# Error function of Functional Principal Component Regression is itself a curve. Why?

I have data indexed from 0 to 54. Using 6 principal components that explains 99% variation of data I was not expecting the error function to exhibit such a clear shape? By error function from following formula that is: $$y_t(x) = \hat{\mu}(x) + \sum_{k=1}^{10} \hat{\phi}_k(x) \hat{\beta}_{k,t} + \epsilon_t(x)$$

Where $$\hat{\phi}_k(x)$$ are principal component vectors and $$\hat{\beta}_{k,t}$$ are respective scores.

To my understanding, If I conduct PCA that can explain 99% of the data, then any functional curve in my data can be reconstructed using principal component vectors and scores. This is in fact not the case in my analysis and I believe it is coming from the fact that the error function itself is a curve or perhaps a principal component vector of higher K than 10? Why is this happening? Increasing K to 20 even can not get rid of this shape in the error.

With a little more clarity, you should probably redefine your model in the following format. Let $$i$$ represent the $$i$$th object $$(i=1,2,\ldots,n)$$ and $$j$$ represent the $$j$$th $$(j=1,2,\ldots,m)$$ principal component (PC) extracted from the correlation or covariance matrix, and let $$\mathbf{F}$$ be the $$n \times m$$ matrix of PC scores for the $$m$$ components.
$$y_i = \beta_0 + \sum_{j=1}^m \beta_j f_{ij} + e_i$$
where $$\beta_j$$ is the regression coefficient for the $$j$$th PC and $$f_{ij}$$ is the $$i$$th PC score from the $$j$$th PC.
In regression terminology, your $$\phi_k(x)$$ is $$\beta_j$$ and your $$\beta_{k,t}$$ is $$f_{ij}$$.