#  A note on non-simply connected rational homotopy models

Syunji Moriya
Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University,
3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585 Japan<EMAIL_ADDRESS>
This work is partially supported by JSPS KAKENHI Grant Number JP17K14192.

##  1\. Introduction

Gómez-Tato–Halperin-Tanré [4] proposed a non-simply connected generalization
of the rational homotopy theory of Quillen [7] and Sullivan [9]. They adopted
local systems of commutative differential graded algebras (CDGA’s) as
algebraic models of spaces and proved that the fiberwise rationalization of a
space with arbitrary fundamental group is recovered from the local system.
They also established an equivalence between homotopy categories by making use
of their minimal models. As another rational model for a non-simply connected
space, a $\pi_{1}$-equivariant CDGA was introduced by Katzarkov-Pantev-Toën
[5] and Pridham [6]. This CDGA can be defined as an algebra of (polynomial) de
Rham forms with coefficients in a large semi-simple local system. This model
is also studied in [10] in terms of dg-categories with emphasis on recovering
homotopy invariants. For example, the rational homotopy groups and Postnikov
invariants are algebraically recovered from this equivariant CDGA model under
some assumption (see also [6]). Both of these algebraic models have their own
advantages. The local system models have clear correspondence with fibrations
over $K(\pi,1)$-spaces, and the equivariant CDGA models admit a notion of
minimal model which is very similar to Sullivan’s one. So direct comparison of
these models might be important. In this note, we illustrate a part of such
comparison in a simple example. Concretely speaking, we present an example of
local system of CDGA’s and compute the corresponding equivariant CDGA model
and see how some homotopy invariants are recovered from the equivariant model
with this example. Especially, we present a relatively small complex which
computes a local coefficient cohomology of the corresponding space by making a
small use of dg-categories. Our example is a non-nilpotent version of an
example in [4]. Some parts of the computation may be generally useful for
translating these models into each other.

##  2\. Preliminary

In this section, we briefly recall basic notions from other papers. One can
find their details in the references.
We suppose that all vector spaces in this note, possibly with some extra
structures such as a grading, differential, and product are defined over a
fixed base field $\mathsf{k}$ of characteristic $0$. Complexes are non-
negatively and cohomologically graded. $\otimes=\otimes_{\mathsf{k}}$ denotes
the tensor product over $\mathsf{k}$. A commutative differential graded
algebra (in short, CDGA) is a complex $A$ equipped with a unital, associative
and graded commutative multiplication $A\otimes A\to A$ commuting with the
differential. A map of CDGA’s is a chain map which preserves the product.
Let $K$ be a simplicial set and $C$ a category. A local system on $K$ in $C$
is a functor from the opposite category of simplexes of $K$ to $C$, so it
consists of a family of objects $F_{\sigma}$ of $C$ labeled by
$\sigma\in\sqcup_{n\geq 0}K_{n}$ and maps (or morphisms) of $C$ labeled by
simplicial maps. A map of local systems is a natural transformation. A local
system in the category of CDGA’s and their maps is called shortly a local
system of CDGA’s. We use freely the terms about local systems in [4]. To ease
a term, by a local system of vector spaces, we mean a locally constant local
system in the category of (ungraded) vector spaces and $\mathsf{k}$-linear
maps. For a local system of CDGA’s $B$ and a local system of vector space $V$,
we define the local system $B\otimes V$ (of complexes) by $(B\otimes
V)_{\sigma}=B_{\sigma}\otimes_{\mathsf{k}}V_{\sigma}$ with the obvious maps,
where $V_{\sigma}$ is regarded as a complex concentrated in degree 0. Let
$A_{K}$ ($=A_{PL,K}$ in the notation of [4].) be the local system of CDGA’s
defined by $A_{K,\sigma}=A_{PL}(\Delta^{|\sigma|})$, the CDGA of polynomial de
Rham forms on the standard simplex of the same dimension as $\sigma$. As in
[4], an $A_{K}$-algebra is a map $A_{K}\to B$ of local systems of CDGA’s such
that $B$ is extendable and $H^{*}(B)$ is locally constant. The authors of [4]
proposed 1-connected $A_{K}$-algebras of finite type as algebraic models of
non-simply connected spaces. For a Kan fibration $p:E\to K$ of simplicial
sets, they defined an $A_{K}$-algebra $\mathcal{F}(E,p)$ by setting
$\mathcal{F}(E,p)_{\sigma}=A_{PL}(E\times_{K}\Delta^{|\sigma|})$ where the
pullback is taken for the diagram $E\stackrel{{\scriptstyle
p}}{{\to}}K\stackrel{{\scriptstyle\sigma}}{{\leftarrow}}\Delta^{|\sigma|}$,
and proved that this $A_{K}$-algebra recovers the fiberwise rational homotopy
type of $E$ if $K$ is a $K(\pi_{1}(E),1)$-space and $p$ is 1-connected, and
$\mathsf{k}=\mathbb{Q}$ (under a finiteness assumption). We call an
$A_{K}$-algebra which is connected with $\mathcal{F}(E,p)$ by a zigzag of
quasi-isomorphisms of $A_{K}$-algebras, a local system model of $E$.
We shall recall another algebraic model from [5, 6, 10]. Let $\pi$ be a
(abstract) group. We say a representation of $\pi$ is semi-simple if it is
isomorphic to a direct sum of finite dimensional irreducible representations.
The pro-reductive completion $\pi^{red}$ is the affine group scheme whose
finite dimensional representations are in one-to-one correspondence with the
finite dimensional semi-simple representations of $\pi$ in a natural manner.
Let $\mathcal{O}(\pi^{red})$ denotes the coordinate ring of $\pi^{red}$ (or
Hopf algebra representing $\pi^{red}$). A $\pi$-CDGA is a CDGA equipped with a
$\mathsf{k}$-linear action of $\pi$ which commutes with the differential and
product. A map of $\pi$-CDGA’s is a map of underlying CDGA’s which commutes
with the action of $\pi$ and a map of $\pi$-CDGA’s is said to be a quasi-
isomorphism if it induces an isomorphism on the cohomology. We say that a
$\pi$-CDGA is semi-simple if its underlying representation is semi-simple. Let
$X$ be a pointed connected simplicial set. We sometimes identify
representations $V$ of $\pi_{1}(X)$ with the local systems of vector spaces on
$X$ which associate the vector space $V$ to each simplex and the action of an
element of $\pi_{1}(X)$ to each coface map. (This identification is made
explicit in the case of $X=T^{2}$ in next section.) This identification
naturally extends to the equivalence between the category of representations
and their homomorphisms and the category of local systems of vector spaces and
their maps, which sends a homomorphism $f:V\to V^{\prime}$ to the map given by
$f$ on each simplex. For a local system $V$ of vector spaces on $X$, we define
a complex $C^{*}_{PL}(X,V)$ by $C^{*}_{PL}(X,V)=\Gamma(A_{PL,X}\otimes V)$,
where $\Gamma$ denotes the global section functor given in Definition 1.1 of
[4]. (This is the same as the complex given in Definition 3.4.2 of [10].) We
regard $\mathcal{O}(\pi_{1}(X)^{red})$ as a local system of $X$ by the right
translation. We define a $\pi_{1}(X)$-CDGA $A_{red}(X)$ as the complex
$C^{*}_{PL}(X,\mathcal{O}(\pi_{1}(X)^{red}))$ equipped with the product
induced by those of polynomial forms and of $\mathcal{O}(\pi_{1}(X)^{red})$
and the action of $\pi_{1}(X)$ induced by the left translation on
$\mathcal{O}(\pi_{1}(X)^{red})$ (see Definitions 3.3.3 and 3.4.4 of [10]).
Since the translation on $\mathcal{O}(\pi_{1}(X)^{red})$ is semi-simple,
$A_{red}(X)$ also has a semi-simple action of $\pi_{1}(X)$. By this feature,
$A_{red}(X)$ has a minimal model similar to the original case of Sullivan.
There exists a quasi-isomorphism $\mathcal{N}\to A_{red}(X)$ of
$\pi_{1}(X)$-CDGA such that the underlying CDGA of $\mathcal{N}$ is minimal
(in the sense of Sullivan, see e.g. [2] or [3]) and $\mathcal{N}$ is semi-
simple. We call $\mathcal{N}$ the equivariant minimal model of $X$. In this
note, we will deduce results on equivariant minimal models using Lemma 2.2
below instead of the definition of $A_{red}(X)$.
To extract homotopical information such as the natural action of $\pi_{1}(X)$
on $\pi_{n}(X)\otimes_{\mathbb{Z}}\mathsf{k}$ and local coefficient cohomology
from the equivariant minimal model, we use dg-categories and extensions of
their objects. We shall recall some related notions. A dg-category is a
category whose sets of morphisms are complexes and the composition
$\mathrm{Hom}_{C}(V^{\prime},V^{\prime\prime})\otimes\mathrm{Hom}_{C}(V,V^{\prime})\to\mathrm{Hom}_{C}(V,V^{\prime\prime})$
is a chain map, see [8] or [10]. An isomorphism in $C$ is an invertible
morphism (of degree 0) which is a cocycle. We say a dg functor $F:C\to
C^{\prime}$ between dg-categories is a quasi-equivalence if it induces an
equivalence $Z^{0}C\to Z^{0}C^{\prime}$ between the categories of 0-th
cocycles and quasi-isomorphisms between complexes of morphisms. An extension
in a dg-category $C$ is a pair of maps $V^{\prime}\stackrel{{\scriptstyle
p}}{{\to}}\tilde{V}\stackrel{{\scriptstyle q}}{{\to}}V$ of degree 0 such that
$dp=0,dq=0,qp=0$, and a splitting exists. Here, a splitting of $(p,q)$ is a
pair of maps
$V^{\prime}\stackrel{{\scriptstyle\alpha}}{{\leftarrow}}\tilde{V}\stackrel{{\scriptstyle\beta}}{{\leftarrow}}V$
of degree 0 satisfying $\alpha\beta=0,\alpha p=id_{V^{\prime}},q\beta=id_{V}$,
and $p\alpha+\beta q=id_{\tilde{V}}$. We call the class $[\alpha d\beta]\in
H^{1}(\mathrm{Hom}_{C}(V,V^{\prime}))$ the extension class of $(p,q)$. The
following lemma is stated in [8] and the proof is easy and omitted.

######  Lemma 2.1 ([8]).

Let $C$ be a dg-category and $V^{\prime}\stackrel{{\scriptstyle
p_{i}}}{{\to}}V_{i}\stackrel{{\scriptstyle q_{i}}}{{\to}}V$ an extension in
$C$ with a splitting
$V^{\prime}\stackrel{{\scriptstyle\alpha_{i}}}{{\leftarrow}}V_{i}\stackrel{{\scriptstyle\beta_{i}}}{{\leftarrow}}V$
for $i=1,2$. If the extensions $V_{1},V_{2}$ have the same extension class,
$V_{1}$ and $V_{2}$ are isomorphic. An isomorphism $V_{1}\to V_{2}$ is given
by $p_{2}\alpha_{1}+\beta_{2}q_{1}-p_{2}\gamma q_{1}$ where
$\gamma\in\mathrm{Hom}^{0}_{C}(V,V^{\prime})$ is a chain with
$d\gamma=\alpha_{2}d\beta_{2}-\alpha_{1}d\beta_{1}$. ∎

We shall recall the completion of a dg-category due to Simpson [8]. A Maurer-
Cartan (MC-) element on an object $V$ of a dg-cateory $C$ is an element
$\eta\in\mathrm{Hom}^{1}_{C}(V,V)$ satisfying $d\eta+\eta^{2}=0$. Let
$\bar{C}$ be a dg-category defined as follows. An object of $\bar{C}$ is a
pair $(V,\eta)$ of an object of $C$ and an MC-element on it. The complex
$\mathrm{Hom}_{\bar{C}}((V,\eta),(V^{\prime},\eta^{\prime}))$ of morphisms is
the same as $\mathrm{Hom}_{C}(V,V^{\prime})$ as a graded vector space, but its
differential is given by $d_{\bar{C}}f=d_{C}f+\eta^{\prime}f-(-1)^{|f|}f\eta$.
$C$ is identified with a full sub dg-category of $\bar{C}$ via the inclusion
$V\mapsto(V,0)$. Any class $[\omega]\in
H^{1}(\mathrm{Hom}_{\bar{C}}((V^{\prime},\eta^{\prime}),(V,\eta)))$ comes from
an extension. Concretely speaking, an extension which induces the class is
given by $(V^{\prime},\eta^{\prime})\to\left(V^{\prime}\oplus
V,\left[\begin{array}[]{cc}\eta^{\prime}&\omega\\\
0&\eta\end{array}\right]\right)\to(V,\eta)$ with an obvious splitting. Let
$\widehat{C}\subset\bar{C}$ be the full sub dg-category spanned by objects
obtained by successive extensions of objects of $C$. The dg-category
$\widehat{C}$ is called the completion of $C$ and has a universal property,
see [8]. For a simplicial set $X$, let $T_{PL}(X)$ be the dg-category defined
as follows. The objects of $T_{PL}(X)$ are the finite dimensional local
systems of vector spaces on $X$ and the complex of morphisms is given by
$\mathrm{Hom}_{T_{PL}(X)}(V,V^{\prime})=C_{PL}(X,\mathfrak{Hom}(V,V^{\prime}))$
where $\mathfrak{Hom}(V,V^{\prime})$ is the natural hom-local system with
$\mathfrak{Hom}(V,V^{\prime})_{\sigma}=\\{\mathsf{k}\text{-linear maps
:}V_{\sigma}\to V^{\prime}_{\sigma}\\}$. Let $T^{ss}_{PL}(X)\subset T_{PL}(X)$
be the full sub dg-category spanned by semi-simple local systems. By the
universal property, we have quasi-equivalences
$\widehat{T^{ss}_{PL}(X)}\simeq\widehat{T_{PL}(X)}\simeq T_{PL}(X)$. Let $B$
be a semi-simple $\pi_{1}(X)$-CDGA. We define dg-categories $\mathsf{T}^{ss}B$
and $\mathsf{T}B$ as follows. The objects of $\mathsf{T}^{ss}B$ are the finite
dimensional semi-simple representations of $\pi_{1}(X)$ and
$\mathrm{Hom}_{\mathsf{T}^{ss}B}(V,V^{\prime})$ is given by the complex of
invariants $(B\otimes\mathfrak{Hom}(V,V^{\prime}))^{\pi_{1}(X)}$ for the
diagonal action, with restriction of the standard differential on the tensor
product. Here, $\mathfrak{Hom}(V,V^{\prime})$ is the hom-representation given
by the conjugate action on the space of linear maps $V\to V^{\prime}$ and
regarded as a complex concentrated in degree 0. We set
$\mathsf{T}B=\widehat{\mathsf{T}^{ss}B}$. The dg-categories $C=T_{PL}^{ss}(X)$
and $\mathsf{T}^{ss}B$ have a natural tensor (dg-)functor $\otimes:C\times
C\to C$. For $T_{PL}^{ss}(X)$, it is given by the tensor of local systems on
objects and the tensor of the product of forms with the tensor of maps on
morphisms. For $\mathsf{T}^{ss}B$, it is defined similarly using the product
on $B$.

######  Lemma 2.2 (Lemma 3.3.1 and Theorem 3.4.5 of [10]).

Under the above notations, for any semi-simple $\pi_{1}(X)$-CDGA $B$ which is
connected to $A_{red}(X)$ by a zigzag of quasi-isomorphisms between semi-
simple $\pi_{1}(X)$-CDGA’s, we have zigzags of quasi-equivalences

$\mathsf{T}^{ss}B\simeq\mathsf{T}^{ss}A_{red}(X)\simeq
T^{ss}_{PL}(X),\quad\mathsf{T}B\simeq\mathsf{T}A_{red}(X)\simeq T_{PL}(X).$

We can take the left zigzag so that it preserves the tensor functors on the
dg-categories and induces the equivalence between the category of semi-simple
representations and the one of semi-simple local systems given in former part
of this section on the categories of 0-th cocycles. ∎

So, for a given finite dimensional representation $V$ of $\pi_{1}(X)$ (or
local system on $X$), if we obtain the equivariant minimal model $\mathcal{N}$
of $X$ and an object $(V_{0},\eta)$ in $\mathsf{T}\mathcal{N}$ corresponding
to $V$ through the zigzag in this lemma, we can compute the local coefficient
cohomology $H^{*}(X,V)$ as the cohomology of $((\mathcal{N}\otimes
V_{0})^{\pi_{1}(X)},d_{\eta}=d+\eta)$. In general, $V_{0}$ is (isomorphic to)
the semi-simplification of $V$. In other words, let $0=V^{0}\subset
V^{1}\subset\cdots\subset V^{k}=V$ be a sequence of $\pi_{1}(X)$-invariant
subspaces such that $V^{i+1}/V^{i}$ is semi-simple for each $0\leq i\leq k-1$,
and $V_{0}$ is isomorphic to $\oplus_{0\leq i\leq k-1}V^{i+1}/V^{i}$. In next
section, we illustrate how MC-elements and representations which are not semi-
simple correspond in the case of very simple group
$\pi=\mathbb{Z}\times\mathbb{Z}$.

##  3\. MC-elements corresponding to representations of
$\mathbb{Z}\times\mathbb{Z}$

Throughout the rest of this note, we set $\pi=\mathbb{Z}\times\mathbb{Z}$, and
let $g_{i}$ denote a fixed generator of the $i$-th component of $\pi$ for
$i=1,2$. We use the realization of the $K(\pi,1)$-space

$T^{2}=[0,1]^{2}/\\{(0,t_{2})\sim(1,t_{2}),\ (t_{1},0)\sim(t_{1},1)\\}.$

$T^{2}$ consists of one 0-cell, two 1-cells $\sigma_{1}=\\{(t_{1},0)\\}$,
$\sigma_{2}=\\{(0,t_{2})\\}$ and one 2-cell $\tau=\\{(t_{1},t_{2})\\}$. (We
are formally working with simplicial sets so these cells are subdivided into
simplexes implicitly.) Furthermore, we always identify a representation $V$ of
$\pi$ with the local system $L$ of vector spaces on $T^{2}$ defined as
follows. We set $L_{\sigma}=V$ for each cell (or simplex) $\sigma$, and the
face map $d_{0}$ corresponding to the inclusion $\\{1\\}\to[0,1]$ is given by
the action of $g_{i}$ on $\sigma_{i}$ for $i=1,2$, and the other face map
$d_{1}$ is the identity. We use the same notations for the corresponding local
systems as the representations in what follows. We shall make explicit the
correspondence between MC-elements and representations in our case.

######  Lemma 3.1.

Let $(V,r_{i})$, $(V^{\prime},r^{\prime}_{i})$ be two representations of
$\pi$, where $r_{i}\in GL(V)$ and $r^{\prime}_{i}\in GL(V^{\prime})$ denote
the action of $g_{i}$ for $i=1,2$. We cosider a 1-cocycle in
$C_{PL}(T^{2},\mathfrak{Hom}(V,V^{\prime}))$ of the form of
$\omega=f_{1}dt_{1}+f_{2}dt_{2}$ where $f_{i}$ is a $\mathsf{k}$-linear map
$V\to V^{\prime}$ which is constant for the variables $t_{1},t_{2}$. A
representation $(\tilde{V},\tilde{r}_{i})$ which has the extension class
$[\omega]$ is given by $\tilde{V}=V^{\prime}\oplus V$ as a vector space with

$\tilde{r}_{i}=\left[\begin{array}[]{cc}r^{\prime}_{i}&-r^{\prime}_{i}f_{i}\\\
0&r_{i}\end{array}\right].$

A splitting
$V^{\prime}\stackrel{{\scriptstyle\alpha}}{{\leftarrow}}\tilde{V}\stackrel{{\scriptstyle\beta}}{{\leftarrow}}V$
is given by

$\alpha=\biggl{[}id_{V^{\prime}}\ \
-(t_{1}f_{1}+t_{2}f_{2})\biggr{]},\qquad\beta=\left[\begin{array}[]{c}\displaystyle
t_{1}f_{1}+t_{2}f_{2}\vspace{2mm}\\\ id_{V}\end{array}\right].$

###### Proof.

Since $\omega$ is a global section, we have

$r^{\prime}_{j}f_{i}r_{j}^{-1}=f_{i}$

for $i=1,2$ and $j=3-i$, which implies the commutativity of $\tilde{r}_{1}$
and $\tilde{r}_{2}$. Similarly, we can verify that $\alpha$ is well-defined as
follows.

$\begin{split}r^{\prime}_{1}(\alpha|_{t_{1}=1})\tilde{r}_{1}^{-1}&=r^{\prime}_{1}\biggl{[}id_{V^{\prime}}\
\ -(1\cdot
f_{1}+t_{2}f_{2})\biggr{]}\left[\begin{array}[]{cc}(r^{\prime}_{1})^{-1}&f_{1}r_{1}^{-1}\vspace{2mm}\\\
0&r_{1}^{-1}\end{array}\right]\\\ &=\biggl{[}id_{V^{\prime}}\ \
r^{\prime}_{1}f_{1}r_{1}^{-1}-r^{\prime}_{1}(f_{1}+t_{2}f_{2})r_{1}^{-1}\biggr{]}\\\
&=\biggl{[}id_{V^{\prime}}\ \
-t_{2}r^{\prime}_{1}f_{2}r_{1}^{-1}\biggr{]}=\alpha|_{t_{1}=0}.\end{split}$

The verification on $\beta$ is similar and the rest of conditions on the
splitting is easily verified. ∎

######  Example 3.2.

In the following examples (1), (2), all representations are supposed to have
trivial action of $g_{2}$, and expressed as the action of $g_{1}$.

  1. (1)

We consider the 2-dimensional representation
$V_{1}=\left[\begin{array}[]{cc}c&e\\\ 0&c\end{array}\right]$. By Lemma 3.1,
this is isomorphic to the object $V_{2}=\left(\left[\begin{array}[]{cc}c&0\\\
0&c\end{array}\right],\eta=\left[\begin{array}[]{cc}0&-(e/c)dt_{1}\\\
0&0\end{array}\right]\right)$ in $\widehat{T_{PL}(T^{2})}$. By Lemma 2.1, an
isomorphism $V_{1}\to V_{2}$ is given by

$\left[\begin{array}[]{cc}1&(e/c)t_{1}\\\ 0&1\end{array}\right].$

  2. (2)

We consider the 3-dimensional representation
$V_{3}=\left[\begin{array}[]{ccc}c&e&h\\\ 0&c&f\\\ 0&0&c\end{array}\right]$. A
cocycle representing the class of the extension $V_{1}\to
V_{3}\to(\mathsf{k},c)$ is given by

$-\left[\begin{array}[]{cc}c&e\\\
0&c\end{array}\right]^{-1}\left[\begin{array}[]{c}h\\\
f\end{array}\right]dt_{1}=-c^{-2}\left[\begin{array}[]{c}ch-ef\\\
cf\end{array}\right]dt_{1}.$

The pushforward by the isomorphism in (1) sends this cocycle to the cocycle
$-c^{-2}\cdot{}^{t}[ch-ef(1-t_{1}),\ cf]dt_{1}$, where ${}^{t}(-)$ denotes the
transposition. Since ${}^{t}[t_{1}(1-t_{1}),\ 0]$ belongs to

$\left(C^{*}_{PL}\left(T^{2},\left[\begin{array}[]{cc}c&0\\\
0&c\end{array}\right]\right),d_{\eta}=d+\eta\right)$

and $d_{\eta}{}^{t}[t_{1}(1-t_{1}),\ 0]={}^{t}[1-2t_{1},\ 0]dt_{1}$, last
cocycle is cohomologous to
$-c^{-2}\cdot{}^{t}[ch-ef/2,\ cf]dt_{1}$. So $V_{3}$ is isomorphic to

$V_{4}=\left(\left[\begin{array}[]{ccc}c&0&0\\\ 0&c&0\\\
0&0&c\end{array}\right],-c^{-2}\left[\begin{array}[]{ccc}0&ce&ch-ef/2\\\
0&0&cf\\\ 0&0&0\end{array}\right]dt_{1}\right)$

in $\widehat{T_{PL}(T^{2})}$.

  3. (3)

We consider the 3-dimensional representation $V_{5}$ on which $g_{1}$ and
$g_{2}$ act by

$\left[\begin{array}[]{ccc}c_{1}&e_{1}&0\\\ 0&c_{1}&0\\\
0&0&c_{1}\end{array}\right],\text{\ and\
}\left[\begin{array}[]{ccc}c_{2}&0&e_{2}\\\ 0&c_{2}&0\\\
0&0&c_{2}\end{array}\right],$

respectively. Similarly to (2), we see that $V_{5}$ is isomorphic to

$\left(V^{ss}_{5},\left[\begin{array}[]{ccc}0&-(e_{1}/c_{1})dt_{1}&-(e_{2}/c_{2})dt_{2}\\\
0&0&0\\\ 0&0&0\end{array}\right]\right)$

in $\widehat{T_{PL}(T^{2})}$ where $V^{ss}_{5}$ is the 3-dimensional
representation on which $g_{i}$ acts by the scalar multiplication of $c_{i}$
for $i=1,2$.

##  4\. An example of local system of CDGA’s

We shall define our main example of $A_{T^{2}}$-algebra $\mathcal{L}$ which is
a slight generalization of Example 6.7 of [4]. We set $(\Lambda
Z,d)=(\Lambda(x,y,z,w,u),d)$ with $x,\ y,\ z$ cocycles of degree 3, $w$ a
cocycle of degree 6, $\deg u=5,\ du=yz$. We set
$\mathcal{L}_{\sigma}=A_{T^{2},\sigma}\otimes(\Lambda Z,d)$ for each cell
$\sigma$. To define a local system, it is enough to define face maps for the
cell structure given in the beginning of previous section. Let $a_{i},b_{i}$
be units in $\mathsf{k}$ for $i=1,2$. Let $d_{j}$ be the degeneracy map
corresponding to the map $\\{1-j\\}\subset[0,1]$ for $j=0,1$ as before. The
map $d_{0}$ on $\sigma_{1}$ is given by $d_{0}(x)=a_{1}x,\
d_{0}(y)=b_{1}y,d_{0}(z)=a_{1}z,\ d_{0}(w)=a_{1}b_{1}(w+xy),$ and
$d_{0}(u)=a_{1}b_{1}u$. $d_{0}$ on $\sigma_{2}$ is given by
$d_{0}(x)=a_{2}(x+z),\ d_{0}(y)=b_{2}y,\ d_{0}(z)=a_{2}z,\
d_{0}(w)=a_{2}b_{2}w,$ and $d_{0}(u)=a_{2}b_{2}u$. We set $d_{1}=id$ on
$\sigma_{i}$ for $i=1,2$. The map
$d_{ij}:\mathcal{L}_{\tau}\to\mathcal{L}_{\sigma_{i}}$ corresponding to the
inclusion $[0,1]\to[0,1]^{\times 2},\
t\mapsto\left\\{\begin{array}[]{cc}(t,1-j)&(i=1)\\\
(1-j,t)&(i=2)\end{array}\right.$ is given by the following formulas for
$i=1,2,j=0,1$. $d_{i0}(y)=b_{3-i}y,\ d_{i0}(z)=a_{3-i}z,\
d_{i0}u=a_{3-i}b_{3-i}u$ for $i=1,2$, and $d_{10}(x)=a_{2}(x+z),\
d_{20}(x)=a_{1}x$, and $d_{10}(w)=a_{2}b_{2}(w-t_{1}yz-dt_{1}u),\
d_{20}(w)=a_{1}b_{1}(w+xy)$, and $d_{i1}=id$ (see Figure 1). This is well-
defined as a 1-connected $A_{T^{2}}$-algebra of finite type since this is
extendable for some subdivision of $T^{2}$, but this is not an
$A_{T^{2}}$-minimal model in the sense of Definition 3.5 of [4], where we put
$A=A_{T^{2}}$. For our purpose, this local system model works well enough.
In the rest of the note $X$ denotes the space corresponding to $\mathcal{L}$
through the equivalence of [4]. More precisely, the pointed simplicial set $X$
is the image of the $A_{T^{2}}$-minimal model of $\mathcal{L}$ by the
realization functor $\langle\ \rangle$ defined there. To compute the
equivariant minimal model of $X$, we first give the equivariant minimal model
of $T^{2}$. The following lemma is a special case of Theorem 4.2.3 of [10]. We
give a shorter proof for this case here.

######  Lemma 4.1.

The equivariant minimal model of $T^{2}$ is given by the $\pi$-CDGA
$\Lambda(s_{1},s_{2})$ with $s_{1},s_{2}$ cocycles of degree 1 with trivial
action.

###### Proof.

Let $\mathcal{N}$ be the equvariant minimal model of $T^{2}$. Since the semi-
simplicity of representations is preserved by extension of scalars, we may
assume that $\mathsf{k}$ is algebraically closed. Since commutative matrixes
are triangularizable simultaneously, the irreducible representations of $\pi$
are 1-dimensional. By straightforward computation, $H^{*}(T^{2},V)=0$ for a
non-trivial irreducible representation $V$ and clearly
$H^{*}(T^{2},V)=\Lambda(s_{1},s_{2})\otimes V$ for a trivial $V$. This means
that the map $(\mathcal{N}^{\pi}\otimes V)^{\pi}\to(\mathcal{N}\otimes
V)^{\pi}(\simeq C^{*}_{PL}(T^{2},V))$ induced by the inclusion
$\mathcal{N}^{\pi}\to\mathcal{N}$ is a quasi-isomorphism for any irreducible
$V$, where $\pi$ acts on the subalgebra of invariants $\mathcal{N}^{\pi}$
trivially. Since $\mathcal{N}$ is semi-simple, we see that the inclusion
$\mathcal{N}^{\pi}\to\mathcal{N}$ is a quasi-isomorphism. If $V$ is a
1-dimensional trivial representation, $\mathcal{N}^{\pi}$ is isomorphic to the
endomorphism algebra $\mathrm{Hom}_{\mathsf{T}^{ss}\mathcal{N}}(V,V)$ by
definition, which in turn, is quasi-isomorphic to $A_{PL}(T^{2})$ as a CDGA by
Lemma 2.2. These observations imply the claim. ∎

######  Remark 4.2.

In the proof of Lemma 4.1 and the proof of Theorem 4.3 below, we apparently
use dg-categories. This is mainly for efficiency of notations and we may write
down these proofs using a concrete description of
$(\mathbb{Z}\times\mathbb{Z})^{red}$ and the standard interpretation of
representations as comodules over
$\mathcal{O}((\mathbb{Z}\times\mathbb{Z})^{red})$ instead of dg-categories
based on the same referred facts. In the algebraic description of the
cohomology of $X$ with local coefficients given below, we need dg-categories
(or some similar algebraic objects) more essentially since it is based on the
right zigzag of Lemma 2.2.

We define a semi-simple $\pi$-CDGA $\mathcal{M}$. We set
$\mathcal{M}=\Lambda(s_{1},s_{2},\bar{x},\bar{y},\bar{z},\bar{w},\bar{u})$
with $s_{1},s_{2}$ cocycles of degree 1, $\bar{y},\ \bar{z}$ cocycles of
degree 3, $\deg\bar{x}=3,\ d\bar{x}=s_{1}\bar{z}$, $\deg\bar{w}=6,\
d\bar{w}=s_{1}\bar{x}\bar{y}-s_{1}s_{2}\bar{u},\ \deg\bar{u}=5,$ and
$d\bar{u}=\bar{y}\bar{z}$. The action of $\pi$ on $\mathcal{M}$ is given by
$g_{i}\cdot\bar{x}=a_{i}\bar{x}$, $g_{i}\cdot\bar{y}=b_{i}\bar{y}$,
$g_{i}\cdot\bar{z}=a_{i}\bar{z}$, $g_{i}\cdot\bar{w}=a_{i}b_{i}\bar{w}$, and
$g_{i}\cdot\bar{u}=a_{i}b_{i}\bar{u}$ for $i=1,2$.
$\pi_{i}(X)$ has a topologically defined natural action of $\pi=\pi_{1}(X)$
(see e.g. [11]). We denote by $\pi_{i}(X)^{\vee}$ the linear dual of
$\pi_{i}(X)\otimes_{\mathbb{Z}}\mathsf{k}$ or the dual representation by abuse
of notations.

######  Theorem 4.3.

With the above notations, the equivariant minimal model of $X$ is (isomorphic
to) $\mathcal{M}$.

###### Proof.

Let $F$ be a fiber of the fibration $p:X\to T^{2}$ naturally associated to the
map $A_{T^{2}}\to\mathcal{L}$. The topologically defined action of $\pi$ on
$\pi_{i}(X)^{\vee}$ is identified with the action on the linear dual
$\pi_{i}(F)^{\vee}$ defined by lifting paths. In turn, this action is
identified with the action on the generators of $(\Lambda Z,d)$ induced by the
local system. So we have the isomorphisms of representations of $\pi$

$(\oplus_{i}\pi_{i}(X))^{\vee}\cong(\oplus_{i}\pi_{i}(F))^{\vee}\cong\mathsf{k}\langle
x,y,z,w,u\rangle.$

For example, the action on $\pi_{3}(X)^{\vee}$ is identified with the
representation

$g_{1}\mapsto\left[\begin{array}[]{ccc}a_{1}&0&a_{1}\\\ 0&b_{1}&0\\\
0&0&a_{1}\end{array}\right],\quad
g_{2}\mapsto\left[\begin{array}[]{ccc}a_{2}&0&0\\\ 0&b_{2}&0\\\
0&0&a_{2}\end{array}\right]$

(for the coordinate ${}^{t}[z\ y\ x]$) through this isomorphism. The semi-
simplification of this representation is given by the action of the diagonal
part of these matrixes. By this and similar observations for other degree,
together with Theorem 4.1.3 of [10] and Lemma 4.1, we see that the equivariant
minimal model of $X$ has generators with the same degree and action of $\pi$
as $\mathcal{M}$. So all we need to care is the differential. Let $L_{1}$ be
the 1-dimensional representation of $\pi$ given by $g_{i}\cdot v=a_{i}^{-1}v$
for $i=1,2$. Similarly, let $L_{2}$ be the 1-dimensional representation given
by $g_{i}\cdot v=b_{i}^{-1}v$. Let $S$ be a set of pairs $(k,l)$ of non-
negative integers such that the representation $L^{k}_{1}\otimes L^{l}_{2}$
with $(k,l)\in S$ appears exactly once in each isomorphism class of the
representations of the form $L_{1}^{k^{\prime}}\otimes L_{2}^{l^{\prime}}$
with $k^{\prime},l^{\prime}\geq 0$. Put $U=\oplus_{(k,l)\in S}L_{1}^{k}\otimes
L_{2}^{l}$. We define a commutative product on $U$ by the map

$(L_{1}^{k}\otimes L_{2}^{l})\otimes(L_{1}^{k^{\prime}}\otimes
L_{2}^{l^{\prime}})\cong L_{1}^{k+k^{\prime}}\otimes L_{2}^{l+l^{\prime}}\cong
L_{1}^{k^{\prime\prime}}\otimes L_{2}^{l^{\prime\prime}}$

where $(k^{\prime\prime},l^{\prime\prime})\in S$ and the first isomorphism is
the transposition and the second is given by $v_{1}^{k+k^{\prime}}\otimes
v_{2}^{l+l^{\prime}}\mapsto v_{1}^{k^{\prime\prime}}\otimes
v_{2}^{l^{\prime\prime}}$ for some fixed generators $v_{i}\in L_{i}$. Let
$\mathcal{N}$ be the equivariant minimal model of $X$ and $\mathcal{L}_{min}$
the $A_{T^{2}}$-minimal model of $\mathcal{L}$. We have the following quasi-
isomorphisms of semi-simple $\pi$-CDGA’s

$\mathcal{N}\cong(\mathcal{N}\otimes
U)^{\pi}\stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}C_{PL}^{*}(X,p^{*}U)\stackrel{{\scriptstyle\varphi_{2}}}{{\longleftarrow}}\Gamma(\mathcal{F}(X,p)\otimes
U)\stackrel{{\scriptstyle\varphi_{3}}}{{\longleftarrow}}\Gamma(\mathcal{L}_{min}\otimes
U)\stackrel{{\scriptstyle\varphi_{4}}}{{\longrightarrow}}\Gamma(\mathcal{L}\otimes
U).$

Here,

  * •

in the right four objects, $U$ is regarded as a local system on $T^{2}$ and
$\Gamma$ denotes the global section functor.

  * •

The products on these objects are natural ones. For example, the product on
$C^{*}_{PL}(X,p^{*}U)$ is the tensor of the product of forms and the product
on $U$.

  * •

The action on each object is given by the multiplication of
$a_{i}^{k}b_{i}^{l}$ on the summand tensored with $L_{1}^{k}\otimes
L_{2}^{l}$.

  * •

$\varphi_{1}$ is induced by the quasi-equivalence in Lemma 2.2.

  * •

$\varphi_{2}$ is the obvious inclusion through the identification
$C^{*}_{PL}(X,p^{*}U)=\Gamma(A_{PL,X}\otimes p^{*}U)$, which is actually an
isomorphism since $E\times_{K}\Delta^{|\sigma|}$ in the definition of
$\mathcal{F}(E,p)$ is simply connected.

  * •

$\varphi_{3}$ and $\varphi_{4}$ are induced by quasi-isomorphisms of local
systems $\mathcal{L}_{min}\to\mathcal{F}(X,p)$ and
$\mathcal{L}_{min}\to\mathcal{L}$, whose existence is ensured by Theorems 3.12
and 5.10 of [4].

The compatibility of $\varphi_{1}$ with the product and action is ensured by
the latter part of Lemma 2.2. Thus, $\mathcal{N}$ and
$\Gamma(\mathcal{L}\otimes U)$ are quasi-isomorphic. We shall prove that
$\mathcal{M}$ is quasi-isomorphic to $\Gamma(\mathcal{L}\otimes U)$ as a semi-
simple $\pi$-CDGA. By abuse of notations, we denote by
$y\in\Gamma(\mathcal{L}\otimes L_{2}),z\in\Gamma(\mathcal{L}\otimes L_{1})$
and $u\in\Gamma(\mathcal{L}\otimes L_{1}\otimes L_{2})$ the constant sections
to the representing generator tensored with the fixed generators of $L_{i}$.
We define an element $x^{\prime}\in\Gamma(\mathcal{L}\otimes L_{1})$ of degree
3 by $x^{\prime}_{\tau}=(-x+t_{2}z)\otimes v_{1}$. Since

$\begin{split}d_{10}(x^{\prime}_{\tau})&=a_{2}(-(x+z)+1\cdot z)\otimes
a_{2}^{-1}v_{1}=-x\otimes v_{1}=d_{11}(x^{\prime}_{\tau}),\text{ and}\\\
d_{20}(x^{\prime}_{\tau})&=a_{1}(-x+t_{2}z)\otimes
a_{1}^{-1}v_{1}=d_{21}(x^{\prime}_{\tau}),\end{split}$

$x^{\prime}$ is well-defined as a global section. We also have
$d(x^{\prime}_{\tau})=dt_{2}z\otimes v_{1}$. We define an element
$w^{\prime}\in\Gamma(\mathcal{L}\otimes L_{1}\otimes L_{2})$ of degree 6 by
$w^{\prime}_{\tau}=(-w+t_{1}xy-t_{2}dt_{1}u)\otimes v_{1}\otimes v_{2}$. We
see that $w^{\prime}$ is well-defined and $dw^{\prime}_{\tau}=(-dt_{1}xy-
dt_{1}dt_{2}u+t_{2}dt_{1}zy)\otimes v_{1}\otimes
v_{1}=(dt_{1}x^{\prime}y-dt_{1}dt_{2}u)_{\tau}$ by a straightforward
computation. We define a map $F:\mathcal{M}\to\Gamma(\mathcal{L}\otimes U)$ by
$s_{i}\mapsto dt_{i}$ for $i=1,2$, $\bar{\alpha}\mapsto\alpha$ for
$\alpha=y,z$ and $u$, $\bar{x}\mapsto x^{\prime}$, and $\bar{w}\mapsto
w^{\prime}$. Clearly, this is a map of $\pi$-CDGA. Since $d\bar{x}$ is not a
boundary in the subalgebra generated by $s_{1},s_{2},\bar{y},\bar{z}$ and we
have already known the generators of the graded algebra $\mathcal{N}$, we see
that $H^{\leq 4}(F)$ is an isomorphism and $H^{5}(F)$ is a monomorphism.
Similarly, as $d\bar{u}$ and $d\bar{w}$ are not boundaries in the subalgebras
of generated by generators of lower degrees, we conclude that $F$ is a quasi-
isomorphism. ∎

$x$$x$$x$$a_{2}(x+z)$$a_{1}x$$w$$w$$w$$a_{2}b_{2}(w-t_{1}yz-
dt_{1}u)$$a_{1}b_{1}(w+xy)$$d_{0}$$d_{1}$$d_{1}$$d_{0}$ Figure 1. Face maps

Note that the topologically defined action on $\pi_{i}(X)^{\vee}$ of
$\pi=\pi_{1}(X)$ is not compatible with the differential of $\mathcal{M}$ for
any graded linear identification between the generators of $\Lambda Z$ and
those of $\mathcal{M}$ with degree $\geq 3$. Neverthless, we can recover the
action on $\pi_{i}(X)^{\vee}$. Let $V^{i}$ be the space of the generators of
$\mathcal{M}$ of degree $i$ and $\mathcal{M}(i-1)\subset\mathcal{M}$ the
subalgebra generated by elements of degree $\leq i-1$. As in Theorem 1.0.2 of
[10], the semi-simplification of $\pi_{i}(X)^{\vee}$ is isomorphic to $V^{i}$,
and the extension is encoded in the differential. The differential
$d:V^{i}\to\mathcal{M}(i-1)^{i+1}\oplus\mathcal{M}^{1}\otimes V^{i}$ followed
by the projection to $\mathcal{M}^{1}\otimes V^{i}$ is naturally regarded as
an MC-element $\eta$, and $(V^{i},\eta)$ is isomorphic to the object in
$\mathsf{T}\mathcal{M}$ corresponding to the topological action on
$\pi_{i}(X)^{\vee}$ through the equivalence of Lemma 2.2 (see sub-subsections
2.2.1 and 3.3.1 of [10] for the sign convention). The MC-element corresponding
to the differential on $V^{3}$ is given by

$\left[\begin{array}[]{ccc}0&0&s_{1}\\\ 0&0&0\\\ 0&0&0\end{array}\right].$

By Lemma 3.1, this corresponds to the representation
$\left(\left[\begin{array}[]{ccc}a_{1}&0&-a_{1}\\\ 0&b_{1}&0\\\
0&0&a_{1}\end{array}\right],\left[\begin{array}[]{ccc}a_{2}&0&0\\\
0&b_{2}&0\\\ 0&0&a_{2}\end{array}\right]\right)$. Clearly, this is isomorphic
to the action obtained from the original local system in the proof of the
theorem.
A nilpotent model i.e., a CDGA quasi-isomorphic to the usual polynomial de
Rham algebra $A_{PL}(X)=\Gamma(A_{X})$ is given by $\mathcal{M}^{\pi}$. So if
for any pair of non-negative integers $(k,l)\not=(0,0)$,
$a_{i}^{k}b_{i}^{l}\not=1$ holds for at least one of $i=1,2$, we have
$A_{PL}(X)\simeq\Lambda(s_{1},s_{2})$. If $a_{i}$ is not a root of unity for
at least one of $i$, and $b_{i}=a_{i}^{-1}$ for both of $i=1,2$, $A_{PL}(X)$
is quasi-isomorphic to the subalgebra of $\mathcal{M}$ generated by
$s_{1},s_{2},xy,yz,u,w$.
We can obtain an algebraic complex which computes cohomology of $X$ with
coefficients in $V_{3}$ in Example 3.2 using the identification given there.
Let $V_{3}^{ss}$ be the semi-simplification of $V_{3}$ so $g_{1}$ acts on
$V_{3}^{ss}$ via scalar multiplication by $c$ and $g_{2}$ does trivially. The
object in $\mathsf{T}\mathcal{M}$ corresponding to $V_{3}$ is given by simply
replacing $dt_{1}$ in $V_{4}$ with $s_{1}$. So $C^{*}_{PL}(X,V_{3})$ is quasi-
isomorphic to the complex $((\mathcal{M}\otimes V_{3}^{ss})^{\pi},D)$ with the
differential $D$ given by

$D(\omega\otimes{}^{t}[k_{1}\ k_{2}\ k_{3}])=d\omega\otimes{}^{t}[k_{1}\
k_{2}\ k_{3}]+s_{1}\omega\otimes M\,{}^{t}[k_{1}\ k_{2}\ k_{3}]$

where $M=-c^{-2}\left[\begin{array}[]{ccc}0&ce&ch-ef/2\\\ 0&0&cf\\\
0&0&0\end{array}\right]$. Similarly, for $V_{5}$ in the same example,
$C^{*}_{PL}(X,V_{5})$ is quasi-isomorphic to the complex $((\mathcal{M}\otimes
V_{5}^{ss})^{\pi},D^{\prime})$ with the differential

$D^{\prime}(\omega\otimes{}^{t}[k_{1}\ k_{2}\
k_{3}])=d\omega\otimes{}^{t}[k_{1}\ k_{2}\
k_{3}]-(e_{1}/c_{1})s_{1}\omega\otimes{}^{t}[k_{2}\ 0\
0]-(e_{2}/c_{2})s_{2}\omega\otimes{}^{t}[k_{3}\ 0\ 0].$

## References

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  * [2] A. K. Bousfield, V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976) no. 179, ix+94 pp.
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# Networks of Twin Peaks: the Dale Cooper Effect

Harun Šiljak, Trinity College Dublin, Ireland

##  1 The shape of stories

When Kurt Vonnegut speaks of the shape of stories [1], he is in the realm of
calculus, continuous functions describing dynamics of the plot, the highs and
lows of the protagonist’s emotions, the speed at which the story unravels. In
a way, it corresponds to the historical development of calculus as a way to
interpret motion as a function of time. The abscissa is the time, and the
ordinate axis is something important for the hero–emotional state, for
example. Figure 1 provides a detailed illustration from contemporary fantasy
fiction.

Figure 1: Annotated emotional arc ofHarry Potter and the Deathly Hallows, by
JK Rowling (from [2])

There is another shape of a story we are interested in: one of its people, the
world it lives in. That shape does not focus on the actions, but on the
structures. It describes the contents of the carrier bag, like in Ursula K. Le
Guin’s Carrier Bag Theory of Fiction [3], and they are neatly arranged in
graphs, maps, and trees (as the eponymous work by Franco Moretti suggests
[4]). The tools from it come from graph theory and algebra–and once put in
context of arbitrary large networks, the field of research is usually dubbed
network theory [5]. Are there only so many community types, networks of
interactions in life and in literature? What can mathematics tell us about it?
Are networks of characters in fiction anything like our social networks? A
growing body of literature has been dealing with this application of network
theory; investigating epic stories from the past, superhero universes from the
modern era, etc. The more characters exist in a work of fiction, the more
interesting and relevant network theoretic results become: patterns emerge,
clusters and communities, and unexpected characters reveal themselves as
keystone elements of networks: once removed, everything falls apart.

“I would go so far as to say that the natural, proper, fitting shape of the
novel might be that of a sack, a bag. A book holds words. Words hold things.
They bear meanings. A novel is a medicine bundle, holding things in a
particular, powerful relation to one another and to us. ” [4] Here, in this
quote from Le Guin lies our motivation to explore the networks, rather than
forward-driving, sharp-edged plots of Vonnegut. She proceeds: “[I]t’s clear
that the Hero does not look well in this bag. (…) That is why I like novels:
instead of heroes they have people in them.” We are curious what the hero
effect is in a network: are they just like anyone else, do they fit in the
bag? Finally, we erase the linear time of Vonnegut’s plots by observing
networks instead of actions, algebra instead of calculus. The community in the
network exists independently of the passage of time, and it needs to feel
real. Again, Le Guin writes “If, however, one avoids the linear, progressive,
Time’s-(killing)-arrow (…) one pleasant side effect is that science fiction
can be seen as (…) in fact less a mythological genre than a realistic one.”

In this work, we scratch the surface of character networks in the cult TV show
_Twin Peaks_. For the readers who have not seen it, the first season of Twin
Peaks revolves around a murder investigation in the town of Twin Peaks. FBI
agent Dale Cooper comes to town to investigate alongside the local police
department, and the viewers get to see networks of crime, romantic
relationships, and law enforcement entangle and blur the lines between.
Compared to other network theoretic treatises, this network is small scale,
with sixty characters introduced over the course of eight episodes. The
community is not even expected to behave as a social network; the crime-
investigation-driven plot of _Twin Peaks_ was expected to merely shine the
light on those relationships relevant to the murder of Laura Palmer,
hyperfocusing on a network built by the law enforcement officers conducting
the investigation. Not only did the network analysis confirm that the
community is more than just a set of photos pinned to a plywood board at the
Sheriff’s office: we also discovered a new storytelling network phenomenon we
called the Dale Cooper Effect, a phase transition in network structure.

##  2 Graphs and associated tools

A graph is a mathematical object made of vertices and edges (Fig. 2 left),
which in the context of networks are often called nodes and links,
respectively. A pair of vertices can be connected by a single edge or no
edges–e.g. B and E have no edge connecting them, while A and B do. For large
graphs and for algebraic manipulation of them, a commonly used representation
is that of the adjacency matrix (Fig. 2 right). It is a square matrix whose
dimension corresponds to the number of vertices in the graph. As Fig. 2 shows,
if there is an edge between the vertex corresponding to the row, and the one
corresponding to the column, we place a “1” at their intersection in the
matrix. Otherwise, that element is a “0” (no edge). If the graphs have no
loops (i.e. if connecting vertex A to itself via an edge is not an option),
the diagonal elements of the adjacency matrix are all zero. Furthermore, if
the graph is undirected (i.e. if an edge between A and B is interpreted both
as a connection of A with B and B with A), the matrix is symmetric.

Figure 2: A graph and its adjacency matrix

An extension of the concept of the graph is a multigraph, where multiple edges
connecting same two vertices are allowed. An example of this is shown in Fig.
3, where vertices A and B now have two common edges. This maps directly into
the adjacency matrix, and the entries AB and BA are now equal to two. This
multigraph can also be interpreted in terms of weighted graphs: all edges
would again be single, but the one between A and B would have the weight of 2,
while others would have the weight of 1. Sometimes, for example in
collaboration networks where the goal is to assess the distance between
entities represented by nodes (e.g. co-authors in mathematics or co-stars in
movies, c.f. Erdős number, Bacon number), $n$ multiple links between two nodes
can be taken to correspond to the weight of $1/n$.

At this point, it is useful to define the degree of a vertex in a graph: for
undirected graphs it is simply the number of edges connected with it (i.e. the
sum of the corresponding row/column in the adjacency matrix). In the example
in Fig. 2, the degree of nodes A and D is 3, the degree of the node C is 2,
and the degree of nodes B and E is 1.

Figure 3: A multigraph and its adjacency matrix

When drawing conclusions from networks appearing in the world around us, it is
often useful to compare their structure to that of basic network models;
common features stand out and cross over in different contexts, suggesting
general patterns. Two common random network models are the Erdős-Renyi (random
network) and the Barabasi-Albert (preferential attachment) model.

In the Erdős-Renyi model, the adjacency matrix is filled by tossing a biased
coin. Let it be a $200\times 200$ matrix, and let the probability of two
vertices being connected by an edge be $0.1$. The upper triangle of the
adjacency matrix is then populated entry by entry with the result of tossing a
coin which has 10% chance of heads (1), and 90% tails (0). After the upper
triangle is filled, it is symmetrically copied into the lower triangle,
finalising the undirected Erdős-Renyi graph generation.

The Barabasi-Albert model recognises the observation that many networks in the
real world are built with the mechanism of preferential attachment. When a new
node is added to such a network, the links it forms are not drawn with equal
probability from the set of all nodes already in the network. The nodes that
are already well-connected with other nodes will have more chance in
attracting the new node. Let our network once again have 200 nodes; in the
process of preferential attachment we will be adding one node at a time,
assign 4 links connecting it to existing nodes in the network, and end the
process once all 200 nodes are added. The probability that the newly added
node will establish a link with a particular existing node is proportional to
the existing node’s degree.

In Figure 4 we depict the complementary cumulative distribution function of
the degrees in two network model examples we described. The plot represents
the probability that a node in the network has degree $X$ greater than some
value $x$. For an $n$-node random network, the degree of a node is a function
of $n-1$ coin tosses and hence is a function of scale: 50% of the nodes will
be expected to have, in our numerical example, degree lower than $200\cdot
0.1=20$, and 50% will be expected to be over it. However, for the preferential
attachment network, no such scale exists as the network generative process
could have continued beyond $n$ nodes (this is why we often refer to these
networks as scale-free). The complementary cumulative distribution function
for such a network is theoretically a straight line in a log-log scale plot
(here we present the actual plot from 200-node network to show the deviations
seen at small scales for a single instance of a network). While the
“preferential attachment results in power law” maxime often holds, interesting
exceptions have been observed–a famous one is the citation distribution of
Physical Review papers demonstrated by Sid Redner to follow a log-normal
distribution [6].

Figure 4: Examples of degree distributions for random networks

Another measure of interest for us in this note is that of assortativity. Are
nodes with large/small degrees more likely to be connected to other vertices
with large/small degrees in a network? In real-world social network, it is
expected to see more popular nodes of the network to cluster together, while
in networks such as the internet, webpages with few links to other webpages
are more likely to be connected to well-linked hubs. The former is the case of
a positive assortativity coefficient, and the latter of a negative one. Random
networks, such as ones created by processes introduced earlier in this
section, are expected to have a neutral (zero) assortativity.

The common reason to investigate networks in works of fiction is to compare
their statistical properties to those of real networks. For example, in [7]
the authors apply the network theory tools to study networks of characters in
the epic narratives of The Beowulf, The Táin Bó Cúailnge and The Iliad. The
network of the Iliad was found to have features of realistic networks such as
positive assortativity, scale-free behaviour. For Beowulf, these properties
are achieved once the protagonist, Beowulf, is removed from the
story–Beowulf’s disproportionate degree of interaction with the world
otherwise skews it. For The Táin Bó Cúailnge, the key to reaching realistic
network properties has been removing one-off interactions between the six
protagonists of the epic and the remainder of the world. There is always a
component of “historicity evaluation” when discussing the studies of
epics–assessing the realness of social networks in them is a factor in
thinking about their genesis. For _Twin Peaks,_ we have no such concerns; it
is a fantasy world for which we do not expect to be a snapshot of the society.
However, we are curious to see what the storytelling structure reveals.

##  3 Twin Peaks

To investigate the networks of interaction in _Twin Peaks,_ we analysed the
transcripts from season 1 of the show (total of 8 episodes)[8]. The choice to
restrict the dataset to season 1 was motivated by the evolution of the script
and cast in season 2, followed by the constraints passage of time put on
season 3 (filmed two decades after the first two). Season 1 was hypothesised
to be the smallest homogeneous unit appropriate for analysis. The adjacency
matrix was formed chronologically: every time a new character appears, a new
node is added to the graph. For every scene two characters share screentime
in, a new edge is added to the graph, allowing multiple edges between the same
pair of nodes. If three characters appear in the scene, all pairs get a new
edge.

Figure 5: Adjacency matrices from Season 1 of Twin Peaks: graph (left) and
multigraph (right)

The resulting matrix is graphically presented in Fig. 5: our visualisation is
similar to that of the Les Miserables Co-occurrence [9]–the difference is in
ordering, as we preserve the order of appearance in the matrix. The left
representation corresponds to the binary interpretation of the graph seen in
Fig. 2: characters either share screentime, or they don’t, the quantity of
time is irrelevant. Red squares correspond to edges of the graph, i.e.
characters that are connected. On the right, the shade of red changes with
respect to the number of scenes characters have in common (i.e. interpretation
akin to Fig. 3)–brighter means more interactions.

We recognise that the characteristic “cross” in the middle of the matrix,
dividing it into four quadrants is the protagonist of the show, _Special Agent
Dale Cooper_. Cooper appears 36 minutes into the pilot (first episode of the
show), and he is the median character: $\approx 30$ characters are introduced
before him, and $\approx 30$ after him (as we stated already, the adjacency
matrix is filled chronologically, so $n$th row/column of it correspond to the
$n$th character to appear in the show). If we now divide the characters into
two categories–those appearing before Cooper (denoted BC in the remainder of
this paper) and those after Dale (AD). The brightest points in the right
adjacency matrix are usually either couples, or the Twin Peaks Police
Department combined with Cooper. Graphically, this is shown in Fig. 6, while
the most popular nodes and links are tabulated in Table 1.

Figure 6: The graphical representation of the Twin Peaks network: the black nodes are BC, the red are AD | The most connected characters | # | The most repeated links | #
---|---|---|---|---
1 | Dale Cooper | 35 | Cooper & Truman | 47
2 | Sheriff Truman | 35 | Truman & Lucy | 16
3 | Deputy Andy | 20 | Cooper & Hawk | 15
4 | Donna | 19 | Truman & Hawk | 14
5 | Bobby | 18 | Donna & James | 13
Table 1: Popular links and nodes in the network: all of them are in the BC
part. The most connected AD character appears as the 18th in the list; the top
repeated link which includes an AD character appears as the 13th in the list.

In Fig. 7 we discuss the relationship between networks of BC characters and AD
characters, what we call _The Dale Cooper Effect_. This cartoon version of the
adjacency matrix aims to show how BC is a closely knit community–the part of
the adjacency matrix corresponding to it (top left quadrant) is densely filled
with connections. On the other end, the AD characters interact among
themselves very little, the bottom right quadrant is very sparse in
connections. It is by virtue of interacting with BC characters that AD
characters are a part of the network (bottom left and top right quadrants).

Figure 7: The Dale Cooper Effect

This suggests a mechanism by which the two networks are spliced together.
After the nodes of BC network are set up, and Dale Cooper is introduced, new
AD nodes are added with a preferential attachment to BC: a newly added node
has a much higher probability of being connected to the BC core than fellow AD
periphery. Also, the density of connections is indicative: the full graph has
236 edges, while the BC graph (in this treatise, we include Cooper in BC) has
128. Doubling the number of characters (i.e. adding AD to BC) hence results in
doubling the number of edges (linear scaling).

The mechanism of network growth in the previous paragraph inspires a dive into
the degree distribution of the network. Cooper and Sheriff Truman appear
together in many scenes and share most of their contacts: their degree is the
same, and rather high (35 in the entire network, i.e. more connected with more
than 1/2 of other characters; 20 in the BC network, growing the fraction to
2/3 of other characters). Motivated by the findings from Beowulf and the Táin
where adjustments are needed to make the protagonists more realistic, we were
curious to see if such high degree nodes as Cooper and Truman are expected in
networks like these.

Figure 8 presents our findings. First, we note that the BC degree distribution
is similar to one of a random Erdos Renyi network seen in 4, and not too far
off from a log-normal distribution (whose surprising appearance in some
preferential attachment networks we already mentioned). For the BC+AD network,
we examine significant deviation from the shape: the curve appears to be a
superposition of three segments: one for low degrees ($x<10$), one for
moderately high degrees ($x<20$) and one for very high degrees ($x>30$). This
is the effect of the mechanism by which AD characters are connected: they
dominate the low degree segment of the distribution and shift it leftwards,
while contributing to an increase in degree for Cooper and Truman (the very
high degrees data point in the plot), pushing them rightwards. From here we
deduce that Truman and Cooper are not abnormally well connected in the BC
network, but in the BC+AD network, they are more popular than they would be in
a random network.

This makes sense as Truman and Cooper drive the plot in the AD introduction
period, and are often the first point of contact for new characters. This is
mirrored in the assortivity measures for BC and BC+AD networks: BC
assortativity is approximately zero (neutral), while for BC+AD the
assortativity is negative: less connected characters in AD are connected to
popular BC characters.

Following the ideas from [7], we proceed with studying the options of
character removal. The first question is about the effect of removing
characters that appear in a single scene (correction similar to that applied
to the _Táin Bó Cúailnge_ in [7]). This is the pruned network in Fig. 8: the
removed nodes are predominantly in the AD group (20 removed from AD, and 4
from BC), so it is not surprising that the pruned network gets closer to the
BC degree distribution. However, the anomaly of Cooper and Truman remains. As
it turns out, the pruned network collapses onto BC if one of the two dominant
nodes is removed: a case for “this town ain’t big enough for both of us”. In a
way, removing Cooper could be justified as removing the outsider (or _Beowulf_
, following on [7]) from the local network of Twin Peaks.

Finally, given the importance of the law enforcement characters in a show
driven by crime investigation, an interesting test is whether the BC+AD
network remains connected after the members of the FBI and the Twin Peaks
Police Department are removed. The answer is yes–it still remains a single
graph (with the exception of the few characters whose appearance in the show,
if limited to Season 1, is tied exclusively to the police investigation) with
Benjamin Horne as its central figure, tying together the young population of
Twin Peaks through his daughter Audrey, and the criminal milieu through his
business connections.

This process of “police removal” is interesting from the clustering point of
view. Networks can often be split into meaningful clusters of nodes based on
the connections between them. Sometimes this split is ambiguous and it is hard
to deal with borderline cases, but they give a general idea of the communities
existing in the network. Using the Louvain algorithm [10] we perform
clustering of both the network before and after removal of police officers in
Twin Peaks, with the results shown in Fig. 9. The blue cluster on the left
represents the cluster consisting of the police and people primarily
interacting with the police, either as suspects or witnesses. Once the police
officers are removed from the network, some nodes are left disconnected, and
the rest of the nodes that had connections with other clusters join those
clusters.

Figure 8: Degree distribution in the Twin Peaks networks compared to random
networks

Figure 9: Clusters in the Twin Peaks network before (left) and after (right)
removing the law enforcement

##  4 Final Remarks

It is hard to draw conclusions from small data and somewhat arbitrary
conventions applied in its analysis. Given that we are in the realm of art,
creation, and interactions, one alternative to the approach taken in this work
would have been resorting to the actor network theory (despite the fact that
Latour would object to observing the actor network as a mathematical network
[11].) Is the dead body of Laura Palmer a character? Is the photo of Laura
Palmer, and is it the same one? How about the Roadhouse?

Furthermore, this study largely ignored the multiplicity of common appearances
of characters in scenes. For instance, in our count, Truman and Cooper share
almost 50 common appearances, often in scenes that involve other characters as
well. This makes them heavily correlated and they may appear as a single
entity, a strongly connected pair. Studying correlations of this sort is a
natural extension of the work presented here.

The one takeaway that stands out and presents a statistically robust result is
The Dale Cooper Effect. The sharp demarcation between the two networks
embodied by the protagonist introduced as a median character is the peculiar
shape of story. Do similar structures emerge in other works of fiction? What
is the fundamental relationship of The Dale Cooper Effect with storytelling?
Those questions guide our future work.

## Acknowledgements

The idea of this paper came from a talk I prepared for the conference _Beyond
Life and Death: Twin Peaks at Thirty._ Discussions with Jackie Brown,
Anastasia Tsukanova, Tom O’Dea, and Dennis McNulty made the paper much better
than it originally was. Support from Science Foundation Ireland (SFI), co-
funded under the European Regional Development Fund under Grant Number
13/RC/2077\\_P2 is acknowledged.

## References

  * [1] Vonnegut, K.. Shapes of stories (video), available online at https://www.youtube.com/watch?v=oP3c1h8v2ZQ
  * [2] Reagan, A.J., Mitchell, L., Kiley, D., Danforth, C.M. and Dodds, P.S., 2016. The emotional arcs of stories are dominated by six basic shapes. EPJ Data Science, 5(1), pp.1-12. Available online at https://epjdatascience.springeropen.com/articles/10.1140/epjds/s13688-016-0093-1
  * [3] Le Guin, U.K., 1986. The Carrier Bag Theory of Fiction, available online at https://usa.anarchistlibraries.net/library/ursula-k-le-guin-the-carrier-bag-theory-of-fiction.lt.pdf
  * [4] Moretti, F., 2005. Graphs, maps, trees: abstract models for a literary history. Verso.
  * [5] Barabasi, A.L., 2014. Network science book. Available online at http://networksciencebook.com/
  * [6] Redner, S., 2005. Citation statistics from 110 years of Physical Review. arXiv preprint physics/0506056.
  * [7] Mac Carron, P. and Kenna, R., 2014, January. Network analysis of Beowulf, the Iliad and the Táin Bó Cúailnge. In Sources of mythology: Ancient and contemporary myths. Proceedings of the Seventh Annual International Conference on Comparative Mythology (15–17 May 2013, Tuebingen) (pp. 125-141). Available online at http://users.complexity-coventry.org/~kenna/MacCarronKenna.pdf
  * [8] Twin Peaks Episode Transcripts, available online at http://www.glastonberrygrove.net/texts/
  * [9] Bostock, M. 2012. Les Miserables Co-occurrence, available online at https://bost.ocks.org/mike/miserables/
  * [10] Newman, M.E. and Girvan, M., 2004. Finding and evaluating community structure in networks. Physical review E, 69(2), p.026113.
  * [11] Latour, B., 1996. On actor-network theory: A few clarifications. Soziale welt, pp.369-381. Available online at http://transnationalhistory.net/interconnected/wp-content/uploads/2015/05/Latour-Actor-Network-Clarifications.pdf


# Kernel Learning for Explainable Climate Science

Vidhi Lalchand Dept. of Physics
University of Cambridge
Cambridge, UK  Kenza Tazi  Dept. of Engineering
University of Cambridge
Cambridge, UK  British Antarctic Survey
Cambridge, UK  Talay M Cheema  Dept. of Engineering
University of Cambridge
Cambridge, UK  Richard E. Turner  Dept. of Engineering
University of Cambridge
Cambridge, UK  Scott Hosking  British Antarctic Survey
Cambridge, UK

###### Abstract

The Upper Indus Basin, Himalayas provides water for 270 million people and
countless ecosystems. However, precipitation, a key component to hydrological
modelling, is poorly understood in this area. A key challenge surrounding this
uncertainty comes from the complex spatial-temporal distribution of
precipitation across the basin. In this work we propose Gaussian processes
with structured non-stationary kernels to model precipitation patterns in the
UIB. Previous attempts to quantify or model precipitation in the Hindu Kush
Karakoram Himalayan region have often been qualitative or include crude
assumptions and simplifications which cannot be resolved at lower resolutions.
This body of research also provides little to no error propagation. We account
for the spatial variation in precipitation with a non-stationary Gibbs kernel
parameterised with an input dependent lengthscale. This allows the posterior
function samples to adapt to the varying precipitation patterns inherent in
the distinct underlying topography of the Indus region. The input dependent
lengthscale is governed by a latent Gaussian process with a stationary
squared-exponential kernel to allow the function level hyperparameters to vary
smoothly. In ablation experiments we motivate each component of the proposed
kernel by demonstrating its ability to model the spatial covariance, temporal
structure and joint spatio-temporal reconstruction. We benchmark our model
with a stationary Gaussian process and a Deep Gaussian processes.

##  1 Motivation

The Indus River is one of the longest rivers in Asia, sustaining the
livelihoods of over 268 million people [Wester et al., 2019]. The river and
its tributaries provide fresh water for drinking, domestic usage, industrial
processes, and agriculture through the world’s largest contiguous irrigation
system [Basharat, 2019]. The river also delivers most of Pakistan’s
electricity through hydropower plants [Nie et al., 2021] and supports
countless ecosystems and biodiversity hotspots [Xu et al., 2019].

Of all the rivers originating in the Himalayas, the Indus depends most
strongly on water from snow and glacier found in the Upper Indus Basin (UIB)
[Lutz et al., 2014]. Over 60% of the Indus’ annual flow is attributed to the
springtime melt of the snowpack and glaciers [Immerzeel et al., 2010]. As
climate change progresses, these drivers are expected to be replaced by
precipitation. This change will lead to on average less but more extreme
variations in river flow and in turn more floods, landslides, and droughts
[Huss et al., 2017].

With no robust adaption measures, financial losses and profound socio-economic
consequences including food and water scarcity, mass migration and violent
conflict are projected [Wester et al., 2019, Huss et al., 2017]. The scale of
these ramifications is still unknown with the largest source of uncertainty
attributed to precipitation. [Li et al., 2016, Wulf et al., 2016, Remesan and
Holman, 2015, Meng et al., 2014, Andermann et al., 2011].

Figure 1: Left: Shaded-relief image of the topography of the Upper Indus Basin
with natural water bodies overlaid. Right: Three precipitation regimes in the
UIB identified through kmeans clustering. Regime names were chosen by the
authors.

Previous research modelling and predicting precipitation in the Hindu Kush
Karakoram Himalayan region are often qualitative or include assumptions and
simplifications which cannot be resolved at lower resolutions [Dahri et al.,
2016]. For example, the regional climates models from Coordinated Regional
Climate Downscaling Experiment for South Asia regularly overestimates
historical precipitation in the Himalayas by over 100% for both winter and
summer [Sanjay et al., 2017]. These models are also computationally expensive,
lack error propagation, and generate large model-dependent uncertainty.

Gaussian processes (GPs) offer a versatile and interpretable way of studying
and predicting precipitation in this area. Crucially, they offer two unique
advantages which preclude several other modelling tools. First, GP estimates
provide principled uncertainty quantification by design. One can derive
concrete prediction intervals through the posterior predictive distribution;
predictions accompanied by robust uncertainty estimates can be critical for
downstream decision making. Secondly, GPs are a flexible prior of functions
and allow one to encode specific inductive biases through kernel construction.
For instance, one can encode properties like periodicity, smoothness or
spatial heterogeneity through careful specification of the covariance kernel.

This work concerns interpretable kernel constructions for precipitation
modelling in the UIB. We account for non-stationarity and spatial
heterogeneity through a spatio-temporal kernel. In the next section, we give a
brief overview of the GP framework along with non-stationary kernels. Section
3 presents experimental results from the case-study where we leverage spatio-
temporal and non-stationary kernels.

##  2 Background

GPs are a powerful probabilistic and non-parametric tool for modelling
functions. They are fully specified by a mean and covariance function where
the latter controls the inductive bias and support of functions under the
prior. The choice of the covariance function (alternatively, kernel or kernel
function) and in turn selecting the hyperparameters of the covariance function
is jointly referred to as the model selection problem [Rasmussen and Williams,
2006] in GPs. A large cross-section of Gaussian process literature uses
universal kernels like the squared exponential (SE) kernel along with
automatic relevance determination (ARD) in high-dimensions. The SE-ARD kernel
is a translation-invariant stationary kernel which gives infinitely smooth and
differentiable samples in function space. The ARD framework operates by
pruning away extraneous dimensions through contracting their inverse-
lengthscales. The SE-ARD framework is the most commonly reported baseline for
Gaussian process regression tasks. The SE-ARD kernel is given by,

$k(\bm{x},\bm{x}^{\prime})_{\text{SE-
ARD}}=\sigma^{2}_{f}\exp\Bigg{\\{}-\dfrac{1}{2}\sum_{d=1}^{D}\dfrac{(x_{d}-x^{\prime}_{d})^{2}}{\ell_{d}^{2}}\Bigg{\\}}$
(1)

where $\bm{x},\bm{x}^{\prime}\in X$ are high-dimensional inputs
$X\equiv\\{\bm{x}_{i}\\}_{i=1}^{N}$, each $\bm{x}_{i}\in\mathbb{R}^{D}$ and
$\\{\ell_{d}\\}_{d=1}^{D}$ denotes a scalar lengthscale per dimension and
$\sigma^{2}_{f}$ is usually a scalar amplitude.

Given observations $(X,\bm{y})=\\{{\bm{x}_{i},y_{i}\\}}_{i=1}^{N}$ where
$y_{i}$ are noisy realizations of some latent function values $f$ corrupted
with Gaussian noise, $y_{i}=f(\bm{x}_{i})+\epsilon_{i}$,
$\epsilon_{i}\in\mathcal{N}(0,\sigma_{n}^{2})$, let
$k_{\theta}(\bm{x}_{i},\bm{x}_{j})$ denote a positive definite covariance
function parameterised with hyperparameters $\theta$. The generative model
governing the data is given by,

$\displaystyle f$ $\displaystyle\sim\mathcal{GP}(0,k_{\theta}(.,.))$
$\displaystyle p(\bm{y}|f)$
$\displaystyle=p(y_{i}|f_{i})=\prod_{i=1}^{N}\mathcal{N}(f_{i},\sigma^{2}_{n})$
(2)

A Gaussian noise setting which we assume through out yields a closed-form
marginal likelihood $p(\bm{y}|\theta)$. In the standard set-up the negative
log marginal likelihood serves as the loss function against which $\theta$ is
optimised, we call this procedure ML-II. The SE-ARD kernel with ML-II
inference serves as the baseline model to benchmark performance for the
variants proposed in this work. Learning occurs through adaptation of the
hyperparameters ($\theta$) of the covariance function. In the case of the SE-
ARD kernel described above,
$\theta=\\{\sigma^{2}_{f},\\{\ell_{d}\\}_{d=1}^{D},\sigma^{2}_{n}\\}$.

Figure 2: Timeseries drawn from each precipitation regime shown in Figure 1
(top: Gilgit, middle: Ngari, bottom: Khyber). Note that there are no obvious
trends in average or extreme precipitation during this period.

$\displaystyle p(\bm{y}|\theta)$ $\displaystyle=\int
p(\bm{y}|f)p(\bm{f}|\theta)df$ (3)
$\displaystyle=\int\mathcal{N}(0,K_{f})\mathcal{N}(f,\sigma_{n}^{2}\mathbb{I})df$
(4) $\displaystyle=\mathcal{N}(0,K_{f}+\sigma_{n}^{2}\mathbb{I})$

The matrix $K_{f}$ denotes the kernel $k_{\theta}(\cdot,\cdot)$ evaluated at
inputs $X$.

This works considers the non-stationary counterpart of the squared-exponential
kernel, the Gibbs kernel [Gibbs, 1998] for the task of spatial precipitation
modelling. In contrast to the SE-ARD kernel proposed above, the Gibbs kernel
introduces input dependent lengthscales in each dimension. Instead of a
single-point estimate $\ell_{d}$ per dimension we can consider
$\ell_{d}(\bm{x})$, a lengthscale function. One can choose a parametric form
for the dependence of the lengthscale on the inputs in each dimension but very
often this dependence is not known a priori. The Gibbs kernel for multi-
dimensional inputs is given by,

$\displaystyle k_{\text{Gibbs}}(\bm{x}_{i},\bm{x}_{j})$
$\displaystyle=\prod_{d=1}^{D}\sqrt{\dfrac{2\ell_{d}(\bm{x}_{i})\ell_{d}(\bm{x}_{j})}{\ell_{d}^{2}(\bm{x}_{i})+\ell_{d}^{2}(\bm{x}_{j})}}\times$
(5) $\displaystyle\hskip
56.9055pt\exp\left\\{-\sum_{d=1}^{D}\dfrac{(x_{i}^{(d)}-x_{j}^{(d)})^{2}}{\ell_{d}^{2}(\bm{x}_{i})+\ell_{d}^{2}(\bm{x}_{j})}\right\\}$

Previous work, Heinonen et al. [2016] considers a formulation where the
lengthscale function is modelled non-parameterically with a latent Gaussian
process defined on the same inputs.

$\displaystyle\hat{\ell}$
$\displaystyle=\log(\ell_{d})\sim\mathcal{GP}(0,k_{\ell}(.,.))$ (6)
$\displaystyle\hat{\ell}(\bm{x})$
$\displaystyle=\log(\ell_{d}(\bm{x}))\sim\mathcal{N}(0,K_{\ell})$ (7)

The hierarchical formulation makes the posterior and marginal likelihood
analytically intractable. However, in the Gaussian likelihood setting one can
consider a maximum-a-posteriori (MAP) solution [Kersting et al., 2007,
Heinonen et al., 2016] by maximising,

$\displaystyle\ell_{\text{MAP}}$
$\displaystyle=\operatorname*{argmax}_{\ell}\log
p(\bm{y}|\hat{\ell})p(\hat{\ell})$ (8)
$\displaystyle=\operatorname*{argmax}_{\ell}\log\mathcal{N}(\bm{y}|0,K_{f}+\sigma^{2}_{n}\mathbb{I})\mathcal{N}(\hat{\ell}|0,K_{\ell})$

where $\ell_{\text{MAP}}$ is a vector of the size of the training inputs $X$
(note that it is typical to work with the log of the marginal likelihood to
avoid numerical underflow). Extrapolating the non-parametric lengthscale to
test inputs entails estimating $p(\ell_{\star}|\ell_{\text{MAP}},\bm{y})$
which we approximate by
$\mathbb{E}(\ell_{\star}|\ell_{\text{MAP}})=K_{\star\ell}K_{\ell}^{-1}\ell_{\text{MAP}}$
(expectation of a conditional Gaussian).

Figure 3: Predicting the precipitation trend with a locally periodic kernel.
The dotted line demarcates the training and test regimes. The error bars
denote 95$\%$ confidence intervals around the predicted mean. Some outliers
remain uncaptured by the Gaussian prediction intervals, indicating mild non-
Gaussianity in the targets. In the bottom plot we regress the outputs in log-
space and visualise the 1d log-normal distribution with the respective
quantiles.

Once the MAP solution has been learnt at the training inputs, the posterior
predictive
$p(\bm{f}^{\star}|\bm{y},\ell_{\text{MAP}},\ell_{\star})\sim\mathcal{N}(\mu^{\star},\Sigma^{\star})$
is given by,

$\displaystyle\mu^{\star}$ $\displaystyle=K_{\star
f}^{T}(K_{f}+\sigma^{2}_{n}\mathbb{I})^{-1}\bm{y}$
$\displaystyle\Sigma^{\star}$ $\displaystyle=K_{\star\star}-K_{\star
f}^{T}(K_{f}+\sigma^{2}_{n}\mathbb{I})^{-1}K_{f\star}$ (9)

where $K_{f}$ is based on the evaluation of eq. 5 on the training inputs
(using $\ell_{\text{MAP}}$) and $K_{\star f}$ is based on the evaluation of
$k_{\text{Gibbs}}$ on test and training inputs using $\ell_{\text{MAP}}$ and
$\mathbb{E}(\ell_{\star}|\ell_{\text{MAP}})$ respectively.

The latent GP parameterisation allows for extremely flexible modelling where
samples from the posterior function space ‘adapt’ to the varying spatial
dynamics inherent in the data.

##  3 Case-study: Accurate Precipitation modelling in the UIB

We conduct three experiments to highlight different features of the kernel
composition. In the spatial regression task, we benchmark the non-stationary
spatial covariance kernel based on the Gibbs construction against stationary
baselines. The temporal extrapolation task uses a locally periodic kernel to
fit the univariate precipitation dynamics at single spatial points over time.
The spatio-temporal task considers an additive kernel with components acting
on the spatial and temporal slice of the inputs to model dynamics over space
and time. The models are implemented using GPyTorch [Gardner et al., 2018] and
data is plotted using xarray [Hoyer and Hamman, 2017].

Data: in this case study, we use the 5th ECMWF Reanalysis (ERA5) dataset
[Hersbach et al., 2020]. Reanalysis blends historical observations from
surface, sonde and satellite measurements with numerical weather forecasting
models. Through this data assimilation, the models create a past record of
historical climate at high temporal and spatial resolution. ERA5 runs from
1959 to the present day over 0.25° grid and assimilates data from a large
number of sources. Each datapoint represents the average monthly precipitation
in mm/day over a gridbox.

Precipitation in the UIB is complex and this is reflected in the ERA5 dataset.
In this area, precipitation is driven by two major atmospheric events: the
Indian Summer Monsoon (ISM) and the western fronts (Westerlies). The ISM
brings rain from June to September. The ISM reaches the south-eastern UIB
first and has a decreasing contribution to the annual rainfall in the North-
West direction.

The Westerlies are strongest in the winter from December to April, peaking in
March [Dahri et al., 2016]. The relative contribution of westerly fronts
increases from the South-East to the North-West of the basin. As a
consequence, the eastern UIB receives up to 70% of its annual rainfall from
the summer whereas the western UIB receives 40-60% of its precipitation during
the winter [Dahri et al., 2016].

The complexity of this distribution is illustrated in Figure 1 and 2 where
three timeseries are sampled from the three characteristic precipitation
regimes identified through K-means clustering.

Figure 4: ERA5 data and 2D precipitation modelling over inputs (lat, lon) for
the non-stationary Gibbs kernel, and two-layer DGP and shallow GP with
stationary kernels. Table 1 provides test predictive performance metrics over
10 splits of the data.

###  3.1 Spatial Regression

We model precipitation in the UIB at a single point in time focusing on
accurately capturing the spatial variation of precipitation. The stationary
model and the deep GP uses a 2D SE-ARD kernel learning stationary
hyperparameters (with ML-II inference) which are constant across space. In the
Gibbs formulation (eq. 5) we learn a lengthscale per spatial point using MAP
estimation ( as described in section 2). Figure 4 shows the best mean
prediction from 10 splits for each of the baselines. The 2-layer deep GP
[Damianou and Lawrence, 2013] with skip connections performs worse than the
standard baseline in terms of prediction error. The high-precipitation areas
in the Khyber zone are best captured by the Gibbs kernel. It is interesting to
note that the predictive intervals for the DGP yield higher predictive
densities for test data despite inferior predictive means. The Gibbs
formulation gives the best trade-off in terms of reconstruction error assessed
against ground-truth and quality of predictive uncertainty.

Metric | SE-ARD | DGP (L2) | Gibbs
---|---|---|---
RMSE | 0.353 ± 0.014 | 0.603 ± 0.018 | 0.271 ± 0.014
NLPD | 0.406 ± 0.063 | 0.023 ± 0.011 | 0.019 ± 0.016

Table 1: 2D Spatial regression on precipitation data over the entire UIB over
the month of Jan, 2000. We report Root Mean Squared Error (RMSE) and Negative
Log Predictive Density (NLPD) on held-out test data over 10 splits.

###  3.2 Temporal Extrapolation

We regress on univariate time inputs (monthly observations) to capture the
local periodicity exhibited in the precipitation dynamics. In order to
moderate the changing amplitudes of the peaks we add flexibility by
multiplying with a SE-ARD kernel (acting on univariate time inputs). The
periodic kernel we use is given by,

$k_{\text{PER}}(\bm{x}_{i},\bm{x}_{j})=\sigma^{2}_{f}\exp\left\\{-\dfrac{2\sin^{2}({\pi|\bm{x}_{i}-\bm{x}_{j}|/p})}{\ell^{2}}\right\\}$
(10)

Figure 3 depicts the prediction performance over train and test where we train
on years 1979-2002. The predictions under the log-normal distribution ensure
positivity (as desired), further the heavy-tailed non-Gaussian prediction
intervals capture the outlying values which elude the symmetric Gaussian
intervals. The test RMSE and NLPD for the fits are given by (0.5536 vs.
0.5328) and (1.3248 vs. 0.9721).

###  3.3 Spatio-Temporal Regression

In this experiment we consider the task of predicting dynamics across space
and time (3D inputs) for a 1 month ahead forecast. We train on January to
April 2000 and test on the month of May. We wanted to constrain the training
data set to a moderate size so as to execute exact GP inference for the
baseline case.

For the stationary (shallow) model, the spatio-temporal kernel is formed by
adding together the spatial and temporal components with one important
innovation. In the earlier sections the temporal component acted solely on the
time dimension while the spatial component acted solely on the spatial
dimensions (latitude and longitude). Denoting the time and spatial coordinates
as $\bm{x}^{\text{t}},\bm{x}^{\text{lat}},\bm{x}^{\text{lon}}$ respectively,
the kernel for the stationary model is given by,

$\displaystyle k_{\text{stat.}}(\bm{x}_{i},\bm{x_{j}})=$
$\displaystyle\underbrace{k_{\text{SE-
ARD}}(({\bm{x}^{\text{lat}}_{i}},{\bm{x}^{\text{lon}}_{i}}),({\bm{x}^{\text{lat}}_{j}},{\bm{x}^{\text{lon}}_{j}}))\times
k_{\text{PER}}({\bm{x}^{t}_{i}},{\bm{x}^{t}_{j}})}_{\text{temporal}}$
$\displaystyle+\underbrace{k_{\text{SE-
ARD}}(({\bm{x}^{\text{lat}}_{i}},{\bm{x}^{\text{lon}}_{i}}),({\bm{x}^{\text{lat}}_{j}},{\bm{x}^{\text{lon}}_{j}}))}_{\text{spatial}}$
(11)

We weave the spatial dimensions into the temporal kernel by making the RBF
component dependent on the latitude and longitude, this allows the
amplitude/scale of the periodic component to vary for different spatial
regions. Note that this is still a stationary kernel. For the non-stationary
extension in this setting we keep the temporal component identical to the
stationary formulation eq. 11 but replace the spatial component with the Gibbs
kernel eq. 5.

$\displaystyle k_{\text{non-stat.}}(\bm{x}_{i},\bm{x_{j}})=$
$\displaystyle\underbrace{k_{\text{SE-
ARD}}(({\bm{x}^{\text{lat}}_{i}},{\bm{x}^{\text{lon}}_{i}}),({\bm{x}^{\text{lat}}_{j}},{\bm{x}^{\text{lon}}_{j}}))\times
k_{\text{PER}}({\bm{x}^{t}_{i}},{\bm{x}^{t}_{j}})}_{\text{temporal}}$
$\displaystyle+\underbrace{k_{\text{Gibbs}}(({\bm{x}^{\text{lat}}_{i}},{\bm{x}^{\text{lon}}_{i}}),({\bm{x}^{\text{lat}}_{j}},{\bm{x}^{\text{lon}}_{j}}))}_{\text{spatial}}$
(12)

where $k_{\text{Gibbs}}$ is parameterised by 2 non-parametric lengthscale
processes - $\ell_{\text{lat}}({\bm{x}^{\text{spatial}}})$ and
$\ell_{\text{lon}}({\bm{x}^{\text{spatial}}})$ acting on the 2d spatial
dimensions as inputs. We enclose the regression results for the 1-month ahead
forecast based on the baseline and non-stationary construction in fig. 5.

We do not present results for the deep GP in this sections as they were found
to yield substandard results in section 3.1.

Figure 5: 3D regression for spatio-temporal precipitation modelling. The red
line demarcates the training and test months. The non-stationary kernel
yielded a an average test RMSE of 0.9426 across three runs vs. 1.1086 for the
stationary kernel.

##  4 Conclusions and further work

We demonstrate kernel composition in modelling the spatial and temporal
precipitation dynamics over the UIB. In the spatial setting, the non-
stationary kernel does much better than a deep GP and the stationary baseline.
More comprehensive and complex experiments will provide answers to how these
methods can be best be used in ‘real world’ settings and, scientifically, what
drives precipitation in the UIB.

Further investigation will seek to better understand historical precipitation
distribution in this area. In particular, a detailed analysis of the Gibbs
hyperparameter functions are needed over longer training and testing
intervals. This would require making the non-stationary formulation compatible
with sparse GPs relying on inducing points [Titsias, 2009]. Future iterations
will also consider more features that are predictive of precipitation such as
elevation, slope and large scale atmospheric variables. The Gibbs
hyperparamaters will give us insight into where and how these features are
influencing rain and snowfall in basin. Model comparison metrics like BIC
(Bayesian information criterion) will help understand more complex non-
stationary constructions in light of model complexity.

Next, precipitation change under different climate scenarios will be
considered. ERA5 climatic variables will be swapped with global climate model
outputs. Although not explored in this paper, the non-stationarity of
precipitation with respect to climate change will become more important in the
future. This non-stationarity should be in part captured by the large scale
atmospheric variables previously mentioned. However, a non-stationary kernel
could also have a role to play here.

Finally, the Bayesian nature of these models will give a more accurate and
informed quantification of uncertainty than regional climate models that uses
ensemble spread as proxy for uncertainty. In the non-stationary setup, the
hyperaparemeter distributions could also point scientists to the most
significant drivers of prediction variability.

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