Exponential weighted average and its deviation elimination

Exponentially weighted average

For a data set in the form of sequence , A method of statistics and obtaining continuous curves is exponential weighted average . In particular , For a data sequence
$${\theta }_1,{\theta }_2,\cdots ,{\theta }_m$$
If you connect these points directly , The resulting curve will contain a lot of noise , Messy and irregular ：

So let's recalculate the value of each point , Make
$$\begin{gathered} v_0=0 \\ {v}_1=\beta v_0+(1-\beta ){\theta }_1 \\ \cdots \\ {v}_i=\beta v_{i-1}+(1-\beta ){\theta }_i \\ \cdots \\ {v}_m=\beta v_{m-1}+(1-\beta ){\theta }_m \end{gathered}$$
Through this averaging method , We are actually right $1\sim i$ The data are weighted average , And the weight is from $i$ To $1$ Decreasing exponentially , The closer to the current index, the higher the weight . An empirical estimate is $v_i$ Probably represents the former $\frac{1}{1-\beta }$ Average the data , Because the weight of samples beyond this range has been relatively small .

Parameters $\beta$ What affects is the decay rate of the weight of the previous samples ,$\beta$ The closer the 1, The slower the sample weight decays , The more samples we include , At this time, the curve will be smoother , But it cannot reflect the effect of the current point in time , It usually lags behind . and KaTeX parse error: Undefined control sequence: \bata at position 1: \̲b̲a̲t̲a̲ The closer the 0, The faster the sample weight decays , The curve will fluctuate more violently , But it is very sensitive to current data , The response is very prompt .

The following figure for $\beta =0.9$（ Red ） and $\beta =0.98$（ green ） When we get the curve ：

Deviation elimination

in application , The curve we get actually deviates a little from the curve in the figure above , If $\beta =0.98$, What we actually get should be the purple line ：

This is because in the early stage of index weighted average , Our initial value $v_0=0$ It also occupies a great weight , The front end of the curve is pulled down , Until the middle and later stages of the curve ,$v_0$ The weight attenuation of is low enough , The purple line gradually coincides with the green line .

To avoid this , You can add another term to the previous operation , obtain
$$v_i=\frac{\beta v_{i-1}+(1-\beta ){\theta }_i}{1-{\beta }^i}$$
Because the initial item $v_0=0$ stay $v_i$ The weight of is ${\beta }^i$, So divide by $1-{\beta }^i$ You can rule out $v_0$ The impact , Get a reasonable curve .