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 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 $$v$$.
 * discrete (probability mass function $$p_v$$) and
 * continuous (probability density function $$p_v$$)

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: The moving average is defined by:
 * 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 $$p_v(x):=p_0(x-v)$$
 * 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 $$p_v$$
 * $$MA(v):= \int_{T} p_v(x) \cdot f(x) \, dx $$

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