.. _uncertainty:


********************
Temporal Uncertainty
********************

*Temporal* is related to time, which is a (quantitative) continuous variable, while 
*uncertainty* refers to situations involving unknown information.

.. Like with most continuous variables, however, we typically think in time as a discrete variable 
.. because time is rounded when measured (a person is 20 years old and not between 20 and 21). 


* (see also :ref:`Time and dating <time>`)


|


Monte-Carlo simulation tests the significance of the observed fluctuations in the context 
of uncertainty in the calibration curve and [archaeological] sampling. 



|

Probability and uncertainty
===========================

We can apply a probabilistic approach when dealing with uncertainty. 
In this case, uncertainty is considered a measure quantifying the likelihood that events will occur. 



Different approaches for assigning probabilities include:

* the proportion of a particular outcome to all possible outcomes (classical)
* the long-run relative frequency of the probability for an outcome to occur
* certain degree of belief (subjective)

|

Probability space
-----------------

A *probability space* defines an occurrence or event as a subset of that space. 


For example, if the probability space is the interval :math:`[0,1]`, with the :ref:`uniform distribution <unif>` then the interval 
:math:`[ \frac{1}{3}, 1 ]` is an event that represents a randomly chosen number between :math:`0` and :math:`1` 
turning out to be at least :math:`\frac{1}{3}`. 



|


Probability distributions
=========================

A *probability distribution* describes the associated probability of possible outcomes for a random variable :math:`X`. 

For a single variable, the data is allocated in univariate distributions, and we use multivariate distributions 
to infer on multiple parameters. 


..  Given a probability space, an *event* is defined to be a (sufficiently nice) 




|


Binomial distribution
---------------------

A binomial distribution is a continuous distribution that describes a binomial experiment where 
there are two possible outcomes.

The probability mass function (p.m.f.) corresponding to the binomial distribution 
represents the probability of :math:`x` successes in a binomial experiment 

..  '\binom' not working

:math:`P(x) \;=\; {n\choose x}  ~p^x (1-p)^{n-x}`


where :math:`P(x)` is the distribution function for the probability of success for :math:`x = 0, 1, 2, \dots, n`. 


|



+-------------------------------------+
|  .. image:: ../images/binomial.png  |
|     :width: 350                     |
+-------------------------------------+

A plot of the binomial distribution for :math:`n=30` and with different :math:`p` values.


|


    .. note:: 
    
        The Bernoulli distribution is a special case of the binomial distribution 
        when :math:`n = 1`. 

|


    .. note:: 
    
        The Beta distribution models the probability :math:`(p)` of success rather than the number of occurrences :math:`(x)`,  
        and it represents all the possible values of :math:`(p)` when we do not know what that probability is.



|


.. _trap:


Trapezoidal distribution
------------------------


Trapezoidal distributions seems appropriate for modeling the duration and the form of a phenomenon with a growth-stage, 
relative stability, and decline. These three parameters are not necessarily similar and an occurrence can have, for example, 
a long development and abrupt decay. 

The probability density function of the trapezoidal distribution for :math:`a \leqslant b \leqslant c \leqslant d` is


:math:`f(x \mid a,b,c,d) = \begin{cases}U \bigl(\frac{x-a}{b-a}\bigr) \quad a \leqslant x < b\\U \qquad b \leqslant x < c\\U \bigl(\frac{d-x}{d-c}\bigr) \quad c \leqslant x < d\\ 0 \quad \text{elsewhere}\end{cases}` 


where :math:`U = 2(d + c - b - a)^{-1}` corresponds to the uniform distribution. 


..  :math:`U \bigl(\frac{x-a}{b-a}\bigr)` 
..  :math:`U  \quad   b \leqslant x < c` 



..  \begin{cases}
..  	U \bigl(\frac{x-a}{b-a}\bigr) \\
..  	U  \quad   b \leqslant x < c \\
..  \end{cases}



|

                                      
+------------------------------------------+
|  .. image:: ../images/trapezoidal1.png   |
|     :width: 400                          |
+------------------------------------------+


A trapezoidal distribution where decline is longer than growth. 


..      .. todo:: 
..      
..         Probability density function of the generalized trapezoidal distribution. 





|

.. _unif:


Uniform distribution
++++++++++++++++++++

The uniform distribution is a special case of a trapezoidal distribution with a constant probability. 

The probability density function of the continuous uniform distribution for :math:`a \leqslant x \leqslant b`

:math:`f(x) \;=\; \frac{1}{b-a}`

and :math:`f(x) = 0` iff :math:`x<a` or :math:`x>b`

|

                                      
+---------------------------------------+
|  .. image:: ../images/uniform45.png   |
|     :width: 300                       |
+---------------------------------------+

Uniform distribution plot for :math:`P(x_1 < X < x_2)`.


..                                        
..  +------------------------------------+
..  |  .. image:: ../images/uniform.png  |
..  |     :width: 300                    |
..  +------------------------------------+
..                                        

|



    .. note:: 
    
        There is also a discrete version of the uniform distribution 
        that is a generalization of the Bernoulli distribution. 



|


Normal distribution
-------------------

The normal distribution --or "bell curve"-- is the probability distribution for a normal 
random variable :math:`X \sim N (\mu,\sigma^2)`. 
The normal distribution is also called *Gaussian* distribution due to C.F. Gauss who described 
this distribution in mathematical terms. 


The normal distribution is the most important distribution in statistics, and 
play a crucial role in statistical inference. 
This is partly because it approximates well the distributions of many types of variables. 

The exact form of the distribution depends on the values of the mean :math:`\mu` and 
the standard deviation :math:`\sigma` parameters. 

The probability density function of a normal random variable :math:`-\infty < x < \infty` is:

:math:`f(x) \;=\; \frac{1}{{\sigma \sqrt {2\pi } }}\;e^{{{ - \frac12 \left( \frac{x - \mu }{\sigma} \right)^2 }  }}`

where :math:`e \approx 2.7183` (Euler's number), and :math:`\pi \approx 3.1416` (Pi number). 


    .. note:: 
    
        A special case of the normal distribution with mean :math:`\mu = 0` and 
        standard deviation :math:`\sigma = 1` is the *standard normal distribution* :math:`Z`. 
        Any arbitrary normal distribution can be converted to :math:`Z`.


|


+----------------------------------------+
|  .. image:: ../images/gaussiansds.png  |
|     :width: 320                        |
+----------------------------------------+

Normal distributions with different variances :math:`\sigma^2` and same mean :math:`\mu`. 

.. +-------------------------------------+
.. |  .. image:: ../images/gaussian.png  |
.. |     :width: 320                     |
.. +-------------------------------------+


|


    .. todo:: 
    
        Expectations of the distributions.






|


Notation
========

To study temporal uncertainty, the following notation is adopted: 


|


:math:`\varOmega =` range of time

:math:`\tau =` time span of existence

:math:`\Delta \tau =` duration of :math:`\tau`

:math:`t_i =` a given portion of time

:math:`\Delta t_i =` duration of :math:`t_i`

:math:`\varphi =` temporal resolution

:math:`e =` event or occurrence

:math:`P, p =` probability

|

..  plus in Appendix A (from Crema, 2012)...


..  :math:`d =` duration of :math:`e` 
..  
..  :math:`b =` numerical index of temporal blocks within :math:`\tau` 
..  
..  
..      .. note:: 
..      
..          :math:`d` is rounded to the value of the temporal resolution, and :math:`b` ranges 
..          from :math:`1` to :math:`\Delta t_i/\varphi`, which are the edges of the time span. 

|






.. _aora:


Aoristic analysis
=================

Aoristic analysis is based on the creation of a series of these artificial divisions of 
the range of time with a fixed value, called time blocks, and the definition of their probability of existence. 


|


.. _tapq:


Boundaries of existence
-----------------------


Limits of dating for events are the earliest and the latest time the event may have happened. 
These upper and lower bounds are termed

* *terminus ante quem* (TAQ) or "limit before which"
* *terminus post quem* (TPQ) or "limit after which"


|


Temporal resolution
-------------------

Temporal resolution :math:`\varphi` refers to the duration of time blocks, and  
with aoristic analysis :math:`\varphi` is fixed.




Aoristic sum
------------

A proxy for evaluating change in the total counts of events across time is the *aoristic sum*, which is the sum of probabilities for each time block.

The aoristic sum is computed accross the sum of probabilities for events in a single portion of time :math:`t_i`.

    .. code-block:: 
    
                                                                 
                                                                 
            t1  t2  t3                                        
             ^   ^   ^                                        
        a    |   |   |                                        
        b    |   |   |                                        
        c    |   |   |                                        
             ^   ^   ^                                        
             aoristic                                         
               sums                                           
                                                                 
                                                                 



    .. todo:: 
    
        Aoristic sum explained in R...



..   :lineno-start: 85390



|


Probability of existence :math:`t_i`
====================================


The probability of existence :math:`p[t_i]` of an event :math:`e` in a range of time :math:`\varOmega` is the probabilty distribution

:math:`p[t_i] = \Delta t_i / \Delta \tau`

where :math:`\tau` is the time span of existence, :math:`\Delta` is for the duration of a given portion of time :math:`t_i` or of :math:`\tau`. 




To compute :math:`p[t_i]`, Crema (2012, p. 447) takes the following example (years are BC):

    .. code-block:: 
    
                                                                         
                                                                         
        350 ---- 300 ---- 250
             t1       t2
        
             342 ---- 288
                 tau
                                                                 
                                                                 


Hence, :math:`\Delta \tau` in this case is :math:`54`.


..  The probabilities of existence of inscriptions 8 and 5 chronological periods are 



|

Time-block boundaries
---------------------


To retrieve the minimum probability :math:`P[\varphi_\alpha]` for the boundaries of the first and last time-blocks, 
we define :math:`\varphi_\alpha = 1` where 

:math:`\phi_\alpha / \Delta_\tau = 1/54 = P[\phi_\alpha] = .018` (approx)


Hence, the probability of existence for :math:`t_1`

:math:`P[t_1] = P[\phi_\alpha] \cdot 42`  

(i.e. :math:`342 - 300`)


:math:`P[t_1] = .018 \cdot 42  =  .78`


The probability of existence for for :math:`t_2`

:math:`P[t_2] = P[\phi_\alpha] \cdot 12`  

(i.e. :math:`300 - 288`)

:math:`P[t_2] = .018 \cdot 12  =  .22`


Etc.



|

*Assumption* (among others): Any equally long portion of time within the time span :math:`\tau` has the same probability of existence. 



|


Probabilities of existence for time blocks
==========================================


The probability of existence for each time block (Appendix A, Crema, 2012)

:math:`P_b \;=\; {\sum_{i=0}^{\pi-1} \sum_{j=1}^{d/\varphi} \theta (i+j,\;b)} / {\pi}`

where :math:`\pi` is the number of all possible permutations, 
and :math:`b` is a numerical index of temporal blocks within :math:`\tau`.

On the other hand, the number of all possible permutations :math:`\pi` equals :math:`{(\Delta \tau - d)}/{\varphi + 1}` 
where :math:`d` is the duration of :math:`e`.

    .. note:: 
    
        :math:`d` is rounded to the value of the temporal resolution, and :math:`b` ranges 
        from :math:`1` to :math:`\Delta t_i/\varphi`, which are the edges of the time span. 



|


Temporal resolution for time blocks
-----------------------------------

The duration of time blocks that is :math:`\varphi` in the denominator for computing 
:math:`\pi` is known as the *temporal resolution*, while the numerator expression for 
the probability of existence for each time block is defined as

.. :math:`\theta (i+j,\;b)` is defined as

:math:`\theta (i+j,\;b) = \begin{cases}1 \quad \text{if}\; i+j = b  \\ 0 \quad \text{if}\; i+j \neq b \end{cases}`




.. :math:`P_b =` probability of existence for each time block 

.. :math:`\pi =` number of all possible permutations; i.e. :math:`{(\Delta \tau - d)}/{\varphi + 1}`



    .. note:: 
    
	The two extremes of :math:`P_b`are :math:`1/\pi` to :math:`1`.


|


Rate of change
--------------

Rate of change refers to transition probabilities like increase/stability/decrease as with the :ref:`trapezoidal <trap>` distribution.

Rate of chage =  :math:`\frac{(e~t_{i+1} - e~t_i)}{\varphi}`


An example of a transition probabilities are increase :math:`=0.4`; stability :math:`=0.5` and decrease :math:`=0.1`




|




.. _cprex:


Computing the probability of existence
======================================

We use function ``prex()`` to compute the probability of existence. 




|


.. function:: prex

Function usage
--------------

.. code-block:: R

    # probability of existence
    R> prex(x, taq, tpq, vars, bins = NULL, cp, aoristic = TRUE, weight = 1, 
            DF, out, plot = FALSE, main = NULL, ...)
    

|




Parameters
----------
 

- `x:`

   list or data frame object of variables and observations. 


- `taq:`

   TAQ or *terminus ante quem*
 

- `tpq:`

   TPQ or *terminus post quem*
 

- `vars:`

   boundaries of existence in ``x`` (vector for TAQ and TPQ) 


- `bins:`

   length of the break (integer) 


- `cp:`

   chronological phase (optional)


- `aoristic:`

   return aoristic sum? (logical)


- `weight:`

   weight to observations


- `DF:`

   return also data frame with observations? Ignored for plot (logical and optional)


- `out:`

   number of outliers to omit (integer or vector where first entry id for latest date)


- `plot:`

   plot the results? (logical and optional)


- `main:`

   plot's main title (optional)



- `...:`

   additional optional parameters


|

Details
+++++++

In case that `bins` is ``NULL``, then the time breaks take the chronological period specified in `cp`, which by
default is ``"bin5"`` or a five-periods model for the ``EDH`` dataset; the other built-in option is ``"bin8"`` for eight
chronological periods. Argument `cp` is open for other chronological phases as long as they are recorded as a list object.


|


For example, the (default) aoristic sum of ``EDH`` inscriptions in the Roman province 
of ``Iudaea`` computed with ``prex()`` 

    .. code-block:: R
    
       
       # get inscriptions from Iudaea in EDH data base
       R> iud <- get.edh(search="inscriptions", province="Iud")
       
       # 5-chronological periods
       R> prex(x=iud, taq="not_before", tpq="not_after", cp="bin5")
       #  Arch     Class      Hell       Rom       Byz 
       # 0.000     0.000  1337.904 13405.017     0.000 
       
       # 8-chronological periods
       R> prex(x=iud, taq="not_before", tpq="not_after", cp="bin8")
       #  Arch     Class      Hell      ERom      MRom      LRom      EByz      LByz 
       #0.0000    0.0000 1337.9040 2396.4529 1200.5623  320.5379    0.0000    0.0000 

|


Plotting aoristic sum
---------------------

To visualize the aoristic sum from the Roman province of Iudaea or ``iud`` as bar plots with function ``prex()`` 
and two types of chronological phases 

    .. code-block:: R
    
       
       #  five chronological phases
       R> barplot(prex(x=iud, taq="not_before", tpq="not_after", cp="bin5"), 
       +          main="Aoristic sum Iudaea (5-periods)" )
       
       # eight chronological phases
       R> barplot(prex(x=iud, taq="not_before", tpq="not_after", cp="bin8"), 
       +          horiz=TRUE, las=1, main="Aoristic sum Iudaea (8-periods)" )


that produce 


|

+------------------------------------------------+------------------------------------------------+
|  .. image:: ../images/AoristicSumIUDBins5.png  |  .. image:: ../images/AoristicSumIUDBins8.png  |
|     :width: 250                                |     :width: 275                                |
+------------------------------------------------+------------------------------------------------+




|

    .. seealso:: 
    
        :ref:`Relative dating of Roman inscriptions <EDHrd>`




|




.. :math:`\theta =` (cf. eqs. [2] and [3])






|





.. meta::
   :description: Probability and Uncertainty
   :keywords: temporal uncertainty, Chronology