R作图 Binomial Distribution（±）

二项分布和负二项分布，图示如下：

 （1）二项分布Rcode

> N <- 100000
> n <- 100
> p <- 0.9
> x <- rbinom(N, n, p)
> hist(x,
+      xlim = c(min(x), max(x)),
+      probability = T,  #概率函数，默认为频率函数
+      nclass =  max(x) - min(x) +1,
+      col =  'lightblue',
+      main = "Binomial Distribution, n=100, p=.9" )
> lines(density(x,
+               bw=1,  #控制函数的峰度？？
+               kernel = 'gaussian'),  #默认的函数拟合是高斯函数
+       col = 'blue',
+       lwd = 3)  #可以控制线宽

（2）负二项分布Rcode

> N <- 100000
> n <- 100
> p <- 0.9
> x <- rnbinom(N, n, p)
> hist(x,
+      xlim = c(min(x), max(x)),
+      probability = T,
+      nclass =  max(x) - min(x) +1,
+      col =  'lightblue',
+      main = "Negative Binomial Distribution, n=100, p=.9" )
> lines(density(x,
+               bw=1,
+               kernel = 'gaussian'),
+       col = 'blue',
+       lwd = 3)

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函数简单解释：

(1) Gaussian function

In mathematics, a Gaussian function is a function of the form:

for some real constants a, b , c, and e ≈ 2.718281828(Euler's number)

（http://en.wikipedia.org/wiki/Gaussian_function）

 

(2) Histograms

Description

The generic function hist computes a histogram of the given data values. If plot=TRUE, the resulting object of class "histogram" is plotted by plot.histogram, before it is returned.

（http://127.0.0.1:12956/library/graphics/html/hist.html）

Usage

hist(x, ...)

## Default S3 method:
hist(x, breaks = "Sturges",
     freq = NULL, probability = !freq,
     include.lowest = TRUE, right = TRUE,
     density = NULL, angle = 45, col = NULL, border = NULL,
     main = paste("Histogram of" , xname),
     xlim = range(breaks), ylim = NULL,
     xlab = xname, ylab,
     axes = TRUE, plot = TRUE, labels = FALSE,
     nclass = NULL, warn.unused = TRUE, ...)

Arguments

x	a vector of values for which the histogram is desired.
breaks	one of: a vector giving the breakpoints between histogram cells, a single number giving the number of cells for the histogram, a character string naming an algorithm to compute the number of cells (see ‘Details’), a function to compute the number of cells. In the last three cases the number is a suggestion only.
freq	logical; if TRUE, the histogram graphic is a representation of frequencies, the counts component of the result; if FALSE, probability densities, component density, are plotted (so that the histogram has a total area of one). Defaults to TRUE if and only if breaks are equidistant (and probability is not specified).
probability	an alias for !freq, for S compatibility.
include.lowest	logical; if TRUE, an x[i] equal to the breaks value will be included in the first (or last, for right = FALSE) bar. This will be ignored (with a warning) unless breaks is a vector.
right	logical; if TRUE, the histogram cells are right-closed (left open) intervals.
density	the density of shading lines, in lines per inch. The default value of NULL means that no shading lines are drawn. Non-positive values of density also inhibit the drawing of shading lines.
angle	the slope of shading lines, given as an angle in degrees (counter-clockwise).
col	a colour to be used to fill the bars. The default of NULL yields unfilled bars.
border	the color of the border around the bars. The default is to use the standard foreground color.
main, xlab, ylab	these arguments to title have useful defaults here.
xlim, ylim	the range of x and y values with sensible defaults. Note that xlim is not used to define the histogram (breaks), but only for plotting (when plot = TRUE).
axes	logical. If TRUE (default), axes are draw if the plot is drawn.
plot	logical. If TRUE (default), a histogram is plotted, otherwise a list of breaks and counts is returned. In the latter case, a warning is used if (typically graphical) arguments are specified that only apply to the plot = TRUE case.
labels	logical or character. Additionally draw labels on top of bars, if not FALSE; see plot.histogram.
nclass	numeric (integer). For S(-PLUS) compatibility only, nclass is equivalent to breaks for a scalar or character argument.
warn.unused	logical. If plot=FALSE and warn.unused=TRUE, a warning will be issued when graphical parameters are passed to hist.default().
...	further arguments and graphical parameters passed to plot.histogram and thence to title and axis (if plot=TRUE).

 

(3)Kernel Density Estimation

Description

The (S3) generic function density computes kernel density estimates. Its default method does so with the given kernel and bandwidth for univariate observations.

（http://127.0.0.1:12956/library/stats/html/density.html）

Usage

density(x, ...)
## Default S3 method:
density(x, bw = "nrd0", adjust = 1,
        kernel = c("gaussian", "epanechnikov", "rectangular",
                   "triangular", "biweight",
                   "cosine", "optcosine"),
        weights = NULL, window = kernel, width,
        give.Rkern = FALSE,
        n = 512, from, to, cut = 3, na.rm = FALSE, ...)

Arguments

x	the data from which the estimate is to be computed.
bw	the smoothing bandwidth to be used. The kernels are scaled such that this is the standard deviation of the smoothing kernel. (Note this differs from the reference books cited below, and from S-PLUS.) bw can also be a character string giving a rule to choose the bandwidth. See bw.nrd. The default, "nrd0", has remained the default for historical and compatibility reasons, rather than as a general recommendation, where e.g., "SJ" would rather fit, see also V&R (2002). The specified (or computed) value of bw is multiplied by adjust.
adjust	the bandwidth used is actually adjust*bw. This makes it easy to specify values like ‘half the default’ bandwidth.
kernel, window	a character string giving the smoothing kernel to be used. This must be one of "gaussian", "rectangular", "triangular", "epanechnikov", "biweight", "cosine" or "optcosine", with default "gaussian", and may be abbreviated to a unique prefix (single letter). "cosine" is smoother than "optcosine", which is the usual ‘cosine’ kernel in the literature and almost MSE-efficient. However, "cosine" is the version used by S.
weights	numeric vector of non-negative observation weights, hence of same length as x. The default NULL is equivalent to weights = rep(1/nx, nx) where nx is the length of (the finite entries of) x[].
width	this exists for compatibility with S; if given, and bw is not, will set bw to width if this is a character string, or to a kernel-dependent multiple of width if this is numeric.
give.Rkern	logical; if true, no density is estimated, and the ‘canonical bandwidth’ of the chosen kernel is returned instead.
n	the number of equally spaced points at which the density is to be estimated. When n > 512, it is rounded up to a power of 2 during the calculations (as fft is used) and the final result is interpolated by approx. So it almost always makes sense to specify n as a power of two.
from,to	the left and right-most points of the grid at which the density is to be estimated; the defaults are cut * bw outside of range(x).
cut	by default, the values of from and to are cut bandwidths beyond the extremes of the data. This allows the estimated density to drop to approximately zero at the extremes.
na.rm	logical; if TRUE, missing values are removed from x. If FALSE any missing values cause an error.
...	further arguments for (non-default) methods.