Abundance

x除以最大的值就是相对丰度

Skewness

Skewness is a fundamental statistical measure used to describe the asymmetry of the probability distribution of a real-valued random variable. It provides insights into the direction and extent of the deviation from a symmetric distribution.

Key Aspects of Skewness:

1. Definition:

   • Skewness is the third standardized moment of a distribution.
   • It is calculated as the average of the cubed deviations of the data from its mean, standardized by the standard deviation raised to the third power.

2. Types of Skewness:

   • Zero Skewness: Indicates a symmetric distribution where the mean, median, and mode are all equal.
   • Positive Skewness (Right-Skewed): The tail on the right side of the distribution is longer or fatter. In this case, the mean is greater than the median.
   • Negative Skewness (Left-Skewed): The tail on the left side of the distribution is longer or fatter. Here, the mean is less than the median.

3. Interpretation:

   • Skewness values close to zero suggest a nearly symmetric distribution.
   • Positive values indicate right-skewed distributions, while negative values indicate left-skewed distributions.
   • The magnitude of the skewness value reflects the degree of asymmetry.

4. Applications:

   • Finance: Used to analyze the distribution of returns on investments, helping investors understand the potential for extreme outcomes.
   • Economics: Assists in examining income distributions, enabling economists to assess income inequality.
   • Natural Sciences: Describes the distribution of experimental data in scientific research.

5. Considerations:

   • Skewness is just one aspect of distribution shape and should be considered alongside other statistical measures like kurtosis for a comprehensive understanding.
   • For small sample sizes, the estimation of skewness can be unreliable.

In essence, skewness is a statistical tool for understanding the asymmetry of data distributions, with wide-ranging applications in various fields such as finance, economics, and the natural sciences.

If x contains missings and these are not removed, the skewness is NA. Otherwise, write xi for the non-missing elements of x, n for their number, μ for their mean, s for their standard deviation, and mr =∑i (xi −μ) ^ r /n for the sample moments of order r.

Joanes and Gill (1998) discuss three methods for estimating skewness:

Type 1: g1 = m3 / m2 ^ (3/2). This is the typical definition used in many older textbooks. Type 2: G1 = g1 sqrt(n(n−1)) /(n−2). Used in SAS and SPSS. Type 3: b1 = m3 /s^3 = g1 ((n−1)/n) ^ (3/2) . Used in MINITAB and BMDP.

All three skewness measures are unbiased under normality.

Kurtosis is a statistical measure that describes the "tailedness" of the probability distribution of a real-valued random variable. In simpler terms, it indicates the extent to which the tails of the distribution differ from those of a normal distribution.

Key Points about Kurtosis:

1. Definition:

   • Kurtosis is the fourth standardized moment of a distribution.
   • It is calculated as the average of the squared deviations of the data from its mean, raised to the fourth power, standardized by the standard deviation raised to the fourth power.

2. Types of Kurtosis:

   • Mesokurtic: Distributions with kurtosis similar to that of the normal distribution (kurtosis value of 3). The tails of a mesokurtic distribution are neither particularly fat nor particularly thin.
   • Leptokurtic: Distributions with positive kurtosis greater than 3. These distributions have "fat tails" and a sharp peak, indicating more frequent large deviations from the mean than a normal distribution.
   • Platykurtic: Distributions with kurtosis less than 3. These distributions have "thin tails" and a flatter peak, indicating fewer large deviations from the mean than a normal distribution.

3. Excess Kurtosis:

   • Often, kurtosis is reported as "excess kurtosis," which is the kurtosis value minus 3. This adjustment makes the kurtosis of a normal distribution equal to 0.
   • Positive excess kurtosis indicates a leptokurtic distribution, while negative excess kurtosis indicates a platykurtic distribution.

4. Interpretation:

   • High kurtosis in a data set is an indicator that data has heavy tails or outliers. This can affect the performance of statistical models and methods that assume normality.
   • Low kurtosis indicates that the data has light tails and lacks outliers.

5. Applications:

   • In finance, kurtosis is used to describe the distribution of returns of an investment. A high kurtosis indicates a higher risk of extreme returns.
   • In data analysis, kurtosis helps in understanding the shape of the data distribution and identifying potential outliers.

6. Calculation in R:

   • The kurtosis() function in the e1071 package can be used to calculate kurtosis in R.
   • Alternatively, kurtosis can be calculated manually using the formula:

 kurtosis <- sum((data - mean(data))^4) / ((length(data) - 1) * sd(data)^4) - 3

kurtosis is a statistical measure for understanding the shape of a data distribution, particularly the behavior of its tails. It is widely used in various fields, including finance, data analysis, and statistics.

If x contains missings and these are not removed, the kurtosis is NA.

Otherwise, write xi for the non-missing elements of x, n for their number, μ for their mean, s for their standard deviation, and mr = ∑i (xi −μ) ^ r /n for the sample moments of order r.

Joanes and Gill (1998) discuss three methods for estimating kurtosis:

Type 1: g2 = m4/m2 ^ 2 −3. This is the typical definition used in many older textbooks. Type 2: G2 = ((n+1)*g2 +6)∗(n−1)/((n−2)(n−3)). Used in SAS and SPSS. Type 3: b2 = m4 /s ^ 4 −3 = (g2 +3)(1−1/n) ^ 2 −3. Used in MINITAB and BMDP.

Only G2 (corresponding to type = 2) is unbiased under normality.