Reconciliation with Immutable Forecasts

class: center, middle, inverse, title-slide

.title[
# Reconciliation<br> with Immutable<br> Forecasts
]
.author[
### Anastasios Panagiotelis
]
.date[
### December 1, 2022
]

---

class:left

# Joint work with

Bohan Zhang <img src="img/bohan.jpeg" width="120"/>
--

Yanfei Kang <img src="img/yanfei.jpeg" width="120"/>
--

Feng Li <img src="img/feng.jpeg" width="120"/>

---
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# Reconciliation background

---

# Hierarchical Time Series

- Multivariate forecasting problem.
--

- Variables follow linear constraints.
--

- Forecast store level sales and aggregates.

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---

# Problem context

- Have .emphcol[base] forecasts at all nodes. How can we make these .emphcol[coherent]?
--

- Bottom Up (Schwarzkopf, Tersine, and Morris, 1988) ignores top nodes.
--
  
  - Noise in bottom level transferred to top level.
--
  
- Top Down (Gross and Sohl, 1990) ignores bottom nodes.
--

- Discards information.
  - Can bias even unbiased base forecasts.

---

# Reconciliation

- Base forecasts stacked in an `$n$`-vector `$\hat{\mathbf{y}}$`.
--

- .emphcol[Reconciled] forecasts `$\tilde{\mathbf{y}}$` given by
`$$\tilde{\mathbf{y}}=\mathbf{S}\left(\mathbf{S}'\mathbf{W}\mathbf{S}\right)^{-1}\mathbf{S}'\mathbf{W}\hat{\mathbf{y}}$$`
--

- Different choices of `$\pmb{W}$` give rise to different methods.
--

- The original is `$\pmb{W}=\pmb{I}$` (Hyndman, Ahmed, Athanasopoulos, and Shang, 2011).

---

# What is `$\pmb{S}$`?

.pull-left[
<div id="htmlwidget-af834fa16c50f1cc26cb" style="width:100%;height:300px;" class="widgetframe html-widget"></div>
<script type="application/json" data-for="htmlwidget-af834fa16c50f1cc26cb">{"x":{"url":"ImmutReco_files/figure-html//widgets/widget_unnamed-chunk-2.html","options":{"xdomain":"*","allowfullscreen":false,"lazyload":false}},"evals":[],"jsHooks":[]}</script>

`$$\pmb{y}=\pmb{S}\pmb{\color{orange}{b}}$$`
]

---

# What is `$\pmb{S}$`?

.pull-left[
<div id="htmlwidget-76187493613ca854e99d" style="width:100%;height:300px;" class="widgetframe html-widget"></div>
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`$$\pmb{y}=\pmb{S}\pmb{\color{orange}{b}}$$`
]
.pull-right[
`$\mathbf{S}=\begin{pmatrix}1&1&1&1\\1&1&0&0\\0&0&1&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}$`
]

---

# Why optimal? Minimises expected loss

MinT method (Wickramasuriya, Athanasopoulos, and Hyndman, 2019) minimises

`$$E\left[(\pmb{y}-\tilde{\pmb{y}})'(\pmb{y}-\tilde{\pmb{y}})\right]$$`

if `$\pmb{W}$` is set to the inverse forecast error covariance of the base forecasts.
--

- Subject to linearity and constraint that unbiasedness of base forecasts is preserved.

---

# Why optimal? Always improves loss

.pull-left[

- Reconciliation also minimises

`$$(\tilde{\pmb{y}}-\hat{\pmb{y}})'\pmb{W}(\tilde{\pmb{y}}-\hat{\pmb{y}})$$`
- Subject to `$\tilde{\pmb{y}}$` being coherent. 
- Guarantee `$(\pmb{y}-\hat{\pmb{y}})'\pmb{W}(\pmb{y}-\hat{\pmb{y}})\geq(\pmb{y}-\tilde{\pmb{y}})'\pmb{W}(\pmb{y}-\tilde{\pmb{y}})$`
]

.pull-right[![](img/geo.png)]

---

# With non negativity

- For data that cannot be negative can minimise

`$$(\tilde{\pmb{y}}-\hat{\pmb{y}})'\pmb{W}(\tilde{\pmb{y}}-\hat{\pmb{y}})$$`

- Subject to the coherence constraint.
- Subject to the constraint that `$\tilde{y}_i\ge 0$` for all `$i$`.
--

- Solved by QP using a block pivoting algorithm (Wickramasuriya, Turlach, and Hyndman, 2020).
--

- Theory does not follow through, but useful in practice.

---
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# Motivation

---

# Joint loss function

- Everything we know about why reconciliation works applies to a .emphcol[joint] loss function.
--

- Forecast accuracy for individual variables may become worse after reconciling.
--

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---

# Unbiased top down

- For 3-variable hierarchy `$y_T=y_A+y_B$`, consider

`$$\begin{pmatrix}
\hat{y}_T\\\hat{y}_A\\\hat{y}_T-\hat{y}_A\end{pmatrix}\mbox{ or }\begin{pmatrix}\hat{y}_T\\\hat{y}_T-\hat{y}_B\\\hat{y}_B\end{pmatrix}$$`

or an average of these (Hollyman, Petropoulos, and Tipping, 2021).

- Top level remains unchanged (or .emphcol[immutable]) and unbiasedness is preserved

---

# Our contribution

- Suppose we want to keep forecasts of a subset of variables .emphcol[immutable].
--
  
  - When is this feasible?
  - How should we choose immutable series?
--

- We provide a conclusive answer to the first question and some suggestions (with an empirical example) for the second.

---
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# Is it feasible?
---

# Feasbility check

- Cannot keep Region 1, Store 1A and Store 1B immutable

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---

# Basis series

- Choose any representation `$\pmb{y}=\pmb{S}\pmb{b}$`.
--

- Let `$j_1,j_2,\dots,j_m$` be the indices of a set of proposed immutable series
--

- Construct a matrix `$\pmb{S}_{\{j\}}$` using the rows of `$\pmb{S}$` corresponding to the indices `$j_1,j_2,\dots,j_m$`.
--

- If `$\pmb{S}_{\{j\}}$` has full (row) rank, then `$y_{j_1},y_{j_2},\dots,y_{j_m}$` can be kept immutable.

---

# Example

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]
.pull-right[
`$\mathbf{S}=\begin{pmatrix}1&1&1&1\\ \color{orange}1&\color{orange}1&\color{orange}0&\color{orange}0\\0&0&1&1\\\color{orange}1&\color{orange}0&\color{orange}0&\color{orange}0\\\color{orange}0&\color{orange}1&\color{orange}0&\color{orange}0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}$`
]

---

# Example

.pull-left[
<div id="htmlwidget-c030113b65e9a0caafa8" style="width:100%;height:300px;" class="widgetframe html-widget"></div>
<script type="application/json" data-for="htmlwidget-c030113b65e9a0caafa8">{"x":{"url":"ImmutReco_files/figure-html//widgets/widget_unnamed-chunk-7.html","options":{"xdomain":"*","allowfullscreen":false,"lazyload":false}},"evals":[],"jsHooks":[]}</script>

]
.pull-right[
`$\mathbf{S}=\begin{pmatrix}\color{orange}1&\color{orange}1&\color{orange}1&\color{orange}1\\ 1&1&0&0\\0&0&1&1\\\color{orange}1&\color{orange}0&\color{orange}0&\color{orange}0\\\color{orange}0&\color{orange}1&\color{orange}0&\color{orange}0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}$`
]

---

# Reconciliation

Find `$\tilde{\pmb{y}}$` to minimise

`$$(\tilde{\pmb{y}}-\hat{\pmb{y}})'\pmb{W}(\tilde{\pmb{y}}-\hat{\pmb{y}})$$`

subject to
--

- Coherence constraints
- Immutability constraints
- Non negativity constraints

---
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# Choosing immutable series

---

# Application

- Sales data from Chinese online retailer
--

- Consider "Food" sales as top level.
  - There are 40 "middle" level categories.
  - There are 1905 "bottom" level categories.
--

- Many series are intermittent.
- Promotions are very important

---

# Top level

![](img/total_sale-1.png)

---

# Setup

- Base models
--

- Intermittent Series (more than 60% zeroes) use simple exponential smoothing.
  - Regression (predictors are strength of promotion) with ARIMA errors and Box Cox transformation for other series.
--

- Immutable series
--

- Top level
  - Series with longer histories (more than 1 year)

---

# Results

<table class=" lightable-minimal" style='font-family: "Trebuchet MS", verdana, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;"> Level </th>
   <th style="text-align:right;"> Base </th>
   <th style="text-align:right;"> C </th>
   <th style="text-align:right;"> CN </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CIN </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Top (I) </td>
   <td style="text-align:right;"> 2.94 </td>
   <td style="text-align:right;"> 2.75 </td>
   <td style="text-align:right;"> 2.77 </td>
   <td style="text-align:right;"> 2.94 </td>
   <td style="text-align:right;"> 2.94 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Middle (M) </td>
   <td style="text-align:right;"> 2.66 </td>
   <td style="text-align:right;"> 2.39 </td>
   <td style="text-align:right;"> 2.40 </td>
   <td style="text-align:right;"> 2.43 </td>
   <td style="text-align:right;"> 2.47 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Bottom (M) </td>
   <td style="text-align:right;"> 2.04 </td>
   <td style="text-align:right;"> 1.86 </td>
   <td style="text-align:right;"> 1.83 </td>
   <td style="text-align:right;"> 1.97 </td>
   <td style="text-align:right;"> 1.88 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Intermittent (I) </td>
   <td style="text-align:right;"> 0.11 </td>
   <td style="text-align:right;"> 1.52 </td>
   <td style="text-align:right;"> 1.52 </td>
   <td style="text-align:right;"> 0.11 </td>
   <td style="text-align:right;"> 0.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Older (I) </td>
   <td style="text-align:right;"> 1.08 </td>
   <td style="text-align:right;"> 1.58 </td>
   <td style="text-align:right;"> 1.19 </td>
   <td style="text-align:right;"> 1.08 </td>
   <td style="text-align:right;"> 1.08 </td>
  </tr>
</tbody>
</table>

---

#Conclusions

- Imposing immutability constraints still leads to improvements over base forecasts.
--

- Immutability stabilises forecasting performance of intermittent series in particular.
--

- Imposing immutability does not lead to better accuracy in all series.
--

- Imposing non-negativity constraints generally improves performance.

---

# Open Questions

- Can we choose immutability constraints in a more principled way?
--

- Is this another version of a shrinkage v sparsity question?
--

- When is it possible reconcile in a way that yields Pareto improvements in forecast accuracy?

---

# References
.small[
Gross, C. W. et al. (1990). "Disaggregation methods to expedite product
line forecasting". In: _Journal of Forecasting_ 9.3, pp. 233-254.

Hollyman, R. et al. (2021). "Understanding Forecast Reconciliation".
En. In: _European Journal of Operational Research_ 294.1, pp. 149-160.
ISSN: 0377-2217. DOI:
[10.1016/j.ejor.2021.01.017](https://doi.org/10.1016%2Fj.ejor.2021.01.017).

Hyndman, R. J. et al. (2011). "Optimal Combination Forecasts for
Hierarchical Time Series". En. In: _Computational Statistics Data
Analysis_ 55.9, pp. 2579-2589. ISSN: 0167-9473. DOI:
[10.1016/j.csda.2011.03.006](https://doi.org/10.1016%2Fj.csda.2011.03.006).

Schwarzkopf, A. B. et al. (1988). "Top-down versus bottom-up
forecasting strategies". In: _International Journal of Production
Research_ 26 (11), pp. 1833-1843.

Wickramasuriya, S. L. et al. (2019). "Optimal Forecast Reconciliation
for Hierarchical and Grouped Time Series Through Trace Minimization".
In: _Journal of the American Statistical Association_ 114.526, pp.
804-819. ISSN: 0162-1459. DOI:
[10.1080/01621459.2018.1448825](https://doi.org/10.1080%2F01621459.2018.1448825).

Wickramasuriya, S. L. et al. (2020). "Optimal Non-Negative Forecast
Reconciliation". En. In: _Statistics and Computing_ 30.5, pp.
1167-1182. ISSN: 1573-1375. DOI:
[10.1007/s11222-020-09930-0](https://doi.org/10.1007%2Fs11222-020-09930-0).
]