Derived Braids of Decreasing Products and their Categories
Christopher Tapo

Derived braids have been used to classify categorical structures based on the braid underlying a braided monoidal category \(\mathcal{V}\). With four-strand braids underlying the composition morphisms of tensor products of categories enriched over \(\mathcal{V}\), equality of derived braids has been seen to correspond with the results of Joyal and Street on coherence for braided monoidal categories, namely that diagrams commute when the braids underlying the legs of the diagram are equal. Equality of derived braids can then be seen as a generalization of the Yang-Baxter equation that appears in the work of Joyal and Street. The main result is a proof that decreasing products, which are braids formed from component braids with even numbers of strands, have equivalent derived braids. Plans for future work include interpreting what categorical structures correspond to decreasing products.

§ INTRODUCTION

The work of Joyal and Street in <cit.> shows a relationship between categorical structures and braids, particularly through braided monoidal categories and the Yang-Baxter equation. This relationship is studied further with generalizations of the Yang-Baxter equation as derived braids and their connection to categories enriched over braided monoidal categories as seen in the work of Stefan Forcey and Felita Humes in <cit.>. A set of slides by Stefan Forcey <cit.> focuses on derived braids and includes questions on what sorts of categorical structures might be possible when considering a braid \(x\) with any amount of strands that also satisfies the derived braid equality \(Lx=Rx\). Here we prove a conjecture appearing in the slides of Stefan Forcey on braids called "decreasing products" and show that they obey \(Lx=Rx\). We also state plans for future work on the categorical structures that can be interpreted from decreasing products.

We proceed as follows: First, we review derived braids and how they generalize the Yang-Baxter equation. Next, we define decreasing products and their component braids. Third, we prove that the component braids for decreasing products satisfy the derived braid equality \(Lx=Rx\). Next, we introduce the notions of combing decreasing products and embedding braids which are used to show that entire decreasing products also satisfy \(Lx=Rx\). Finally, we state plans for future work on the categorical interpretations for decreasing products and their derived braids.

§ DERIVED BRAIDS

We shall consider braids with n strands in the braid group \(B_n\). The notion of k-ribbons is introduced in order to simplify the graphical presentation of braids with many strands. We then define left and right derived braids and show that the Yang-Baxter equation arises as the equality of left and right derived braids, or \(Lx=Rx\), for the single crossing of two strands. We end the section with some additional examples of derived braids, including ones that obey \(Lx=Rx\) and others that do not.

§.§ k-ribbons

A k-ribbon is a collection of \(k\) strands that are parallel and adjacent throughout a braid for \(k \in \mathbb{N}\).

A braid can be graphically reduced by representing k parallel strands with k-ribbons, which are represented by single strands with a the value \(k\) printed above the strand. This can be illustrated with the following example:

number of strands=7,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_4, a_3-a_5, a_2-a_4, a_3;

[scale=1.5] at (9,-2) $\cong$;

braid/number of strands=4,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (10,-1.5) braid= a_2;

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,-1) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11,-1) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,-1) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,-1) \(1\);

All braids from this point will be pictured in the reduced form if not already reduced, unless otherwise noted.

§.§ Derived Braids and the Yang-Baxter equation

We restrict our attention to the left and right derived braids as discussed in Forcey's slides <cit.>, but other types of derived braids appear in <cit.>. We then present various examples of derived braids and show whether they satisfy \(Lx=Rx\).

Given a braid \(x \in B_{2n}\) for \(n \in \mathbb{N}\), we define the left derived braid \(Lx\) as the braid in \(B_{3n}\) formed from \(x\) by attaching n identity strands on the right side of \(x\), followed by pairing off strands at the bottom of \(x\) into \(n\) many 2-ribbons resulting in \(2n\) strands (considering 2-ribbons as strands), and then braiding the remaining \(2n\) strands according to the braid \(x\).

An example illustrates this process as follows:

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (2,-1) braid= a_2;

[scale=1.5] at (0,-1.75) For $x=$;
[scale=1.5] at (7,-1.75) , $Lx=$;

braid/number of strands=6,
ultra thick,
braid/strand 1/.style=green,
braid/strand 2/.style=cyan,
braid/strand 3/.style=green,
braid/strand 4/.style=cyan,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (9,0) braid= a_2, a_4, a_3;

[draw=black, dashed] (8.75,-1.25) – (14.25,-1.25);

The dashed line separates the original braid \(x\) from the braiding described in the definition of left derived braid. Also, these braids are already in a graphically reduced form without k-ribbons, however strands for \(x\) are colored to help show the pairing of strands into 2-ribbons, which aren't 2-ribbons graphically as they do not remain parallel in \(x\).

A right derived braid \(Rx\) is defined similarly, except with n identity strands being adjoined on the left.We now include some examples of braids and show whether they obey the equality of left and right derived braids, \(Lx=Rx\).

number of strands=6,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (1.5,0) braid= a_2-a_4;

[scale=1.5] at (0,-0.75) For $x=$;
[scale=1.5] at (10,-0.75) , $Lx\neq Rx$ is shown as;

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2-a_4, a_4-a_7, a_3-a_5, a_4;

[scale=1.5] at (11,-2) $\neq$;

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (12,0) braid= a_5-a_7, a_2-a_5, a_4-a_6, a_5;

[draw=black, dashed] (11.75,-1.25) – (20.25,-1.25);
[draw=black, dashed] (1.75,-1.25) – (10.25,-1.25);

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (1.5,0) braid= a_2, a_1-a_3, a_2, a_2, a_1-a_3;

[scale=1.5] at (0,-2.75) For $x=$;
[scale=1.5] at (8,-2.75) , $Lx=Rx$ is shown as;

braid/number of strands=6,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2, a_1-a_3, a_2, a_2, a_1-a_3, a_4, a_3-a_5, a_2-a_4, a_1-a_3, a_2, a_2, a_3, a_1-a_4, a_2-a_5;

[scale=1.5] at (9.5,-7) $=$;

braid/number of strands=6,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (12,0) braid= a_4, a_3-a_5, a_4, a_4, a_3-a_5, a_2, a_1-a_3, a_2-a_4, a_3-a_5, a_4, a_4, a_3, a_2-a_5, a_1-a_4;

[draw=black, dashed] (11.75,-5.25) – (17.25,-5.25);
[draw=black, dashed] (1.75,-5.25) – (7.25,-5.25);

number of strands=2,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (1.5,-2) braid= a_1;

[scale=1.5] at (0,-2.75) For $x=$;
[scale=1.5] at (6,-2.75) , $Lx=Rx$ is shown as;

braid/number of strands=3,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_1, a_2, a_1;

[scale=1.5] at (5,-2) $=$;

braid/number of strands=3,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (6,0) braid= a_2, a_1, a_2;

[draw=black, dashed] (5.75,-1.25) – (8.25,-1.25);
[draw=black, dashed] (1.75,-1.25) – (4.25,-1.25);

The last example is the Yang-Baxter equation which can now be seen as the equality of left and right derived braids of the two-stranded braid with a single crossing.

§ DECREASING PRODUCTS

Here we define decreasing products and their component braids, followed by a proof that decreasing product components satisfy \(Lx=Rx\). We then introduce the method of combing decreasing products and the notion of embedded braids which will be used to prove our main result.

§.§ Decreasing Product Components

In order to define decreasing products, we need to first introduce their component braids. Decreasing product components are the braids \(b_0, b_1,...,b_k \in B_{2n}\) for \(k=\frac{n}{2}\) if \(n\) is even and \(k=\frac{n-1}{2}\) if \(n\) is odd, shown as follows:

number of strands=2,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (1.5,-2) braid= a_1;

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (5.5,-2) braid= a_2;

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (11.5,-2) braid= a_2;

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (2.5,-5) braid= a_2;

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (10.5,-5) braid= a_2;

[scale=1.5] at (0,-2.75) $b_0=$;
[scale=1.5] at (4,-2.75) , $b_1=$;
[scale=1.5] at (10,-2.75) , $b_2=$;
[scale=1.5] at (0,-5.75)  $b_{\frac{n}{2}}$ ($n$ even) =;
[scale=1.5] at (8,-5.75) , $b_{\frac{n-1}{2}}$ ($n$ odd) =;

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (1.5,-1.5) \(n\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2.5,-1.5) \(n\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5.5,-1.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1] at (6.5,-1.5) \(n-1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1] at (7.5,-1.5) \(n-1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8.5,-1.5) \(1\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11.5,-1.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1] at (12.5,-1.5) \(n-2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1] at (13.5,-1.5) \(n-2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14.5,-1.5) \(2\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2.5,-4.5) \(\frac{n}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3.5,-4.5) \(\frac{n}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4.5,-4.5) \(\frac{n}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5.5,-4.5) \(\frac{n}{2}\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10.5,-4.5) \(\frac{n-1}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11.5,-4.5) \(\frac{n+1}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12.5,-4.5) \(\frac{n+1}{2}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13.5,-4.5) \(\frac{n-1}{2}\);

Decreasing products are defined as the combinations of decreasing product components \(b_{i_1},b_{i_2},...b_{i_k}\) where \(i_j < i_l\) for \(0\le j<l\le k\).

To prove that decreasing products obey \(Lx=Rx\), we start by showing that decreasing product components obey \(Lx=Rx\).

Given a decreasing product component \(b_k \in B_{2n}\), the equality of derived braids \(L(b_k)=R(b_k)\) is satisfied.

Case 1. k=0

\(L(b_k)=R(b_k)\) follows from the Yang-Baxter equation on n-ribbons:

braid/number of strands=3,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_1, a_2, a_1;

[scale=1.5] at (5,-2) $=$;

braid/number of strands=3,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (6,0) braid= a_2, a_1, a_2;

[draw=black, dashed] (5.75,-1.25) – (8.25,-1.25);
[draw=black, dashed] (1.75,-1.25) – (4.25,-1.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(n\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(n\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(n\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(n\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(n\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8,0.5) \(n\);

Case 2. $0<k\le\frac{n-1}{2}$ if n is odd or $0<k<\frac{n}{2}$ if n is even

Here the braid \(b_k\) is of the form:

braid/number of strands=4,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2;

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(p\);

with \(p<q\).

\(L(b_k)=R(b_k)\) follows from the following braid equality:

braid/number of strands=7,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2, a_3, a_5, a_4, a_3;

[scale=1.5] at (9,-3) $=$;

braid/number of strands=7,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (10,0) braid= a_5, a_4, a_2, a_3, a_4;

[draw=black, dashed] (1.75,-2.25) – (8.25,-2.25);
[draw=black, dashed] (9.75,-2.25) – (16.25,-2.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(r\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8,0.5) \(p\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,0.5) \(r\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (15,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (16,0.5) \(p\);

Where \(q-p=r\).

Case 3. k=$\frac{n}{2}$

\(L(b_k)=R(b_k)\) follows from the following braid equality:

braid/number of strands=6,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2, a_4, a_3;

[scale=1.5] at (8,-1.75) $=$;

braid/number of strands=6,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (9,0) braid= a_4, a_2, a_3;

[draw=black, dashed, thick] (1.75,-1.25) – (7.25,-1.25);
[draw=black, dashed, thick] (8.75,-1.25) – (14.25,-1.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(k\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (9,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,0.5) \(k\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14,0.5) \(k\);

Before proving our main result, we will introduce the notions of combing and embedding braids. We now include some examples of decreasing products, each of which obeys \(Lx=Rx\).

number of strands=4,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (2.5,-1) braid= a_2;

[scale=1.5] at (0,-1.75) For $b_2$ ($n=5$) =;
[scale=1.5] at (9,-1.75) , $Lx=Rx$ is shown as;

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2.5,-0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3.5,-0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4.5,-0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5.5,-0.5) \(2\);

braid/number of strands=7,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2, a_3, a_5, a_4, a_3;

[scale=1.5] at (9,-3) $=$;

braid/number of strands=7,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (10,0) braid= a_5, a_4, a_2, a_3, a_4;

[draw=black, dashed] (9.75,-2.25) – (16.25,-2.25);
[draw=black, dashed] (1.75,-2.25) – (8.25,-2.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8,0.5) \(2\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14,0.5) \(2\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (15,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (16,0.5) \(2\);

number of strands=6,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (3.5,-1) braid= a_3, a_2-a_4, a_3, a_3;

[scale=1.5] at (0,-1.75) For $b_1$$b_2$ ($n=5$) =;
[scale=1.5] at (12,-1.75) , $Lx=Rx$ is shown as;

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3.5,-0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4.5,-0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5.5,-0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6.5,-0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7.5,-0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8.5,-0.5) \(1\);

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_3, a_2-a_4, a_3, a_3, a_6, a_5-a_7, a_4-a_6, a_3-a_5, a_4, a_4, a_5;

[scale=1.5] at (11,-6) $=$;

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (12,0) braid= a_6, a_5-a_7, a_6, a_6, a_3, a_2-a_4, a_3-a_5, a_4-a_6, a_5, a_5, a_4;

[draw=black, dashed] (11.75,-4.25) – (20.25,-4.25);
[draw=black, dashed] (1.75,-4.25) – (10.25,-4.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (9,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,0.5) \(1\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (15,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (16,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (17,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (18,0.5) \(1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (19,0.5) \(3\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (20,0.5) \(1\);

§.§ Combing Decreasing Products

Combing braids is a form of braid isotopy yielding equivalent braids in the braid group, as seen in the work of Artin <cit.>. To show that the left and derived braids for a decreasing product are equivalent, we use a method of combing the the derived braids into components that satisfy the equalities for decreasing product components seen in Proposition 1.

Consider a left derived braid \(Lx\) of a decreasing product \(x=b_{i_1}b_{i_2}...b_{i_k}\) as having sections corresponding to the decreasing product components \(b_{i_j}\) for  \(j \in 1,...,k\) as well as sections corresponding to the braiding from the derived braid based on those components, denoted \(b_{i_j}^*\). We can comb each \(b_{i_j}\) of \(Lx\) through sections so that it becomes an equivalent braid \(Lx^{comb}=b_{i_1}b_{i_1}^*b_{i_2}^{comb}b_{i_2}^*...b_{i_k}^{comb}b_{i_k}^*\) where each \(b_{i_j}\) with \(j>1\) is combed through sections transforming the braid component into a combed version denoted \(b_{i_j}^{comb}\). Combing for a right derived braid \(Rx\) is analogous. The process of combing is illustrated below, where the following braids are not shown in the reduced form using k-ribbons.

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_3, a_2-a_4, a_1-a_3-a_5, a_2-a_4, a_3, a_3, a_2-a_4, a_3, a_6, a_5-a_7, a_4-a_6-a_8, a_3-a_5-a_7, a_2-a_4-a_6, a_1-a_3-a_5, a_2-a_4, a_3, a_3, a_2-a_4, a_3-a_5, a_4-a_6, a_5;

[scale=2] at (11.5,-11) $\stackrel{\text{comb}}{\Rightarrow}$;

braid/number of strands=9,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (13,0) braid= a_3, a_2-a_4, a_1-a_3-a_5, a_2-a_4, a_3, a_6, a_5-a_7, a_4-a_6-a_8, a_3-a_5-a_7, a_2-a_4-a_6, a_1-a_3-a_5, a_2-a_4, a_3, a_6, a_5-a_7, a_6, a_3, a_2-a_4, a_3-a_5, a_4-a_6, a_5;

[draw=black, dashed] (12.75,-5.25) – (23,-5.25);
[draw=black, dashed] (0,-5.25) – (10.25,-5.25);

[draw=black, dashed] (12.75,-13.25) – (23,-13.25);
[draw=black, dashed] (0,-8.25) – (10.25,-8.25);

[draw=black, dashed] (12.75,-16.25) – (23,-16.25);
[draw=black, dashed] (0,-16.25) – (10.25,-16.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (0.5,-2.5) \(b_0\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (22.25,-2.5) \(b_0\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (0.5,-6.75) \(b_1\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (22.25,-8.5) \(b_0^*\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (0.5,-12) \(b_0^*\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (22.25,-14.75) \(b_1^{comb}\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (0.5,-19) \(b_1^*\);
[circle, draw=none, fill=white, inner sep=0pt, scale=2] at (22.25,-19) \(b_1^*\);

It is important to note that the combination \(b_1^{comb}b_1^*\) in \(Lx^{comb}\) as pictured above is equal to the right derived braid \(R(b_1)\) over \(b_1 \in B_9\). This is not true in general for combinations of combed components, however, these combinations are equivalent to a left or right derived braid of a decreasing product component with either less or equal strands embedded into \(3n\) strands. This can be checked using the cases in the proof of Proposition 1, while the notion of embedding braids is detailed in the following section.

§.§ Embedded Braids

We shall consider braids in \(B_m\) embedded in \(B_n\) for \(m<n\) by attaching \(n-m\) many identity strands. Let \(b \in B_m\) embedded in \(B_n\) be denoted \(b^m \in B_n\). More decorations could specify the exact manner in which the identity strands are attached, but this is not needed for proving our main result and thus omitted to simplify notation.

The braid appearing in the second case of Proposition 1 and in the first example of Example 3.1.1. can be rewritten using braid isotopy to reveal a Yang-Baxter equation as an embedded braid:

number of strands=7,
ultra thick,
control factor=0,
nudge factor=0,
name prefix=braid,
] at (2,0) braid= a_2-a_5 | a_4 | a_3 | a_4;

[scale=1.5] at (9,-2) $=$;

braid/number of strands=7,
ultra thick,
braid/control factor=0,
braid/nudge factor=0,
name prefix=braid,
] at (10,0) braid= a_2-a_5 | a_3 | a_4 | a_3 ;

[draw=black, dashed, thick] (9.75,-4.25) – (16.25,-4.25);
[draw=black, dashed, thick] (9.75,-1.25) – (16.25,-1.25);
[draw=black, dashed, thick] (1.75,-4.25) – (8.25,-4.25);
[draw=black, dashed, thick] (1.75,-1.25) – (8.25,-1.25);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (2,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (3,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (4,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (5,0.5) \(r\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (6,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (7,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (8,0.5) \(p\);

[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (10,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (11,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (12,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (13,0.5) \(r\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (14,0.5) \(p\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (15,0.5) \(q\);
[circle, draw=none, fill=white, inner sep=0pt, scale=1.25] at (16,0.5) \(p\);

§.§ Theorem for Decreasing Products

We are now ready to prove the main theorem.

Given a decreasing product \(x=b_{i_1}b_{i_2}...b_{i_k} \in B_{2n}\), the equality of derived braids \(Lx=Rx\) is satisfied.

The derived braids of \(x\) are equivalent to an alternating combination of derived braids of decreasing product components embedded in \(B_{3n}\) through combing as follows:

\(L(b_{i_1}b_{i_2}...b_{i_k})=L(b_{i_1})R(b_{i_2}^{comb})^{m_2}...L(b_{i_k}^{comb})^{m_k}\) for odd \(k\), or \(R(b_{i_k}^{comb})^{m_k}\) for even \(k\)

\(R(b_{i_1}b_{i_2}...b_{i_k})=R(b_{i_1})L(b_{i_2}^{comb})^{m_2}...R(b_{i_k}^{comb})^{m_k}\) for odd \(k\), or \(L(b_{i_k}^{comb})^{m_k}\) for even \(k\)

for \(i_j^{'} \le i_j\) and \(n \ge m_j \ge m_l\) for \(j<l\). Note that \(b_{i_1}\) stays at the top of the braid and is never combed, hence \(b_{i_1}\) corresponds with \(L(b_{i_1})\) and \(R(b_{i_1})\). Since \(L(b_{i_1})=R(b_{i_1})\) and \(L(b_{i_j}^{comb})^{m_j}=R(b_{i_j}^{comb})^{m_j}\) for all \(j\) due to Proposition 1, it follows that \(Lx=Rx\) for all \(x=b_{i_1}b_{i_2}...b_{i_k} \in B_{2n}\).

§ FUTURE WORK

With the main result showing that decreasing products always obey \(Lx=Rx\), this is a starting point for future work on finding possible categorical structures related to decreasing products and their derived braids. Additional plans for future work include investigations of similar categorical structures, like tortile tensor categories described in <cit.>, to find other potential connections to derived braids. Regarding the relationship between derived braids and categorical enrichment, other braid constructions are also planned to be investigated alongside decreasing products in order to further generalize ideas on tensor products of categories enriched over a braided monoidal category \(\mathcal{V}\) as seen in <cit.>. While considering derived braids as a generalization of the Yang-Baxter equation, it may be worthwhile to investigate possible applications to braided Frobenius algebras and topological quantum field theories, as well as various topics in knot theory.


# Hierarchical Neural Additive Models for Interpretable Demand Forecasts

Leif Feddersen<EMAIL_ADDRESS>Catherine Cleophas
<EMAIL_ADDRESS>
###### Abstract

Demand forecasts are the crucial basis for numerous business decisions,
ranging from inventory management to strategic facility planning. While
machine learning (ML) approaches offer accuracy gains, their interpretability
and acceptance are notoriously lacking. Addressing this dilemma, we introduce
Hierarchical Neural Additive Models for time series (HNAM). HNAM expands upon
Neural Additive Models (NAM) by introducing a time-series specific additive
model with a level and interacting covariate components.

Covariate interactions are only allowed according to a user-specified
interaction hierarchy. For example, weekday effects may be estimated
independently of other covariates, whereas a holiday effect may depend on the
weekday and an additional promotion may depend on both former covariates that
are lower in the interaction hierarchy.

Thereby, HNAM yields an intuitive forecasting interface in which analysts can
observe the contribution for each known covariate. We evaluate the proposed
approach and benchmark its performance against other state-of-the-art machine
learning and statistical models extensively on real-world retail data. The
results reveal that HNAM offers competitive prediction performance whilst
providing plausible explanations.

###### keywords:

Interpretable Machine-Learning , Neural Networks , Human-Computer-Interaction
, Demand Prediction

††journal: International Journal of Forecasting

[a]organization=Institute for Business, Christian-Albrechts-University Kiel,
country=Germany [*]Corresponding author

We introduce Hierarchical Neural Additive Models (HNAM) for time series
forecasting. These models provide interpretability by clearly attributing how
covariates affect forecasting outcomes while maintaining competitive
predictive accuracy compared to advanced black-box models like Temporal Fusion
Transformers (TFT) and clearly outperforming statistical methods.

HNAM’s inherently explainable architecture, which composes forecasts from
individual covariate effects according to a user-specified interaction
hierarchy, has the potential to improve human-computer interaction in
operational business planning by allowing practitioners to integrate their
expertise and trust into the forecasting process.

##  1 Introduction

Demand forecasting as a problem of time series forecasting is at the heart of
business planning activities (Mentzer and Moon,, 2004) and, therefore,
frequently subject to managerial oversight and intervention (Fildes et al.,,
2009). Contributions on judgmental interventions show that their overall
impact on forecast accuracy is questionable while certainly costly (Fildes and
Goodwin,, 2021), biased and inefficient (Fildes et al.,, 2009), and highly
dependent on the adjuster’s expertise (Sanders and Ritzman,, 1992; Edmundson
et al.,, 1988).

One motivation for adjustments can be misaligned incentives across the
organization (Mello,, 2009). However, a general aversion to algorithmic advice
(Dietvorst et al.,, 2015) is further thought to contribute to
counterproductive human-computer interactions in forecasting tasks (Prahl and
Van Swol,, 2017). One potential driver for algorithm aversion is that human
decision-makers perceive automated algorithms as incapable of capturing the
true demand dynamics (Lehmann et al.,, 2022), or conversely that opaque models
prohibit model understanding, thus hindering trust (Adadi and Berrada,, 2018).

Classical, less opaque time series models such as (Triple) Exponential
Smoothing, ARIMA, and Theta have long been viewed as superior to machine
learning (ML) based forecasting approaches (Makridakis et al.,, 2018). Yet,
recent results from forecasting competitions point to the superiority of ML
methods for forecasting, particularly when time series are rich in covariates
(Makridakis et al.,, 2022). Hence, while the increasing availability of high-
dimensional data calls for complex ML models, their opaqueness could
ultimately prevent organizations from realizing their potential due to lack of
acceptance.

To address this dilemma for time series forecasting, we introduce a novel
approach we term Hierarchical Neural Additive Models (HNAM). HNAM combines the
quasi-unlimited expressivity of neural networks and the interpretable form of
additive models. Its forecasts are additively composed of a level as well as
covariate effects that account for interactions across variables over time. To
that end, HNAM lets a neural network estimate the coefficients of a linear
model where the model’s available information set expands for each covariate
according to a to a user-specified interaction hierarchy. For example, given
the hierarchy _Weekday, Promotion, Holiday_ , the estimated coefficient for
the weekday effect would only depend on past observations, product attributes,
and the weekday itself. The estimated promotion coefficient, one level up the
hierarchy, would crucially account for a weekday-dependent magnitude. Finally,
the effect of a holiday depends on its weekday as well as scheduled promotions
at the same time. The obtained coefficients are finally multiplied by the
feature values. These feature values are one-hot encodings for categorical
values whereas real values are preferably expressed as relative changes to a
baseline (e.g. price deviation from the baseline) to foster sparsity and
concise explanations.

Thereby, coefficients vary for each forecast and each horizon; HNAM estimates
them via a black-box neural network able to model interactions and non-
linearities. Yet, coefficients ultimately interact with the input features,
constraining the model’s expressivity and thereby its potential for
overfitting.

Section 2 motivates the HNAM architecture by discussing related literature on
judgemental forecasting, algorithm aversion, and the impact of a decomposed
transparent display of time series on acceptance and task outcomes. This
section also introduces the existing models that have inspired our proposed
architecture.

Section 3 details the technical aspects and implementation details of HNAM.
Section 4 introduces the three retail demand datasets and outlines the
preprocessing used to benchmark HNAM against prevalent machine learning and
statistical models. Subsequently, Section 6 presents the results of these
benchmarking experiments and illustrates exemplary interpretability outputs.
We derive conclusions from these results in Section 7.

##  2 Motivation and Methodological Background

To motivate the proposed architecture, this section provides a brief overview
on algorithm aversion and links it to judgmental forecasting. This motivation
supports our argument that the interpretability output of HNAM could improve
managerial interaction with its forecasts. Additionally, this section also
provides a methodological background and outlines related approaches, namely
Prophet, Neural Additive Models (NAMs) and Neural Additive Time Series Models
(NATMs).

###  2.1 Algorithm Aversion and Interpretability

Research on the efficacy of algorithmic decision-making and the reluctance of
human experts to accept algorithmic advice is extensive. It traces back to
work by Meehl, (1954) on medical predictions and by Dawes, (1979), who
showcase the superior performance of even basic linear models compared to
human predictions. Despite these findings, practitioners frequently hesitate
to rely on algorithmic advice. This reluctance is coined algorithm aversion in
Dietvorst et al., (2015). Burton et al., (2020) offer a comprehensive
literature review and identify five major factors for algorithm aversion. In
the remainder of this section, we summarize these factors and argue how they
relate to our proposed architecture. We conclude that a lack of compatibility
in how managers and algorithms approach the forecasting task seems to be at
the core of algorithm aversion.

#### False Expectations

Decision-makers’ trust in algorithmic advice is shaped by their own
experiences and external reports, leading to different perceptions compared to
human-generated advice. The desire for social interaction with the advice
source, particularly among confident experts, can make algorithmic assistance
seem unnecessary. People view human errors as random and correctable, while
algorithmic errors are seen as systematic. Even if initially trusting an
algorithm, unavoidable errors may lead decision-makers to preferentially rely
on human decision-making, even despite recognizing the algorithm’s superior
performance. This aspect is largely independent of the model, except for the
obvious goal of avoiding erroneous algorithms.

#### Lack of Perceived Control

In high-stakes situations, decision-makers want influence, independence, and
the option to overrule or engage with algorithmic decisions. This necessitates
the algorithm to be ”behaviorally packaged” (Burton et al.,, 2020, p.5) to
impart meaningfulness. However, the algorithm’s underlying architecture
determines the degree to which it can be presented meaningfully. In this
contribution, we target this by designing HNAM to create composed forecasts.
Instead of adjusting outputs from an opaque architecture, users can exert
control on the level of individual covariate effects.

#### Cognitive Compatibility

While human decision-making processes tend to be marked by instincts,
heuristics, and biases, algorithms rely on deterministic, data-driven logic.
Integrating the two calls for bridging these very different systems.
Algorithmic interpretability could offer a solution, but at present, it only
applies to simpler approaches and not to cutting-edge machine learning
methods. The research presented here relies on the idea that managers tend to
think about forecasts by adding and subtracting the expected impact of ex-ante
known events to a baseline. Therefore, we design HNAM to feature an additive
output layer - thereby matching the world model of the decision-maker. The
preceding layers’ exact logic are still opaque and non-traceable. However, we
consider this aspect to mirror human experience insofar as humans come up with
simple explanations of the world while the neurological processes that
underlie our conscious thinking are phenomenologically in the dark (Nisbett
and Wilson,, 1977).

#### Lack of Incentives

Incorporating algorithmic advice in human decision-making can require
substantial work which requires appropriate incentives. Economic rewards that
encourage high-variance decision-making can deter the application of
supposedly lower-variance algorithmic advice (Dietvorst and Bharti,, 2020).
Social and organizational incentives may not align with the optimization
function of the algorithm (Mello,, 2009). This aspect is largely model-
independent and therefore not within the scope of this paper. However, future
research, particularly focusing on behavioural experiments, could and should
pick up on this aspect.

#### Divergent Rationalities

Humans and algorithms draw different subsets of information from their
surroundings, process them in unique ways, and may have divergent objectives.
In fact, ’broken-leg’ cues (i.e., highly pertinent information that a human
would know to incorporate into their judgment, but which a statistical model
based on historical data would miss) are a common and sensible adjustment
motive (Meehl,, 1954). HNAM’s composed forecasts enable a more precise
injection of extra-model knowledge by clearly differentiating considered
covariates and their effect sizes.

###  2.2 Judgmental Forecasting and Adjustments

Unfortunately, behavioral research on improved acceptance and task outcomes of
additively composed forecasts is scarce. A related case study by Vössing et
al., (2022) finds that a more transparent forecasting interface, which lists
the considered covariates for a forecast, results in more favourable
adjustments. The authors conclude that their transparent user interface allows
managers to _calibrate_ their interventions more precisely and productively.
That is, they become more attuned to the circumstances under which their
extra-model information (or intuition) can, in fact, improve the forecast
whilst abstaining from adjustments otherwise.

Further relevant studies on judgmental forecasts suggest that a decomposed
presentation and processing of time series aids humans in creating their own
forecasts: Both Lawrence et al., (1985) and Carbone and Gorr, (1985), conduct
a comparison of judgmental and statistical forecasts with M1 Competition data
(Makridakis et al.,, 1982). The former study finds that judgmental
extrapolation matches the effectiveness of statistical extrapolation, while
the latter reveals a significantly inferior performance of human judgment. The
critical distinction in the experimental designs is that Lawrence et al.,
(1985) let participants extrapolate the components of the time series,
including trend and seasonality, through a tabular or graphical presentation,
thus facilitating comparisons and the identification of seasonal patterns.

Noticing this discrepancy, Edmundson, (1990) offers an application named
GRAFFECT to structure judgmental forecasting. The authors test this
application using a student sample and the 68 monthly time series from the M1
Competition. GRAFFECT assists users in initially identifying a linear trend
with a potential of one change-point in history and an extrapolation slope
defined by the user. Following this, users discern the perceived seasonal
pattern from the de-trended data. This process can be repeated until users are
satisfied that the residual contains nothing beyond noise. The forecasts
produced through this approach significantly outperform unstructured
judgmental forecasts. While they are also superior to any statistical method
used in the M1 competition, the difference is not statistically significant.

Similarly, Marmier and Cheikhrouhou, (2010) develop a structured approach to
guide forecasters in making adjustments to statistical forecasts based on
expert knowledge of event effects. They identify four types of factors that
can lead to significant and predictable changes in the forecast but are not
accounted for by purely statistical methods: transient factors, transferred
impact factors, quantum jump factors, and trend change factors. The authors
test their approach in two case studies and find that the structured
integration of managerial judgment leads to significant improvements in
forecast accuracy compared to unaided adjustments and purely statistical
forecasts.

Trapero et al., (2013) investigate the effectiveness of judgmental adjustments
to statistical forecasts in the presence of promotional events. The study
finds that judgmental adjustments can improve forecast accuracy when dealing
with promotions, particularly when the statistical model does not explicitly
account for promotional effects. The effectiveness of these adjustments
depends on various factors, such as the product type, promotional
characteristics, and the underlying statistical model. The authors suggest
that structured approaches to judgmental adjustments, which provide guidance
on when and how to adjust based on the characteristics of the promotion and
the statistical model, can lead to improved forecast accuracy.

The discussed research shows that judges perform considerably better when they
handle a time series component-wise and that adjustment quality is contingent
on the product time series and adjacent covariates. We conclude that HNAM’s
composed forecasts promise firstly a higher acceptance due to a higher
transparency and secondly more precise adjustments as they can happen on the
level of individual covariate effects.

###  2.3 Methodological Background

This section introduces prior work that motivated HNAM. That is, Meta’s
Prophet (Taylor and Letham,, 2018) as a highly interpretable yet arguably
inflexible and narrowly applicable model as well as Neural Additive Models
(NAMs) Agarwal et al., (2021) for the idea of separately processing features
with small neural networks before adding them in a linear model. Finally,
Neural Additive Time Series Models (Jo and Kim,, 2023) follow a similar
approach as ours with some important differences with respect to how sequences
and feature interactions are handled.

#### Prophet

The Prophet algorithm, introduced by Taylor and Letham, (2018) relies on a
seasonal decomposition with covariates to create interpretable forecasts. Its
accessible outputs and usability have generated considerable academic and
practical interest as evidenced by 2455 citations according to Google Scholar
and 17.7k stars on GitHub as per April 2024.

The Prophet forecast model is expressed as

$y(t)=g(t)+s(t)+h(t)+e(t)$

Here, each function corresponds to a unique aspect of the time series data:
$g(t)$ denotes the trend, encapsulating non-periodic changes in the time
series. This is typically modelled as a piece-wise linear or logistic
function. $s(t)$ signifies seasonality, capturing periodic changes. This can
include daily, weekly, or yearly cycles. $h(t)$ accounts for the effects of
holidays and additional covariates. $e(t)$ represents the error term,
capturing any idiosyncratic changes not modelled by the trend, seasonality, or
holiday effects.

Prophet’s ease of use and intuitive model structure makes it a popular choice
for applied forecasting. At the same time, the simple structure can miss
important non-linear effects and interactions. Further, Prophet has to be
individually fitted for each time series, foregoing the benefits of cross-
learning.

#### Neural Additive Models

NAMs, as introduced by Agarwal et al., (2021), represent a class of models
that harness the structure of generalized additive models with the ability to
capture non-linear relationships characteristic of neural networks.

The structure of the model can be denoted as:

$f(x)=\sum_{i=1}^{d}f_{i}(x_{i})$

In this formulation, each term $f_{i}(x_{i})$ represents an individual shallow
neural network that is associated with the $i^{th}$ feature or input, $x_{i}$,
on the response, $f(x)$.

Accordingly, NAMs can capture non-linear effects. Additionally, their
modularity facilitates the independent examination of each feature’s effect on
the outcome. The additive structure inherent in NAMs provides a means to infer
the effect sizes of each feature on the response variable, akin to the
components $g(t)$, $s(t)$, and $h(t)$ in the Prophet model.

#### Neural Additive Time Series Models

Jo and Kim, (2023) propose Neural Additive Time Series Models (NATMs). Similar
to HNAM, NATMs adapt the concept of NAMs for time series forecasting.

NATMs differ from our approach in that they use a small neural net for every
past timestep and for each feature. This makes NATMs potentially
computationally expensive - an aspect the authors address by sharing weights
across time or features. NATMs, in their original formulation, are not
modeling interactions across features and do not handle covariates known in
the future.

#### Ex-post interpretability

Research in interpretable and thereby more acceptable ML produced numerous
tools to create ex-post explanations, such as LIME (Ribeiro et al.,, 2016) or
SHAP (Lundberg and Lee,, 2017). However, their ex-post nature makes these
methods computationally costly. At the same time, they still generate over-
simplified approximations of the actual dynamics, leading scholars to call for
inherently interpretable models (Rudin,, 2019).

###  2.4 Research Gap

From the above, we conclude that present models are either interpretable but
oversimplifying or complex black-boxes that exacerbate algorithm aversion.
NAMs and NATMs provide promising steps towards interpretable yet sufficiently
complex models. Our proposed HNAM architecture builds on these ideas to yield
interpretable forecasts that account for complex temporal dependencies and
feature interactions.

##  3 Proposed Architecture

In describing the proposed model architecture of HNAM, we first present a
conceptual overview and differentiate the roles of covariate types and the
functional components processing them. Afterwards, we describe our proposed
implementation of the functional components in detail.

###  3.1 Covariates and Components

#### Conceptual overview

Figure 1 gives a high-level view on the architecture of HNAM. While all
depicted concepts will be explained in later paragraphs, the general setup is
as follows: Covariates are differentiated with respect to their temporal
availability and whether they should have an individual effect (i.e. be causal
covariates). They are embedded or projected (i.e. translated into a numerical
representation of the model’s internal dimensionality). The level network
employs an Attention block (Vaswani et al.,, 2017) to project an estimated
demand level into the forecasting horizon, using only information about the
past. One Attention-based coefficient network per causal covariate processes
the past, static, and non-causal covariates in addition to the respective
causal covariate and those lower in the interaction hierarchy to derive one
coefficient per timestep and causal covariate. These coefficients are
multiplied with the respective transformed values of the otherwise unprocessed
covariates and all effects are added to the estimated level.

Figure 1: Conceptual Overview of the HNAM architecture

#### Causal covariates

HNAM is designed for demand forecasting situations, where some covariates that
affect demand are known in the past and in advance. Examples of such
covariates are the date (weekday, season, holiday) or planned promotional
activities (price changes, advertisement). We call these _causal covariates_.
To address causal covariates, HNAM requires them to be specified as a
hierarchical list, so that each covariate’s estimated causal effect can depend
on the covariate itself and those lower in the hierarchy. For the exemplary
interaction hierarchy of _Weekday, Promotion, Holiday_ , the model estimates
weekday effects independently of other causal covariates, promotional effects
are estimated contingent on the weekday, and holiday effects are contingent on
both weekday and promotional activity. Accordingly, HNAM considers a matrix
$C$ with one row per covariate, yielding $n_{c}$ rows ordered by the causal
hierarchy, and $T_{h}+T_{f}$ columns for the concatenation of historical and
future timesteps, respectively. Note that our model does not prove or assume
causality in the strict sense but differentiates covariates as to whether an
analyst or manager deems that they _should_ have an individually estimated
effect on demand.

#### Non-causal covariates

There are covariates that are known in the past and future that do not require
individually estimated effects. Typical examples are the absolute and relative
time indices and sinusoidal seasonal representations (Hyndman and
Athanasopoulos,, 2021, Chapter 12.1) that induce a temporal understanding to
each timestep. This is necessary in non-recurrent neural networks (Vaswani et
al.,, 2017). Thereby, non-causal covariates modulate the effects of causal
covariates whilst not having an individually estimated effect. Matrix $T$
stores these mostly time-related covariates across $T_{h}+T_{f}$ columns.

#### Static covariates

Similarly, matrix $S$ stores time-invariant covariates, typically product and
store attributes that remain constant across $T_{h}+T_{f}$ columns.

#### Past covariates

Finally, matrix $P$ contains covariates that are only known in the past across
$T_{f}$ columns. It typically contains the past demand observations alongside
other covariates that are not known in advance. Similar to non-causal and
static covariates, past covariates have no individually estimated effects but
also modulate causal covariate effects. For more convenient indexing below, we
let $P$ also have $T_{h}+T_{f}$ columns where values in the future are 0 and
will not be used for forecasts.

#### Functional Components

In order to produce a forecast vector $y$ of $T_{f}$ forecast steps, the
introduced covariate matrices pass through a level neural network $g$ and
coefficient neural networks $f_{i}$ with one neural network for each of the
$n_{c}$ causal covariates. Further, $t$ is a transformation function that k-1
one-hot encodes categorical causal covariates and standardizes continuous
covariates.

In order for the estimated level and coefficients to be meaningful, we must
mask information accordingly. The level component only observes historical
information and projects a future level for each time step in the forecasting
horizon. In this way, it establishes the baseline demand level and captures
possible trends.

Each coefficient neural network $f_{i}$ observes $S$, $T$, and $P$ completely,
in addition to the $i$th causal covariate as well as causal covariates below
it in the interaction hierarchy. For continuous covariates, it outputs one
coefficient; for categorical covariates it outputs $k-1$ coefficients where
$k$ is their cardinality.

On a high-level, starting indexing at 0, HNAM can therefore be formulated as:

$y=g(S,T,P)+\sum_{i=0}^{n_{c}-1}f_{i}({S,T,P,C[:(i+1)]})\cdot
t({C[i,T_{h}:])}$ (1)

###  3.2 Implementation

In the following, we detail how functional components are implemented. These
encompass how covariates are processed via embeddings and projections, how
covariate interactions are enabled by additively assembling information, and
how temporal dependencies are modeled via Attention layers to determine the
forecast’s level and coefficients.

#### Embeddings and Projections

All covariates are projected into the model’s embedding size, a hyperparameter
that determines the model’s internal dimensionality and thereby
expressiveness. To achieve this, an embedding table handles categorical
covariates. Embeddings are a sparser alternative to one-hot encodings to
translate categorical values into a numerical representation. Continuous
covariates are projected into the model’s embedding size through a linear-
affine transformation.

#### Assembling information tensors

The constant embedding size of embedded covariates allows them to be summed in
the feature dimension. Whenever HNAM requires a certain information set, the
respective covariates are summed and then passed through a layer
normalization.

#### MLP

At multiple points in our architecture, a multi-layer perceptron is
responsible for non-linear processing. The input is projected into an
embedding size that is larger by a user-parameterized factor and then passes
through a Gaussian Error Linear Unit (GELU) activation function for non-
linearity. Subsequently, the representation is projected back with dropout to
the network’s embedding size.

#### Temporal Convolution Layer

This layer expands its input using a 1-dimensional convolution with a kernel
size of 3, accounting for the last two timesteps with zero padding for the two
oldest observations. Post-convolution, a GELU activation and dropout
regularize the output before it is projected back to the embedding size. This
layer is used in Attention blocks described below to extract local features
across adjacent timesteps.

#### Attention Block

Both level and coefficient networks incorporate information from other time
steps globally via multi-head Attention (Vaswani et al.,, 2017) with a
temporal convolution to calculate queries, keys, and values. The respective
implementations are described below.

#### Level Network

The level component is implemented as an Attention block. The query consists
of of $(S[:,T_{h}:]+T[:,T_{h}:])$, i.e., static and temporal information for
the forecast horizon. The keys and values both are $(S+T+P)[:T_{h}]$, i.e.,
static, temporal, and past information in the past time steps.

  * 1.

Query ($Q$), Key ($K$), and Value ($V$) projections are obtained by passing
the input sequence through temporal convolutional layers:

$\displaystyle Q$ $\displaystyle=\text{TemporalConv}((S+T)[:,T_{h}:]),$
$\displaystyle K$ $\displaystyle=\text{TemporalConv}((S+T+P)[:,:T_{h}]),$
$\displaystyle V$ $\displaystyle=\text{TemporalConv}((S+T+P)[:,:T_{h}])$

  * 2.

The sequence is split in the channel dimension into multiple heads and scaled
dot-product Attention is computed for each head:

$\displaystyle Y$ $\displaystyle=\text{Attention}(Q,K,V)$
$\displaystyle=\text{Softmax}\left(\frac{QK^{T}}{\sqrt{d_{k}}}\right)V$

where $d_{k}$ is the dimension of each head.

  * 3.

The attention output is then combined with the output of an MLP after layer
normalization:

$\displaystyle Y^{\prime}$ $\displaystyle=Y+\text{MLP}(\text{LayerNorm}(Y))$

  * 4.

Finally, the Level vector and a Level Embedding in the embedding size are
projected from $Y^{\prime}$:

Level $\displaystyle=\text{Linear}(Y^{\prime})$ Level Emb
$\displaystyle=\text{Linear}(Y^{\prime})$

#### Coefficient Networks

For each causal covariate, we have one coefficient network. A similar
Attention block as described above is at the core of each coefficient network.
Queries, keys and values for the $i$th covariate and coefficient network are
obtained as:

$\displaystyle Q$
$\displaystyle=\text{TemporalConv}((S+T+C[:i+1])[T_{h}:]+\text{Level Emb}),$
$\displaystyle K$
$\displaystyle=\text{TemporalConv}((S+T+P+C[:i+1])[:T_{h}]),$ $\displaystyle
V$ $\displaystyle=\text{TemporalConv}((S+T+P+C[:i+1])[:T_{h}]))$

As described previously, for each causal covariate’s coefficient network, the
values of causal covariates higher in the interaction hierarchy are not
available. We add the Level Embedding obtained earlier to the query to utilize
the extracted information on the level here.

Although each feature is processed in an individual network, we let Attention
scores be computed in parallel by assembling tensors such that the different
causal covariates are stacked in a second batch dimension. The results of the
Attention block are projected back to obtain the coefficients.

##  4 Datasets and Preprocessing

For robust benchmarking, we generate and evaluate forecasts in three different
retail demand datasets. All datasets have multiple covariates alongside the
recorded sales data. Two datasets are publicly available and one stems from an
industry partner.

In the following, we provide an overview of the preprocessing we apply in this
study, highlight how time series are filtered, what covariate features are
used, and how a rolling-forward evaluation is carried out.

###  4.1 Datasets

#### Walmart

This public dataset (University of Nicosia,, 2020) consists of hierarchical
sales data provided by Walmart through the University of Nicosia amid the M5
forecasting competition. It spans 1941 days from 2011 to 2016. The dataset
includes daily item-level sales data of 3049 products and 10 store locations
across three states.

#### Favorita

The Favorita dataset (Favorita,, 2018) is a publicly available dataset
provided by Corporación Favorita, an Ecuadorian-based grocery retailer, for a
forecasting competition. It spans 1684 days from 2013 to 2017 across 4035
products and 54 stores.

#### Retail

We further evaluate models on a dataset provided by a medium-sized retailer.
It spans 2472 days from 2015 to 2022 observing 2910 products across three
stores.

###  4.2 Data Selection

The research documented in this paper focuses on the interpretability aspect
of forecasting to enhance human-computer interaction. To create a fitting
benchmarking environment for this purpose, we select for time series, i.e. a
unique product and store combination, that are likely to be adjusted due to
their economic relevance and for which we can assess the plausibility of
interpretability outputs.

We exclude time series with apparently low, intermittent, or cyclical sales.
All datasets lack pertinent information as to whether a product was out of
stock or not offered at all, making it challenging to create realistic
benchmarks for forecasting consumer demand. Since we can not differentiate
these scenarios without making substantial assumptions, we focus on products
with more stable sales, assuming they were less frequently out of stock.

Further, we argue that daily forecasts of intermittently demanded products are
generally less relevant than non-intermittent ones. For instance, products
with high daily demand and short shelf-lives, such as fruits, vegetables, and
dairy products, typically do not exhibit intermittency. On the other hand,
products with slower turnover rates, often due to higher costs or specificity,
usually have longer shelf-lives, like spices, beverages, and non-food items.
For these products, aggregate forecasts are more relevant, making the problem
of intermittency conveniently disappear.

Based on these arguments, we select time series from the given data sets
according to the following criteria:

  1. 1.

The time series must display non-zero sales for at least 100 days prior to the
first test period.

  2. 2.

The time series must not exhibit more than 100 days of no sales after the
first recorded sale.

  3. 3.

The time series’ median sales must be greater than five during the test
periods and 100 days before.

  4. 4.

The time series must be in the top 500 in terms of sales during the test
periods and 100 days before.

  5. 5.

The time series must be in the top 500 in terms of revenue during the test
periods and 100 days before (not applicable for Favorita since price
information is missing).

  6. 6.

For Favorita, we instead select time series with less than 1% missing
observations.

This selection process yields 221 (Walmart), 287 (Favorita), and 193 (Retail)
time series.

###  4.3 Features

The multivariate models we benchmark utilize a number of covariates. While
most covariates are available in all datasets, some are dataset specific.
Unless indicated otherwise, all covariates are available not only for past
observations but also in the prediction horizon. Table 1 lists the covariates,
differentiates their type as categorical or continuous and indicates their
availability per dataset. In addition to the listed covariates, neural
networks use an absolute and a relative time index as well as sine and cosine
embeddings of the day of the year to induce a temporal sense to the models.

Table 1: Covariates entering multivariate models per dataset.

| M5 Walmart | Favorita | Retail
---|---|---|---
Categorical
Product | ✓ | ✓ | ✓
Store | ✓ | ✓ | ✓
Promotion | ✗ | ✓ | ✓
Weekday | ✓ | ✓ | ✓
Holiday1 | ✓ | ✓ | ✓
SNAP2 | ✓ | ✗ | ✗
Continuous
Sales (only past) | ✓ | ✓ | ✓
Relative Price 3 | ✓ | ✗ | ✓
Oil Price (only past) | ✗ | ✓ | ✗

  * 1

For Favorita, we differentiate local, regional, and national holidays as
separate covariates; for Walmart we differentiate sporting, cultural, and
national events or holidays.

  * 2

The Supplemental Nutrition Assistance Program (SNAP), previously known as the
Food Stamp Program, provides funds specifically for purchasing food to low-
income households. The feature marks a day at which funds are distributed.

  * 3

The percentage deviation of the day’s price from the rolling 20-day mean.

###  4.4 Evaluation

Across all datasets, we use the last five fully available months as test sets.
For each test month, we train neural networks with observations up to two
weeks prior to the test month and use the remaining two weeks prior as
validation set for early stopping.

We generate forecasts with a maximum horizon of two weeks via a sliding window
in each test month without retraining. We fine tune with updated data prior to
each new test month. Similarly, we keep the hyperparameters for the remaining
benchmark models constant per test month but use all data up to the respective
forecast date for training.

We aim to forecast sales with daily granularity. While coarser temporal
aggregations can be sensible depending on the business case, we focus on our
model’s interpretable forecasts based on covariates of daily frequency.
Further, we limit our exploration to forecasting expected sales, as opposed to
providing probabilistic forecasts as required for newsvendor models. While
HNAM supports quantile prediction, we opt for point forecasts for the sake of
clearer error metrics and interpretability outputs in this paper.

##  5 Models

The following provides an overview of the considered benchmarks and
implementation details. To that end, we first describe variants of HNAM
implemented for the various target sets, before, detailing our implementation
of Temporal Fusion Transformers, Prophet, ETS, and SARIMAX.

#### Hierarchical Neural Additive Models

HNAM’s architecture has been outlined in Section 3. There, we discriminated
different types of covariates and introduced the interaction hierarchy applied
for the benchmark data sets.

We use the same hierarchy for all datasets, given that the respective
covariate is available: _Weekday, Relative Price, Promotion, Holiday_. We
select this hierarchy because it feels natural to estimate weekday effects
independently of other modulating covariates and add more sparsely occurring
covariates on top. Generally speaking, we suggest to order covariates by
decreasing frequency of their occurrence to limit the total amount of
covariate interactions.

The Walmart data set does not include information on scheduled promotions.
However, _SNAP_ indicates days where eligible customers received financial
support for groceries. Holidays are differentiated into three separate
covariates in Walmart and Favorita, differentiating their type and regionality
respectively.

#### Temporal Fusion Transformers

Temporal Fusion Transformers (TFT) (Lim et al.,, 2021) is a deep learning
model that employs the transformer architecture for time series forecasting.
Its introductory paper demonstrated TFT outperforming other popular neural
networks for forecasting, including DeepAR (Salinas et al.,, 2020). Like HNAM,
TFT handles and differentiates covariates with respect to their availability
during the forecast horizon. Notably, its attention mechanism and variable
selection networks offer some interpretability that is, however, significantly
less immediate than in our proposed architecture (Feddersen and Cleophas,,
2023).

#### Neural Network Implementations

The neural network-based methods above are implemented in Pytorch Forecasting
(Beitner,, 2020). In pre-experiments, we observed better validation scores
with an embedding size of 32 rather than 16 in both HNAM and TFT. Otherwise,
we use the package’s default implementations and parameters for TFT. We do not
perform automatic hyperparameter tuning due to its exponential computational
costs and because predictive accuracy is not the main proposition of our
proposed architecture. Rather, our aim is to validate whether constraints
imposed on HNAM for interpretability would reduce its predictive accuracy
notably compared to more opaque models like TFT.

Neural network models learn global models for all time series in a dataset
whereas the benchmark methods below fit local models per time series. Global
models can thereby offset their complexity (number of parameters) with a
relatively larger amount of training data (fitting only one model per dataset
instead of one model per time series). Accordingly, we fit the neural network
models with a learning rate of 0.001, weight decay of 0.01 and a batch size of
256. For the first test month, we train up to 300 epochs with early stopping
on the loss in the validation set (i.e., the last two weeks of each training
set) with a patience of 30 epochs and select the best model with respect to
the validation loss. For the remaining test months, neural networks are fine-
tuned for up to 100 epochs, also employing early stopping. The models are
trained on a high-performance cluster node with an NVIDIA V100 GPU.

#### Prophet

Prophet (Taylor and Letham,, 2018) decomposes a time series into trend,
seasonality, and holiday components. The model conveniently handles missing
values and is intended to allow fast human-in-the-loop iterations. We use
Prophet’s Python implementation (Taylor and Letham,, 2017) with standard
parameters. Per time series we validate an additive versus a multiplicative
model in the training data and use the better performing model for generating
forecasts.

#### ETS

Seasonal exponential smoothing (also termed ETS for Error, Trend, Seasonality)
applies exponentially decaying weights to the error, trend, and seasonal
components of univariate time series. We use the statsforecast (Garza et al.,,
2022) AutoETS implementation to automatically determine the additive or
multiplicative configuration of error, trend, and seasonality via the Akaike
Information Criterion.

#### SARIMAX

Seasonal Autoregressive Integrated Moving-Average with Exogenous Regressors
(SARIMAX) models understand the detrended time series as a combination of
autoregressive and moving average processes, alongside a seasonal component
and external regressors. The respective parameters are found with the
statsforecast AutoARIMA implementation.

##  6 Results

###  6.1 Computation Times

Table 2 depicts details on the training process of the neural network models
HNAM and TFT for the first test set of each data set. Although both models run
for similar numbers of epochs until early stopping triggers and achieve
comparable loss scores in the training and validation sets, HNAM requires
considerably less time per epoch. Accordingly, it needs markedly shorter
overall training times than TFT.

As to be expected, the linear models train notably faster, with ETS taking up
to ten minutes, Prophet up to two hours, and SARIMAX up to four hours on an M1
Macbook Pro to generate all forecasts. One should note, however, that a pre-
trained neural network requires less training time for ongoing fine-tuning and
that inference is possible in a few minutes.

Table 2: Training times and loss metrics for HNAM and TFT.

| Walmart | Favorita | Retail
---|---|---|---
| HNAM | TFT | HNAM | TFT | HNAM | TFT
Time per epoch (m:s) | 01:57 | 03:44 | 01:30 | 03:12 | 01:32 | 05:02
Epochs | 258 | 253 | 220 | 261 | 211 | 217
Training time (h:m:s) | 8:23:06 | 15:44:32 | 5:30:00 | 13:55:12 | 5:23:32 | 18:12:14
Training loss | 0.650 | 0.636 | 0.537 | 0.517 | 0.572 | 0.569
Validation loss | 0.601 | 0.669 | 0.518 | 0.500 | 0.543 | 0.550

###  6.2 Aggregate Error Metrics

This section presents aggregate error metrics of the considered models in the
three dataset’s test sets. Negative predictions are truncated at 0. In terms
of error metrics, we consider SMAPE, MAE and RMSE. We select SMAPE for its
intuitive interpretation noting its asymmetric weighing of over- and
underprediction. MAE as standardized with each time series’ standard deviation
provides a common linear and symmetric metric. We chose the equally
standardized RMSE as a loss metric in the training process. Finally, we also
consider the frequency of 1st and 2nd ranks achieved across time series when
ranking models by RMSE per time series.

Table 3: Mean and median error metrics in Walmart

| SMAPE | Std. MAE | Std. RMSE | Rank Freq.
---|---|---|---|---
| $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | 1st | 2nd
HNAM | 0.297 | 0.211 | 0.571 | 0.451 | 4.000 | 1.387 | 76.5% | 23.5%
TFT | 0.310 | 0.222 | 0.601 | 0.485 | 4.304 | 1.611 | 23.5% | 75.6%
PROPHET | 0.412 | 0.307 | 0.838 | 0.653 | 9.590 | 2.897 | 0.0% | 0.5%
SARIMAX | 0.400 | 0.296 | 0.796 | 0.635 | 8.045 | 2.747 | 0.0% | 0.5%
ETS | 0.373 | 0.273 | 0.748 | 0.594 | 7.190 | 2.384 | 0.0% | 0.0%

Table 4: Mean and median error metrics in Retail

| SMAPE | Std. MAE | Std. RMSE | Rank Freq.
---|---|---|---|---
| $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | 1st | 2nd
HNAM | 0.414 | 0.294 | 0.428 | 0.311 | 8.892 | 1.597 | 47.2% | 50.3%
TFT | 0.407 | 0.293 | 0.427 | 0.315 | 8.294 | 1.637 | 52.3% | 46.6%
PROPHET | 0.555 | 0.438 | 0.674 | 0.466 | 22.782 | 3.551 | 0.0% | 0.5%
SARIMAX | 0.587 | 0.465 | 0.715 | 0.493 | 25.649 | 3.983 | 0.0% | 1.0%
ETS | 0.541 | 0.416 | 0.647 | 0.434 | 23.738 | 3.060 | 0.5% | 1.6%

Table 5: Mean and median error metrics in Favorita

| SMAPE | Std. MAE | Std. RMSE | Rank Freq.
---|---|---|---|---
| $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | $\bar{x}$ | $\tilde{x}$ | 1st | 2nd
HNAM | 0.202 | 0.139 | 0.430 | 0.319 | 20.580 | 5.205 | 29.6% | 68.6%
TFT | 0.191 | 0.135 | 0.413 | 0.312 | 17.382 | 4.942 | 70.0% | 30.0%
PROPHET | 0.288 | 0.198 | 0.643 | 0.474 | 98.417 | 11.485 | 0.0% | 0.0%
SARIMAX | 0.282 | 0.204 | 0.637 | 0.471 | 49.997 | 11.361 | 0.0% | 0.3%
ETS | 0.261 | 0.185 | 0.591 | 0.434 | 45.212 | 9.617 | 0.3% | 1.0%

Overall, we note that the neural network models HNAM and TFT clearly
outperform the linear benchmarks. Comparing the two, HNAM performs better in
the Walmart dataset (Table 3), whereas TFT dominates in the Favorita dataset
(Table 5), while Retail shows balanced outcomes slightly favoring TFT (Table
4).

These results confirm that the considerably less flexible HNAM architecture is
competitive with a state-of-the-art model that employs more complex components
and takes considerably longer to train. As an additional interesting finding,
SARIMAX and Prophet, despite utilizing covariates, fare worse than the
univariate ETS, once again confirming ETS as a hard-to-beat benchmark.

###  6.3 Interpretability

After confirming HNAM’s competitive accuracy on a macrolevel, this section
explores specific forecasting instances. This examination will highlight
HNAM’s ability to offer explanations for its forecasts, as well contrasting
its accuracy versus TFT in favorable and infavorable instances.

Figure 2(a) visualises a forecast instance from the Walmart dataset, where
HNAM fares considerably worse than TFT as measured by the largest gap between
the two in SMAPE. Timestep 1802 on the the x-axis marks the first date of the
forecast horizon. The y-axis measures sales quantities. To explain an HNAM
forecast, one first considers the predicted Level (light-blue line). Causal
covariate effects are then added to that Level (colored stacked bars) and
amount to the model’s forecast (dashed black line). In the depicted instance,
the first forecast at timestep 1802 is determined solely by the Level, devoid
of any causal covariate effects. This occurs due to the application of k-1
dummy encoding for categorical variables, where the first category is
represented implicitly by the absence of other categories (all zeros in the
dummy-encoded vector). We state all potentially considered covariates in the
legend, even if they have no effect in the given forecast horizon. Such a
representation clarifies that the covariates would, in fact, be considered if
they had a non-zero effect. The TFT prediction is given as a dashed blue line
for comparison.

Comparatively bad performances of HNAM as observed in Figure 2(a) often occur
after rapid level changes, however, we do not have a concise explanation at
hand that goes beyond suspecting that HNAMs composed structure makes it more
inflexible for picking up such rapid shifts.

Figure 2(b) depicts an instance where HNAM markedly outperforms TFT.
Particularly, the initially demand-reducing effect of a substantial price
increase as well as a subsequent rebound following a slight price decrease is
accurately captured by HNAM. We observe how the additive nature of HNAM’s
predictions arguably make it more prone for producing negative predictions
which would have to be truncated.

Figure 3 depicts an instance where HNAM performs relatively well and
apparently predicts the promotional effects quite well. This instance is an
example of its strength and weakness regarding the clear attribution to
covariates. Considering timestep 1494, the data does not indicate a planned
promotion for that day although it is surrounded by days with active
promotions. Yet, the actual sales remain relatively high which is accurately
predicted by TFT. Assuming this is a regular, learnable pattern, HNAM would
have no chance to do so since the estimated coefficient for promotional
activity for that day is multiplied with zero anyway.

(a) Relatively inaccurate HNAM predictions for one forecast instance in
Walmart

(b) Relatively accurate HNAM predictions for one forecast instance in Walmart

Figure 2: Composed HNAM predictions and TFT predictions versus actuals in
Walmart. Figure 3: Composed HNAM predictions and TFT predictions versus
actuals in Favorita.

##  7 Conclusions and Future Research

#### Accuracy results

This study introduces Hierarchical Neural Additive Models (HNAM) for time
series forecasting emphasising the critical aspect of interpretability. The
accuracy results observed in the computational study demonstrate that HNAM
delivers competitive forecasting accuracy when benchmarked against advanced
models such as Temporal Fusion Transformers (TFT) across multiple datasets.
This is particularly noteworthy as HNAM incorporates design elements aimed at
enhancing interpretability, which could potentially restrict the model’s
expressive power. In addition, HNAMS’s composed forecasts allow for precise
attributions how different covariates impact the forecast outcomes. This
interpretability can be used for designing user interfaces that foster trust
and encourage the productive injection of extra-model knowledge by managers.

#### Possible challenges in practice

Although HNAM marks a big step towards inherently interpretable models for
demand forecasting, we anticipate common problems upon its practical
implementation for decision-support:

While HNAM’s architecture allows a clear mapping of which information the
model used to estimate a covariate effect, one must not confuse this with
proving causality. At least two examples come to mind: In the Retail dataset,
we have two covariates regarding promotional activity: promotion and price
change. Some products have promotions without price changes and vice versa.
Given a product that only has price changes concurrent with promotions, the
estimated price sensitivity might counterintuitively show a negative effect of
the price change if it is of a lesser extent than other price changes during
promotions. One solution would be to drop the promotion covariate entirely or
to use it only when there are no price changes. As a similar example, it is
conceivable that past price increases were a reaction to anticipated increased
demand. Then, the price sensitivity would again imply higher demand for higher
prices. Hence, the usual caveats of statistical modeling apply to HNAM and
estimated effects must be carefully evaluated before concluding any causality
to rest pricing decisions upon.

#### Behavioral validation

The present research rests upon the arguments laid forth in Section 2 to
motivate HNAM’s potential for improving human-computer interaction in the
retail demand prediction context. However, no study to date has clearly
examined the case where analysts work with a black-box versus a composed
forecasting model. Therefore, we plan to conduct and encourage future
behavioral research on analysts’ interactions with explainable models like
HNAM. These would optimally include discussions and quantitative evaluations
with domain-experts as well as larger-scale evaluations with business
students.

#### Architecture

Our HNAM implementation uses several best practices (e.g., layer
normalization, dropout, multi-head Attention), yet it is by no means the
result of an extensive architecture search. We emphasise that the general idea
of HNAM might be implemented in more efficient and more accurate ways.

#### Delayed and anticipatory effects

As alluded to earlier when discussing Figure 3, HNAM in its current
implementation does not natively model delayed or anticipatory effects. That
is, for example, when a promotion affects subsequent periods (Hewage et al.,,
2022), or when customers change their shopping patterns in anticipation of an
event or holiday. Such effects, if suspected, could be captured by adding
suitably engineered covariates with positive and negative shifts. Future
research, however, might further consider neural architectural solutions that
maintain the clear effect attribution of HNAM while inherently modeling such
time-shifted covariate effects.

#### Final Remarks

As organizations contend with data-rich environments and challenges with
algorithm aversion, our results demonstrate that HNAM makes a significant step
towards more effective decision-making through accurate yet explainable
models. We call for behavioral research that refines the concept of algorithm
aversion as to how it may be attenuated through inherently interpretable
models like the one presented here.

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