CRPS-Learning

.title[
# CRPS-Learning
]
.subtitle[
## Jonathan Berrisch, Florian Ziel
]
.author[
### University of Duisburg-Essen
]
.date[
### 2022-06-04
]

---

# Outline

</br>
</br>

1. Multivariate CRPS Learning
  - Smoothing procedures
  - Application to multivariate electricity price forecasts
2. The `profoc` R package
  - Package overview
  - Implementation details
  - Illustrative examples

---

# Multivariate CRPS Learning

We extend most objects by an additional dimension

We utilize the mean Pinball Score over the entire space for hyperparameter optimization (e.g, `$\lambda$`)

Additionally, we extend the **B-Smooth** and **P-Smooth** procedures to the multivariate setting:

- Basis matrices for reducing 
- - the probabilistic dimension from P to l
- - the multivariate dimension from D to m
- Hat matrices
- - penalized smoothing across P and D dimensions

]

*Basis Smoothing*

Represent weights as linear combinations of bounded basis functions:

`\begin{align}
    \underbrace{w_{t,k}}_{D \text{ x } P} = \underbrace{\boldsymbol \varphi^{mult}}_{D\text{ x }m} \underbrace{\boldsymbol \beta_{t,k}'}_{m\text{ x }l} \underbrace{\boldsymbol \varphi^{prob}}_{l\text{ x }P}
\end{align}`

A popular choice: B-Splines

`$\boldsymbol \beta_{t,k}$` is calculated using a reduced regret matrix:

`$\underbrace{\boldsymbol r_{t,k}}_{\text{l x m}} = \underbrace{\boldsymbol \varphi^{\text{prob}}}_{\text{lxP}} \underbrace{\left({\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widetilde{\boldsymbol X}_{t},Y_t)- {\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widehat{\boldsymbol X}_{t},Y_t)\right)}_{\text{PxD}}\underbrace{\boldsymbol \varphi^{\text{mult}}}_{\text{Dxm}}$`

If `$l = P$` it holds that `$\boldsymbol \varphi^{prob} = \boldsymbol{I}$`  (pointwise)

For `$l = 1$` we receive constant weights

]

---

# Multivariate CRPS Learning

Penalized cubic B-Splines for smoothing weights:

Let `$\varphi=(\varphi_1,\ldots, \varphi_L)$` be bounded basis functions on `$(0,1)$` Then we approximate `$w_{t,k}$` by

`\begin{align}
w_{t,k}^{\text{smooth}} = \sum_{l=1}^L \beta_l \varphi_l = \beta'\varphi
\end{align}`

with parameter vector `$\beta$`. The latter is estimated to penalize `$L_2$`-smoothing which minimizes

`\begin{equation}
    \| w_{t,k} - \beta' \varphi  \|^2_2 + \lambda \| \mathcal{D}^{d}  (\beta' \varphi)  \|^2_2
    \label{eq_function_smooth}
\end{equation}`

with differential operator `$\mathcal{D}$`

Computation is easy since we have an analytical solution and we can precompute the hat matrix.

]

</br>
</br>

**Multivariate smoothing core**:

</br>

for( t in 1:T ) {

&nbsp; for( d in 1:D ) {

&nbsp; .grey[...]

&nbsp;&nbsp; `$\boldsymbol w_{t,k} = \mathcal{H^{mult}}  \boldsymbol \varphi^{mult} \beta'_{t,k} \varphi^{prob} \mathcal{H^{prob}}$`

&nbsp; }  .grey[\#h]

} .grey[\#t]

]

---

# Application to Multivariate Settings

Apply CRPS-Learning to (24 dimensional) day-ahead power forecasts

Forecasts from probabilistic neural networks:

Pre-Print to be released soon: Grzegorz Marcjasz, Michał Narajewski, Rafał Weron and Florian Ziel

```r
# 182 obs as burn-in
mod <- online(
  y = Y, # 736 x 24
  experts = experts, # 736, 24, 99, 2
  tau = 1:99 / 100
)
```

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</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>CRPS Values</th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">norm</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">jsu</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">comb</th>
    </tr>
  </thead>
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    <tr><td class="gt_row gt_right">1.448</td>
<td class="gt_row gt_right">1.372</td>
<td class="gt_row gt_right">1.334</td></tr>
  </tbody>
  
  
</table>
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]

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]

---

# Smoothing in Multivariate Setting

Smoothing parameters can be optimized online

Below are parameters chosen for illustrative purposes

```r
mod <- online(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  p_smooth_pr = list(
    lambda = c(20, 50, 100) # 50 selected
  ),
  p_smooth_mv = list(
    lambda = c(20, 50, 100) # 50 selected
  )
)
```

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<table class="gt_table">
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  </thead>
  <thead class="gt_col_headings">
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">norm</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">jsu</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">comb</th>
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  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1.448</td>
<td class="gt_row gt_right">1.372</td>
<td class="gt_row gt_right">1.333</td></tr>
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]

---

# The Profoc Package

#### Probabilistic Forecast Combination - profoc

Available on [Github](https://github.com/BerriJ/profoc) and [CRAN](https://CRAN.R-project.org/package=profoc)

Main Function: `online()` for online learning.
- Works with multivariate and/or probabilistic data
- Implements BOA, ML-POLY, EWA (and the gradient versions)
- Implements many extensions like smoothing, forgetting, thresholding, etc.
- Various loss functions are available 
- Various methods (`predict`, `update`, `plot`, etc.)

]

#### Speed

Large parts of profoc are implemented in C++.

We use `Rcpp`, `RcppArmadillo`, and OpenMP.

We use `Rcpp` modules to expose a class to R
- Offers great flexibility for the end-user
- Requires very little knowledge of C++ code
- High-Level interface is easy to use

]

---

# Profoc - B-Spline Basis

Basis specification `b_smooth_pr` is internally passed to `make_basis_mats()`:

```r
mod <- online(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  b_smooth_pr = list(
    knots = 9,
    mu = 0.5,
    sigma = 1,
    nonc = 0,
    tailweight = 1,
    deg = 3
  )
)
```

Knots are distributed using the generalized beta distribution.

]

Exemplary Basis with 9 Knots:

</br>

]

---

# Profoc - B-Spline Basis

Basis specification `b_smooth_pr` is internally passed to `make_basis_mats()`:

```r
mod <- online(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  b_smooth_pr = list(
    knots = 9,
    mu = 0.3, # NEW
    sigma = 1,
    nonc = 0,
    tailweight = 1,
    deg = 3
  )
)
```

Knots are distributed using the generalized beta distribution.

]

Exemplary Basis with 9 Knots ...

... and `mu` set to `0.3`:

]

---

# Profoc - B-Spline Basis

Basis specification `b_smooth_pr` is internally passed to `make_basis_mats()`:

```r
mod <- online(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  b_smooth_pr = list(
    knots = 9,
    mu = 0.5,
    sigma = 2, # NEW
    nonc = 0,
    tailweight = 1,
    deg = 3
  )
)
```

Knots are distributed using the generalized beta distribution.

]

Exemplary Basis with 9 Knots ...

... and `sigma = 2`:

]

---

# Profoc - Other Online Learning Options

**Forget Regret**

Only taking part of the old cumulative regret into account

Exponential forgetting of past regret

`\begin{align*}
    R_{t,k} & =  R_{t-1,k}(1-\xi) + \ell(\widetilde{F}_{t},Y_i) - \ell(\widehat{F}_{t,k},Y_i) \label{eq_regret_forget}
\end{align*}`

**Forget Past Performance**

Only taking part of the old performance into account

**Initialization**

Initial Weights

Initial Regret

]

**Thresholding**

`\begin{align*}
    \mathcal{F}(x;\phi)   & = \phi/K  + (1-\phi) x ,       \\
    \mathcal{S}(x;\nu)    & = \text{sign}(x)||x|-\nu| ,          \\
    \mathcal{H}(x;\kappa) & = x  \mathbb{1}\{|x|>\kappa \}
\end{align*}`
for some `$\phi\in[0,1]$`, `$\nu\geq0$` and `$\kappa\geq0$`.

**Pre Computed Elements**

Precomputed Regret (also as a share)

Precomputed Loss (also as a share)

**Parametercombinations**

Profoc considers all parameter combinations

Specific combinations can be provided by the user

]

---

# Profoc - Customizing Model Specifications

`profoc` also offers a low-level interface:

- Can be used to provide custom Basis and Hat matrices
- Can be used to manipulate all objects before learning
- `online.R` function can be used as a boilerplate.
  - It also uses the `conline` class which functions as the low-level interface

The right side shows an excerpt of `online.R`

]

```r
# Create a new instance of the conline class
conline_obj <- new(conline)

# Provide the data
model_instance$y <- y
model_instance$experts <- experts
model_instance$tau <- tau

# # Basis matrices for probabilistic smoothing
tmp <- make_basis_mats(
  x = 1:P / (P + 1),
  n = P,
  mu = 0.5,
  sigma = 1,
  nonc = 0,
  tailw = 1,
  deg = 1
)
model_instance$basis_pr <- tmp$basis
model_instance$params_basis_pr <- tmp$params

# Execute online learning
model_instance$learn()
```

]

---

# Profoc - Workflow example (1/2)

```r
# For data prep example see ?profoc::online

model <- online(
  y = matrix(y),
  experts = experts,
  tau = prob_grid,
  p_smooth_pr = list(
    lambda = c(2^-10:10)
  )
)

print(model)
```

```
##  Name        Loss     
##  E1          0.4159400
##  E2          1.2095500
##  Combination 0.3528742
```

```r
plot(model)
```
]

![](data:image/png;base64,#index_files/figure-html/unnamed-chunk-17-1.svg)

```
## Showing Plot 1/2. Press [enter] to continue
```

![](data:image/png;base64,#index_files/figure-html/unnamed-chunk-17-2.svg)

]

---

# Profoc - Workflow example (2/2)

**Predict**

```r
new_y <- matrix(rnorm(1)) # Realized
new_experts <- experts[T, , , drop = FALSE]

# Predict will expand model$predictions
model <- predict(model,
  new_experts = new_experts,
  update_model = TRUE
)
```

The predict method will never change weights.

It appends new_experts to the model if `update_model = TRUE`

Corresponding realizations can be provided using `update()`.

]

**Update**

`update()` calculates new weights based on new realizations.

```r
# Update will update the models weights etc
# if you provide new realizations
model <-
  update(model,
    new_y = new_y
  )

# Predict can also take new_experts
model <-
  update(model,
    new_y = new_y,
    new_experts = new_experts
  )
# The above updates, and predicts
# at the same time
```

]

---

# Wrap-Up

The [<svg aria-hidden="true" role="img" viewBox="0 0 496 512" style="height:1em;width:0.97em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg> profoc](https://profoc.berrisch.biz/) R Package:

Profoc is a flexible framework for online learning.

- It implements several algorithms
- It implements several loss functions
- It implements several extensions
- Its high- and low-level interfaces offer great flexibility

Profoc is fast.

- The core components are written in C++
- The core components utilize OpenMP for parallelization

]

Multivariate Extension:

- Code is available now
- Pre-Print available later this year

Get these slides:

.center[
<center>
<img src="frame.png">
</center>
[https://berrisch.biz/slides/22_06_iccf/](https://berrisch.biz/slides/22_06_iccf/)
]
]

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