- 2d density plot with ggplot2
- 2d Density Plots in Python/v3
- 3D Surface Plots in R
- Density 2d
2d density plot with ggplot2This chapter of the tutorial will give a brief introduction to some of the tools in seaborn for examining univariate and bivariate distributions. You may also want to look at the categorical plots chapter for examples of functions that make it easy to compare the distribution of a variable across levels of other variables. The most convenient way to take a quick look at a univariate distribution in seaborn is the distplot function. By default, this will draw a histogram and fit a kernel density estimate KDE. Histograms are likely familiar, and a hist function already exists in matplotlib. A histogram represents the distribution of data by forming bins along the range of the data and then drawing bars to show the number of observations that fall in each bin. You can make the rug plot itself with the rugplot function, but it is also available in distplot :. When drawing histograms, the main choice you have is the number of bins to use and where to place them. The kernel density estimate may be less familiar, but it can be a useful tool for plotting the shape of a distribution. Like the histogram, the KDE plots encode the density of observations on one axis with height along the other axis:. Drawing a KDE is more computationally involved than drawing a histogram. What happens is that each observation is first replaced with a normal Gaussian curve centered at that value:. Next, these curves are summed to compute the value of the density at each point in the support grid. The resulting curve is then normalized so that the area under it is equal to We can see that if we use the kdeplot function in seaborn, we get the same curve. This function is used by distplotbut it provides a more direct interface with easier access to other options when you just want the density estimate:. The bandwidth bw parameter of the KDE controls how tightly the estimation is fit to the data, much like the bin size in a histogram. It corresponds to the width of the kernels we plotted above. The default behavior tries to guess a good value using a common reference rule, but it may be helpful to try larger or smaller values:. As you can see above, the nature of the Gaussian KDE process means that estimation extends past the largest and smallest values in the dataset. You can also use distplot to fit a parametric distribution to a dataset and visually evaluate how closely it corresponds to the observed data:. It can also be useful to visualize a bivariate distribution of two variables. The easiest way to do this in seaborn is to just use the jointplot function, which creates a multi-panel figure that shows both the bivariate or joint relationship between two variables along with the univariate or marginal distribution of each on separate axes. The most familiar way to visualize a bivariate distribution is a scatterplot, where each observation is shown with point at the x and y values. This is analogous to a rug plot on two dimensions. You can draw a scatterplot with scatterplotand it is also the default kind of plot shown by the jointplot function:.
2d Density Plots in Python/v3
Different types of 2d density chart. This is the two dimension version of the classic histogram. The plot area is split in a multitude of small squares, the number of points in each square is represented by its color. Learn how to customize the color and the bin size of your 2d histogram. Very similar to the 2d histogram above, but the plot area is split in a multitude of hexagons instead of squares. Learn how to customize the color and the bin size of your hexbin chart. Like it is possible to plot a density chart instead of a histogram to represent a distribution, it is possible to make a 2d density plot. Several variations are available using ggplot2 :. Build a hexbin chart with the hexbin package and color it with RColorBrewer. Color and bin size Learn how to customize the color and the bin size of your 2d histogram. Most basic Most basic, default parameters. Color and bin size Learn how to customize the color and the bin size of your hexbin chart. Contour plot. Use the raster geom. Hexbin package Build a hexbin chart with the hexbin package and color it with RColorBrewer. Scatter on top of 2d distribution Add a scatterplot on top of the ggplot2 2d density chart. Related chart types.
3D Surface Plots in R
In statisticskernel density estimation KDE is a non-parametric way to estimate the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen—Rosenblatt window method, after Emanuel Parzen and Murray Rosenblattwho are usually credited with independently creating it in its current form. Its kernel density estimator is. Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov, normal, and others. The Epanechnikov kernel is optimal in a mean square error sense,  though the loss of efficiency is small for the kernels listed previously. The construction of a kernel density estimate finds interpretations in fields outside of density estimation. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning e. Kernel density estimates are closely related to histogramsbut can be endowed with properties such as smoothness or continuity by using a suitable kernel. To see this, we compare the construction of histogram and kernel density estimators, using these 6 data points:. For the histogram, first the horizontal axis is divided into sub-intervals or bins which cover the range of the data. In this case, we have 6 bins each of width 2. If more than one data point falls inside the same bin, we stack the boxes on top of each other. For the kernel density estimate, we place a normal kernel with standard deviation 2. The kernels are summed to make the kernel density estimate solid blue curve. The smoothness of the kernel density estimate is evident compared to the discreteness of the histogram, as kernel density estimates converge faster to the true underlying density for continuous random variables. The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution plotted at the blue spikes in the rug plot on the horizontal axis. The grey curve is the true density a normal density with mean 0 and variance 1. The most common optimality criterion used to select this parameter is the expected L 2 risk functionalso termed the mean integrated squared error :. Many review studies have been carried out to compare their efficacies,        with the general consensus that the plug-in selectors   and cross validation selectors    are the most useful over a wide range of data sets. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate balloon estimator or the samples pointwise estimatorthis produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is said to be relatively difficult. If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h that is, the bandwidth that minimises the mean integrated squared error is . Another modification that will improve the model is to reduce the factor from 1. Then the final formula would be:. IQR is the interquartile range. This approximation is termed the normal distribution approximationGaussian approximation, or Silverman 's rule of thumb. While this rule of thumb is easy to compute, it should be used with caution as it can yield widely inaccurate estimates when the density is not close to being normal. For example, consider estimating the bimodal Gaussian mixture:. The figure on the right below shows the true density and two kernel density estimates—one using the rule-of-thumb bandwidth, and the other using a solve-the-equation bandwidth. The Matlab script for this example uses kde. Knowing the characteristic function, it is possible to find the corresponding probability density function through the Fourier transform formula. Thus the kernel density estimator coincides with the characteristic function density estimator. Namely, M is the collection of points for which the density function is locally maximized. From Wikipedia, the free encyclopedia. For broader coverage of this topic, see Kernel estimation.