# Estimate a periodic time series using least square fit

Hi I was attempting the following question and I am stuck and I don't know how to proceed. From My understanding I first substituted $$\hat{y} = Ax$$ where A is T x P selector matrix , in the minimizing equation. the will have to minimize $$\|A x - y\|^2$$ over $$x$$. which will give us $$\hat{x} = (A^{T}A)^{-1}A^Ty$$

The Problem is problem 15.10 in Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares by Stephen Boyd

Estimating a periodic time series. (See $$\{15.3 .2 .)$$ Suppose that the $$T$$ -vector $$y$$ is a measured time series, and we wish to approximate it with a $$P$$ -periodic $$T$$ -vector. For simplicity, we assume that $$T=K P$$, where $$K$$ is an integer. Let $$\hat{y}$$ be the simple least squares fit, with no regularization, $$i . e .,$$ the $$P$$ -periodic vector that minimizes $$\|\hat{y}-y\|^{2} .$$ Show that for $$i=1, \ldots, P-1,$$ we have $$\hat{y}_{i}=\frac{1}{K} \sum_{k=1}^{K} y_{i+(k-1) P}$$ ] In other words, each entry of the periodic estimate is the average of the entries of the original vector over the corresponding indices.