Moving Average/Basic Approach

Generic Approach to Moving Average


An element   moves in an additive Group (mathematics) or Vector Space V. In a generic approach, we have a moving probability distribution   that defines how the values in the environment of   have an impact on the moving average.

Discrete/continuous Moving Average


According to probability distributions we have to distinguish between a

  • discrete (probability mass function  ) and
  • continuous (probability density function  )

moving average. The terminology refers to probability distributions and the semantics of probability mass/density function describes the distrubtion of weights around the value  . In the discrete setting the   means that   has a 20% impact on the moving average   for  .

Moving/Shift Distributions


If the probility distribution are shifted by   in  . This means that the probability mass functions   resp. probability density functions   are generated by a probability distribution   at the zero element of the additive group resp. zero vector of the vector space. Due to nature of the collected data f(x) exists for a subset  . In many cases T are the points in time for which data is collected. The and the shift of a distribution is defined by the following property:

  • discrete: For all   the probability mass function fulfills   for  
  • continuous: For all probability density function fulfills  

The moving average is defined by:

  • discrete: (probability mass function  )

Remark:   for a countable subset of  

  • continuous probability density function  

It is important for the definition of probability mass functions resp. probability density functions   that the support (measure theory) of   is a subset of T. This assures that 100% of the probability mass is assigned to collected data. The support   is defined as: