This is the last post in my recent series of articles on computing contours in SAS. Last month a SAS customer asked how to compute the contours of the bivariate normal cumulative distribution function (CDF). Answering that question in a single blog post would have resulted in a long article,
Like many other computer packages, SAS can produce a contour plot that shows the level sets of a function of two variables. For example, I've previously written blogs that use contour plots to visualize the bivariate normal density function and to visualize the cumulative normal distribution function. However, sometimes you
When I need to graph a function of two variables, I often choose to use a contour plot. A surface plot is probably easier for many people to understand, but it has several disadvantages when compared to a contour plot. For example, the following statements in SAS/IML Studio displays a
A previous article discussed how to compute probabilities for the bivariate standard normal distribution. The standard bivariate normal distribution with correlation ρ is denoted BVN(0,ρ). For any point (x,y), you can use the PROBBNRM function in SAS to compute the probability that the random variables (X,Y) ~ BVN(0,ρ) is observed
This article shows how to use SAS to compute the probabilities for two correlated normal variables. Specifically, this article shows how to compute the probabilities for rectangular regions in the plane. A second article discusses the computation over infinite regions such as quadrants. If (X,Y) are random variables that are
Can we use Computer Vision (CV) to recognize the identity of over 500 Galapagos sea turtles by using just an image? This was the question asked of SAS by researchers at the Galapagos Science Center (GSC), a joint partnership between the University of North Carolina at Chapel Hill’s (UNC) Center for
I have previously written about how to efficiently generate points uniformly at random inside a sphere (often called a ball by mathematicians). The method uses a mathematical fact from multivariate statistics: If X is drawn from the uncorrelated multivariate normal distribution in dimensiond, then S = r*X / ||X|| has
A SAS programmer wanted to compute the distribution of X-Y, where X and Y are two beta-distributed random variables. Pham-Gia and Turkkan (1993) derive a formula for the PDF of this distribution. Unfortunately, the PDF involves evaluating a two-dimensional generalized hypergeometric function, which is not available in all programming languages.
A previous article shows that you can use the Intercept parameter to control the ratio of events to nonevents in a simulation of data from a logistic regression model. If you decrease the intercept parameter, the probability of the event decreases; if you increase the intercept parameter, the probability of
A previous article shows how to compute the probability density function (PDF) for the multivariate normal distribution. In a similar way, you can compute the density function for the multivariate t distribution. This article discusses the density function for the multivariate t distribution, shows how to compute it, and visualizes
SCUBA diving is one of those activities that should be on your bucket list - everyone should experience it at least once! But where should you go diving? ... Coral reefs are always popular - and they are typically in shallow water, making it an 'easy' dive. But, being in
As an example, when using ordinary least squares regression (OLS), if your response or dependent attribute’s residuals are not normally distributed, your analysis is likely going to be affected and typically not in a good way. The farther away your input data is from normality, the impact on your model
Article co-écrit avec Frédéric Sanchez, directeur de l’Ecole de la Data, et Didier Gaultier, Business & Decision Créée il a juste 2 ans, l’École de la data et de l’IA est l’illustration d’une ambition unique dans le monde des ESN : recruter des talents, former de manière pratique et opérationnelle
This article shows how to estimate and visualize a two-dimensional cumulative distribution function (CDF) in SAS. SAS has built-in support for this computation. Although the bivariate CDF is not used as much as the univariate CDF, the bivariate version is still a useful tool in understanding the probable values of
Cindy Wang's curiosity about the Mandelbrot set led her to draw one using SAS Visual Analytics.
Ranking is a fundamental concept in statistics. Ranks of univariate data are used by statisticians to estimate statistics such as percentiles (quantiles) and empirical distributions. A more advanced use is to compute various rank-based measures of correlation or association between pairs of variables. For example, ranks are used to compute
In a previous article, I showed how to generate random points uniformly inside a d-dimensional sphere. In that article, I stated the following fact: If Y is drawn from the uncorrelated multivariate normal distribution, then S = Y / ||Y|| has the uniform distribution on the unit sphere. I was
A SAS customer asked a great question: "I have parameter estimates for a logistic regression model that I computed by using multiple imputations. How do I use these parameter estimates to score new observations and to visualize the model? PROC LOGISTIC can do the computation I want, but how do
To help visualize regression models, SAS provides the EFFECTPLOT statement in several regression procedures and in PROC PLM, which is a general-purpose procedure for post-fitting analysis of linear models. When scoring and visualizing a model, it is important to use reasonable combinations of the explanatory variables for the visualization. When
Human brains are hardwired to build maps. We navigate the world around us through the creation of mental maps. Maps that assemble abstract landmarks and build spatial relationships between them. If you think about how you navigate through your house, even in the dark, it is quite amazing. Maps have
I've previously written about linear interpolation in one dimension. Bilinear interpolation is a method for two-dimensional interpolation on a rectangle. If the value of a function is known at the four corners of a rectangle, an interpolation scheme gives you a way to estimate the function at any point in
Data tell a story. A purpose of data visualization is to convey that story to the reader in a clear and impactful way. Sometimes you can let the data "speak for themselves" in an unadorned graphic, but sometimes it is helpful to add reference lines to a graph to emphasize
Computer vision can augment radiologists and make the image interpretation process cheaper, faster and more accurate. The ultimate goal is to achieve a better patient outcome facilitated by the use of computer vision.
One of the strengths of the SAS/IML language is its flexibility. Recently, a SAS programmer asked how to generalize a program in a previous article. The original program solved one optimization problem. The reader said that she wants to solve this type of problem 300 times, each time using a
In a scatter plot that displays many points, it can be important to visualize the density of the points. Scatter plots (indeed, all plots that show individual markers) can suffer from overplotting, which means that the graph does not indicate how many observations are at a specific (x, y) location.
A few years ago Mandelbrot sets and fractals were all the rage! (Am I showing my age? Hahaha!) I thought creating some plots of this type of data would be a good way to sharpen my SAS programming skills, and it would make a nice/interesting example to help teach people
Knowing how to visualize a regression model is a valuable skill. A good visualization can help you to interpret a model and understand how its predictions depend on explanatory factors in the model. Visualization is especially important in understanding interactions between factors. Recently I read about work by Jacob A.
A quadratic form is a second-degree polynomial that does not have any linear or constant terms. For multivariate polynomials, you can quickly evaluate a quadratic form by using the matrix expression x` A x This computation is straightforward in a matrix language such as SAS/IML. However, some computations in statistics
An analyst was using SAS to analyze some data from an experiment. He noticed that the response variable is always positive (such as volume, size, or weight), but his statistical model predicts some negative responses. He posted the data and asked if it is possible to modify the graph so
Statisticians often emphasize the dangers of extrapolating from a univariate regression model. A common exercise in introductory statistics is to ask students to compute a model of population growth and predict the population far in the future. The students learn that extrapolating from a model can result in a nonsensical
