101 lines
3.2 KiB
ReStructuredText
101 lines
3.2 KiB
ReStructuredText
Metrics
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==================
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* **Expected value/Espérance**: Noted :math:`\mathbb{E}[X]`, is a **theorical value**. For example, when playing coin
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flipping, the expected value for getting heads or tails is 0.5.
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Variance
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------------------
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Variance can be seen as the expected squared deviation from the expected value of a random variable :math:`X`.
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.. math::
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\mathbb{V}[X]=\mathbb{E}[X-\mathbb{E}[X]]^2=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])^2}{n}=\mathrm{Cov}(X,X)
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Covariance
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------------------
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Covariance is a way to quantify the relationship between two random variables :math:`X` and
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:math:`Y` (`source <https://www.youtube.com/watch?v=qtaqvPAeEJY>`_). Covariance **DOES NOT**
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quantify how strong this correlation is! If covariance is:
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Positive
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For large values of :math:`X`, :math:`Y` is also taking large values
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Negative
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For large values of :math:`X`, :math:`Y` is also taking low values
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Null
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No correlation
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.. math::
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\mathrm{Cov}(X,Y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])(y_i - \mathbb{E}[Y])}{n}
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Standard deviation
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-----------------------
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Standard deviation provides a way to interprete the variance using the unit of :math:`X`.
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.. math::
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\sigma=\sqrt{\mathbb{V}[X]}
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Standard Error of the Mean
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-----------------------------
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Standard Error of the Mean (SEM) quantifies the error that is potentially made when computing the mean.
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.. math::
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\mathrm{SEM}=\sigma_X^{-}=\sqrt{\frac{\mathbb{V}[X]}{n}}=\frac{\sigma}{\sqrt{n}}
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Here is how to interpret it.
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If :math:`n=1`, the error is at most :math:`\sqrt{\mathbb{V}[X]}=\sigma_X` which is the standard deviation or :math:`X`.
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The more :math:`n` increases, the lower the error becomes.
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The greater :math:`\mathbb{V}[X]`, the greater :math:`n` needs to be to ensure a low :math:`\mathrm{SEM}`.
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More infos in `this video <https://www.youtube.com/watch?v=BwYj69LAQOI>`_.
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If it is still unclear, see the following R code:
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.. literalinclude:: code/sem.R
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:language: R
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Output example:
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.. code-block:: console
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----- Experiment 1 -----
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Means SD: 1.22
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SEM 1.26
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----- Experiment 2 -----
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Means SD: 1.26
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SEM 1.26
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----- Experiment 3 -----
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Means SD: 1.27
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SEM 1.26
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Degree of Freedom
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--------------------
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The degree of freedom is a quantity defined for a given computation.
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It corresponds to the number of parameters that are allowed to vary in that computation.
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In other words, how many varying values are contributing to the computation.
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For example, when computing the mean of a random variable :math:`X={x_1,...,x_n}`, there are :math:`n` parameters
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that are allowed to change in the following formula:
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.. math::
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\overline{x}=\frac{\sum_{i=1}^n x_i}{n}
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Thus, the degree of freedom in this computation is :math:`n`.
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When computing the standard deviation of :math:`X`, we have:
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.. math::
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\hat{\sigma}_x=\frac{\sum_{i=0}^n (x_i-\overline{x})^2}{n}
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In this case, the degree of freedom is :math:`n-1`. As the mean is already known, only :math:`n-1`
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of the :math:`x_i` are allowed to vary. By knowing :math:`n-1` of the :math:`x_i`, we can deduce the last
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one as follow:
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.. math::
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\overline{x}=\frac{(\sum_{i=1}^{n-1} x_i) + x_n}{n} \Longrightarrow x_n = n\overline{x} - (\sum_{i=1}^{n-1} x_i)
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