# Ensemble Forecasts: Probabilistic Seasonal Forecasts Based on a Model Ensemble

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Data

#### 2.2. Probabilistic Models

#### 2.2.1. A First Probabilistic Forecast Model

#### 2.2.2. Information Gain Over Climatology

#### 2.3. Improved Probabilistic Models

#### 2.3.1. Bias-Corrected Probabilistic Model

#### 2.3.2. Climatological Variance Adjusted Probabilistic Models

#### 2.3.3. Mean Adjusted Forecast RMSE Adjusted Probabilistic Models

#### 2.4. Autoregressive Models

#### 2.4.1. Autoregressive Climatology

#### 2.4.2. Combined GCM-Autoregressive Forecast Model

#### 2.4.3. Auto-Regressive Weights

- computing the climatology using EWMA (${R}_{cwe}$, ${R}_{hwe}$)
- updating the weights in the online regression using EWMA (${R}_{cew}$, ${R}_{hew}$)
- combining methods 1 and 2 (${R}_{cww}$, ${R}_{hww}$)

## 3. Results

#### 3.1. Non-Auto-Regressive Probabilistic Models

#### 3.2. Auto-Regressive Models

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Estimating the skill of models predicting the temperature for November 2004 at a grid point on the equatorial Atlantic ocean (12${}^{\circ}$ S, 356${}^{\circ}$ E). The probability density ${p}_{{h}_{0}}$ is a naive normal (Gaussian) distribution constructed using the 9 CFSv2 hindcasts $\{{g}_{1},\dots ,{g}_{9}\}$ at a single point l for the date t; the bias-corrected (mean shifted) version is ${p}_{{h}_{1}}$. The historical observations $o{({t}^{\prime},l)}_{{t}^{\prime}\in {M}_{t}}$, restricted to the same calendar month as t, at l are used to construct ${p}_{{c}_{0}}$. The bias-corrected mean of the forecasts and the standard deviation of the climatology is used to build ${p}_{{h}_{2}}$. The information gain is the difference between ${p}_{{h}_{2}}\left(o\right|t,l)$ and ${p}_{{c}_{0}}\left(o\right|t,l)$.

**Figure 2.**The deviation in the predicted ensembles is consistently lower than what is shown in both the climatology ${\sigma}_{c}$ and the error ${RMSE}_{{\widehat{\mu}}_{h}}$, as seen in the temporal averages shown in (

**b**–

**d**) and the spatial average over the non-shaded time period in (

**a**). The ensemble spreads tendency towards lower uncertainty is especially evident in the polar regions, which indicates overconfidence in the predictions for those regions; this over-confidence persists even after the forecast has been bias-corrected, as in (d). The shaded period is shown for consistency with the other graphs but is omitted from later calculations.

**Figure 3.**The predictive skill of ${c}_{0}$ is used as the baseline against which all the other models are compared because it is based solely on past observations. The spikes in IG only occur in the models that use ${\sigma}_{f}$, ${h}_{0}$ and ${h}_{1}$ so these models are removed in the lower graph to highlight that the models based on other proxies for uncertainty, ${h}_{2}$ and ${h}_{3}$ are not susceptible to these errors and show positive information gain in the later part of the time series. The shaded portion of the time series is omitted from later analysis but is shown here to demonstrate that the IG grows more positive over time.

**Figure 4.**These maps are the averages of NLL and IG for each grid point from February 1999 to January 2009. Climatology can be used to build a fairly good predictor in the tropical oceans, but falters on most landmasses as shown by the higher NLL in (

**a**). The models ${h}_{2}$ and ${h}_{3}$ which incorporate the hindcasts and the climatology mostly do better in regions such as the ocean where ${c}_{0}$ already does well, but they also show small gains on land, especially in the northern hemisphere. As seen in (

**b,c**), ${h}_{3}$ does slightly better in the ENSO region but otherwise ${h}_{3}$ and ${h}_{2}$ are mostly indistinguishable.

**Figure 5.**While all the regression based probabilistic forecast models (${R}_{cee}$ ${R}_{cwe}$ ${R}_{cew}$ ${R}_{cww}$ ${R}_{hee}$ ${R}_{hwe}$ ${R}_{hew}$ ${R}_{hww}$) have very similar skill, the combined models (${R}_{hee}$ ${R}_{hwe}$ ${R}_{hew}$ ${R}_{hww}$) are consistently more skilled, especially as the data becomes positive. Each time series is the spatial average of the global NLL at each observation time. The shaded region is not used in further analysis.

**Figure 6.**The auto-regressive model yields small improvements on land over the simpler model shown in Figure 4. Figure 6c shows that ${R}_{hee}$ is generally more skilled than ${R}_{cee}$ over land, especially in the ENSO region. While there is a lack of skill in the Arctic and Antarctic, the difference is very small. As with Figure 4, these maps are the temporal average between 1999 and 2009 as that is the period when the IG improves. (

**a**) $\mu (IG({R}_{cee},{c}_{0}))$. (

**b**) $\mu (IG({R}_{hee},{c}_{0}))$. (

**c**) $\mu (IG({R}_{hee},{R}_{cee}))$.

**Figure 7.**(

**a**) shows that, at first, climatology is weighed more heavily (positively) than the forecast coefficient (γ) in (

**d**); (

**e**) illustrates how that contribution wanes over time, replaced by the a stronger contribution from forecast, especially in the ENSO region, in (

**h**). This shift indicates that the improved skill over time shown in (h) is likely due to the added information forecasts provide. The changes in the contributions of the 2-month and 3-month lookbacks are mostly negligible, as seen in the lack of strong visible differences between (

**b,f**) and (

**c,g**).

**Table 1.**Gaussian probabilistic models were constructed using the listed parameters for the mean and standard deviation. The IG of the model is computed relative to the baseline model ${c}_{0}$ and the table reports the IG averaged over space and time between 1999 and 2009. ${h}_{2}$ and ${h}_{3}$ are the most skilled models because they have largest positive mean. Models are statistically distinguishable from each other if they differ by at least 0.01 bit at an α level of 0.05 .

IG of $\mathcal{N}(\mathbf{\mu},\mathbf{\sigma})$ Relative to ${\mathit{c}}_{\mathbf{0}}$ | ||||
---|---|---|---|---|

Model | Param. | Mean | Median | |

${c}_{0}$ | ${\mu}_{c}$ | ${\sigma}_{c}$ | - - - - | - - - - |

${h}_{0}$ | ${\mu}_{h}$ | ${\sigma}_{h}$ | –5.602 | –0.410 |

${h}_{1}$ | ${\widehat{\mu}}_{h}$ | ${\sigma}_{h}$ | –0.858 | 0.151 |

${h}_{2}$ | ${\widehat{\mu}}_{h}$ | ${\sigma}_{c}$ | 0.140 | 0.035 |

${h}_{3}$ | ${\widehat{\mu}}_{h}$ | ${RMSE}_{{\widehat{\mu}}_{h}}$ | 0.169 | 0.037 |

**Table 2.**Gaussian probabilistic models were constructed using the regression’s predicted value q as the mean. ${\sigma}_{cr}$ or ${\sigma}_{fr}$, for climatology only and forecast inclusive regressions, were used as the standard deviation. The IG of the model is computed relative to the baseline model ${c}_{0}$ and the table reports the IG averaged over space and time between 1999 and 2009. While ${R}_{hee}$ is the most skilled based on mean and median IG, the scores between the various forecast inclusive models are very similar. Models are statistically distinguishable from each other if the differ by at least 0.01 bit at an α level of 0.05.

IG $\mathcal{N}(\widehat{\mathit{q}},RMSE)$ Relative to ${\mathit{c}}_{\mathbf{0}}$ | |||||
---|---|---|---|---|---|

Model | Hindcasts | EWMA Weighted | Mean | Median | |

Climatology | Regression | ||||

${R}_{cee}$ | no | no | no | 0.112 | –0.053 |

${R}_{cwe}$ | no | yes | no | 0.095 | –0.084 |

${R}_{cew}$ | no | no | yes | 0.095 | –0.056 |

${R}_{cww}$ | no | yes | yes | 0.067 | –0.074 |

${R}_{hee}$ | yes | no | no | 0.200 | 0.005 |

${R}_{hwe}$ | yes | yes | no | 0.190 | –0.011 |

${R}_{hew}$ | yes | no | yes | 0.151 | –0.014 |

${R}_{hww}$ | yes | yes | yes | 0.137 | –0.0033 |

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**MDPI and ACS Style**

Aizenman, H.; Grossberg, M.D.; Krakauer, N.Y.; Gladkova, I.
Ensemble Forecasts: Probabilistic Seasonal Forecasts Based on a Model Ensemble. *Climate* **2016**, *4*, 19.
https://doi.org/10.3390/cli4020019

**AMA Style**

Aizenman H, Grossberg MD, Krakauer NY, Gladkova I.
Ensemble Forecasts: Probabilistic Seasonal Forecasts Based on a Model Ensemble. *Climate*. 2016; 4(2):19.
https://doi.org/10.3390/cli4020019

**Chicago/Turabian Style**

Aizenman, Hannah, Michael D. Grossberg, Nir Y. Krakauer, and Irina Gladkova.
2016. "Ensemble Forecasts: Probabilistic Seasonal Forecasts Based on a Model Ensemble" *Climate* 4, no. 2: 19.
https://doi.org/10.3390/cli4020019