Moving Average/Basic Approach

Generic Approach to Moving Average

An element $v\in V$ moves in an additive Group (mathematics) or Vector Space V. In a generic approach, we have a moving probability distribution $P_v$ that defines how the values in the environment of $v\in V$ 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 $p_v$) and
• continuous (probability density function $p_v$)

moving average. The terminology refers to probability distributions and the semantics of probability mass/density function describes the distrubtion of weights around the value $v\in V$. In the discrete setting the $p_v(x)=0.2$ means that $x$ has a 20% impact on the moving average $MA(v)$ for $v$.

Moving/Shift Distributions

If the probility distribution are shifted by $v$ in $V$. This means that the probability mass functions $p_v$ resp. probability density functions $p_v$ are generated by a probability distribution $p_0$ 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 $T\subseteq V$. 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 $x\in V$ the probability mass function fulfills $p_v(x):=p_0(x-v)$ for $v\in T$
• continuous: For all probability density function fulfills $p_v(x):=p_0(x-v)$

The moving average is defined by:

• discrete: (probability mass function $p_v$)

$MA(v):=\sum _{x\in T}{p}_v(x)\cdot f(x)$

Remark: $p_v(x)>0$ for a countable subset of $V$

• continuous probability density function $p_v$

$MA(v):=\underset{T}{\int }{p}_v(x)\cdot f(x)dx$

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

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(p_v):=\overline{\{x\in V\mid p_v(x)>0\}}\subset T.$