-
Notifications
You must be signed in to change notification settings - Fork 5
/
presentation.tex
1809 lines (1541 loc) · 68.1 KB
/
presentation.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
\documentclass{beamer}
\usepackage{multirow}
\usepackage{pgfpages}
%\setbeameroption{show notes}
\setbeameroption{show notes on second screen=right}
\mode<presentation> {
\usetheme{Warsaw}
\setbeamercovered{transparent}
}
\usepackage{graphicx}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{colortbl}
\usepackage{tikz}
\usepackage{pbox}
\usepackage{tcolorbox}
%\pgfdeclareimage[height=0.5cm]{le-logo}{logo-irisa}
%\logo{\pgfuseimage{le-logo}}
\setbeamertemplate{headline}{}
\setbeamertemplate{footline}[]
\AtBeginSection[]{
\begin{frame}
\vfill
\centering
\begin{beamercolorbox}[sep=8pt,center,shadow=true,rounded=true]{title}
\usebeamerfont{title}\insertsectionhead\par%
\end{beamercolorbox}
\vfill
\end{frame}
}
\setbeamertemplate{footline}[frame number]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[]
{Scalable Models of Probabilistic Forecasting with Fuzzy Time Series}
%\subtitle {}
\author[]
{Petr\^onio C\^andido de Lima e Silva\\\vspace{0.5cm} \\{\small Advisor: Frederico Gadelha Guimar\~aes} \\{\small Co-Advisor: Hossein Javedai Sadaei}}
\institute[]
{
Machine Intelligence and Data Science (MINDS) Lab\\
Graduate Program in Electrical Engineering - PPGEE\\
Federal University of Minas Gerais - UFMG\\
Belo Horizonte - MG - Brazil
}
\date[Belo Horizonte, 2019]
\begin{document}
\input{definitions.tex}
\beamertemplatenavigationsymbolsempty
\linespread{1.0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
\note[itemize]{
\item Good afternoon guys, it is a pleasure be here today. Thank you for attending this presentation.
\item In particular I would like to thank the board of referees and also would ask you my apologies for the inconveniences that happened
\item My apologies for my voice, I just had a flu past week and my voice did not recovered yet.
\item Well, Let's start!
\item Today I will present the developments of my research titled "..." highlighting the most important achievements
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Schedule}
\tableofcontents
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Introduction - Motivation}
\linespread{2}
How to represent the forecasting uncertainty in point forecasts?
\begin{itemize}
\item Uncertainties:
\begin{itemize}
\item Intrinsic / Ontological;
\item Extrinsic / Empirical;
\end{itemize}
\item Interval and Probabilistic Forecasting
\end{itemize}
\end{frame}
\note[itemize]{
\item Time Series Forecasting is one of the most important activities of the human kind
\item Forecasting is spread over several knowledge fields, engineering, economy, etc.
\item However, despite its importance, not so many people take in consideration the uncertainty inherent to the forecasting
\item There are several kinds of uncertainties, but we will focus in the tow most important
\item Intrinsic is proper to the stochastic processes, its non-determinism, randomness, etc.
\item In the other hand, the Extrinsic uncertainty originates outside the process, is caused by instrumentation, measurement, data storage, etc
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Introduction - Uncertainty}
\linespread{2}
Why it is important to represent the uncertainty?
\begin{itemize}
\item Decision Making
\item Risk Management
\item Transparency
\item Robustness
\item Reliability
\end{itemize}
\end{frame}
\note[itemize]{
\item To know the uncertainty of the forecasts is essential for Risk Management
\item
\item It is essential to create applications with
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Introduction - Uncertainty}
\linespread{2}
Modern Applications of Probabilistic Forecasting
\begin{itemize}
\item Dynamic and Robust Optimization
\item Intelligent Management
\item Investment Portfolio
\item Smart Grids and Environments
\end{itemize}
\end{frame}
\note[itemize]{
\item These are some examples of modern applications of probabilistic
\item all these applications needs of forecasting tools and also to know the uncertainty of these forecasts to improve the decision making, turning it more clever, more secure, robust
\item the key idea is: insted of to react to the uncertainty, be prepared for it
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Introduction}
\linespread{2}
Some Application Demands:
\begin{itemize}
\item Big Data scalability;
\item Computational performance;
\item Auditability
\item Explainability
\end{itemize}
\end{frame}
\note[itemize]{
\item One of the key points of the Big Data is velocity, therefore low cost computational methods are desired
\item European Union
\item New regulatory barriers are being imposed, especially on European Union [hics guidelines for trustworthy AI. Technical report, European Commissionon High-Level Expert Group on Artificial Intelligence (AI HLEG), 2019]
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Introduction}
\linespread{2}
Why Fuzzy Time Series?
\begin{itemize}
\item Accuracy;
\item Low computational cost;
\item Explainability;
\item Data Driven / Non Parametric;
\item Simplicity;
\item Flexibility;
\end{itemize}
\end{frame}
\note[itemize]{
\item FTS put together several features desirable for such kind of application
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Main Objectives}
\linespread{1.5}
\begin{enumerate}
\item[1] Review FTS methods and probabilistic forecasting and identify extension opportunities on known methods;
\item[2] Propose interval and probabilistic forecasting approaches based on the Fuzzy Time Series methods;
\item[3] Improve the scalability of the proposed FTS methods in order to enable it to deal with big time series
\item[4] Propose a method for Hyperparameter Optimization - DEHO;
\item[5] Extend the proposed methods to enable multivariate time series;
\end{enumerate}
\end{frame}
\note[itemize]{
\item The objectives sought during this research
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fuzzy Time Series}
\note[itemize]{
\item We start by defining what are the Fuzzy Time Series
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Definitions and notations}
\begin{table}[]
\centering
\begin{tabular}{|c|m{7cm}|} \hline
$Y \in \mathbb{R}^1$ & the univariate time series \\ \hline
$y(t) \in Y$ & an individual instance of $Y$ \\ \hline
$T \in \mathbb{R}^1$ & the total length of $Y$ \\ \hline
$t \in T$ & the time index \\ \hline
$U$ & the Universe of Discourse of $Y$ \\ \hline
$\ulvar$ & the linguistic variable, the group of fuzzy sets created for $Y$ \\ \hline
$\ufset \in \ulvar$ & the individual fuzzy sets in $\lvar$ \\ \hline
$F$ & the fuzzyfied time series \\ \hline
$f(t) \in F$ & an individual instance of $F$ \\ \hline
\end{tabular}
\end{table}
\end{frame}
\note[itemize]{
\item This symbols are used hereinafter to define some common time series and FTS concepts
\item CAPITAL letters usually refer to sets
\item LOWER letters usually refer to instances
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series}
\includegraphics[width=\textwidth,height=8cm]{figures/fts_simplified.pdf}
\end{frame}
\note[itemize]{
\item The general FTS approach is simple and it is based on these main stages
\item The first FTS method were published in 1993 by Song and Chissom and used matrices to represent the temporal transitions between the fuzzy sets
\item Chen published in 1996 an improvement of the Song and Chissom method, which is based on rules - the FLRG
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series - Training Procedure}
\includegraphics[width=\textwidth,height=8cm]{figures/fts_training.pdf}
\end{frame}
\note[itemize]{
\item Over the years several other improvements were proposed, extending FTS methods to a more complex framework
\item This figure represents a consensus method based on the most important contributions of the last years.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{HOFTS and WHOFTS Model Parameters}
\scriptsize
\begin{center}
$LHS \rightarrow RHS$
IF $F(t)$ is $A_i$ THEN $F(t+1)$ WILL BE $A_j$ OR $A_k$ ...
\end{center}
\begin{columns}
\column{0.5\textwidth}
\begin{block}{Example FLRG set ($k=6$, $\Omega=1$)}
$$
\begin{array}{rcl}
A_1 & \rightarrow & A_1 \\
A_2 & \rightarrow & A_2,\ A_3 \\
A_3 & \rightarrow & A_3,\ A_4 \\
A_4 & \rightarrow & A_3,\ A_4,\\
& & \ A_5 \\
A_5 & \rightarrow & A_6 \\
A_6 & \rightarrow & A_5,\ A_6 \\
\end{array}
$$
\end{block}
\begin{block}{HOFTS Defuzzyfication}
$$
\begin{array}{rcl}
mp_j & = & \displaystyle |RHS|^{-1}\sum_{i \in RHS} w_i \cdot c_i \\
\\
\hat{y}(t+1) & = & \displaystyle \frac{\sum_{j \in K} \mu_j\cdot mp_j}{\sum_{j \in K} \mu_j}
\end{array}
$$
\end{block}
\column{0.5\textwidth}
\begin{block}{Example Weighted FLRG Set ($k=6$, $\Omega=1$)}
$$
\begin{array}{rcl}
A_1 & \rightarrow & (1.0)\; A_1 \\
A_2 & \rightarrow & (0.667)\; A_2,\ (0.333)\; A_3 \\
A_3 & \rightarrow & (0.75)\; A_3,\ (0.25)\; A_4 \\
A_4 & \rightarrow & (0.167)\; A_3,\ (0.667)\; A_4,\\
& & \ (0.167)\; A_5 \\
A_5 & \rightarrow & (1.0)\; A_6 \\
A_6 & \rightarrow & (0.333)\; A_5,\ (0.667)\; A_6 \\
\end{array}
$$
\end{block}
\begin{block}{WHOFTS Defuzzyfication}
$$
\begin{array}{rcl}
mp_j & = & \sum_{i \in RHS} w_i \cdot c_i \\
\\
\displaystyle \hat{y}(t+1) & = & \displaystyle \frac{\sum_{j \in K} \mu_j \cdot mp_j}{\sum_{j \in K} \mu_j}
\end{array}
$$
\end{block}
\end{columns}
\end{frame}
\note[itemize]{
\item The rule-based knowledge representation is easy to read and explain
\item Each rule is composed by a precedent (LHS), and the consequent (RHS) and can be interpreted as...
\item These figures show the difference between HOFTS and WHOFTS rules and their defuzzyfication processes.
\item Each weight is the normalized frequency of each fuzzy set in the consequent
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series - Forecasting Procedure}
\includegraphics[width=\textwidth,height=8cm]{figures/fts_forecasting.pdf}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series Models}
\includegraphics[width=\textwidth,height=8cm]{figures/fts_taxonomy.pdf}
\end{frame}
\note[itemize]{
\item CUSTOMIZABLE
\item In this figure we can see a little sample of the FTS flexibility
\item There is a wide variety of components for FTS
\item The most important parameters are the Time Behavior, Order and partitioning scheme
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series Models - Scope}
\begin{table}[]
\centering
\begin{tabular}{|c|c|} \hline
\textbf{Time Behavior} & Time Invariant \\ \hline
\textbf{Partitioning Scheme} & Grid \\ \hline
\textbf{Order} & High Order \\ \hline
\textbf{Fuzzyfication} & Holistic \\ \hline
\textbf{Knowledge Mode} & Rule Based \\ \hline
\textbf{Fuzzy Sets} & Conventional \\ \hline
\textbf{Transformations} & None \\ \hline
\end{tabular}
\caption{FTS scopes of this research}
%\label{tab:my_label}
\end{table}
\begin{itemize}
\item Consensus methods:
\begin{enumerate}
\item High Order Fuzzy Time Series (HOFTS)
\item Weighted High Order Fuzzy Time Series (WHOFTS)
\end{enumerate}
\end{itemize}
\end{frame}
\note[itemize]{
\linespread{2}
\item With such variety of components we had to impose limits in our research scope, or the research would be endless
\item This table show some imposed constraints in the FTS components that make the research scope more strict
\item Instead of reference several methods, each one with few improvements we propose the consensus methods to simplify the method comparations
\item The consensus methods were proposed as an aggregation of the most important contributions for the FTS in the last years
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{HOFTS and WHOFTS Hyperparameters}
\small
\begin{table}[]
\centering
\begin{tabular}{|c|l|c|} \hline
\textbf{Alias} & \textbf{Parameter} & \textbf{Type} \\ \hline
$k$ & Number of partitions & $\mathbb{N}^+$ \\ \hline
$\Omega$ & Order & $\mathbb{N}^+ \geq 1$ \\ \hline
$\Pi$ & Partitioning scheme & $\Pi: Y \rightarrow \ulvar$ \\ \hline
$L$ & Lag indexes & $\mathbb{N}^+$ \\ \hline
$\mu$ & Membership function & $\mu: U \rightarrow [0,1]$ \\\hline
$\alpha$ & $\alpha$-cut & $[0,1]$ \\ \hline
\end{tabular}
\caption{HOFTS and WHOFTS hyperparameters}
\label{tab:hyperparameters}
\end{table}
\end{frame}
\note[itemize]{
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Time Series}
\linespread{2}
\begin{itemize}
\item How to represent the uncertainty of FTS forecasts?
\begin{itemize}
\item No interval or probabilistic methods!
\end{itemize}
\item How the uncertainty grows as the forecasting horizon increases?
\end{itemize}
\end{frame}
\note[itemize]{
\item Our experiments showed that FTS has accuracy comparable with the most known forecasting methods.
\item The main drawback of the FTS until this work is that they just forecast points;
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interval Forecasting}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Interval Forecasting}
\begin{itemize}
\item Prediction Intervals - \cite{Chatfield1993}
$$
\interval \qquad P(\underline{l} \leq y(t+1) \leq \overline{u}) = 1 - \alpha \qquad \alpha \in (0,1)
$$
\item Mean-Variance Models - \cite{Chatfield2001}
\begin{equation}
\begin{array}{rcl}
\intvl & = & [\underline{\mu - z_{\alpha/2}\sigma_\epsilon}\ ,\ \overline{\mu + z_{\alpha/2}\sigma_\epsilon}] \\
& & \\
\mu &=& \mathbb{E}[Y_{t+1}|Y_t,Y_{t-1},...] \\
\sigma_\epsilon & = & \sqrt{VAR[\epsilon]} \qquad
\epsilon \sim \mathcal{N}(0,1) \\
z_{\alpha/2} & = & \Phi((1- \alpha)/2)
\end{array}
\end{equation}
\end{itemize}
\end{frame}
\note[itemize]{
\item Prediction intervals are composed by a lower and upper bounds
\item the PI is expected to contain the future value with a confidence level alpha
\item Mean variance depends on the residual normality, z is the standard score $\ = \frac{x - \mu}{\sigma}$ function
\item $\Phi$ is the cumulative standard normal distribution
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Interval Forecasting}
\linespread{1.5}
\begin{itemize}
\item Inter-quantile Interval
$$
\begin{array}{rcl}
\intvl & = & [\underline{Q(\alpha/2)}, \overline{Q(1-\alpha/2)}] \\
Q(\tau) & = & \arg\min_x\{ x \in U \ |\ \tau < F(x) \} \\
\tau & \in & (0,1)
\end{array}
$$
\item Quantile Auto-Regression - QAR(p)
\begin{itemize}
\item \cite{Koenker2006}, \cite{Takeuchi2006}, \cite{Hansen2006}
\end{itemize}
\end{itemize}
$$
\begin{array}{rcl}
Q_{y(t)}(\tau | y(t-1),\ldots) & = & \min_\theta \sum_{i=1}^n \rho_\tau (y(t) - y(i)\theta \\
\rho_\tau(u) & = & u(\tau - \mathbf{1}(u < 0))
\end{array}
$$
\end{frame}
\note[itemize]{
\item The interquantile interval can be defined with a confidence level $\alpha$
\item The quantile function uses the cumulative distribution function to find the value that best represent the quantile
\item When CDF is not available it is necessary to minimize the Pinball Loss Function to find the quantile
\item QAR is an autoregressive model that fits parameters that infer the best value for a given quantile $\tau$
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Interval Forecasting - Many Steps Ahead}
\linespread{2}
\begin{itemize}
\item $H \in \mathbb{N}^+$: the forecasting horizon
\item How uncertainty grows as $H \rightarrow \infty$ ?
\item Mean-Variance
\begin{itemize}
\item $\sigma_\epsilon^h = (1 + h\beta)\sigma_\epsilon^1$, for some smoothing value $\beta \in (0,1)$
\end{itemize}
\item Quantile Auto-Regression - QAR(p)
\begin{itemize}
\item A particular model must be fitted for each $h \in H$
\end{itemize}
\end{itemize}
\end{frame}
\note[itemize]{
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Evaluating Interval Forecasts}
\linespread{2}
\begin{itemize}
\item Coverage, Sharpness, Resolution, Pinball Score
\item Winkler Score
$$
WS(\alpha, y(t), \intvl(t)) = \left\{ \begin{array}{ccl}
\delta & if & \underline{l} \leq y \leq \overline{u} \\
\delta + 2(\underline{l} - y)/\alpha & if & y < \underline{l} \\
\delta + 2(y - \overline{u})/\alpha & if & \overline{u} < y
\end{array} \right.
$$
\end{itemize}
\end{frame}
\note[itemize]{
\item Intervals can be evaluated using several perspectives, as its hit rate (cov), average range (shp), range variation (res) or proximity to an especific quantile (ps)
\item Winkler Score try to condensate all these measures
\item In the equation is possible to see all these components
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Interval Fuzzy Time Series - $[\mathbb{I}]$FTS}
\linespread{2}
\begin{itemize}
\item Represents the total fuzzy uncertainty contained in the consequent of the fuzzy rules
\item The interval contains minimum and maximum possible values;
\item Can be adapted to work with all rule-based FTS methods;
\item Only changes the Forecasting Procedure;
\end{itemize}
\end{frame}
\note[itemize]{
\item To initially bring the FTS to the interval forecasting field we propose the IFTS method
\item The FTS methods use FLRG to forecast and each FLRG can have many RHS sets, the real value can be at any point inside that fuzzy sets
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$[\mathbb{I}]$FTS Deffuzification procedure}
\linespread{1.5}
\begin{itemize}
\item Each chosen FLRG will generate an interval
\item Each interval is composed by the RHS fuzzy set bounds
$$
\begin{array}{lcr}
\mathbb{I}^i_{min} & = & \min( \underline{A_1}, ..., \underline{A_k} ) \\
& & \\
\mathbb{I}^i_{max} & = & \max( \overline{A_1}, ..., \overline{A_k} ) \\
& & A_1, ..., A_k \in RHS
\end{array}
$$
\item The final forecast is the weighted sum of the intervals by the membership degree of the LHS
$$
\mathbb{I}(t+1) = \frac{\sum_{j \in A} \mu_i \mathbb{I}^j}{\sum_{j \in A} \mu_j} = \frac{\sum_{j \in A} [\mu_j\underline{\mathbb{I}^j_{min}} , \mu_j\overline{\mathbb{I}^j_{max}}] }{\sum_{j \in A} \mu_j}
\label{eqn:ifts}
$$
\end{itemize}
\end{frame}
\note[itemize]{
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Weighted $[\mathbb{I}]$FTS}
\begin{itemize}
\item The bounds of the intervals are the weighted sum of the bounds of the RHS fuzzy sets
$$
\begin{array}{ccc}
\intvl_{min} & = & \sum_{j \in RHS} w_j \cdot \underline{\ufset} \\
& & \\
\intvl_{max} & = & \sum_{j \in RHS} w_j \cdot \overline{\ufset}
\end{array}
\label{eqn:weighted_iminimax}
$$
\item Final forecast
$$
\mathbb{I}(t+1) = \frac{\sum_{j \in A} \mu_i \mathbb{I}^j}{\sum_{j \in A} \mu_j} = \frac{\sum_{j \in A} [\mu_j\underline{\mathbb{I}^j_{min}} , \mu_j\overline{\mathbb{I}^j_{max}}] }{\sum_{j \in A} \mu_j}
\label{eqn:ifts}
$$
\end{itemize}
\end{frame}
\note[itemize]{
\item The step 4 mix all intervals of the chosen FLRG's weighted by their membership values
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{PWFTS Forecasting Sample}
\includegraphics[width=\textwidth,height=5cm]{figures/ifts_sample_onestep.png}
\end{frame}
\note[itemize]{
\item This figure compares the intervals generated by IFTS and WIFTS methods for one step ahead forecasting
\item WIFTS is sharper but with less coverage
\item IFTS is less sharp but with high coverage
\item the length of the intervals is directly impacted by the number of fuzzy sets
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\ifts$ Many Steps Ahead}
\includegraphics[width=\textwidth]{figures/ifts_many_steps.pdf}
\end{frame}
\note[itemize]{
\item This diagram show how the many steps ahead intervals are calculated
\item The approach always consider the extremum values, or the worst cases scenarios - if you consider that the best case is when real value fall in the middle of the interval
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\ifts$ Many Steps Ahead}
\includegraphics[width=\textwidth,height=6cm]{figures/ifts_sample_manystep.png}
\end{frame}
\note[itemize]{
\item This figure shows a sample of the 10 steps ahead interval forecasting
\item This figure reveals how the uncertainty of IFTS propagates over the forecasting horizon
\item The main drawback is that the intervals often reach the bounds of the UoD
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Probabilistic Forecasting}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Probabilistic Forecasting}
\linespread{1.5}
\begin{itemize}
\item PDF/CDF over the Universe of Discourse
$$
P: U \rightarrow (0,1) \qquad \sum_{x \in U} P(x) = 1
$$
\begin{itemize}
\item $P(y(t+1)|Y)$: the probability of an $y(t+1) \in U$ given $Y$
\end{itemize}
\item Mean-Variance and QAR methods can generate $P(y(t+1)|Y)$ by stacking intervals $\intvl(t+1)$ with different values of $\alpha$
\end{itemize}
\end{frame}
\note[itemize]{
\item Interval forecasting provide a partial view of the uncertainty, its upper and lower bounds
\item Probabilistic forecasting give us a more detailed view of the uncertainty - a probability for each point of the UoD
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Probabilistic Forecasting}
\begin{itemize}
\item Bayesian Structural Time Series (BSTS) \cite{Scott2014}
$$
\begin{array}{rlc}
y(t) = & Z_ts_t + \epsilon_t & \epsilon_t \sim \mathcal{N}(0, H_t) \\
s_t = & T_t s_{t-1} + R_t\eta_t & \eta_t \sim \mathcal{N}(0, Q_t)
\end{array}
$$
$$
P(y(t+1)|\Theta,Y) = \int_U \int_\Theta P(y|\theta,Y)P(Y|\theta)P(\theta) dy d\theta
$$
\item Each parameter $\theta \in \Theta$, $\Theta= \{Z_t, T_t, R_t\}$, is treated as a probability distribution $P(\theta|Y)$
\item Markov Chain Monte Carlo methods - MCMC - \cite{Hastings1970}
\begin{itemize}
\item Estimation of parameters $P(\theta|Y)$, $\forall \theta \in \Theta$
\item Inference of $P(y(t+1)|\Theta,Y)$
\end{itemize}
\end{itemize}
\end{frame}
\note[itemize]{
\item Baysian Statiscs is gaining more attention in recent years due its ability to represent uncertainty in models parameters
\item BSTS can be used to describe the great majority of the statistical time series forecasting models, as ARIMA, Wolt Winters, etc.
\item However it has an high cost for estimation and inference
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Probabilistic Forecasting}
\linespread{2}
\begin{itemize}
\item k-Nearest Neighbors (kNN) with Kernel Density Estimation (KDE)
$$
\begin{array}{}
P(x) = (nh)^{-1} \sum_{i \in Y} K\left(\frac{x - i}{h}\right) \\
\int_{-\infty}^{+\infty}K(u) du = 1
\end{array}
$$
\item Ensemble Forecasts - \cite{Gneiting2008}
\begin{itemize}
\item Same method with different parameters
\end{itemize}
\item Ensemble Learning - \cite{Mohammed2016}
\begin{itemize}
\item Distinct methods, parameters and even data
\end{itemize}
\end{itemize}
\end{frame}
\note[itemize]{
\item \cite{Krzysztofowicz2001}: should quantify the total uncertainty that remains about the predictand, conditional on all information utilized on the forecasting process
\item \cite{Gneiting2008}: an ensemble prediction system consists of multiple runs of numerical weather prediction models, which differ in the initial conditions
\item \cite{Leutbecher2008}: The ultimate goal of ensemble forecasting is to predict quantitatively the probability density of the state of the atmosphere at a future time
\item \cite{Smith2001}: In practice, ensemble forecasting is a Monte Carlo approach to estimating the probability density function (PDF) of future model states given uncertain initial conditions
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Evaluating Probabilistic Forecasts}
Continuous Ranked Probability Score\cite{Gneitinga}
$$
CRPS(F,x) = \int_{-\infty}^{+\infty} (F(y) - \mathbf{1}\{y \geq x\})^2 dy
$$
$$
CRPS(F,x) = \frac{1}{N} \sum_{t=1}^{N} \int_{-\infty}^{+\infty} (F_t(y) - \mathbf{1}\{y \geq x_t\})^2 dy
$$
\end{frame}
\note[itemize]{
\item Generalizes the Mean Average Error for probabilistic forecasts
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ensemble FTS}
\linespread{2}
\begin{itemize}
\item Objective:
\begin{itemize}
\item Represent the uncertainty of the FTS parameters ($k$, $\Omega$)
\end{itemize}
\item Set of FTS models with different parameters
\item KDE is used to smooth the set of forecasts into a probability distribution
\end{itemize}
\end{frame}
\note[itemize]{
\item Our second proposed model was the first attempt in the FTS literature to perform probabilistic forecasting
\item it is based on an homogeneous ensemble of FTS methods
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ensemble FTS - Training Procedure}
\includegraphics[width=\textwidth,height=8cm]{figures/ensemblefts_training.pdf}
\end{frame}
\note[itemize]{
\item This diagram shows the traning of the EnsembleFTS
\item The main parameters are the range of Orders and partitions
\item For each combination of orders and partitions an FTS model will be trained and stored in the ensemble
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ensemble FTS - Forecasting Procedure}
\includegraphics[width=\textwidth,height=8cm]{figures/ensemblefts_forecasting.pdf}
\end{frame}
\note[itemize]{
\item This diagram shows the forecasting process
\item The input sample is passed to each ensemble model generating the set of crisp forecasts
\item The selection remove extreme values, considering just an alpha interquantile interval
\item Then the values are smoothed in a probability distribution using a KDE
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ensemble FTS - Sample}
\includegraphics[width=\textwidth,height=8cm]{figures/ensemblefts_sample_onestep.png}
\end{frame}
\note[itemize]{
\item This is a sample for one step ahead probabilistic forecasting
\item The intervals are generated from the PDF using alpha levels
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ensemble FTS - Sample}
\includegraphics[width=\textwidth,height=8cm]{figures/ensemblefts_sample_manystep.png}
\end{frame}
\note[itemize]{
\item This is a sample for many steps ahead probabilistic forecasting
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Limitations of the proposed methods}
\linespread{2}
\begin{table}[]
\begin{tabular}{|c|p{4.5cm}|p{4.5cm}|} \hline
& \begin{center}
$\ifts$
\end{center}& \begin{center}
EnsembleFTS
\end{center} \\ \hline
Pros & \begin{tabular}{p{4cm}}
$\bullet$ Fuzzy Uncertainty \\
$\bullet$ Flexibility
\end{tabular}
&
\begin{tabular}{p{4cm}}
$\bullet$ Interval Forecasting \\
$\bullet$ Probabilistic Forecasting
\end{tabular}
\\ \hline
Cons & \begin{tabular}{p{4cm}}
$\bullet$ No confidence level \\
$\bullet$ No probabilistic forecasting
\end{tabular}
&
\begin{tabular}{p{4cm}}
$\bullet$ High computational cost \\
$\bullet$ Not parsimonious
\end{tabular} \\ \hline
\end{tabular}
\end{table}
\end{frame}
\note[itemize]{
\item The computational experiments showed that IFTS and EnsembleFTS has the same forecasting skills than the standard Mean/Variance, QAR and BSTS methods
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Probabilistic Weighted Fuzzy Time Series - PWFTS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Probabilistic Weighted Fuzzy Time Series - PWFTS}
\linespread{2}
\begin{itemize}
\item Integrated method for point, interval and probabilistic forecasting
\item New rule based model - Fuzzy Temporal Pattern Group
\item Represent both the extrinsic uncertainty (fuzzy sets) and ontologic uncertainty (empirical probabilities)
\end{itemize}
\end{frame}
\note[itemize]{
\item PWFTS is the main achievement of this research and I will give more attention on it
\item Aims to be the all-in-one method
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Empirical Probabilities}
\includegraphics[width=\textwidth]{figures/pwfts_fuzzyfrequency.pdf}
$$
\begin{array}{ccc}
P(\ufset) = \frac{\sum_{y \in U} \mu_{\ufset}(y)}{\sum_{\ufset \in \ulvar}Z_{\ufset}} & & Z_{\ufset} = \sum_{y \in U} \mu_{\ufset}(y)
\end{array}
$$
\end{frame}
\note[itemize]{
\item Traditional probabilities are measured over crisp events, using integer counting
\item Fuzzy events do not have integer counts, but partial counts - its membership grades
\item Fuzzy frequencies are the membership degrees of the crisp values
\item the probability of a fuzzy set is the normalized sum of its fuzzy occurrences - or the sum of its memberships
\item $Z_{\ufset}$ is called Partition Function
\item the idea is to generalize the probability of the fuzzy set using the available sample
\item spread the measured occurrences over the shape and area of the membership function
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fuzzy Empirical Probabilities}
\begin{itemize}
\item Probability of a crisp value $y \in U$ given a fuzzy set $\ufset \in \ulvar$:
$$
P(y | \ufset) = P(\ufset) \cdot \frac{\mu_{\ufset}(y)}{Z_{\ufset}}
$$
\item Total probability of a crisp value $y \in U$:
$$
P(y,\ulvar) = \sum_{\ufset \in \ulvar} P(y | \ufset)\cdot P(\ufset)
$$
\end{itemize}
\end{frame}
\note[itemize]{
\item If the empirical probability of a fuzzy set is measured by the membership of crisp events, the opposite is also possible
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{PWFTS Rule Model}
\linespread{2}
$$
\begin{array}{rcl}
\pi_1 \cdot A_1 & \rightarrow & w_{11} \cdot A_1, ..., w_{1k} \cdot A_k \\
\ldots & \ldots & \ldots \\
\pi_k \cdot A_k & \rightarrow & w_{k1} \cdot A_1, ..., w_{kk} \cdot A_k
\end{array}
$$
\begin{itemize}
\item $\pi_j$: \textit{Unconditional} empirical probabilities, or the probability $P(LHS)$, $\qquad\sum \pi_j = 1$
\item $w_{ji}$: \textit{Conditional} empirical probabilities, or the probability $P(RHS|LHS)$, $\qquad\sum w_{ji} = 1$
\end{itemize}
\end{frame}
\note[itemize]{
\item Fuzzy frequencies are the membership degrees of the crisp values
\item The training process just store the frequencies and calculate the empirical probabilities with them
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{PWFTS Rule Sample}
\centering
\begin{tabular}{rcrrr}
$0.005\cdot A0$ & $\rightarrow$ & $0.4\cdot A0$, & $0.6\cdot A1$ & \\
$0.05\cdot A1$ & $\rightarrow$ & $0.05\cdot A0$, & $0.6\cdot A1$, & $0.35\cdot A2$ \\
$0.11\cdot A2$ & $\rightarrow$ & $0.1\cdot A1$&, $0.6\cdot A2$, & $0.3\cdot A3$ \\
$0.14\cdot A3$ & $\rightarrow$ & $0.15\cdot A2$, & $0.6\cdot A3$, & $0.25\cdot A4$ \\
$0.15\cdot A4$ & $\rightarrow$ & $0.2\cdot A3$, & $0.55\cdot A4$, & $0.25\cdot A5$ \\
$0.1\cdot A5$ & $\rightarrow$ & $0.2\cdot A4$, & $0.55\cdot A5$, & $0.25\cdot A6$ \\
$0.12\cdot A6$ & $\rightarrow$ & $0.2\cdot A5$, & $0.6\cdot A6$, & $0.2\cdot A7$ \\
$0.09\cdot A7$ & $\rightarrow$ & $0.25\cdot A6$, & $0.55\cdot A7$, & $0.2\cdot A8$ \\
$0.06\cdot A8$ & $\rightarrow$ & $0.25\cdot A7$, & $0.6\cdot A8$, & $0.15\cdot A9$ \\
$0.02\cdot A9$ & $\rightarrow$ & $0.6\cdot A8$, & $0.4\cdot A9$ & \\
\end{tabular}
\end{frame}
\note[itemize]{
\item In this figure we can see a sample of a PWFTS model, with 10 fuzzy sets
\item In the left side are the precedent fuzzy sets with their unconditional probabilities
\item In the right side are the consequent fuzzy sets with their conditional probabilities
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{PWFTS Rule Sample}
\includegraphics[width=\textwidth]{figures/pwfts_rules_firstorder.png}
\end{frame}
\note[itemize]{
\item This figure show a graphic representation of the previous PWFTS model
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Mixture Distributions}
\linespread{1.5}
How to transform the PWFTS model in a probabilistic forecast?
\begin{itemize}
\item Each rule is a conditional probability distribution
\item Each input crisp value "activates" one or more rules, with a different degree (its membership value)
\item Smooth the probability distributions of each rule given:
\begin{itemize}
\item Membership value
\item Unconditional proability (\textit{a priori})
\end{itemize}
\end{itemize}
\end{frame}
\note[itemize]{
\item
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Mixture Distributions}
How to transform the PWFTS model in a probabilistic forecast?
$$
\begin{array}{c}
P(y) = \sum \omega_j \cdot P_j(y) \\
\\
P_j: U \rightarrow [0,1] \\
\\
\sum \omega_j = 1
\end{array}
$$
\end{frame}
\note[itemize]{
\item Smoothing distribution, is the same of mixing distributions
\item The mixture distributions is a way to mix several distributions with the same sampling universe
\item each probability distribution has a weight
\item The total probability of a point is the weighted sum of its probability on each distribution