qq图

qq图 A normal Q-Q plot of randomly generated, independent standard exponential data, (X ~ Exp(1)). This Q-Q plot compares asample of data on the vertical axis to astatistical population on the horizontal axis. The points follow a strongly nonlinear pattern, suggesting that the data are not distributed as a standard normal (X ~ N(0,1)). The offset between the line and the points suggests that the mean of the data is not 0. The median of the points can be determined to be near 0.7 A normal Q-Q plot comparing randomly generated, independent standard normal data on the vertical axis to a standard normal population on the horizontal axis. The linearity of the points suggests that the data are normally distributed. A Q-Q plot of a sample of data versus aWeibull distribution. The deciles of the distributions are shown in red. Three outliers are evident at the high end of the range. Otherwise, the data fit the Weibull(1,2) model well. A Q-Q plot comparing the distributions of standardized daily maximum temperatures at 25 stations in the U.S. state of Ohioin March and in July. The curved pattern suggests that the central quantiles are more closely spaced in July than in March, and that the March distribution is skewedto the right compared to the July distribution. The data cover the period 1893–2001. [1]在统统学中～QQ统 ;Q代表分位数 Quantile,是一统通统出分位比统率画数来两个概 分布的统形方法。首先统定统统度～点区(x,y)统统于第一分布个(x统)的分位和第二分数个 布(y统)相同的分位。因此出的是一含的曲统～统统统统统。数画条参数参数区个数 如果被比统的分布比统相似～统其两个QQ统近似地位于y = x上。如果分布统性相两个 统～统QQ统上的点近似地落在一直统上～但不一定是条并y = x统统。条QQ统同统可以 用统一分布的位置。来估个参数 A Q-Q plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale, and skewness are similar or different in the two distributions. Q-Q plots can be used to compare collections of data, or theoretical distributions. The use of Q-Q plots to compare two samples of data can be viewed as a non-parametric approach to comparing their underlying distributions. A Q-Q plot is generally a more powerful approach to doing this than the common technique of comparing histograms of the two samples, but requires more skill to interpret. Q-Q plots are commonly used to [2][3]compare a data set to a theoretical model. This can provide an assessment of "goodness of fit" that is graphical, rather than reducing to a numerical summary. Q-Q plots are also used to compare two theoretical distributions to each other.[4] Since Q-Q plots compare distributions, there is no need for the values to be observed as pairs, as in a scatterplot, or even for the numbers of values in the two groups being compared to be equal. The term "probability plot" sometimes refers specifically to a Q-Q plot, sometimes to a more general class of plots, and sometimes to the less commonly used P-P plot. The probability plot correlation coefficient is a quantity derived from the idea of Q-Q plots, which measures the agreement of a fitted distribution with observed data and which is sometimes used as a means of fitting a distribution to data: see later. 目统 [统藏] • 1 定统和统构 • 2 Interpretation • 3 Plotting positions o 3.1 Expected value of the order statistic o 3.2 Median of the order statistics o 3.3 Heuristics o 3.4 Filliben's estimate • 4 See also • 5 注统 • 6 参考统料 • 7 外部统接 [统统]定统和统构 Q-Q plot for first opening/final closing dates ofWashington State Route 20, versus a normal [5]distribution. Outliers are clearly present in the upper right corner. A Q-Q plot is a plot of the quantiles of two distributions against each other, or a plot based on estimates of the quantiles. The pattern of points in the plot is used to compare the two distributions. 在建的主要步统是一 构个QQ位的统算或统要统制。如果一或统的一 数估个两个个QQ 统是基于一理统分布统统累统分布函;个与数CDF,～所有位的定统～是唯一可以通统数 反相的累统分布函。如果一理统率分布不统统的累统分布函是其中的分布统数个概数两个 行比统～所以部分的位不能定统～所以一分量可能被统制。如果数插QQ 统是根据据数～ 有多分位统的使用。 个数估QQ号数估插称统制形成统统位统必统统或被统统制位置。 A simple case is where one has two data sets of the same size. In that case, to make the Q-Q plot, one orders each set in increasing order, then pairs off and plots the corresponding values. A more complicated construction is the case where two data sets of different sizes are being compared. To construct the Q-Q plot in this case, it is necessary to use an interpolated quantile estimate so that quantiles corresponding to the same underlying probability can be constructed. [4]More abstractly, given two cumulative probability distribution ?1?1functions F and G, with associatedquantile functions F and G (the inverse function of the CDF is the quantile function), the Q-Q plot draws the qth quantile of F against the qth quantile of G for a range of values of q. Thus, the Q-Q plot is 2a parametric curve indexed over [0,1] with values in the real plane R. [统统]Interpretation The points plotted in a Q-Q plot are always non-decreasing when viewed from left to right. If the two distributions being compared are identical, the Q-Q plot follows the 45? line y = x. If the two distributions agree after linearly transforming the values in one of the distributions, then the Q-Q plot follows some line, but not necessarily the line y = x. If the general trend of the Q-Q plot is flatter than the line y = x, the distribution plotted on the horizontal axis is more dispersed than the distribution plotted on the vertical axis. Conversely, if the general trend of the Q-Q plot is steeper than the line y = x, the distribution plotted on the vertical axis is more dispersed than the distribution plotted on the horizontal axis. Q-Q plots are often arced, or "S" shaped, indicating that one of the distributions is more skewed than the other, or that one of the distributions has heavier tails than the other.Although a Q-Q plot is based on quantiles, in a standard Q-Q plot it is not possible to determine which point in the Q-Q plot determines a given quantile. For example, it is not possible to determine the median of either of the two distributions being compared by inspecting the Q-Q plot. Some Q-Q plots indicate the deciles to make determinations such as this possible. The slope and position of a linear regression between the quantiles gives a measure of the relative location and relative scale of the samples. If the median of the distribution plotted on the horizontal axis is 0, the intercept of a regression line is a measure of location, and the slope is a measure of scale. The distance between medians is another measure of relative location reflected in a Q-Q plot. The "probability plot correlation coefficient" is the correlation coefficient between the paired sample quantiles. The closer the correlation coefficient is to one, the closer the distributions are to being shifted, scaled versions of each other. For distributions with a single shape parameter, theprobability plot correlation coefficient plot (PPCC plot) provides a method for estimating the shape parameter – one simply computes the correlation coefficient for different values of the shape parameter, and uses the one with the best fit, just as if one were comparing distributions of different types. Another common use of Q-Q plots is to compare the distribution of a sample to a theoretical distribution, such as the standard normal distribution N(0,1), as in a normal probability plot. As in the case when comparing two samples of data, one orders the data (formally, computes the order statistics), then plots them [3]against certain quantiles of the theoretical distribution. [统统]Plotting positions The choice of quantiles from a theoretical distribution has occasioned much discussion. A natural choice, given a sample of size n, is k / n for k = 1, ..., n, as these are the quantiles that the sampling distribution realizes. Unfortunately, the last of these, n / n, corresponds to the 100th percentile – the maximum value of the theoretical distribution, which is often infinite. To fix this, one may shift these over, using (k ? 0.5) / n, or instead space the points evenly in the uniform distribution, using k / (n + 1). This last has been argued to be the definitive [6]position by some, though other choices have been suggested, both formal and heuristic. [统统]Expected value of the order statistic In using a normal probability plot, the quantiles one uses are the rankits, the quantile of the expected value of the order statistic of a standard normal distribution. More generally, Wilk–Shapiro uses the expected values of the order statistics of the given distribution; the resulting plot and line yields thegeneralized least squares estimate for location and scale (from the intercept and slope of the fitted [3]line). Although this is not too important for the normal distribution (the location and scale are estimated by the mean and standard deviation, respectively), it can be useful for many other distributions. However, this requires calculating the expected values of the order statistic, which may be difficult if the distribution is not normal. [统统]Median of the order statistics Alternatively, one may use estimates of the median of the order statistics, which one can compute based on estimates of the median of the order statistics of a uniform distribution and the quantile function of the distribution; this was [3]suggested by ;Filliben 1975,. This can be easily generated for any distribution for which the quantile function can be computed, but conversely the resulting estimates of location and scale are no longer precisely the least squares estimates, though these only differ significantly for n small. [统统]Heuristics For the quantiles of the comparison distribution typically the formula k/(n + 1) is used. Several different formulas have been used or proposed as symmetrical plotting positions. Such formulas have the form (k ? a)/ (n + 1 ? 2a) for some value of a in the range from 0 to 1/2, which gives a range between k/(n + 1) and (k ? 1/2)/n. Other expressions include: [7],(k ? 0.3)/(  n + 0.4). [8],(k ? 0.3175)/(  n + 0.365). [9],(k ? 0.326)/(  n + 0.348). [10],(k ? ?)/(  n + ?). [11],(k ? 0.375)/(  n + 0.25). [12],(k ? 0.4)/(  n + 0.2). [13],(k ? 0.44)/(  n + 0.12). [14],(k ? 0.567)/(  n ? 0.134). [15],(k ? 1)/(  n ? 1). For large sample size, n, there is little difference between these various expressions. [统统]Filliben's estimate The order statistic medians are the medians of the order statistics of the distribution. These can be expressed in terms of the quantile function and the order statistic medians for the continuous uniform distribution by: N(i) = G(U(i)) where U(i) are the uniform order statistic medians and G is the quantile function for the desired distribution. The quantile function is the inverse of the cumulative distribution function (probability that X is less than or equal to some value). That is, given a probability, we want the corresponding quantile of the cumulative distribution function. James J. Filliben ;Filliben 1975, uses the following estimates for the uniform order statistic medians: The reason for this estimate is that the order statistic medians do not have a simple form. [统统]See also 相统的统基共享统源, QQ 统 , Probit analysis was developed by Chester Ittner Bliss in 1934. , Shapiro–Wilk test 注统[统统] 1. ^ Gnanadesikan, R.; Wilk, M.B., Probability plotting methods for the analysis of data, Biometrika. Biometrika Trust. 1968, 55 (1): 1–17, PMID 5661047. 2. ^ Gnanadesikan (1977) p199. 3.03.13.23.33.^ ;Thode 2002, Section 2.2.2, Quantile-Quantile Plots, p. 21, 4.04.14.^ ;Gibbons & Chakraborti 2003, p. 144, 5. ^ SR 20 – North Cascades Highway – Opening and Closing History. North Cascades Passes. Washington State Department of Transportation [2009-02-08]. 6. ^ Makkonen, Lasse, Bringing Closure to the Plotting Position Controversy, Communications in Statistics – Theory and Methods. January 2008, 37 (3): 460–467,doi:10.1080/94 7. ^ Benard & Bos-Levbach (1953). 8. ^ Engineering Statistics Handbook: Normal Probability Plot – Note that this also uses a different expression for the first & last points. [1] cites the original work by ;Filliben 1975,. This expression is an estimate of the medians of U.(k) 9. ^ Distribution free plotting position , Yu & Huang 10. ^ A simple (and easy to remember) formula for plotting positions; used in BMDP statistical package. 11. ^ This is ;Blom 1958,’s earlier approximation and is the expression used in MINITAB. 12. ^ Cunane (1978). 13. ^ This plotting position was used by Irving I. Gringorten (Gringorten (1963)) to plot points in tests for the Gumbel distribution. 14. ^ Larsen, Currant & Hunt (1980). 15. ^ Used by Filliben (1975), these plotting points are equal to the modes of U.(k) [统统]参考统料 , Template:NIST-PD ,Blom, G., Statistical estimates and transformed beta variables, New York: John Wiley and Sons. 1958 ,Chambers, John; William Cleveland, Beat Kleiner, and Paul Tukey, Graphical methods for data analysis, Wadsworth. 1983 ,Cleveland, W.S. (1994) The Elements of Graphing Data, Hobart Press ISBN 0-9634884-1-4 ,Filliben, J. J., The Probability Plot Correlation Coefficient Test for Normality, Technometrics. American Society for Quality. February 1975, 17 (1): 111–117,doi:10.2307/1268008. ,Gibbons, Jean Dickinson; Chakraborti, Subhabrata, Nonparametric statistical inference. 4th, CRC Press. 2003, ISBN 978 0 82474052 8,Gnanadesikan, R. (1977) Methods for Statistical Analysis of Multivariate Observations, Wiley ISBN 0-471-30845-5. ,Thode, Henry C., Testing for normality, New York: Marcel Dekker. 2002, ISBN 0-8247-9613-6