Analytics 
++++++++++++++++++++++++++++++++++++++++++++++++++++


Exploratory Data Analysis (EDA)
------------------------------------------------------------------------------------------

In this section we are going to see a brief description of different concepts as mean, standard deviation and so on, 
you also will find notebook(s) to replicate in python.



Slides
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

* `EDA <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/EDA_.pdf>`_



**Levels of measurement**  allow us classify the data according to the information nature and its  mathematical richness 
in terms of operations, the herarchy is the following:

 * Ratio
 * Interval
 * Ordinal
 * Nominal

This level of classification give us flexbility and precision in our analysis, consider the  income of persons, you could collect ordinal ranges as 
*high income*, *Medium income*  annd *lower income* and so, but this not allow us the degree of differences and lesser to known the exact value. Unlike if you  
collect the numerical values, with this information  you could construct the ranges aforementioned and gain precision over the collected information.


Temperature measured in Celsious is not  a intervale variable given that zero not indicate absence of temperature, also the definition of
zero differ in Farenheith regarding celcious. Unlike zero meters and zero feets means the same; absence of distance!.

*inverval variables* have an arbitary zero point and *ratio varibales* have an absolute zero point.
IQ  is an example of interval scale, a IQ test of 80 and another of 160 not imply that that one person  twice smarter  that another, but you can said that 
a person is twice loaded of another.


This is important to carry out adecuately statistical description or analysis, for instance the coeffcieint of variation
only have sense in *ratio* varaibles.   





Frequencies - categorical data
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Categorical data as countries name, kind of bacteria (gram positive or negative), 
or in sport base ball teams, could be described with *absolute* and *relative* frequencies.

Example: World bank data have a classification of countries by its level of income.

+----------+------------------------------+---------------------------------+
| Category | Absolute frequency $(n_{i})$ | Relative frequency $(freq_{i})$ |
+----------+------------------------------+---------------------------------+
|    HIC   |              82              |              0.378              |
+----------+------------------------------+---------------------------------+
|    LMC   |              54              |              0.249              |
+----------+------------------------------+---------------------------------+
|    UMC   |              54              |              0.249              |
+----------+------------------------------+---------------------------------+
|    LIC   |              26              |              0.120              |
+----------+------------------------------+---------------------------------+
|    INX   |               1              |              0.005              |
+----------+------------------------------+---------------------------------+


.. image:: ./images/bar-chart.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %




We can also repsent the distribution of some numerical value in each category, for instance:

+----------+-----------------------------------+
| Category | Relative share of population 2020 |
+----------+-----------------------------------+
| HIC      | 15.624                            |
+----------+-----------------------------------+
| INX      | 0.365                             |
+----------+-----------------------------------+
| LIC      | 8.556                             |
+----------+-----------------------------------+
| LMC      | 39.982                            |
+----------+-----------------------------------+
| UMC      | 35.473                            |
+----------+-----------------------------------+



.. image:: ./images/pie-popDist.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %




Mean
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Used as central tendency measure is sensible to outliers (very tiny or bigger values...)

.. math::
    :nowrap:

    \begin{equation}
        \bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_{i}
    \end{equation}



Median and Quartiles
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Quartiles split the data in four equal groups, therefore 
each slice group a $\frac{1}{4}$ of data.
Median is the data point that split the data in two halves, finally
percentiles split the data in $100$ equal groups.

* Q1 = P25 
* Q2 = P50
* Q3 = P75


The following table summary some values aforementioned:

+-------+----------------------+
|       | Life expectancy 2020 |
+-------+----------------------+
| count |          209         |
+-------+----------------------+
|  mean |         72,31        |
+-------+----------------------+
|  std  |         7,483        |
+-------+----------------------+
|  min  |        52,777        |
+-------+----------------------+
|  25%  |        66,797        |
+-------+----------------------+
|  50%  |        72,889        |
+-------+----------------------+
|  75%  |        78,041        |
+-------+----------------------+
|  max  |        85,498        |
+-------+----------------------+

Standard deviation and Interquantile Range
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::
    :nowrap:

    \begin{equation}
        \sigma^{2} = \frac{\sum_{i=1}^{N} (x_{i} - \bar{x})^{2}}{N}
    \end{equation}


.. math::
    :nowrap:

    \begin{equation}
        \sigma^{2} = \frac{\sum_{i=1}^{N} (x_{i} - \bar{x})^{2}}{N-1}
    \end{equation}


Why it is important this distiction? statistics give us; procedures, techniques and mathematical formulas to take better
decisions with data this imply that we need draw the most information possible our data, and in this case the second formula
give us a better approximation to the population parameter that describe the variability.
See here `Simulation <https://github.com/it-ces/PUBLIC-AI/blob/main/Probability/Simlab/Variance_estimators(Biased_unbiased).ipynb>`_




Coefficient of variation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Useful when need compare population with different characteristics, for instance 
the life expectancy in different geographical unities grouped by level of income.

+---------------------------+-------+--------+-------+-------+
| Income   Level (Category) | Count |  Mean  |  Std  | CV(%) |
+---------------------------+-------+--------+-------+-------+
|            HIC            |   75  | 79,127 | 3,987 | 5,039 |
+---------------------------+-------+--------+-------+-------+
|            INX            |   1   | 71,095 |  NaN  |  NaN  |
+---------------------------+-------+--------+-------+-------+
|            LIC            |   26  | 62,106 | 4,805 | 7,737 |
+---------------------------+-------+--------+-------+-------+
|            LMC            |   54  | 67,839 | 5,642 | 8,317 |
+---------------------------+-------+--------+-------+-------+
|            UMC            |   53  | 72,246 | 4,232 | 5,858 |
+---------------------------+-------+--------+-------+-------+


According to the table the category LIC and HIC have considerably differences in means.


Histogram
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This represent graphically the distribution of variable itself according to ranges,
this histogram shows the life expectancy in some countries in the world.

The height of each bar is porportional to the number of data points that are in the 
respective range, for instance....

+------+---+---+---+----+---+---+----+
| data | 1 | 2 | 4 | 10 | 8 | 5 | 12 |
+------+---+---+---+----+---+---+----+

+---------+---------+--------+
|  Ranges |  Values | Height |
+---------+---------+--------+
|  (1-5)  | 1,2,4,5 | 0,57   |
+---------+---------+--------+
|  (6-10) |   8,10  | 0,29   |
+---------+---------+--------+
| (11-16) |    12   | 0,14   |
+---------+---------+--------+


Note that column Height  could sum up 1.




.. image:: ./images/Histogram.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %




Boxplot
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


There are three important things to remark in this graph, p25, p50, p75,  we need sort the values of the variables
from smallest to largest.
The wide of box is denominated as Interquantile range.







Box plot also give us visual information about *skewness* of the distribution:

Skewness  is a measure of the asymetry:




* In *poisitive skew (skewed to the right)*, The right tail is longer, the mass of the distribution is concentrated on the left of the figure.


.. |pic1| image:: ./images/GDP-percapita.jpg
   :width: 45%
.. |pic2| image:: ./images/boxplot-gdp2020.jpg
   :width: 45%
|pic1| |pic2|


* In *negative skew (skewed to the left)*,  the left tail is longer; the mass  of the distribution is 
Concentrated on the right of the figure.

.. |pic3| image:: ./images/hist-acc_clean.jpg
   :width: 45%
.. |pic4| image:: ./images/boxplot-acc_clean.jpg
   :width: 45%
|pic3| |pic4|


Boxplot is also very usefult to compare numerical variables over groups.

.. image:: ./images/boxplot-incomelevel.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %





.. code::

    from scipy.stats import norm
    norm.ppf(0.25) # Give us the value of the 25 centil  Q1
    norm.ppf(0.75) # Give us the value of the 75 centil  Q3



..  code::
    
    
    norm.cdf(norm.ppf(0.25))


The interquantile range (IQR) is $Q_{3} - Q_{1}$ and for normal distribution is  $\approx 1.34$ 

Now cosider the following:


Now we are searching two values as decision criteria to values more extreme of them as outliers values:

.. math::
    :nowrap:

    \begin{equation}
     Q_{3} +  \lambda   IQR
    \end{equation}

    \begin{equation}
     Q_{1} - \lambda IQR
    \end{equation}

Reemplacing the values we need select a $\lambda$ value:

.. math::
    :nowrap:

    \begin{equation}
    0.67 + \lambda 1.34
    \end{equation}

    \begin{equation}
    - (0.67 + \lambda 1.34)
    \end{equation}



If we select $\lambda = 1.5$ then the probability of getting a *outlier value* is 0.0073.

.. code::

    norm.cdf(-2.68)*2 # The probability of be a outlier....





Normality
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Notes about normality...


.. image:: ./images/normalHistogram.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %


Test of normality is important to decide the statistics used
to describe data or to use parametric or no parametric statistics, for instance
uses mean with no-normal variables could lead to misleading conclusions.

We also need normality for uses ANOVA and compare groups, for instance when you need 
test if the treatment A , B , C  or D present significative difference among a group of patients.


there are numercial and graphically methods to test data normality, some
of those are:

* Shapiro-Wilk test
* Kolgomorov-Sirnov test
* Histogram
* Q-Q plot


where used each one?

uses Shapiro-Wilk for small  sample sizes (<50)
uses Kolgomorov-Sirnov with sizes greater than  or equal to  50.
in both test $H_{0}$ indicate normality.


The histogram must be as the presented above, the Q-Q plot or Quantile-Quantile plot graph 
expected vs observed quantiles in standardize data with standardize normal distribution.
For normal data the points must be looks like a line.

In Q-Q plot we are comparing theoretical agains empirical quantiles of a 
variable. If the variable come from that distribution Q-Q plot is a straigth
line.

.. |nhist| image:: ./images/normalHistogram.jpg
   :width: 45%
.. |qqplotnorm| image:: ./images/qqplot-normal.jpg
   :width: 45%
|nhist| |qqplotnorm|


The following is a not normal especifically a exponential distribution:



.. |exphist| image:: ./images/exponentialHistogram.jpg
   :width: 45%
.. |qqplotexp| image:: ./images/qqplot-exponential.jpg
   :width: 45%
|exphist| |qqplotexp|


Analytical methods are sensitive to sample size.
assuming that $X \sim N()$ with small samples $p-value$ tend to be greater, and with
larges samples $p-value$ tend to be small.


Rule: if you have sample size $< 30$ uses Q-Q plot.





So far, we conclude that the descriptive 
statistics aim to communicate the major quantity of information in the simplest way.



    




Line chart
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Sometimes we need decribes variables as a time series.

.. image:: ./images/time-serie.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %







EDA laboratories I
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here we find some laboratories to practice.


* `EDA Categorical <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/EDA_population(BIT).ipynb>`_
* `EDA Numerical <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/EDA_lifeExp_(BIT).ipynb>`_
* `Relative position <https://github.com/it-ces/PUBLIC-AI/blob/main/short-labs-pandas/Quantile(BIT).ipynb>`_
* `Practical lab <https://github.com/it-ces/PUBLIC-AI/blob/main/short-labs-pandas/LaboratoryPandas(ONE).ipynb>`_
* `Matplotlib lab <https://github.com/it-ces/PUBLIC-AI/blob/main/short-labs-pandas/introduction%20to%20matplotlib.ipynb>`_




Covariance
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~






Correlation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Measure the degree of linearity among two variables, take values among
-1 and 1, there is not correlation among variables when the coefficient is equal to zero.

.. |pic5| image:: ./images/gdp-ruralpop.jpg
   :width: 45%
.. |pic6| image:: ./images/gdp-co2.jpg
   :width: 45%
|pic5| |pic6|

For this image put here the notebook, where two important pearson and spearman.



+----------+----------------------+--------------------------------+---------------------+
|          |                      | Percentage of rural population | Co2 tons per capita |
+----------+----------------------+--------------------------------+---------------------+
| GDP 2020 |  Pearson coefficient |             -0,634             |        0,722        |
+----------+----------------------+--------------------------------+---------------------+
|          | Spearman Coefficient |             -0,712             |        0,907        |
+----------+----------------------+--------------------------------+---------------------+



Here there are a plot of correlation matrix...


EDA laboratories II
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


* `Correlation Matrix <https://github.com/it-ces/PUBLIC-AI/blob/main/short-labs-pandas/Correlation_matrix(BIT).ipynb>`_
* `Mutual Information (MI) <https://github.com/it-ces/PUBLIC-AI/blob/main/short-labs-pandas/Correlation_matrix(BIT).ipynb>`_
* `Normality test <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/EDA_population(BIT).ipynb>`_
* `Contigency Table <https://github.com/it-ces/labHealth/blob/master/Health_Second_Class.ipynb>`_




Matrix correlation could be useful...

Preprocessing
-------------------------------------------------------


Standarization 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. math::
    :nowrap:

    \begin{equation}
        z_{i} = \frac{x_{i} - \bar{x}}{\sigma_{x}}
    \end{equation}

where $\bar{x}$ is the mean, and $\sigma$ is standard deviation, this very useful given 
that $z \sim (0, 1)$ follow a distribution with mean zero and standard deviation of one.

You can see it from:


.. math::
    :nowrap:

    \begin{equation}
        \bar{z} =  \frac{1}{n}  \sum_{i=1}^{n} \left( \frac{x_{i} - \bar{x}}{ \sigma_{x} } \right)
    \end{equation}


note that $\sum_{i=1}^{n}  (x_{i} - \bar{x}) = n\bar{x}  - n\bar{x} = 0$ therefore $\bar{z} = 0$.



Now consider the variance of the variable $z$:

.. math::
    :nowrap:

    \begin{equation}
        \sigma_{z} = \frac{\sum (z_{i} -  \bar{z})^{2}}{n} 
    \end{equation}


reemplacing $\bar{z}=0$  and $z_{i} = \frac{x_{i} - \bar{x}}{\sigma_{x}}$ then:



.. math::
    :nowrap:

    \begin{equation}
     \sigma_{z} =  \frac{1}{n} \sum  \left(   \frac{ x_{i} - \bar{x} }{\sigma_{x}}  \right)^{2}    =  \frac{1}{\sigma_{x}^{2}} \frac{\sum (x_{i} - \bar{x})^{2}}{n}.
    \end{equation}


Given that $\sigma_{z}^{2}  = \frac{\sum (x_{i} - \bar{x})^{2}}{n}$, This show us that $\sigma_{z} = 1$.






.. note::

    `(See lab) <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/Z_standard(BIT).ipynb>`_






Normality 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

What it is normal? 

We are going to say that $X$ is normal as; $X \sim N(\mu, \sigma)$ Discovered by abraham de moivre.




Standard Normal distribution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

$X \sim N(0,1)$.

.. image:: ./images/normaldist.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %

The before distribution have $\mu=0$ and $\sigma = 1$.

our guide text..

There are a family of nornal distributions described by its mean and standard deviation.

its $x$ is described as:


.. math::
    :nowrap:

    \begin{equation}
    \frac{1}{ \sqrt{2 \pi \sigma^{2}}} \exp{ \left( \frac{-(x-\mu)^{2}}{2\sigma^{2}} \right)}
    \end{equation}



.. image:: ./images/normales.jpg
   :height: 800px
   :width:  800px
   :scale: 50 %


Each curve is described by its median and standard deviation.






Laboratories
*************************************************************************************************





Handling categorical variables
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this section we are going to work with categories in Machine learning, using important methos as OneHotEncoder.

.. note::
    
    `(See lab) <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/Encoding(Sklearn_pandas).ipynb>`_





Prediction perfromance assestmet
---------------------------------------------------------------------------------

OCD occurs whent you cant stop of thinking about something,
but i could i sure that i have OCD?, 
you can visit your professional health, and take several test to diagnose,
but what happens if the test is wrong? or what happend if the test said that not and
your thoughts are persistent!



imagine a similar situation with a major degree of certain, whichs means?
this means that there are a golden standard to diagnose, and you are testing a cheaper method, 
think in diagnose of covid-19 there are some situations

#. Test say **not** and i'm really **not have** covid-19
#. Test say **not** and i'm reall **have** covid-19
#. Test say **yes** and i'm really **have** covid-19
#. Test say **yes** and i'm really **dont have** covid-19



This could be summaries in:






Ok!, then how good fit this test?,  there are some metrics that i could study:

specificity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::
    :nowrap:

    \begin{equation}
        P( \hat{-} \mid - )
    \end{equation}

In other words, you can understand as the probability of get a negative result given that
the persons not have a the condition.

sensitivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::
    :nowrap:

    \begin{equation}
        P( \hat{+} \mid + )
    \end{equation}

This is the probability of get a test positive result when really exist the condition.


All this measures are built on the **confusion matrix**

Confusion Matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+---------------+---------------------+-----------------------+
|               |  Disease (Positive) | No-disease (Negative) |
+---------------+---------------------+-----------------------+
| Test Positive |  True Positive (TP) |  False Positive (FP)  |
+---------------+---------------------+-----------------------+
| Test Negative | False Negative (FN) |   True Negative (TN)  |
+---------------+---------------------+-----------------------+

The confusion matrix present the different combinations of possible results.


Accuracy
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~




.. math:: 
    :nowrap:

    \begin{equation}
       \frac{ TP  + TN}{(FP + FN + TP + FN)}
    \end{equation}





.. warning:: 
    
    This measure is biased when there are unbalanced classes


is a unbiased measure of performance given that is affected by unbalanced classes.



.. math:: 
    :nowrap:

    \begin{equation}
        \text{Type 1} = \frac{FP}{TN + FP}
    \end{equation}

 
.. math:: 
    :nowrap:

    \begin{equation}
        \text{Type 2} = \frac{FN}{TP+FN}
    \end{equation}




Now, we are going to see in probability terms to have major intuition of the some measures.

.. math::
    :nowrap:

    \begin{equation}
    Sesintivity  = recall = \frac{TP}{TP+FN} = \frac{ P( + \cap \hat{+})}{P(+)} = P(\hat{+} \mid +)
    \end{equation}

.. math::
    :nowrap:

    \begin{equation}
    PPV = precision = \frac{TP}{TP+FP} = \frac{P(+ \cap \hat{+})}{P(\hat{+})}  = P(+ \mid \hat{+})
    \end{equation}

 

.. note::

    `(See lab) <https://colab.research.google.com/drive/1M8AzYbuaGZOheAg7KTYUu5-c9kziIYP2?usp=sharing>`_



Bayes theorem
-------------------------------------------------------------------------

Think a moment what is the relationship behind precision and recall or
recall and predicted positive values?.



.. math::
    :nowrap:

    \begin{equation}
        P(A \mid B) = \frac{P(A \cap B)}{P(B)}
    \end{equation}

.. math::
    :nowrap:

    \begin{equation}
        P(B \mid A) = \frac{P( A \cap B)}{P(B)}
    \end{equation}


Note that $P(  A \cap B) = P(A \mid B)P(B)$ and $P(A \cap B ) = P(B \mid A ) P(A)$, therefore

.. math::
    :nowrap:

    \begin{equation}
    P(A \mid B) P(B)  = P(B \mid A) P(A)
    \end{equation}

We can write:
    
    .. math::
        :nowrap:

        \begin{equation}
            P(A \mid B ) = \frac{P(B \mid A) P(A) }{P(B)}
        \end{equation}



How a model could be snesible and not precise?

the patient $X$, have a diagnose confirmed $+$ or $-$ and the results test as $\hat{+}$ or $\hat{-}$
thus a patient with confirmed diagnose and a negative result test is 
denoted as: $X_{+}^{\hat{-}}$, Analize the following situation:

.. math::
    :nowrap:

    \begin{equation}
    X_{+}^{\hat{+}} \quad X_{-}^{\hat{+}} \quad X_{-}^{\hat{+}} \quad X_{+}^{\hat{+}} \quad X_{-}^{\hat{+}} \quad X_{-}^{\hat{-}}
    \end{equation}

Notice that the test have a sensitivity of 100\%, and whics is the value of precision?
consider that a false positive case is  positive test when really is a negative confirmed diagnosed.


We could be interested in a model that the two properties
 



 how is related precision and specificity?..


.. note::

    Take in mind that precision and specificity are related
    by False Negatives.



You could uses take an average two ponder recall and precision, 
however note that the mean not differentiated of balanced (model 4) and extreme cases (models 1,2,3).


+----------+--------+-----------+-------+-----------+
|          | Recall | Precision |  Mean | F1 -score |
+----------+--------+-----------+-------+-----------+
|  Model 1 |   0,3  |    0,8    |  0,55 |    0,44   |
+----------+--------+-----------+-------+-----------+
|  Model 2 |   0,8  |    0,3    |  0,55 |    0,44   |
+----------+--------+-----------+-------+-----------+
| Model 3  |  0,12  |    0,98   |  0,55 |    0,21   |
+----------+--------+-----------+-------+-----------+
|  Model 4 |  0,52  |    0,58   |  0,55 |    0,55   |
+----------+--------+-----------+-------+-----------+
|  Model 5 |  0,83  |    0,91   |  0,87 |    0,87   |
+----------+--------+-----------+-------+-----------+
|  Model 6 |  0,08  |    100    | 50,04 |    0,16   |
+----------+--------+-----------+-------+-----------+


Threshold to profit - risk
-------------------------------------------------------------

What level of precision and sensitivity could maximize the profit?





Training and Test
--------------------------------------------------------------------

A simplified **workflow** could be described as; 


A common way to build machine learning models is assest the performance 
of the model to solve  a particular task; for instance classification of a new message
in *spamn* or *jam*. 

in ML is important to known that the evaluation of a model is a large task, 
that include the metric that quantify the model performance until the dataset selected,


.. note::

    `See lab <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/K_fold_Simple_Stratified(Sklearn).ipynb>`_











Crossvalidation
-------------------------------------------------------------------------

K-fold
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Over a dataset $D$ make $k$ partitions(folds) $(d_{1}, d_{2}, d_{3}, ..., d_{k})$  where $d_{i} \cap d_{j}  = \emptyset$
$,\forall i \neq j$, and $\bigcup_{i=1}^{k} d_{i} = D$.  The goal is select the $p$ partition to evaluate the training made it
with another $k-1$ partitions. This process is repeated with each partition $(d_{i})$  of the $k$ as validation dataset.
Now defining $\Xi(d_{j})$  as the model performance in the dataset $d_{j}$ then 
the overall assesment of the model will be:

.. math::
    :nowrap:

    \begin{equation}
        \frac{1}{K} \sum_{i=1}^{K} \Xi(d_{i}) 
    \end{equation}


Stratified K-fold
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given a more realistic scenario,  each partition have the target variable in roughly the same
approxiamate proportion that original dataset....


.. note::

    `See lab <https://github.com/it-ces/PUBLIC-AI/blob/main/Analytics/K_fold_Simple_Stratified(Sklearn).ipynb>`_





Hyperparameter tunning
------------------------------------------------------------------------------


It is important note that grid search not is the only way to 
search the hyperameters in a model, for instance GA's also can be used see here.


In machine learning algorithms the learning process is affected by some parameter, this means that
there are paraemters that are not learned by the algorithm, si no por el contario the user must specified, 
for instance learning rate in multilayer perceptron. Therefore the objective is find 
the set of parameters that optimize the algorithm.



Grid search
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This approach consist in test all the possible combinations of parameters defined by the users for instance
test the epochs:$(100, 120, 300)$ and the learning rates of $(0.05 , 0.01, 0.001)$ that
give us a total of $9$ combinations,

In this case, how will be selected the number of epocs and the value of the learning rate?

An overall metric performance will be obtanaied using ``cross-validation`` and the set of parameters
with higher value  will be selected $\max \Xi(X | \alpha, epochs)$.


this approach is computationally expensive, and not guaranteed reach a optimal solution,
there are some other techniques as ``random search```, ``bayesian  optimization```, ``evolutionary techniques``, among others...



Feature selection
--------------------------------------------------------------------------------


there are a many techniques to select variables,




Feature enginiring
---------------------------------------------------------------------------------



Logarithm transformation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A symmetric distribution, mean and median are equal, therefore in the central half of data have a lesser
value and the other halve have a greater value.

Skewed is presented in some variables for instance in Gross Product Domestic (GDP), namely *right skewed*,
this mean that only a minor fraction of countries have a huge GDP! otherwise. Another example could be
the time that a person is in hospital, and a transformation aim to be more symmetric distribution.

How you believe that is the behavior of the retirement age?..


.. code::

    import numpy as np
    np.log(10) - np.log(1)
    np.log(100) - np.log(10)
    np.log(1000) - np.log(100)

what happend whith the difference?, this behavior is more adecuate in right-skewed variables.
in left-skewed variables you could uses quadratic or cubic transformation why?.