Introduction

We define total snowfall as $s(d,y)$, known for all days $d \in D_y = \{1,2,...,n_y\}$ and years $y \in Y = \{199X, ..., 2025\}$. We are given a set of N models $\{m_1, m_2, ..., m_N\}$, for which each model has a $m_i(d,y) = \hat{s}_i(d,y)$, where $\hat{s}$ approximates total snowfall.

Naive model

We simply identify the model that has historically shown the lowest error across all days. For any model $m_i$, we can define

We can then call our naive prediction:

where

Year-agnostic performance weighted model

Instead of blindly accepting one model, we can weight the outputs of each of the models based on how trustworthy they've been in the past. The softmax function gives us a clean, tunable way to weight the impact of the errors of each of the models. We define

where $E_i = \textrm{Total Model Error}(i)$, and thus calculate

$\beta$ here is an informative parameter that goes from uniform weighting ($\beta \rightarrow 0$) to winner-take-all ($\beta\rightarrow \infty$).

Yearly performance weighted model

One issue here is that we treat all years as equal. However, years where all models performed badly should matter less for grading than years where most models were more accurate. [MORE TO COME]

Yearly correlation weighted model

Given different years may follow different patterns based on macros factors (e.g., El Nino, etc), we can preferentially weight models that performed well in the years that have been most similar to this year. [MORE TO COME]