# How does forecast skill score change when seasonality in the forecast quantity is removed?

Given RMSE skill score $$s$$: $$$$\label{eq:msess} s = 1-\frac{\text{RMSE}(f,x)}{\text{RMSE}(r,x)},$$$$ where $$f$$, $$r$$, and $$x$$ are forecasts of interest, reference forecasts, and observations, respectively. Suppose $$x$$ has a known multiplicative seasonal component $$c$$ (which is strictly positive), i.e., $$x = c\times k$$, and $$k$$ is the deseasonalized quantity, and assume reference forecasts are generated using the seasonally adjusted persistence model, the RMSEs can thus be written as: $$\begin{eqnarray} \text{RMSE}(f,x) &=& \sqrt{\frac{1}{N}\sum_{t=1}^{N}\left(f_{t}-x_t\right)^2} \\ \text{RMSE}(r,x) &=& \sqrt{\frac{1}{N}\sum_{t=1}^{N}\left(x_{t-1}\cdot c_t/c_{t-1}-x_t\right)^2}. \end{eqnarray}$$ When does the skill score $$s$$ equal to $$s'$$, namely, the skill score computed based on the forecasts of the deseasonalized quantity? $$$$s' = 1-\frac{U}{V},$$$$ where $$\begin{eqnarray} U &=& \sqrt{\frac{1}{N}\sum_{t=1}^{N}\left(\frac{f_t-x_t}{c_t}\right)^2},\\ V &=& \sqrt{\frac{1}{N}\sum_{t=1}^{N} \left(\frac{x_{t-1}}{c_{t-1}}-\frac{x_t}{c_t}\right)^2}. \end{eqnarray}$$

My current approach is to assume the seasonal component has $$m$$ discrete states, namely, $$c^{(1)}$$, $$\cdots$$, $$c^{(m)}$$, so that $$U^2$$ can be rewritten as: $$\begin{eqnarray} U^2 & = & \frac{1}{N}\left\{ \sum_{t\in\mathcal{N}_1}\left(\frac{f_t-x_t}{c^{(1)}}\right)^2 + \sum_{t\in\mathcal{N}_2}\left(\frac{f_t-x_t}{c^{(2)}}\right)^2 + \cdots + \sum_{t\in\mathcal{N}_m}\left(\frac{f_t-x_t}{c^{(m)}}\right)^2 \right\} \\ &=& \frac{1}{N}\left\{\frac{1}{\left(c^{(1)}\right)^2} \sum_{t\in\mathcal{N}_1}\left(f_t-x_t\right)^2 + \frac{1}{\left(c^{(2)}\right)^2}\sum_{t\in\mathcal{N}_2}\left(f_t-x_t\right)^2 + \cdots + \frac{1}{\left(c^{(m)}\right)^2}\sum_{t\in\mathcal{N}_m}\left(f_t-x_t\right)^2 \right\} \\ &=& \frac{1}{N}\left\{\frac{|\mathcal{N}_1|}{\left(c^{(1)}\right)^2} \mathbb{E}\left[(f-x)^2\big| c=c^{(1)}\right] + \frac{|\mathcal{N}_2|}{\left(c^{(2)}\right)^2} \mathbb{E}\left[(f-x)^2\big|c= c^{(2)}\right] + \cdots + \frac{|\mathcal{N}_m|}{\left(c^{(m)}\right)^2} \mathbb{E}\left[(f-x)^2\big| c=c^{(m)}\right] \right\}, \end{eqnarray}$$ where $$\mathcal{N}_1$$, $$\cdots$$, $$\mathcal{N}_m$$ are the sets of data index that satisfy events $$c_t=c^{(1)}$$, $$\cdots$$, $$c_t=c^{(m)}$$, respectively. The notation $$|\mathcal{N}_1|$$ denotes the cardinality of $$\mathcal{N}_1$$. If the expected squared error of forecast $$f$$ is same for all $$c$$, i.e., $$$$\mathbb{E}\left[(f-x)^2\big| c = c^{(1)}\right] = \mathbb{E}\left[(f-x)^2\big| c = c^{(2)}\right] =\cdots = \mathbb{E}\left[(f-x)^2\big| c = c^{(m)}\right] = \mathbb{E}\left[(f-x)^2\right] ,$$$$ then $$$$U^2 = \frac{1}{N}\left\{\frac{|\mathcal{N}_1|}{\left(c^{(1)}\right)^2} + \frac{|\mathcal{N}_2|}{\left(c^{(2)}\right)^2} + \cdots \frac{|\mathcal{N}_m|}{\left(c^{(m)}\right)^2} \right\}\mathbb{E}\left[(f-x)^2\right].$$$$ Similarly, the $$V^2$$ can be written as: $$$$\notag V^2 = \frac{1}{N}\left\{\frac{|\mathcal{N}_1|}{\left(c^{(1)}\right)^2} + \frac{|\mathcal{N}_2|}{\left(c^{(2)}\right)^2} + \cdots \frac{|\mathcal{N}_m|}{\left(c^{(m)}\right)^2} \right\}\mathbb{E}[\left(r-x\right)^2],$$$$ if the squared reference forecast error, $$(r-x)^2$$, is independent of the seasonal component. As a result, $$\begin{eqnarray} s' &= & 1-\frac{\sqrt{ \frac{1}{N}\left\{\frac{|\mathcal{N}_1|}{\left(c^{(1)}\right)^2} + \frac{|\mathcal{N}_2|}{\left(c^{(2)}\right)^2} + \cdots \frac{|\mathcal{N}_m|}{\left(c^{(m)}\right)^2} \right\}}\sqrt{\mathbb{E}\left[(f-x)^2\right]} }{\sqrt{ \frac{1}{N}\left\{\frac{|\mathcal{N}_1|}{\left(c^{(1)}\right)^2} + \frac{|\mathcal{N}_2|}{\left(c^{(2)}\right)^2} + \cdots \frac{|\mathcal{N}_m|}{\left(c^{(m)}\right)^2} \right\}} \sqrt{\mathbb{E}\left[(r-x)^2\right]}} \\ \notag &= &1-\frac{\sqrt{\mathbb{E}\left[(f-x)^2\right]} }{ \sqrt{\mathbb{E}\left[(r-x)^2\right]}} \\ &=& 1-\frac{\text{RMSE}(f,x)}{\text{RMSE}(r,x)} = s, \end{eqnarray}$$ if both the squared forecast error and squared reference forecast error are independent of the seasonal component. I'm not sure if this approach is valid.