsar population. On the one hand, this significantly higher fraction of highly magnetized neutron stars could be due to inherently different source populations, e.g. neutron star formation as a result of an electron capture supernova (Nomoto 1984; Podsiadlowski et al. 2004) or accretion induced collapse (Nomoto 1984; Taam & van den Heuvel 1986; Nomoto & Kondo 1991). On the other hand, selection effects could be hindering detection of isolated highly magnetized neutron stars. Effects include X-ray absorption, radio dispersion and/or quenching of radio emission beyond the death line, and some of these effects are taken into account in population synthesis models (see, e.g. Faucher-Giguère & Kaspi 2006; Popov et al. 2010). Note that for SXPs, their rotational energy loss rate $$\dot{E}$$ ($$=4\pi ^2I\skew4\dot{P}/P^3$$ and using equation 1) is ∼4 × 1025 erg s−1(B/1013 G)2(P/100 s)−4. In addition, magnetars and high-B pulsars have a short spin-down time-scale ( ≈ 103-105 yr; see Figs 1 and 4) and are young (age on the same order as their spin-down time-scale). Meanwhile, the high-mass companion of SXPs has a much longer lifetime (∼107 yr), which allows us to study the pulsars at later stages in their life. In other words, it is only because of accretion that we are able to detect these highly magnetized neutron stars when they are older than when they are young and isolated (see also Chashkina & Popov 2012). This may suggest that magnetic field decay in neutron stars occurs slower than previously thought (Pons, Miralles & Geppert 2009) and could be the result of the magnetic field extending into the superconducting neutron star core, where the field decay time-scale is much longer (Glampedakis et al. 2011). Importantly, this scenario where all pulsars are drawn from the same source population would resolve the neutron star birthrate problem (Keane & Kramer 2008; Watters & Romani 2011) and support a unified picture of neutron stars (Keane & Kramer 2008; Kaspi 2010; Popov et al. 2010).

## SUMMARY

Using recent measurements of a large number of spin period time derivatives $$\skew4\dot{P}$$ for pulsars in high-mass binaries in the Small Magellanic Cloud (Klus et al. 2013b), we examined the torque implications for the standard disc accretion model (Ghosh & Lamb 1979b) and model of Kluźniak & Rappaport (2007) and for neutron star magnetic fields. We find several interesting results. Many of the 42 SXPs are near spin equilibrium, where torques that increase and decrease the spin period are balanced, and thus SXPs are ESPs; previously only four pulsars have been found to possibly be near spin equilibrium. The standard disc accretion model links SXPs (which have high-mass companion stars) with pulsars at very short spin periods (which have low-mass companions); this is demonstrated by the shaded region in Fig. 4. The strong magnetic field of many SXPs is above the QED field and links them with magnetars and strong B radio/X-ray pulsars. The decay of magnetic field in SXPs occurs after 106 yr, suggesting magnetic field penetration into the superconducting pulsar core. It is possible there are many more superstrong magnetic field pulsars in the Galaxy that remain as yet undetected.

We thank the anonymous referee for helpful comments. WCGH and NA acknowledge support from the Science and Technology Facilities Council (STFC) in the UK. HK acknowledges an STFC studentship.

## REFERENCES

Alpar
M. A.
Cheng
A. F.
Ruderman
M. A.
Shaham
J.
Nature
,
1982
, vol.
300
pg.
728

Archibald
A. M.
, et al.  .
Science
,
2009
, vol.
324
pg.
1411

Baykal
A.
Stark
M. J.
Swank
J. H.
ApJ
,
2002
, vol.
569
pg.
903

Bhattacharya
D.
van den Heuvel
E. P. J.
Phys. Rep.
,
1991
, vol.
203
pg.
1

Bhattacharya
D.
Wijers
R. A. M. J.
Hartman
J. W.
Verbunt
F.
A&A
,
1992
, vol.
254
pg.
198

Bildsten
L.
, et al.  .
ApJS
,
1997
, vol.
113
pg.
367

Chashkina
A.
Popov
S. B.
New Astron.
,
2012
, vol.
17
pg.
594

Chevalier
R. A.
ApJ
,
1989
, vol.
346
pg.
847

Coe
M. J.
Edge
W. R. T.
Galache
J. L.
McBride
V. A.
MNRAS
,
2005
, vol.
356
pg.
502

Cui
W.
Smith
B.
ApJ
,
2004
, vol.
602
pg.
320

Davidson
K.
Ostriker
J. P.
ApJ
,
1973
, vol.
179
pg.
585

Esposito
P.
Israel
G. L.
Sidoli
L.
Mason
E.
Rodríguez Castillo
G. A.
Halpern
J. P.
Moretti
A.
Götz
D.
MNRAS
,
2013
, vol.
433
pg.
2028

Faucher-Giguère
C.-A.
Kaspi
V. M.
ApJ
,
2006
, vol.
643
pg.
332

Geppert
U.
Page
D.
Zannias
T.
A&A
,
1999
, vol.
345
pg.
847

Ghosh
P.
Lamb
F. K.
ApJ
,
1979a
, vol.
232
pg.
259

Ghosh
P.
Lamb
F. K.
ApJ
,
1979b
, vol.
234
pg.
296

Glampedakis
K.
Jones
D. I.
Samuelsson
L.
MNRAS
,
2011
, vol.
413
pg.
2021

Goldreich
P.
Reisenegger
A.
ApJ
,
1992
, vol.
395
pg.
250

Gunn
J. E.
Ostriker
J. P.
Nature
,
1969
, vol.
221
pg.
454

Haensel
P.
Urpin
V. A.
Yakovlev
D. G.
A&A
,
1990
, vol.
229
pg.
133

B.
Patruno
A.
ApJ
,
2011
, vol.
738
pg.
L14

Ikhsanov
N. R.
MNRAS
,
2012
, vol.
424
pg.
39

Kaspi
V. M.
,
2010
, vol.
107
pg.
7147

Keane
E. F.
Kramer
M.
MNRAS
,
2008
, vol.
391
pg.
2009

Klus
H.
Bartlett
E. S.
Bird
A. J.
Coe
M.
Corbet
R. H. D.
Udalski
A.
MNRAS
,
2013a
, vol.
428
pg.
3607

Klus
H.
Ho
W. C. G.
Coe
M. J.
Corbet
R. H. D.
Townsend
L. J.
MNRAS
,
2013b

preprint (arXiv:1311.4343)
Kluźniak
W.
Rappaport
S.
ApJ
,
2007
, vol.
671
pg.
1990

Lyne
A. G.
Graham-Smith
F.
Pulsar Astronomy
,
1998
2nd edn
Cambridge
Cambridge Univ. Press
Manchester
R. N.
Hobbs
G. B.
Teoh
A.
Hobbs
M.
AJ
,
2005
, vol.
129
pg.
1993

Mereghetti
S.
A&AR
,
2008
, vol.
15
pg.
225

Ng
C.-Y.
Kaspi
V. M.
Göğüş
E.
Belloni
T.
Ertan
Ü.
AIP Conf. Proc. 1379, Astrophysics of Neutron Stars 2010
,
2011
New York
Am. Inst. Phys.
pg.
60

Nomoto
K.
ApJ
,
1984
, vol.
277
pg.
791

Nomoto
K.
Kondo
Y.
ApJ
,
1991
, vol.
367
pg.
L19

Patruno
A.
ApJ
,
2010
, vol.
722
pg.
909

Patruno
A.
B.
D'Angelo
C.
ApJ
,
2012
, vol.
746
pg.
9

Ph.
Langer
N.
Poelarends
A. J. T.
Rappaport
S.
Heger
A.
Pfahl
E.
ApJ
,
2004
, vol.
612
pg.
1044

Pons
J. A.
Perna
R.
ApJ
,
2011
, vol.
741
pg.
123

Pons
J. A.
Miralles
J. A.
Geppert
U.
A&A
,
2009
, vol.
496
pg.
207

Popov
S. B.
Pons
J. A.
Miralles
J. A.
Boldin
P. A.
Posselt
B.
MNRAS
,
2010
, vol.
401
pg.
2675

Pringle
J. E.
Rees
M. J.
A&A
,
1972
, vol.
21
pg.
1

Rappaport
S.
Joss
P. C.
Nature
,
1977
, vol.
266
pg.
683

Reig
P.
Torrejon
J. M.
Blay
P.
MNRAS
,
2012
, vol.
425
pg.
595

Riggio
A.
Burderi
L.
di
Salvo T.
Papitto
A.
D'A
A.
Iaria
R.
Menna
M. T.
A&A
,
2011
, vol.
531
pg.
A140

Romani
R. W.
Nature
,
1990
, vol.
347
pg.
741

Ruderman
M. A.
Sutherland
P. G.
ApJ
,
1975
, vol.
196
pg.
51

Shakura
N.
Postnov
K.
Kochetkova
A.
Hjalmarsdotter
L.
MNRAS
,
2012
, vol.
420
pg.
216

Spitkovsky
A.
ApJ
,
2006
, vol.
648
pg.
L51

Taam
R. E.
van den Heuvel
E. P. J.
ApJ
,
1986
, vol.
305
pg.
235

Tauris
T.
Science
,
2012
, vol.
335
pg.
561

Townsend
L. J.
Coe
M. J.
Corbet
R. H. D.
Hill
A. B.
MNRAS
,
2011
, vol.
416
pg.
1556

Wang
Y.-M.
A&A
,
1987
, vol.
183
pg.
257

Wang
Y.-M.
ApJ
,
1995
, vol.
449
pg.
L153

Wang
Y.-M.
ApJ
,
1996
, vol.
465
pg.
L111

Watters
K. P.
Romani
R. W.
ApJ
,
2011
, vol.
727
pg.
123

White
N. E.
Zhang
W.
ApJ
,
1997
, vol.
490
pg.
L87

Woods
P. M.
Thompson
C.
Lewin
W. H. G.
van der Klis
M.
Compact Stellar X-ray Sources
,
2006
Cambridge
Cambridge Univ. Press
pg.
547


text:
## Prealgebra (7th Edition)

$-3(5-2b)-4b=2b-15$
$-3(5-2b)-4b\rightarrow$ Use the distributive property. $=-15+6b-4b\rightarrow$ Combine like terms. $=2b-15$


text:
# How do you use the rational root theorem to find the roots of 4x^3 - 3x^2 + x - 2 = 0?

Sep 22, 2015

Use the rational zero theorem to find one root. Then factor (or divide) and find the other roots.

#### Explanation:

Possible rational roots: $\pm 1 , \pm 2 , \pm \frac{1}{2} , \pm \frac{1}{4}$

$1$ is a root, so $x - 1$ is a factor on the polynomial.
By division or by stepwise analysis,

$4 {x}^{3} - 3 {x}^{2} + x - 2 = 4 {x}^{2} \left(x - 1\right) + {x}^{2} + x - 2$

$= 4 {x}^{2} \left(x - 1\right) + x \left(x - 1\right) + 2 x - 2$

$= 4 {x}^{2} \left(x - 1\right) + x \left(x - 1\right) + 2 \left(x - 1\right)$

$= \left(x - 1\right) \left(4 {x}^{2} + x + 2\right)$

The other two roots are roots of $4 {x}^{2} + x + 2 = 0$, which may be found by completing the square or by the quadratic formula.

The roots of the original equation are:

$1 , \frac{- 1 + \sqrt{31} i}{8} , \frac{- 1 - \sqrt{31} i}{8}$


text:
# Random change of time

## Stochastic processes derived by varying the rate of time’s passage, which is more convenient than I imagined

🏗 Various notes on a.e. continuous monotonic random changes of index in order build new processes.

In Warping and registration problems you try to align two or more processes; this can sometimes be an alignment problem, but not not necessarily.

## To explore

Lamperti representation for continuous state branching processes,

Ogata’s time rescaling: Intensity estimation for point processes uses this as a statistical test.

Relation to e.g. martingale transforms.

## Subordinator

A subordinator is a non-decreasing Levy Process taking values on the reals. AFAICS this is precisely a Gamma process. Curiously, upon giving that definition, many proceed to immediately assert that such a process is a model for a random change of time. This is not insane per se, but doesn’t have much narrative momentum, as a Gamma process can model a bunch of other things than time, and it is a weird specialisation in a field that normally tends to excessive generality.

🏗 I should explain why one would bother doing such an arbitrary thing as changing time though; Basically it is because a time-changed Lévy process is still a Lévy process.

## Point process rate transform

As used in point process residual goodness of fit tests.

A summary in (Vere-Jones and Schoenberg 2004):

Knight (Knight 1970) showed that for any orthogonal sequence of continuous local martingales, by rescaling time for each via its associated predictable process, we form a multivariate sequence of independent standard Brownian motions. Then Meyer (Meyer 1971) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process $${N_i ; i = 1, 2, . . . , n}$$, the multivariate point process obtained by rescaling each $$N_i$$ according to its compensator is a sequence of independent Poisson processes, each having intensity 1. Since then, alternative proofs and variations of this result have been given by (Brown and Nair 1988; Kurtz 1980; Papangelou 1972; Aalen and Hoem 1978; Brémaud 1972) Papangelou (Papangelou 1972) gave the following interpretation in the univariate case:

Roughly, moving in $$[0, \infty)$$ so as to meet expected future points at a rate of one per time unit (given at each instant complete knowledge of the past), we meet them at the times of a Poisson process. […]

Generalizations of Meyer’s result to point processes on $$\mathbb{R}^d$$ have been established by (Nair 1990; Merzbach and Nualart 1986; Schoenberg 1999). In each case, the method used has been to focus on one dimension of the point process, and rescale each point along that dimension according to the conditional intensity.

## Going Multivariate

As seen in, e.g. (Barndorff-Nielsen, Pedersen, and Sato 2001). How does multivariate time work then?

Aalen, Odd O., and Jan M. Hoem. 1978. “Random Time Changes for Multivariate Counting Processes.” Scandinavian Actuarial Journal 1978 (2): 81–101. https://doi.org/10.1080/03461238.1978.10419480.

Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

Baddeley, A., R. Turner, J. Møller, and M. Hazelton. 2005. “Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67 (5): 617–66. https://doi.org/10.1111/j.1467-9868.2005.00519.x.

Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. 2001. “Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability 33 (1): 160–87. https://doi.org/10.1017/S0001867800010685.

Barndorff-Nielsen, Ole E, and Albert Shiryaev. 2010. Change of Time and Change of Measure. Vol. 13. Advanced Series on Statistical Science & Applied Probability. WORLD SCIENTIFIC. https://doi.org/10.1142/7928.

Brémaud, Pierre. 1972. “A Martingale Approach to Point Processes.” University of California, Berkeley.

Brown, Timothy C., and M. Gopalan Nair. 1988. “A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes.” Journal of Applied Probability 25 (1): 210–14. https://doi.org/10.2307/3214247.

Caballero, M. E., and L. Chaumont. 2006. “Conditioned Stable Lévy Processes and the Lamperti Representation.” Journal of Applied Probability 43 (4): 967–83. http://www.maths.bath.ac.uk/~ak257/pssMp/CC.pdf.

Chaumont, Loïc, Henry Pantí, and Víctor Rivero. 2013. “The Lamperti Representation of Real-Valued Self-Similar Markov Processes.” Bernoulli 19 (5B): 2494–2523. https://doi.org/10.3150/12-BEJ460.

Cheng, R. C. H., and M. A. Stephens. 1989. “A Goodness-of-Fit Test Using Moran’s Statistic with Estimated Parameters.” Biometrika 76 (2): 385–92. https://doi.org/10.1093/biomet/76.2.385.

Cox, D. R. 1955. “Some Statistical Methods Connected with Series of Events.” Journal of the Royal Statistical Society: Series B (Methodological) 17 (2): 129–57. https://doi.org/10.1111/j.2517-6161.1955.tb00188.x.

Çinlar, Erhan. 1980. “On a Generalization of Gamma Processes.” Journal of Applied Probability 17 (2): 467–80. https://doi.org/10.2307/3213036.

Giesecke, K., H. Kakavand, and M. Mousavi. 2008. “Simulating Point Processes by Intensity Projection.” In Simulation Conference, 2008. WSC 2008. Winter, 560–68. https://doi.org/10.1109/WSC.2008.4736114.

Haslinger, Robert, Gordon Pipa, and Emery Brown. 2010. “Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking.” Neural Computation 22 (10): 2477–2506. https://doi.org/10.1162/NECO_a_00015.

Knight, Frank B. 1970. “An Infinitesimal Decomposition for a Class of Markov Processes.” The Annals of Mathematical Statistics 41 (5): 1510–29. https://doi.org/10.1214/aoms/1177696796.

Kurtz, Thomas G. 1980. “Representations of Markov Processes as Multiparameter Time Changes.” The Annals of Probability 8 (4): 682–715. https://doi.org/10.1214/aop/1176994660.

Lamperti, John. 1958. “On the Isometries of Certain Function-Spaces.” Pacific J. Math 8 (3): 459–66. http://msp.org/pjm/1958/8-3/pjm-v8-n3-p09-s.pdf.

———. 1967. “Continuous-State Branching Processes.” Bull. Amer. Math. Soc 73 (3): 382–86. http://bfi.cl/papers/Lamperti%201967%20-%20Continuous%20state%20branching%20processes.pdf.

———. 1972. “Semi-Stable Markov Processes. I.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 22 (3): 205–25. https://doi.org/10.1007/BF00536091.

Leonenko, Nikolai N, Mark M Meerschaert, René L Schilling, and Alla Sikorskii. 2014. “Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics 6 (1). https://doi.org/10.1685/journal.caim.483.

Madan, Dilip B., and Eugene Seneta. 1990. “The Variance Gamma (V.G.) Model for Share Market Returns.” The Journal of Business 63 (4): 511–24. http://finance.martinsewell.com/stylized-facts/distribution/MadanSeneta1990.pdf.

Merzbach, Ely, and David Nualart. 1986. “A Characterization of the Spatial Poisson Process and Changing Time.” The Annals of Probability 14 (4): 1380–90. https://doi.org/10.1214/aop/1176992378.

Meyer, P. A. 1971. “Demonstration simplifiee d’un theoreme de Knight.” In Séminaire de Probabilités V Université de Strasbourg, 191–95. Lecture Notes in Mathematics. Springer Berlin Heidelberg. https://doi.org/10.1007/BFb0058859.

Nair, M. Gopalan. 1990. “Random Space Change for Multiparameter Point Processes.” The Annals of Probability 18 (3): 1222–31. https://doi.org/10.1214/aop/1176990743.

Papangelou, F. 1972. “Integrability of Expected Increments of Point Processes and a Related Random Change of Scale.” Transactions of the American Mathematical Society 165: 483–506. https://doi.org/10.1090/S0002-9947-1972-0314102-9.

Rao, C. R., and Y. Wu. 2001. “On Model Selection.” In Institute of Mathematical Statistics Lecture Notes - Monograph Series, 38:1–57. Beachwood, OH: Institute of Mathematical Statistics. http://projecteuclid.org/euclid.lnms/1215540960.

Schoenberg, Frederic. 1999. “Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications 81 (2): 155–64. https://doi.org/10.1016/S0304-4149(98)00098-2.

Schoenberg, Frederic Paik. 2002. “On Rescaled Poisson Processes and the Brownian Bridge.” Annals of the Institute of Statistical Mathematics 54 (2): 445–57. https://doi.org/10.1023/A:1022494523519.

Seneta, Eugene. 2007. “The Early Years of the Variance-Gamma Process.” In Advances in Mathematical Finance, edited by Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, 3–19. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser. https://doi.org/10.1007/978-0-8176-4545-8_1.

Vere-Jones, David, and Frederic Paik Schoenberg. 2004. “Rescaling Marked Point Processes.” Australian & New Zealand Journal of Statistics 46 (1): 133–43. https://doi.org/10.1111/j.1467-842X.2004.00319.x.


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# Degrees of freedom and conversion of a system of O.D.E.'s into 1 O.D.E.

1. Sep 18, 2011

### alk

I am wrighting a short introduction to dynamical systems, and I can't seem to understand when are we allowed to talk about the "degrees of freedom" of a dynamical system.

A system of one degree of freedom can be described by a 2nd order ordinary differential equation of the form

$\ddot{x}$=f(x,t)______(1) (mechanics $\rightarrow$ Newton's 2nd law)

and this equation can be converted into a system of two O.D.E.'s of the form

$\dot{x}_{1}$=$x_{2}$__________(2a)
$\dot{x}_{2}$=f($x_{1}$,t)_______(2b)

I'm thinking that if and when this conversion is invertible, then starting from eq's (2) as the description of a dynamical system of 2 dimensions, one can say that this dynamical system has one degree of freedom.

The trouble is that I don't know if I'm correct, and I can't find the theorem (if there exists one) that states the conditions under which the conversion of a system of O.D.E.'s into one (or even more) O.D.E. of higher order (lets say 2nd), is possible.

More specifically, Langrangian and Hamiltonian mechanics implies that the dimension of a dynamical system should be an even number. Am I right to think that this may be one of the conditions?