Weighted moving average derived from parabola approximation

Let us have five points at $$x = -2,-1,0,1,2$$ with ordinates equal to $$y_i$$, I want to derive the formula for $$a_0$$, such that the parabola $$y = a_0 + a_1x + a_2x^2$$ fits the points the best in ordinary least square sense.

$$(y_{-2}-a_0 +2a_1-4a_2)^2 + (y_{-1} - a_0 + a_1 -a_2)^2 + (y_{0} -a_0)^2 + (y_{1} -a_0 - a_1 - a_2)^2 + (y_{2} - a_0 - 2a_2 - 4a_2)^2 \to min \ \text{w.r.t. } a_0, a_1 \text{and } a_2.$$

I know the answer $$a_0 = \frac{1}{35} [ -3 y_{-2} + 12 y_{-1} + 17 y_0 + 12 y_1 -3 y_2]$$. It was given on a lecture and the proof was left as an exercise.

Background of the question is time series weighted moving average smoothing. We substitute every $$y$$ with $$a_0$$ to get a smoother time series.

However I don't understand how we can derive it not taking three derivatives, getting 3 linear equations and solving them. But looks like there is a simpler proof for $$a_0$$ particularly that I don't see. If we denote every summand being squared as $$S_i$$ then we get a system of linear equations.

$$\begin{cases} S_{-2} + S_{-1} + S_0 + S_1 + S_2 = 0 \\ 2S_{-2}+S_{-1} - S_1 - 2S_2 = 0 \\ 4S_{-2}+S_{-1}+S_{1}+4S_2 = 0 \end{cases}$$

But is there a faster proof? The original slide is given below for reference I don't know about faster, but I managed to simplify the task by exploiting a few symmetries. Very likely I am missing something major, for this is not really my cup of coffee.

We are looking for a weighted average, so it is clear that the solution must have the form $$\hat{a}_0=L(y_{-2},\ldots,y_2)=\sum_{i=-2}^2q_iy_i,$$ where the constants $$q_i$$ are the weights. Clearly we must have $$\sum_{i=-2}^2q_i=1\tag{1}$$ for this to be an average.

• The weighted average is immune to a change of direction of time. In other words, for all inputs $$(y_i)_{-2\le i\le 2}$$, we must have $$L(y_2,y_1,y_0,y_{-1},y_{-2})=L(y_{-2},y_{-1},y_0,y_1,y_2).$$ For this to be possible we must have the symmetries $$q_{-1}=q_1$$ as well as $$q_{-2}=q_2$$.
• If we add a quadratic term to the input, this will not affect the value of $$\hat{a}_0$$, for such a change will be absorbed by $$\hat{a}_2$$. In other words, we must have $$L(4,1,0,1,4)=0$$ as that input matches perfectly with $$y_i=i^2$$. Together with the symmetries this gives us the equation $$q_1+4q_2=0.\tag{2}$$

So far so good. We have three unknowns, $$q_0,q_1,q_2$$ and two equations, $$(1)$$ and $$(2)$$. To solve for them we need something extra. Here I'm sad to say I had to brute force it. Consider a pulse input $$(y_i)_{-2\le i\le 2}=(0,0,1,0,0)$$. That is $$y_0=1$$ and the rest all zero. Again by symmetry, the best quadratic approximation has no linear term, so we test this against $$y_i=a_0+a_2i^2$$. The optimal $$a_0$$ will then be $$L(0,0,1,0,0)=q_0$$ allowing us to solve the problem.

The sum of squared errors will then be $$E(a_0,a_2)=(1-a_0)^2+2(a_2+a_0)^2+2(4a_2+a_0)^2=1-2a_0+5a_0^2+20a_0a_2+34a_2^2.$$ Solving the system $$\partial E/\partial a_0=0, \partial E/\partial a_2=0$$ is easy, and yields $$a_0=17/35, a_2=-1/7.\tag{3}$$ As the only critical point that must yield the optimum.

So we can conclude that $$q_0=17/35$$, whence $$(1)$$ together with the symmetries gives $$q_1+q_2=9/35\tag{4}.$$ Combining this with $$(2)$$ leads to the given solution.

• Knowing from elsewhere on the site that the OP is also studying algebra I could not resist checking how far symmetries alone will take us. Unfortunately didn't quite take us the distance, but I was too far gone to stop :-( Anyway, the estimate is linear, so it is natural to consider a single pulse as the input. – Jyrki Lahtonen Sep 1 at 18:01
• Updated my profile info, I appreciate your trait, giving the answer considering the asker's background. – Lada Dudnikova Sep 3 at 8:52