理解exponential weighted || 指数

Exponentially weighted moving average

average is a single data point;

moving average is a sequence of data point;

weighted moving average means the process calculating average 给不同元素不同权重;

exponential weighted moving average 指计算权重的方式是指数方式;

先看这四个概念中最核心的average，求average的目的在物理意义上是smoothing the data: reduce the noise in noisy time-series data.

EWMA formula

$S_t=ax_t+(1-a)S_{t-1}$

最新值给予权重 $a$;

前值给予权重 $1-a$;

指数体现在前值中。

The degree of mixing is controlled by the parameter $a$ (value between 0-1).

If a=10% (small), most of the contribution will come from the previouse value of the signal. In this case, “smoothing” will be very strong.

If a=90%(large), most of the contribution will come from the current value of the signal. In this case, “smoothing” will be minimum.

Exponential

If $n$ is a positive integer and $x$ is any real number, then $x^n$ corresponds to repeated mutiplication
$x^n=x*x*...*x$
We can call this “x raised to the power of n”;

or “x to the power of n”;

or “x to the n”;

Here , x is the base and n is the exponent or the power.

An exponential function can describe growth or decay.
$g(x)=(\frac{1}{2})^x$
is an example of exponential decay.
$f(x)=2^x$
is an example of exponential growth.

In the exponential growth of $f(x)$ , the function doubles every time you add one to its input $x$;

In the exponential decay of $g(x)$, the function shrinks in half every time you add one to its input $x$.

The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay.

指数方程用于描述增长或衰减的速度。

References