The equation for Exponential Smoothing is defined as: Exponential Smoothing equation
When we increase a, the red line will be shifted to the right. How come does that happen while the red line is being restricted by given dataset? Thanks in advance.
For data given by $\{x_n\}$, the exponential smoothing output is written: $$ s_t = \alpha x_t + (1-\alpha)s_{t-1} $$ where $s_0 = x_0$ and $\alpha\in(0,1)$.
Notice that when $\alpha\rightarrow 0$, we get $s_t \rightarrow s_{t-1} = x_0$, meaning the output will be a constant that pays no attention to the data. When $\alpha\rightarrow 1$, we get $s_t\rightarrow x_t $, i.e. no smoothing is being done (we are just copying the data).
As for the shifting, notice that every output point is a weighted average of the current point $x_t$ and an average encapsulating the previous points $s_{t-1}$. This means that the more we weight $s_{t-1}$, the more we are weighting the previous time points. Thus, decreasing $\alpha$ (which increases the weight on $s_{t-1}$) will "shift the curve left", because it is forcing the output at time $t$ to better match the values at the previous time points. Increasing $\alpha$ will have the opposite effect.