This file gives the elementary information on calculation of basic Statistics.
ˉz=1nn∑i=1zi
The median m of the set of values is a value such as half of the total observations are below and half is above.
If the size n is odd, it is the value of the z(n+12) where z(i) is the value of the i th observation (when ordered in increasing order).
If n is even, we can take z(n2)+z(n2+1)2.
The median is less sensitive than the mean to extreme values.
The quartiles are the values which divide the samples as follows:
We can generalize to any proportion p.
The p−quantile denoted qp is a value which divides the samples such as a proportion p of the individuals are below the quantile.
The median is q12. The lower quartile is q14 and the upper quartile is q34.
The dispersion summaries try to measure how the data are spread.
max
q_{\frac{3}{4}}-q_{\frac{1}{4}}
The variance measures the average distance between each individual to the mean:
\frac{1}{n}\sum_{i=1}^n(z_i-\bar{z})^2
It is sometimes more convenient to use the equivalent formula
\frac{1}{n}\sum_{i=1}^nz_i^2-\bar{z}^2 = \bar{z^2}-\bar{z}^2
Note that for statistical reasons, one often prefers to use
\frac{1}{n-1}\sum_{i=1}^n(z_i-m)^2
for the variance. The two formulas give close results when n is large.
To have a measure in the same unit as the variable, one often consider the standard deviation.
\sqrt{\frac{1}{n}\sum_{i=1}^n(z_i-\bar{z})^2}
To have a good idea of the distribution of a variable, one can compute the histogram.
The idea is
Normalized histogram rescales the ordinate such as the total surface is equal to 1.
We can represent the cumulated histogram. It is a function which computes, for each value, the proportion of individuals below this value. It can be written as
F(z_c) =\frac{1}{n}\sum_{i=1}^n 1\!\!\!1_{]z_{i},+\infty]}(z_c)
where 1\!\!\!1_A is the indicator function of the set A:
1\!\!\!1_A(x)=\left\{\begin{array}{ccc}1 &\textrm{ if } & x\in A\\ 0 & \textrm{ otherwise } & \end{array} \right.
If we inverse the two axes, we obtain the quantile function which gives, for each value p, the quantile q_p.
q(p) = F^{-1}(p)
In mine, we often consider the ore function T(z_c) = 1-F(z_c)
Indeed, it gives the proportion of the data which are above a cut-off.
Q(z_c) =\frac{1}{n}\sum_{i=1}^n z_i1\!\!\!1_{]z_{i},+\infty]}(z_c)
m(z_c)=\frac{Q(z_c)}{T(z_c)}
We just represent the Metal with respect to the Ore for various cut-off values z_c.
B(z_c) = Q(z_c)-z_cT(z_c)
Now we consider two variables:
and we will study their relationship.
We can compute the covariance between the two vectors z^{(1)} and z^{(2)}.
\textrm{cov}(z^{(1)},z^{(2)}) = \frac{1}{n}\sum_{i=1}^n (z^{(1)}_i-\bar{z}^{(1)})(z^{(2)}_i-\bar{z}^{(2)})
where \bar{z}^{(j)} is the mean of the variable z^{(j)} with j=1,2.
The covariance depends on the scale of z^{(1)} and z^{(2)}. In order to have a scale invariant measure, we can use the correlation coefficient \rho = \frac{\textrm{cov}(z^{(1)},z^{(2)})}{\sqrt{\textrm{var}(z^{(1)})\textrm{var}(z^{(2)})}}
The correlation coefficient lies within [-1,1].
When it is equal to -1 or 1, the variables are linked by a linear relationship
z^{(2)}=a.z^{(1)}+b
where the sign of a corresponds to the sign of \rho.
When \rho=0, we say that the variables are uncorrelated. But they can still have a link (not linear).
When we have several variables z^{(1)},\dots,z^{(p)}, we can compute their covariance matrix \Sigma which stores the covariances between each pair of variable.
\Sigma = \left[ \begin{array}{cccc} \textrm{var}(z^{(1)}) & \textrm{cov}(z^{(1)},z^{(2)}) &\dots & \textrm{cov}(z^{(1)},z^{(p)})\\ \textrm{cov}(z^{(2)},z^{(1)}) & \textrm{var}(z^{(2)}) & \dots & \textrm{cov}(z^{(2)},z^{(p)})\\ \vdots & \vdots & \ddots & \vdots \\ \textrm{cov}(z^{(p)},z^{(1)}) & \textrm{cov}(z^{(p)},z^{(2)})&\dots & \textrm{var}(z^{(p)})\\ \end{array}\right]
Note that this matrix is symmetric.
If the variables (centered by their means) are stored in a matrix Z_c (one column per variable), then
\Sigma = \frac{1}{n} Z_c^TZ_c where ^T designates the transposition.
In other words, Z_c^T is the matrix where each line is a variable.
We can represent the scatter plot between the two variables (only isotopic samples are represented).
Here the relation could be considered as linear. Let’s try to find the coefficents of the regression line.
We can model the relationship between z^{(1)} and z^{(2)} by using a linear regression. model z^{(2)}=az^{(1)}+b + R where R is a residual.
We try to find (a,b) by minimizing the sum of the squared difference between z^{(2)} and az^{(1)}+b:
||R||^2 =\sum_{i=1}^n(z^{(2)}_i - (az^{(1)}_i+b))^2.
We can show that the coefficients a and b can be estimated by
\hat a = \frac{\textrm{cov}(z^{(1)},z^{(2)})}{\textrm{var}(z^{(1)})}
and b by
\hat b = \bar{z}^{(2)}-\hat a\bar{z}^{(1)}
When we have several variables x^{(1)},\dots,x^{(p)} to explain an interest variable y we can also use a linear regression
y=\sum_{j=1}^p \beta_j x^{(j)} + \beta_0 + R
Note that for convenience, we will rewrite the relation
y=\sum_{j=0}^p \beta_j x^{(j)}+R
where the variable x^{(0)} is equal to 1.
Last, we can rewrite more compactly
y = \beta^T X +R
where \beta = \left[\begin{array}{c}\beta_0 \\ \vdots \\ \beta_p\end{array}\right]
and X is the table with all the observations. The first column contains 1’s and then each column is a variable X = \left[\begin{array}{cccc} 1 & x^{(1)} & \dots & x^{(p)}\end{array}\right]
As in the simple linear regression case, we will try to minimize
||R||^2=||y-\beta^TX||^2
We can show that
\hat\beta = (X^TX)^{-1}X^Ty
To represent the two variables, we can perform a 2d histogram.
Then we could look at the histogram of z_2 for a given class of z_1.
For instance, if we consider the 3rd class, z_1\in[1.14,1.67] :
It shows the conditional distribution of z_2 knowing that z_1\in[1.14,1.67].
In the same spirit, we can consider the conditional mean (mean of z_2 for different class of z_1).
It is named conditional mean (or regression).