3		& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0
\end{bNiceArray}
\end{equation}

\begin{equation}
\scalebox{0.75}{
\NiceMatrixOptions{code-for-first-row = \scriptstyle \color{blue},
code-for-first-col = \scriptstyle \color{blue}
\begin{bNiceArray}{*{3}{c}|*{3}{c}|*{3}{c}}[first-row,first-col]
& 1,1 	& 1,2	& 1,3	& 2,1 	& 2,2	& 2,3	& 3,1 	& 3,2	& 3,3	\\
1		& 1	& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0	\\
2		& 0	& 0	& 0	& 0	& 1	& 0	& 0	& 0	& 0	\\
3		& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0
\end{bNiceArray}
\; \text{;} \;
\scalebox{0.75}{
\NiceMatrixOptions{code-for-first-row = \scriptstyle \color{blue},
code-for-first-col = \scriptstyle \color{blue}
\begin{bNiceArray}{*{3}{c}|*{3}{c}|*{3}{c}}[first-row,first-col]
& 1,1 	& 1,2	& 1,3	& 2,1 	& 2,2	& 2,3	& 3,1 	& 3,2	& 3,3	\\
1		& 0	& 0	& 0	& 1	& 1	& 0	& 0	& 0	& 0	\\
2		& 1	& 1	& 0	& 0	& 0	& 0	& 0	& 0	& 0	\\
3		& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0
\end{bNiceArray}
\; \text{;} \;
\scalebox{0.75}{
\NiceMatrixOptions{code-for-first-row = \scriptstyle \color{blue},
code-for-first-col = \scriptstyle \color{blue}
\begin{bNiceArray}{*{3}{c}|*{3}{c}|*{3}{c}}[first-row,first-col]
& 1,1 	& 1,2	& 1,3	& 2,1 	& 2,2	& 2,3	& 3,1 	& 3,2	& 3,3	\\
1		& 1	& 1	& 0	& 0	& 0	& 0	& 0	& 0	& 0	\\
2		& 0	& 0	& 0	& 1	& 1	& 0	& 0	& 0	& 0	\\
3		& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0	& 0
\end{bNiceArray}
\end{equation}

Step 5: There are no new bilabelled graph diagrams to consider in this step as $A$ does not have any loops.

In total, this gives us $60$ $(3,0)$–bilabelled graph diagrams that are appropriate for $A$.

We have implicitly applied Step 6 in calculating the $A$-homomorphism matrices
during Steps 1 through 5 inclusive since we had to convert
each $(3,0)$–bilabelled graph diagram into a
$(2,1)$–bilabelled graph diagram first before applying the functor $\mathcal{F}^{G}$
to obtain each $A$-homomorphism matrix.
Hence we move onto the second half of Step 7:
we remove all duplicate matrices and the all zero matrices.
We see that we are left with $31$ matrices in the spanning set.
After this,
to obtain the weight matrix for an $\Aut(A) \cong S_2$-equivariant linear
layer function from
$(\mathbb{R}^3)^{\otimes 2}$ to
we would weight each of the $31$ matrices in the spanning set and then add them together.

It is important to highlight that the vector space
\Hom((\mathbb{R}^{3})^{\otimes 2},
\mathbb{R}^{3})
in which
\Hom_{S_2}((\mathbb{R}^{3})^{\otimes 2},
\mathbb{R}^{3})
lives, is of dimension $27$; hence, there must be linear dependencies amongst the $31$
spanning set elements that we have found.
However, determining these linear dependencies in general
at the graph level
a priori
is left for future work.

We can also find the elements of the spanning set for
\Hom_{S_2}((\mathbb{R}^{3})^{\otimes 2},
\mathbb{R}^{3})
for the other two embeddings of $S_2$ in $S_3$,
given by $\Aut(B)$ and $\Aut(C)$.
For $\Aut(B)$, we perform the permutation $(13)$ on each index of the rows and columns
of the spanning set matrices found for $\Aut(A)$, and for $\Aut(C)$,
we perform the permutation $(23)$ instead.

§ ADDING FEATURES AND BIASES

§.§ Features

We have assumed throughout that the feature dimension for all of the layers
that appear in a graph automorphism group equivariant neural network is one.
We can adapt all of the results that

we have shown for the case where the feature dimension of the layers is
greater than one.

Let $G$ be a graph having $n$ vertices, and suppose that an $r$-order tensor
has a feature space of dimension $d_r$.  We now wish to find a spanning set for

\begin{equation} \label{HomGraphfeatures}
\Hom_{\Aut(G)}((\mathbb{R}^{n})^{\otimes k} \otimes \mathbb{R}^{d_k},
(\mathbb{R}^{n})^{\otimes l} \otimes \mathbb{R}^{d_l})
\end{equation}

in the standard basis of $\mathbb{R}^{n}$.

The spanning set can be found by adapting the result given in
(<ref>), and is
\begin{equation} \{X_{\bm{H}, i, j}^G \mid [\bm{H}]
\in \mathcal{G}(k,l), i \in [d_l], j \in [d_k]\} \end{equation}
where now, if
$\bm{H} \coloneqq (H, \bm{k}, \bm{l})$, then $X_{\bm{H}, i, j}^G$ is the $(n^l
\times d_l) \times (n^k \times d_k)$ matrix that has $(I,i,J,j)$–entry given
by the number of graph homomorphisms from $H$ to $G$ such that $\bm{l}$ is
mapped to $I$ and $\bm{k}$ is mapped to $J$, and is $0$ otherwise.

§.§ Biases

Including bias terms in the layer functions of an $\Aut(G)$-equivariant neural
network is also possible.  If we consider a learnable linear layer in
$\Hom_{\Aut(G)}((\mathbb{R}^{n})^{\otimes k}, (\mathbb{R}^{n})^{\otimes l})$,
Pearce-Crump, 2022 shows that the $\Aut(G)$-equivariance of the bias function,
$\beta : ((\mathbb{R}^{n})^{\otimes k}, \rho_k) \rightarrow
((\mathbb{R}^{n})^{\otimes l}, \rho_l)$, needs to satisfy

\begin{equation} \label{AutGbiasequiv}
c = \rho_l(g)c
\end{equation}

for all $g \in \Aut(G)$ and for all $c \in (\mathbb{R}^{n})^{\otimes l}$.

Since any $c \in (\mathbb{R}^{n})^{\otimes l}$ satisfying (<ref>)
can be viewed as an element of $\Hom_{\Aut(G)}(\mathbb{R},
(\mathbb{R}^{n})^{\otimes l})$, to find the matrix form of $c$, all we need to
do is to find a spanning set for $\Hom_{\Aut(G)}(\mathbb{R},
(\mathbb{R}^{n})^{\otimes l})$.

But this is simply a matter of applying Theorem <ref>, setting $k
= 0$.


$\displaystyle P_{N}L_{f}\geq-2\tau_{0}\|f-f^{*}\|^{2}_{L_{2}}+c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{2})\tau^{2}\|f-f^{*}\|_{L_{2}}^{2}.$

Choosing $\tau_{0}<c_{16}Q_{\cal H}(2\tau)\rho(0,t_{2})\tau^{2}/4$, we have,
$P_{N}L_{f}>0.$ But the empirical minimizer $\hat{f}$ satisfies
$P_{N}L_{\hat{f}}\leq 0$. This implies, together with choosing
$\mathscr{A}(N)=N^{-1/4+\iota}$, that with probability at least
$\mathscr{P_{c}}=\mathscr{P}_{1,c}-\mathscr{P}_{2,c}$,

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max(2\omega_{Q}({\cal F}-{\cal
F},N,Q_{\cal H}(2\tau)^{\frac{3}{2}},Q_{\cal H}(2\tau)),\mathscr{A}(N)),$

where

$\displaystyle\mathscr{P_{c}}=1-c_{9}Q_{\cal
H}(2\tau)^{1-\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}e^{-c_{10}Q_{\cal
H}(2\tau)^{1+\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}}$ $\displaystyle-$
$\displaystyle N{\rm
exp}\left(-\frac{(N^{\frac{1}{2}+2\iota}\tau_{0})^{\eta}}{C_{1}}\right)-{\rm
exp}\left(-\frac{N^{1+4\iota}\tau_{0}^{2}}{C_{2}(1+NV)}\right)-{\rm
exp}\left(-\frac{N^{4\iota}\tau_{0}^{2}}{C_{3}}{\rm
exp}\left((1-\eta)^{\eta}\frac{(N^{\frac{1}{2}+2\iota}\tau_{0})^{\frac{\eta(1-\eta)}{2}}}{C_{4}2^{\eta}}\right)\right).$

We now prove part 2. When $\|f-f^{*}\|_{L_{2}}\geq\mathscr{A}(N)$, using Lemma
2.2 we have

$\displaystyle{\mathbb{P}}\left(\left\lvert\frac{1}{N}\sum_{i=1}^{N}l^{\prime}(\xi_{i})(f-f^{*})(X_{i})-\mathbb{E}\left[l^{\prime}(\xi)(f-f^{*})\right]\right\rvert\geq\tau_{0}\|f-f^{*}\|_{L_{2}}^{2}\right)$
$\displaystyle\leq$ $\displaystyle\frac{2^{\eta_{2}+3}}{(d_{2}\log
N\tau_{0}\|f-f^{*}\|_{L_{2}}^{2})^{\frac{1-\eta_{2}}{\eta_{1}}}}N(N\tau_{0}\|f-f^{*}\|_{L_{2}}^{2})^{-(1+d_{1}(\eta_{2}-1))}+8N(N\tau_{0}\|f-f^{*}\|_{L_{2}}^{2})^{-(1+d_{2}c^{\prime})}$
$\displaystyle+2e^{-\frac{(N\tau_{0}\|f-f^{*}\|_{L_{2}}^{2})^{2-2d_{1}}(d_{2}\log(N\tau_{0}\|f-f^{*}\|_{L_{2}}^{2}))^{1/\eta_{1}}}{9N}}$
$\displaystyle\leq$
$\displaystyle\frac{2^{\eta_{2}+3}}{(d_{2}\log(N\tau_{0}\mathscr{A}(N)^{2}))^{\frac{1-\eta_{2}}{\eta_{1}}}}N(N\tau_{0}\mathscr{A}(N)^{2})^{-(1+d_{1}(\eta_{2}-1))}+8N(N\tau_{0}\mathscr{A}(N)^{2})^{-(1+d_{2}c^{\prime})}$
$\displaystyle+2e^{-\frac{(N\tau_{0}\mathscr{A}(N)^{2})^{2-2d_{1}}(d_{2}\log(N\tau_{0}\mathscr{A}(N)^{2}))^{1/\eta_{1}}}{9N}}\equiv\mathscr{P}_{2,c}.$
(95)

We will choose $d_{1}$ suitably to allow $\mathscr{A}(N)$ to decrease with $N$
as fast as possible while ensuring $\lim_{N\to\infty}\mathscr{P}_{2,c}\to 0$.
Combining (67), and (70), with probability at least
$\mathscr{P}_{1,c}-\mathscr{P}_{2,c}$, for every $f\in{\cal F}$ that satisfies
$\|f-f^{*}\|_{L_{2}}\geq\max(2\omega_{Q},\mathscr{A}(N))$, we get

$\displaystyle P_{N}L_{f}\geq-2\tau_{0}\|f-f^{*}\|^{2}_{L_{2}}+c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{2})\tau^{2}\|f-f^{*}\|_{L_{2}}^{2}.$

Choosing $\tau_{0}<c_{16}Q_{\cal H}(2\tau)\rho(0,t_{2})\tau^{2}/4$, we have,
$P_{N}L_{f}>0.$ But the empirical minimizer $\hat{f}$ satisfies
$P_{N}L_{\hat{f}}\leq 0$. This implies, together with choosing
$\mathscr{A}(N)=N^{-(1-1/\eta_{2})/4+\iota}$, $d_{1}=1/(1+\eta_{2})$,
$d_{2}=(\eta_{2}-1)/(\eta_{2}+1)$, and $\iota<(1-1/\eta_{2})/4$, that with
probability at least $\mathscr{P}_{c}=\mathscr{P}_{1,c}-\mathscr{P}_{2,c}$,

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max(2\omega_{Q}({\cal F}-{\cal
F},N,\zeta_{1},\zeta_{2}),N^{-(1-1/\eta_{2})/4+\iota}),$

where

$\displaystyle\mathscr{P}_{c}$ $\displaystyle=1-c_{9}Q_{\cal
H}(2\tau)^{1-\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}e^{-c_{10}Q_{\cal
H}(2\tau)^{1+\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}}-\frac{2^{\eta_{2}+3}\tau_{0}^{-\frac{2\eta_{2}}{1+\eta_{2}}}}{\left(\log\left(\tau_{0}N^{\frac{1}{2}+\frac{1}{2\eta_{2}}+2\iota}\right)/2\right)^{\frac{1-\eta_{2}}{\eta_{1}}}}N^{-\frac{4\iota\eta_{2}}{1+\eta_{2}}}$
$\displaystyle-8\tau_{0}^{-\frac{2\eta_{2}}{1+\eta_{2}}}N^{-\frac{4\iota\eta_{2}}{1+\eta_{2}}}-2e^{-\frac{\tau_{0}^{\frac{2\eta_{2}}{1+\eta_{2}}}\left(\log\left(\tau_{0}N^{\frac{1}{2}+\frac{1}{2\eta_{2}}+2\iota}\right)/2\right)^{1/\eta_{1}}}{9}N^{\frac{4\iota\eta_{2}}{1+\eta_{2}}}}.$

Note that Proposition B.2 is exactly same as Theorem 3.2 except for the fact
one needs $\ell$ to be strongly convex in $[-t_{0},t_{0}]$ instead of
$[-t_{2},t_{2}]$ where $t_{2}$ is of the order
$\mathcal{O}((2\tau+1/\sqrt{Q_{{\cal H}}(2\tau)})\|\xi\|_{L_{2}})$. So now we
will show that empirical minimizer $\hat{f}\in{\cal F}$ satisfies
$\|\hat{f}-f^{*}\|_{L_{2}}\leq\max(\|\xi\|_{L_{2}},2\omega_{Q})$ with high
probability. One has the following result from [Men18]:

$\displaystyle\\{h-f^{*}:h\in{\cal F},\|h-f^{*}\|_{L_{2}}\geq
R\\}\subset\\{\lambda(f-f^{*}):\lambda\geq 1,f\in{\cal
F},\|f-f^{*}\|_{L_{2}}=R\\}.$ (96)

######  Lemma B.10 ([Men18, Lemma 5.6]).

When (88) is true, if
$\|f-f^{*}\|_{L_{2}}\geq\max(\|\xi\|_{L_{2}},2\omega_{Q})$, and $\lambda\geq
1$, then

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\ell^{\prime\prime}(\widetilde{\xi}_{i})(\lambda(f-f^{*}))^{2}(X_{i})\geq\lfloor\lambda\rfloor
c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}\max\left(\|\xi\|_{L_{2}}^{2},4\omega_{Q}^{2}\right).$
(97)

######  Lemma B.11.

With probability at least $1-\mathscr{P}_{2,c}$ with $\tau_{0}=c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}/4$, we have

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max(\|\xi\|_{L_{2}},2\omega_{Q}).$

Proof. From (94), with probability at least $1-\mathscr{P}_{2,c}$ we have,

$\displaystyle\left\lvert\frac{1}{N}\sum_{i=1}^{N}l^{\prime}(\xi_{i})(f-f^{*})(X_{i})-\mathbb{E}\left[l^{\prime}(\xi)(f-f^{*})\right]\right\rvert\leq\tau_{0}\|f-f^{*}\|_{L_{2}}^{2}.$

To make the dependency of $\mathscr{P}_{2,c}$ on $\tau_{0}$ explicit, we use
the notation $\mathscr{P}_{2,c,\tau_{0}}$ to denote $\mathscr{P}_{2,c}$ for
this proof. If $\|f-f^{*}\|_{L_{2}}\leq\max(\|\xi\|_{L_{2}},2\omega_{Q})$,
choosing $\tau_{0}=c_{16}Q_{\cal H}(2\tau)\rho(0,t_{0})\tau^{2}/4$, with
probability at least $1-\mathscr{P}_{2,c,c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}/4}$, for the same $\lambda\geq 1$ as in Lemma
B.10, we have,

$\displaystyle\left\lvert\frac{1}{N}\sum_{i=1}^{N}l^{\prime}(\xi_{i})(\lambda(f-f^{*}))(X_{i})-\mathbb{E}\left[l^{\prime}(\xi)(\lambda(f-f^{*}))\right]\right\rvert\leq\frac{c_{16}\lambda
Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}}{4}\max(\|\xi\|_{L_{2}},2\omega_{Q})^{2}.$

If $\|f-f^{*}\|_{L_{2}}=\max(\|\xi\|_{L_{2}},2\omega_{Q})$, and $\lambda\geq
1$, we also have

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\ell^{\prime\prime}(\widetilde{\xi}_{i})(\lambda(f-f^{*}))^{2}(X_{i})-\left\lvert\frac{1}{N}\sum_{i=1}^{N}l^{\prime}(\xi_{i})(\lambda(f-f^{*}))(X_{i})-\mathbb{E}\left[l^{\prime}(\xi)(\lambda(f-f^{*}))\right]\right\rvert$
$\displaystyle\geq$ $\displaystyle\lfloor\lambda\rfloor c_{16}Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}\max\left(\|\xi\|_{L_{2}}^{2},4\omega_{Q}^{2}\right)-\frac{c_{16}\lambda
Q_{\cal
H}(2\tau)\rho(0,t_{0})\tau^{2}}{4}\max(\|\xi\|_{L_{2}},2\omega_{Q})^{2}$
$\displaystyle>$ $\displaystyle 0.$

So by (96), and Lemma B.10, the empirical minimizer $\hat{f}$ satisfies,

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max(\|\xi\|_{L_{2}},2\omega_{Q}).$

Proof. [Proof of Theorem 3.2] Combining Lemma B.11 with the two parts of
Proposition B.2 gives us Theorem 3.2.

######  Corollary B.2.

For the convex ERM procedure, under Assumptions 2.1, with condition (b)
replaced by Assumption 3.1 with $p=8$, for some $0<\iota<\frac{1}{2}$ and
$r,\mu$ and $\tau_{0}$ same as in Theorem 3.2, for sufficiently large $N$, we
have

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max\left(N^{-\frac{1}{2}+\iota},2\omega_{Q}({\cal
F}-{\cal F},N,Q_{\cal H}(2\tau)^{\frac{3}{2}},Q_{\cal H}(2\tau))\right)$ (98)

with probability at least (for some constants $c_{9},c_{10},\tilde{C}_{2}>0$)

$\displaystyle 1-c_{9}Q_{{\cal
H}}(2\tau)^{1-\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}e^{-c_{10}Q_{{\cal
H}}(2\tau)^{1+\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}}-\tilde{C}_{2}N{\rm
exp}\left(-(N^{2\iota}\tau_{0})^{\eta}/M_{1}\right).$

Proof. [Proof of Corollary B.2] The proof is same as Corollary 3.1 and hence
we omit it here.

##  Appendix C Proofs of Section 4.1

######  Lemma C.1 (Lemma 6.4 of [Men15]).

If $W=(w_{i})_{i=1}^{d}$ is a random vector on $\mathbb{R}^{d}$, then for
every integer $1\leq k\leq d$,

$\displaystyle\mathbb{E}\left[\sup_{t\in\sqrt{k}B_{1}^{d}\cap
B_{2}^{d}}\left\langle W,t\right\rangle\right]\leq
2\mathbb{E}\left[\left(\sum_{i=1}^{k}{w_{i}^{*}}^{2}\right)^{\frac{1}{2}}\right],$

where $(w_{i}^{*})_{i=1}^{d}$ is a monotone non-increasing reaarangement of
$(|w_{i}|)_{i=1}^{d}$.

######  Lemma C.2.

Let $w_{1},w_{2},\cdots,w_{d}$ are independent copies of a mean-zero, variance
1 random variable $w\sim\textsc{sw}(\eta)$. Then for all $p\geq 1\wedge\eta$,
$\|w\|_{L_{p}}\leq K_{1}p^{\frac{1}{\eta}}$ for some constant $K_{1}>0$. Then
for every $1\leq k\leq d$,

$\displaystyle\mathbb{E}\left[\left(\sum_{i=1}^{k}{w_{i}^{*}}^{2}\right)^{\frac{1}{2}}\right]\leq\sqrt{2k}K_{1}\left(\log(ed)\right)^{1/\eta}.$

Proof. [Proof of Lemma C.2] For $1\leq j\leq d$, and $p\geq 2$,

$\displaystyle{\mathbb{P}}(w_{j}^{*}\geq
t)\leq\binom{d}{j}{\mathbb{P}}^{j}(|w|>t)\leq\binom{d}{j}\left(\frac{\|z\|_{L_{p}}}{t}\right)^{jp}.$

Setting $t=uK_{1}\left(\log(ed/j)\right)^{1/\eta}$ and $p=\log(ed/j)$, we get

$\displaystyle{\mathbb{P}}\left(w_{j}^{*}\geq
uK_{3}\right)\leq\left(\frac{1}{u}\right)^{j\log(ed/j)},$ (99)

where $K_{3}=K_{1}\left(\log(ed/j)\right)^{1/\eta}$. Using (99) we will bound
$\mathbb{E}\left[{w_{j}^{*}}^{2}\right]$. For some $v$,

$\displaystyle\mathbb{E}\left[{w_{j}^{*}}^{2}\right]=$
$\displaystyle\int_{0}^{\infty}{\mathbb{P}}({w_{j}^{*}}^{2}>u)du$
$\displaystyle=$
$\displaystyle\int_{0}^{v}{\mathbb{P}}({w_{j}^{*}}^{2}>u)du+\int_{v}^{\infty}{\mathbb{P}}({w_{j}^{*}}^{2}>u)du$
$\displaystyle\leq$ $\displaystyle
v+\int_{0}^{\infty}{\mathbb{P}}({w_{j}^{*}}^{2}>u+v)du$ $\displaystyle\leq$
$\displaystyle
v+\int_{0}^{\infty}\left(\frac{K_{3}}{\sqrt{u+v}}\right)^{j\log(ed/j)}du$
$\displaystyle=$ $\displaystyle
v-K_{5}\left[\frac{(u+v)^{1-j\log(ed/j)/2}}{j\log(ed/j)/2-1}\right]_{0}^{\infty}\quad\quad[\text{where
}K_{5}=K_{3}^{j\log(ed/j)}]$ $\displaystyle=$ $\displaystyle
v+K_{5}\left[\frac{v^{1-j\log(ed/j)/2}}{j\log(ed/j)/2-1}\right].$

To minimize the upper bound on $\mathbb{E}\left[{w_{j}^{*}}^{2}\right]$ we
choose

$v=K_{5}^{\frac{2}{j\log(ed/j)}}=K_{3}^{2}=K_{1}^{2}\left(\log(ed/j)\right)^{2/\eta}.$

and get

$\displaystyle\mathbb{E}\left[{w_{j}^{*}}^{2}\right]\leq
2K_{1}^{2}\left(\log(ed/j)\right)^{2/\eta}.$

For any $1\leq k\leq d$, using Jensen’s inequality,

$\displaystyle\mathbb{E}\left[\left(\sum_{i=1}^{k}{w_{i}^{*}}^{2}\right)^{\frac{1}{2}}\right]\leq\left(\sum_{i=1}^{k}\mathbb{E}\left[{w_{i}^{*}}^{2}\right]\right)^{\frac{1}{2}}\leq\left(\sum_{i=1}^{k}2K_{1}^{2}\left(\log(ed/i)\right)^{2/\eta}\right)^{\frac{1}{2}}\leq\sqrt{2k}K_{1}\left(\log(ed)\right)^{1/\eta}.$

Proof. [Proof of Proposition 4.1] In order to provide a bound on
$\|\hat{f}-f^{*}\|_{L_{2}}$, we need to compute the order of
$\omega_{\mu}({\cal F}_{R}-{\cal F}_{R},\tau Q_{{\cal H}_{R}}(2\tau)/16)$.
Based on Lemma C.1 and C.2 it is easy to see that, in a similar way to
[Men15],

$\mathbb{E}\left[\sup_{f\in{\cal F}_{R}\cap s{\cal
D}_{f^{*}}}\left\lvert\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\epsilon_{i}(f-f^{*})(X_{i})\right\rvert\right]\leq\begin{cases}c_{1}K_{1}R\left(\log
ed\right)^{\frac{1}{\eta}}&\quad(R/s)^{2}>d/4,\\\ c_{2}K_{1}s\sqrt{d}&\quad
u\leq(R/s)^{2}\leq d/4,\\\ \end{cases}$

where $c_{1},c_{2}$ are constants. Hence, following similar steps as in the
proof of [Men15, Lemma 4.6], we have

$\omega_{\mu}({\cal F}_{R}-{\cal F}_{R},\tau Q_{{\cal
H}_{R}}(2\tau)/16)\leq\begin{cases}\frac{c_{3}R}{\sqrt{\mu}}\log\left(ed\right)^{\frac{1}{\eta}}&\text{if
$\mu\leq c_{1}d,$}\\\ 0&\text{if $\mu>c_{1}d$}.\end{cases}$

From (C), choosing $r=1-2\iota$ by Theorem 3.1, for sufficiently large $N$, we
have with probability at least

$\displaystyle 1-\tilde{C}_{1}\frac{N^{1-2\iota}Q_{\cal
H}(2\tau)c^{\frac{1}{\eta_{1}}}}{4}{\rm
exp}(-N^{2\iota\eta_{1}})-\tilde{C}_{2}N{\rm
exp}\left(-\frac{(N^{2\iota}\tau_{0})^{\eta}}{M_{1}}\right),$

we have

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max\left(\frac{2c_{3}R\log(ed)^{\frac{1}{\eta}}}{\sqrt{Q_{\cal
H}(2\tau)}c^{\frac{1}{2\eta_{1}}}}N^{-\frac{1}{2}+\iota},N^{-\frac{1}{2}+\iota}\right).$

##  Appendix D Proofs of Section 4.2

######  Lemma D.1.

Let $\\{X^{\prime}_{i}\\}_{i=1}^{\mu}$ be an $\operatorname*{\texttt{iid}}$
sample with independent coordinates $X^{\prime}_{i,j}\sim L(\eta_{3},d_{j})$
and let $w$ be a random vector with coordinates

$\displaystyle
w_{j}=\frac{1}{\sqrt{\mu}}\sum_{i=1}^{\mu}X^{\prime}_{i,j}\qquad
j=1,2,\cdots,d.$ (101)

Then we have

$\displaystyle\mathbb{P}\left(\left\lvert w_{j}\right\rvert\geq t\right)\leq
C_{3}\left(d_{j}^{\eta_{3}-2p-1}\mu^{1-\frac{\eta_{3}}{2}}t^{\eta_{3}-2p}+d_{j}^{-2}t^{-p}\right),$
(102)

for some constant $C_{3}>0$.

Proof. [Proof of Lemma D.1] Using the symmetry of the distribution of $w_{j}$
we can write,

$\displaystyle\mathbb{P}\left(|w_{j}|\geq t\right)\leq
2\mathbb{P}\left(w_{j}\geq t\right).$ (103)

Setting $p=\eta_{3}-0.5\iota$, using Theorem 2.1 of [Che07] we get for any
$t>0$

$\displaystyle\mathbb{P}\left(w_{j}\geq t\right)\leq
C_{p}t^{-p}\max\left(r_{\mu,p}(t),\left(r_{\mu,2}(t)\right)^{\frac{p}{2}}\right)+{\rm
exp}\left(-\frac{d_{j}^{2}t^{2}}{16\sigma_{X,2}^{2}}\right),$ (104)

where

$C_{1,p}=2^{2p+1}\max\left(p^{p},p^{p/2+1}e^{p}\int_{0}^{\infty}x^{p/2-1}(1-x)^{-p}dx\right),$

and for any $k\in\\{p,2\\}$,

$r_{\mu,k}(t)=\sum_{i=1}^{\mu}\mathbb{E}\left[\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert^{k}\mathbbm{1}\left(\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert\geq\frac{3\sigma_{X,2}^{2}}{td_{j}^{2}}\right)\right].$

Now,

$\displaystyle\mathbb{E}\left[\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert^{p}\mathbbm{1}\left(\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert\geq\frac{3\sigma_{X,2}^{2}}{td_{j}^{2}}\right)\right]=$
$\displaystyle\int_{-\infty}^{\infty}\left\lvert\frac{x}{\sqrt{\mu}}\right\rvert^{p}\mathbbm{1}\left(\left\lvert\frac{x}{\sqrt{\mu}}\right\rvert\geq\frac{3\sigma_{X,2}^{2}}{td_{j}^{2}}\right)\frac{\eta_{3}(|x|d_{j})^{\eta_{3}-1}}{2\left(1+(|x|d_{j})^{\eta_{3}}\right)^{2}}dx$
$\displaystyle=$
$\displaystyle\int_{\frac{3\sigma_{X,2}^{2}\sqrt{\mu}}{td_{j}^{2}}}^{\infty}\left(\frac{x}{\sqrt{\mu}}\right)^{p}\frac{\eta_{3}(xd_{j})^{\eta_{3}-1}}{\left(1+(xd_{j})^{\eta_{3}}\right)^{2}}dx$
$\displaystyle\leq$
$\displaystyle\frac{\eta_{3}}{d_{j}^{2p-\eta_{3}+1}\mu^{\frac{\eta_{3}}{2}}(\eta_{3}-p)}\left(\frac{3\sigma_{X,2}^{2}}{t}\right)^{p-\eta_{3}}$
$\displaystyle\leq$ $\displaystyle
C_{2}d_{j}^{\eta_{3}-2p-1}\mu^{-\frac{\eta_{3}}{2}}t^{\eta_{3}-p},$ (105)

where $C_{2}$ is a constant which depends on $\eta_{3}$ and $p$. Then

$\displaystyle r_{\mu,p}(t)\leq
C_{2}d_{j}^{\eta_{3}-2p-1}\mu^{1-\frac{\eta_{3}}{2}}t^{\eta_{3}-p}.$

The term $r_{\mu,2}(t)$ can similarly be bounded as follows:

$\displaystyle
r_{\mu,2}(t)=\sum_{i=1}^{\mu}\mathbb{E}\left[\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert^{2}\mathbbm{1}\left(\left\lvert\frac{X^{\prime}_{i,j}}{\sqrt{\mu}}\right\rvert\geq\frac{3\sigma_{X,2}}{td_{j}^{2}}\right)\right]\leq\frac{\sigma_{X,2}^{2}}{d_{j}^{2}}.$
(106)

Using (105), and (106), from (104) we get

$\displaystyle\mathbb{P}\left(\left\lvert w_{j}\right\rvert\geq t\right)\leq
C_{3}\left(d_{j}^{\eta_{3}-2p-1}\mu^{1-\frac{\eta_{3}}{2}}t^{\eta_{3}-2p}+d_{j}^{-2}t^{-p}\right),$

for some constant $C_{3}>0$.

######  Lemma D.2.

Let $w_{1},w_{2},\cdots,w_{d}$ are independent copies of a random variable
such that (102) is true for all $j=1,2,\cdots,d$, i.e.,

$\displaystyle\mathbb{P}\left(\left\lvert w_{j}\right\rvert\geq t\right)\leq
C_{3}\left(d^{\eta_{3}-2p-1}\mu^{1-\frac{\eta_{3}}{2}}t^{\eta_{3}-2p}+d^{-2}t^{-p}\right)\quad
j=1,2,\cdots,d,$

for $\eta_{3}>2+2\iota$, and $p=\eta_{3}-0.5\iota$. Let
$\\{w_{j}^{*}\\}_{j=1}^{d}$ be the non-increasing arrangement of
$\\{|w_{j}|\\}_{j=1}^{d}$. Then for every $1\leq k\leq d$,

$\displaystyle\mathbb{E}\left[\left(\sum_{i=1}^{k}{w_{i}^{*}}^{2}\right)^{\frac{1}{2}}\right]\leq
C_{6}\sqrt{k}\left(d^{\eta_{3}/(2p)-1/2+1/p}d_{1}^{\eta_{3}/2-p-\eta_{3}/p+3/2-2/p}\mu^{1/2-\eta_{3}/4}+d^{1/p}d_{1}^{-2/p}\right),$

for some constant $C_{6}>0$ which depends on $\eta_{3}$ and $p$.

Proof. [Proof of Lemma D.2] First note that we have

$\displaystyle\mathbb{P}\left({w_{1}^{*}}^{2}\geq
t\right)=\mathbb{P}\left({w_{1}^{*}}\geq\sqrt{t}\right)\leq\sum_{j=1}^{d}\mathbb{P}\left(\left\lvert
w_{j}\right\rvert\geq\sqrt{t}\right)\leq
C_{3}\left(dd_{1}^{\eta_{3}-2p-1}\mu^{1-\eta_{3}/2}t^{\eta_{3}/2-p}+dd_{1}^{-2}t^{-p/2}\right).$

Now, using (102), for any $v>0$ (to be chosen later), we have

$\displaystyle\mathbb{E}\left[{w_{1}^{*}}^{2}\right]=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}({w_{1}^{*}}^{2}\geq t)dt$
$\displaystyle\leq$ $\displaystyle
v+\int_{0}^{\infty}\mathbb{P}({w_{1}^{*}}^{2}\geq t+v)dt$ $\displaystyle\leq$
$\displaystyle
v+\int_{0}^{\infty}C_{3}\left(dd_{1}^{\eta_{3}-2p-1}\mu^{1-\eta_{3}/2}(t+v)^{\eta_{3}/2-p}+dd_{1}^{-2}(t+v)^{-p/2}\right)dt$
$\displaystyle\leq$ $\displaystyle
v+C_{4}\left(dd_{1}^{\eta_{3}-2p-1}\mu^{1-\eta_{3}/2}v^{\eta_{3}/2-p+1}+dd_{1}^{-2}v^{1-p/2}\right),$

where $C_{4}=C_{3}\max(1/(p-1-\eta_{3}/2),1/(p/2-1))$. Choosing
$v=d^{2/p}d_{1}^{-4/p}$, we get

$\displaystyle\mathbb{E}\left[{w_{1}^{*}}^{2}\right]\leq
C_{5}\left(d^{\eta_{3}/p-1+2/p}d_{1}^{\eta_{3}-2p-2\eta_{3}/p+3-4/p}\mu^{1-\eta_{3}/2}+d^{2/p}d_{1}^{-4/p}\right),$

where $C_{5}=C_{4}+1$. Using Jensen’s inequality, we have

$\displaystyle\mathbb{E}\left[\left(\sum_{j=1}^{k}{w_{j}^{*}}^{2}\right)^{\frac{1}{2}}\right]$
$\displaystyle\leq\left(\sum_{j=1}^{k}\mathbb{E}\left[{w_{j}^{*}}^{2}\right]\right)^{\frac{1}{2}}\leq\left(k\mathbb{E}\left[{w_{1}^{*}}^{2}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C_{6}\sqrt{k}\left(d^{\eta_{3}/(2p)-1/2+1/p}d_{1}^{\eta_{3}/2-p-\eta_{3}/p+3/2-2/p}\mu^{1/2-\eta_{3}/4}+d^{1/p}d_{1}^{-2/p}\right),$

where $C_{6}=\sqrt{C_{5}}$, thereby completing the proof.

Proof. [Proof of Proposition 4.2] We start by obtaining a bound on the term
$\omega_{\mu}({\cal F}_{R}-{\cal F}_{R},\tau Q_{{\cal H}_{R}}(2\tau)/16)$. Let
$w$ be a random vector with coordinates

$\displaystyle
w_{j}=\frac{1}{\sqrt{\mu}}\sum_{i=1}^{\mu}X^{\prime}_{i,j}\qquad
j=1,2,\cdots,d.$ (107)

Let $\\{w_{j}^{*}\\}_{j=1}^{d}$ be the non-increasing arrangement of
$\\{|w_{j}|\\}_{j=1}^{d}$. Then, we have

$\displaystyle\mathbb{E}\left[\sup_{f\in{\cal F}_{R}\cap
s\mathcal{D}_{f^{*}}}\left\lvert\frac{1}{\sqrt{\mu}}\sum_{i=1}^{\mu}\epsilon_{i}(f-f^{*})(X^{\prime}_{i})\right\rvert\right]\leq\mathbb{E}\left[\sup_{t\in
B_{1}^{d}(2R)\cap
B_{2}^{d}(s)}\left\langle\frac{1}{\sqrt{\mu}}\sum_{i=1}^{\mu}X^{\prime}_{i},t\right\rangle\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[\sup_{t\in B_{1}^{d}(2R)\cap
B_{2}^{d}(s)}\left\langle w,t\right\rangle\right]=s\mathbb{E}\left[\sup_{t\in
B_{1}^{d}(2R/s)\cap B_{2}^{d}(1)}w^{\top}t\right]\leq
2s\mathbb{E}\left[\left(\sum_{j=1}^{(2R/s)^{2}}{w_{j}^{*}}^{2}\right)^{\frac{1}{2}}\right].$

If $(2R/s)^{2}<d$, using Lemma D.2 we get

$\displaystyle\mathbb{E}\left[\sup_{t\in B_{1}^{d}(2R)\cap
B_{2}^{d}(s)}w^{\top}t\right]\leq$ $\displaystyle
4C_{6}R\left(d^{\eta_{3}/(2p)-1/2+1/p}d_{1}^{\eta_{3}/2-p-\eta_{3}/p+3/2-2/p}\mu^{1/2-\eta_{3}/4}+d^{1/p}d_{1}^{-2/p}\right)$
$\displaystyle\leq$ $\displaystyle C_{7}Rd^{1/p+\iota/8},$

when $d_{1}\geq C_{6}^{\prime}$ for some constants $C_{6}^{\prime},C_{7}>0$.

If $(2R/s)^{2}\geq d$,

$\displaystyle\mathbb{E}\left[\sup_{t\in B_{1}^{d}(2R)\cap
B_{2}^{d}(s)}w^{\top}t\right]\leq 2s\sigma_{X,2}\sqrt{d}.$

So when $(2R/s)^{2}\geq d$,

$\displaystyle\mathbb{E}\left[\sup_{f\in{\cal F}_{R}\cap
s\mathcal{D}_{f^{*}}}\left\lvert\frac{1}{\mu}\sum_{i=1}^{\mu}\epsilon_{i}(f-f^{*})(X^{\prime}_{i})\right\rvert\right]\leq\gamma
s,$

for all $s>0$. When $\mu\leq C_{8}d^{1+2/p+\iota/4}$, we have
$(2R/s)\leq\sqrt{d}$ for

$\displaystyle s\geq\frac{C_{7}Rd^{1/p+\iota/8}}{\gamma\sqrt{\mu}}.$

When $\mu>C_{8}d^{1+2/p+\iota/4}$, we have $(2R/s)\geq\sqrt{d}$ for

$\displaystyle s\leq\frac{C_{7}Rd^{1/p+\iota/8}}{\gamma\sqrt{\mu}}.$

Combining the above facts, and choosing $\mu=\frac{N^{r}Q_{\cal
H}(2\tau)c^{\frac{1}{\eta_{1}}}}{4}$, and $r=1-2\iota$ we get

$\displaystyle\omega_{\mu}({\cal F}_{R}-{\cal F}_{R},\tau Q_{{\cal
H}_{R}}(2\tau)/16)\leq\begin{cases}\frac{C_{9}R}{\tau Q_{{\cal
H}_{R}}(2\tau)^{3/2}}d^{1/p+\iota/8}N^{-1/2+\iota}&\text{if $\mu\leq
C_{8}d^{1+2/p}$},\\\ 0&\text{if $\mu>C_{8}d^{1+2/p}$},\end{cases}$

where $C_{9}=32C_{7}/c^{1/(2\eta_{1})}$. Then using part 2 of Theorem 3.1, we
get,

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max\left\\{N^{-\frac{1}{4}\left(1-\frac{1}{\eta_{2}}\right)+\iota},\frac{C_{9}R}{\tau
Q_{{\cal H}_{R}}(2\tau)^{3/2}}d^{1/p+\iota/8}N^{-1/2+\iota}\right\\},$

with probability given by at least (14).

##  Appendix E Proofs of Section 4.3

Proof. [Proof of Proposition 4.3] Since we assumed $X$ to be Gaussian, ${\cal
F}_{R}$ is a $L_{g}$-subGaussian function class for some constant $L_{g}>0$.
So as shown in Section 6.5.2, we have,

$\displaystyle\omega_{Q}({\cal F}-{\cal
F},N,\zeta_{1},\zeta_{2})\leq\begin{cases}\frac{c_{3}(L_{g})R}{\sqrt{N}}\sqrt{\log(ed/N)}&\text{if}N\leq
c_{1}(L_{g})d,\\\ \frac{c_{4}(L_{g})R}{\sqrt{d}}&\text{if}c_{1}(L_{g})d<N\leq
c_{2}(L_{g})d,\\\ 0&\text{if}N>c_{2}(L_{g})d,\\\ \end{cases}$

where $c_{i}(L_{g}),i=1,2,3,4$ are constants dependent on only $L_{g}$. Then
using Corollary B.2, we have

$\displaystyle\|\hat{f}-f^{*}\|_{L_{2}}\leq\max\left(N^{-\frac{1}{2}+\iota},2\frac{c_{3}(L_{g})R}{\sqrt{N}}\sqrt{\log(ed/N)}\right),$

with probability at least (for some constants $c_{9},c_{10},\tilde{C}_{2}>0$)

$\displaystyle
1-c_{9}\epsilon^{1-\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}e^{-c_{10}\epsilon^{1+\frac{1}{\eta_{1}}}N^{\eta_{1}/(1+\eta_{1})}}-\tilde{C}_{2}N{\rm
exp}\left(-(N^{2\iota}\tau_{0})^{\eta}/M_{1}\right).$

##  Appendix F A Note on Condition (c) in Assumption 2.1 and
$\alpha_{N}^{*}(\gamma,\delta)$ in [Men15]

In this section, we discuss the relationship between Condition (c)-(i) of our
Assumption 2.1 and the multiplier process based assumption in [Men15, Equation
2.2 and $\alpha_{N}^{*}(\gamma,\delta)$]. For simplicity, we consider the
following simple model. Let $\\{X_{i}\\}_{i\in\mathbb{Z^{+}}}$ is an
$\operatorname*{\texttt{iid}}$ sequence of symmetric, zero-mean, random
vectors. Let $\\{Y_{i}\\}_{i\in\mathbb{Z^{+}}}$, $Y_{i}\in\mathbb{R}$ denote
the sequence given by $Y_{i}={\theta^{*}}^{\top}X_{i}+\xi_{i}$, where
$\theta^{*}\in B_{1}^{1}(R)$ and $\\{\xi_{i}\\}_{i=1}^{N}$ is an
$\operatorname*{\texttt{iid}}$ sequence and independent of $X_{i}$, $\forall
i$, and $\xi_{i}\sim N(0,\sigma_{1}^{2})$. The function class $\mathcal{F}$ we
consider is $\mathcal{F}\coloneqq{\cal
F}_{R}=\left\\{\langle\theta,\cdot\rangle:\theta\in B_{1}^{d}(R)\right\\}$.
Now let us assume $\frac{1}{N}\sum_{i=1}^{N}X_{i}\xi_{i}$ is heavy-tailed
random vector, with the tail lower bounded by $N{\rm exp}(-M(Nt))$, for some
positive increasing function of $t$, $M(t)$, i.e.,
$\mathbb{P}\left(\left\lvert
N^{-1}\sum_{i=1}^{N}X_{i}\xi_{i}\right\rvert>t\right)\geq M_{3}N{\rm
exp}(-M(Nt))$ for some $M_{3}>0$. Specifically setting $M(t)=t^{\eta},\eta>0$,
and $M(t)=\eta_{2}\log t,\eta_{2}>2$ one recovers (9) and (10). Now, recall
from [Men15] that,

$\displaystyle\alpha_{N}^{*}(\gamma,\delta):=\inf\left\\{s>0:{\mathbb{P}}\left(\sup_{\theta\in
B_{1}^{1}(2R)\cap
B_{2}^{1}(s)}\left\lvert\frac{1}{N}\sum_{i=1}^{N}\xi_{i}X_{i}\theta\right\rvert\leq\gamma
s^{2}\right)\geq 1-\delta\right\\}.$ (108)

Note that, for $s>0$,

$\sup_{\theta\in B_{1}^{1}(2R)\cap B_{2}^{1}(s)}\left\lvert
N^{-1}\sum_{i=1}^{N}\xi_{i}X_{i}\theta\right\rvert=\left\lvert
N^{-1}\sum_{i=1}^{N}\xi_{i}X_{i}\right\rvert\min(2R,s).$

We also have,

$\mathbb{P}\left(\left\lvert\frac{1}{N}\sum_{i=1}^{N}\xi_{i}X_{i}\right\rvert\leq\gamma
s^{2}/\min(2R,s)\right)\leq 1-M_{3}N{\rm exp}\left(-M\left(\gamma
Ns^{2}/\min(2R,s)\right)\right).$

Then, from (108), when $s=\alpha_{N}^{*}(\gamma,\delta)$,

$\displaystyle\delta\geq NM_{3}{\rm exp}\left(-M\left(\gamma
N{\alpha_{N}^{*}(\gamma,\delta)}^{2}/\min(2R,\alpha_{N}^{*}(\gamma,\delta))\right)\right).$

Hence, if we want a non-trivial bound on the generalization error, we need
${\alpha_{N}^{*}(\gamma,\delta)}^{2}\leq N^{-m_{0}}$ for some $m_{0}>0$. Set
$2R>N^{-m_{0}/2}$. When $M(t)\sim t^{\gamma_{2}},\gamma_{2}>0$,
$\frac{1}{N}\sum_{i=1}^{N}\xi_{i}X_{i}$ has a sub-weibull tail. If it has a
polynomially decaying tail, i.e., $M(t)=M_{4}\log t$ for some constant
$M_{4}>0$, then

$\delta\geq NM_{3}{\rm exp}\left(-M_{4}\log\left(\gamma
N^{1-m_{0}/2}\right)\right)=M_{3}\gamma^{-M_{4}}N^{1-(1-\frac{m_{0}}{2})M_{4}}.$

This implies that if $\frac{1}{N}\sum_{i=1}^{N}\xi_{i}X_{i}$ has a
polynomially decaying tail, one gets a polynomial probability statement on the
rate using complexity measure $\alpha_{N}^{*}(\gamma,\delta)$. Note that,
since we are considering $\operatorname*{\texttt{iid}}$ setting, choosing
$m_{0}<1$ would allow $\alpha_{N}^{*}(\gamma,\delta)$ to be of the order of
$N^{-1/2+\iota}$ where $\iota>0$ is a small number. Recall that the rates we
obtain in Theorem 3.1, and 3.2 are for $\beta$-mixing case. Indeed the worse
rates are due to the presence of the third terms on the RHS of (9), and (10) –
one needs to choose $\mathcal{A}(N)$ (used in the proofs of Theorem 3.1,and
3.2) suitably so that the third terms on the RHS of (9), and (10) decay to $0$
as $N\to\infty$.


# Mind the (spectral) gap: How the temporal resolution of wind data affects
multi-decadal wind power forecasts

Nina Effenberger
Cluster of Excellence Machine Learning
University of Tübingen
72076 Tübingen
<EMAIL_ADDRESS>
&Nicole Ludwig
Cluster of Excellence Machine Learning
University of Tübingen
72076 Tübingen
<EMAIL_ADDRESS>&Rachel H. White
Department of Earth, Ocean and Atmospheric Sciences
University of British Columbia
Vancouver, BC Canada V6T 1Z4
<EMAIL_ADDRESS>corresponding author

###### Abstract

To forecast wind power generation in the scale of years to decades, outputs
from climate models are often used. However, one major limitation of the data
projected by these models is their coarse temporal resolution - usually not
finer than three hours and sometimes as coarse as one month. Due to the non-
linear relationship between wind speed and wind power, and the long forecast
horizon considered, small changes in wind speed can result in big changes in
projected wind power generation. Our study indicates that the distribution of
observed 10min wind speed data is relatively well preserved using three- or
six-hourly instantaneous values. In contrast, daily or monthly values, as well
as any averages, including three-hourly averages, are almost never capable of
preserving the distribution of the underlying higher resolution data. Assuming
that climate models behave in a similar manner to observations, our results
indicate that output at three-hourly or six-hourly temporal resolution is high
enough for multi-decadal wind power generation forecasting. In contrast, wind
speed projections of lower temporal resolution, or averages over any time
range, should be handled with care.

##  1 Introduction

Wind is one of the main sources of renewable power and its utilisation is on
the rise in many countries, (e.g. Soares-Ramos et al., 2020). It is therefore
of uttermost importance to have reliable wind power forecasts in the range of
years to decades (i.e. a turbine’s lifetime) for site assessment and reliable
future power supply (Copernicus, 2023). However, the high variability and
stochasticity of weather and wind introduces uncertainty that makes multi-
decadal planning difficult. Furthermore, computational complexity limits the
resolution of forecasts; the temporal and spatial resolution 111Notice that
temporal and spatial resolution are often directly linked e.g. Courant et al.
(1928) of long-term and multi-decadal forecasts are therefore usually much
coarser than that of short-term forecasts (Eyring et al., 2016).

But do the low temporal resolution outputs from e.g. climate models provide
wind speeds representative of the true site-specific high resolution wind
speed distribution? An indicator that this could be the case is the so called
wind power spectral gap (Van der Hoven, 1957). Wind speed variability can be
assessed in the frequency domain in terms of power spectra by (Fourier-)
decomposing high-resolution wind speed observations. This decomposition
reveals an amplitude gap between high and low frequencies where the strong
variability in high frequencies (on the order of seconds to minutes) is
associated with turbulence, while strong low frequency variability (hours to
days) is associated with synoptic weather systems (Van der Hoven,
1957),(Stull, 1988). In the gap between the synoptic weather and turbulence
peaks in the frequency spectrum are frequencies with little variability; this
was found to be in the range of $10^{-3}Hz(\sim 17min)$ to $10^{-4}Hz(\sim
3h)$ by Van der Hoven (1957). However, other research suggests the gap is
smaller (Kang and Won, 2016) or may not exist at all (Larsén et al., 2016).
This spectral gap is described in many research papers (Horvath et al., 2012),
(Kang and Won, 2015), (Larsén et al., 2016), (Lopez-Villalobos et al., 2021)
indicating that observations that fall into the corresponding frequencies do
not add much knowledge. Given the inconsistency of the width of this spectral
gap, the question remains of what temporal resolution we should aim for in
multi-decadal wind power forecasting.

Due to the non-linear relationship between wind speed and wind power, small
changes in the wind speed distribution can lead to significant changes in wind
power generation. However, if the underlying wind speed characteristics are
preserved, lower-resolution data is preferred for multi-decadal forecasting to
reduce required storage space and potentially also computational costs.

Additionally, multi-decadal wind speed projections should also account for
climate change and interannual climate variability (Pryor et al., 2018).
Several studies indicate that climate change will affect average wind speeds
(McVicar et al., 2012), (Tobin et al., 2015)) as well as wind speed
variability (Pryor and Barthelmie, 2010), (Tobin et al., 2016), (Dunn et al.,
2019), (Jeong and Sushama, 2019), (Ringkjøb et al., 2020) which will impact
wind power generation. In general, it must be assumed that current wind
conditions may not be representative of future wind conditions (Jung and
Schindler, 2019); we thus often rely on output from climate models to help
inform about future potential wind power. But which data (resolution) do we
need for meaningful wind speed and power predictions?

There seems to be agreement among researchers that certain temporal
resolutions are too low. It is therefore common to account for additional
variability using so-called downscaling techniques (Pryor and Barthelmie,
2010). In the past, different statistical downscaling approaches have been
introduced, e.g. (Von Storch, 1999), (Tobin et al., 2015), (Shin et al.,
2018), as well as dynamical downscaling using regional climate models such as
CORDEX (Giorgi and Gutowski Jr, 2015), used by e.g. Davy et al. (2018) and
Yang et al. (2022). The question remains, however: what temporal resolution of
wind speed data is high enough?

The output from climate models are often available in various temporal
aggregations, where some datasets represent temporal averages and other
datasets consist of instantaneous values. In the CMIP6 datasets (Eyring et
al., 2016), one of the most widely used set of global circulation models
(GCMs), wind projections are available as temporal means and instantaneous
values (CMIP6 data request, 2016). Currently, however, little focus has been
placed on the choice of data and many studies are performed without explicitly
stating whether averages or instantaneous values are being used.

With this study we show that the type of data (instantaneous vs averages)
influences the wind speed distributions and thus the estimated wind power
generation. We also conduct analysis to determine whether there is a temporal
resolution that is high enough, i.e. for which added temporal resolution
provides little additional information and accuracy. We give recommendations
regarding the choice of data and the temporal resolution that downscaling
techniques, and climate model outputs, should aim for. To do this, we conduct
empirical data analysis using data from eight different mid-latitude sites
across Europe and North America. In Section 2 we describe the data and the
methodology used. Our results are presented in Section 3 and we discuss their
implications in Section 4. Finally, we conclude in Section 5.

##  2 Methods

To investigate how aggregating wind speed data affects the wind speed
distribution we use both parametric and non-parametric approaches. In the main
manuscript we focus on observations of turbine-hub-height winds from four
sites, with locations shown in Figure 1. We use these as our primary data as
they are hub-height data; however, all four sites have a relatively limited
observational period of only a few years (see Table 1). We thus also use 10$m$
wind speeds from an additional four observational met masts from locations
across Germany (see Figure A.1 for locations) that have between $18$ to $34$
years of data available. These data show very similar results to the hub-
height datasets (analysis presented in the supplementary material in Appendix
A). We also use these longer datasets to analyze multi-decadal tendencies of
wind power generation in Section 3. In the following we first give a
description of the data used and then describe our methodology.

### Data

We investigate wind speed observations using open-source high met mast wind
data of four mid-latitude locations. All of the wind speeds are either
measured at wind turbine hub-height directly, i.e. by nacelle anemometers
(sites Penmanshiel and Kelmarsh) or by high met masts (sites NWTC and Owez).
The observation heights are between $59m$ and $116m$ and all of the
measurements are provided as 10min averages, a very common aggregation-level
of wind resource data (Harper et al., 2010). In Table 1 we present static
information of the observation sites. Static information of the longer
datasets (at 10$m$ height) is presented in Table A.1. Unless specified
otherwise, the following abbreviations for the datasets can be found in the
top left corner of figures: a) Kelmarsh, b) Penmanshiel, c) NWTC, d) Owez, e)
Aachen, f) Zugspitze, g) Boltenhagen, h) Fichtelberg.

To bring the data to a format where we can compare different temporal
resolutions, we pre-process the data by excluding all days where at least one
observation is missing. We then average the 10 min wind speed observations to
three-hourly, six-hourly and daily data: the $n$’th wind speed value $w_{n}$
in the time-series averaged over $t$ consecutive time steps is computed as

$w^{t}_{n}=\frac{1}{t}\sum_{i=nt}^{nt+t}w^{1}_{i},$ (1)

where $t=18$ (three-hourly), $36$ (six-hourly) and $144$ (daily) respectively;
for $t=1$ we get the original 10 min resolution time series $w^{1}$. In
addition to calculating averages, we also consider wind speed time series of
lower resolution, which we call instantaneous values. To do so, we use wind
speed measurements every $t$’th time step only and exclude all other wind
speed measurements $w_{i}$ where $i\neq nt$. The results are eight
observations per day (every three hours), four observations per day (every six
hours) and one observation per day respectively.

Figure 1: Locations of the two wind farms in the UK and the tall towers in central North America and the Netherlands. Our hub-height observation locations include one mountainous site (NWTC), one off-shore site (Owez), one coastal site (Penmanshiel) and one site on flat terrain (Kelmarsh). Penmanshiel and Kelmarsh are wind farm sites, their wind speeds are therefore influenced by wake effects of the surrounding turbines (e.g. González-Longatt et al., 2012) and we only used one turbine for our evaluations.  Name |  Country
(Lat, Lon) |  Temporal resolution in min |  Observation period
(Number of days) |  Observation height in $m$ |  Mean wind speed (variance) in $\frac{m}{s}$ |  Data source
---|---|---|---|---|---|---
Kelmarsh |  United Kingdom (52.40, -0.95) |  10 |  2016-2021 (1687) |  78.5 |  6.25 (7.70) |  Plumley (2022a)
Penmanshiel |  United Kingdom (55.90, -2.31) |  10 |  2016-2021 (1522) |  59 |  7.27 (15.30) |  Plumley (2022b)
Owez |  Netherlands (52.61, 4.39) |  10 |  2012-2017 (1148) |  116 |  4.82 (13.02) |  Ramon et al. (2020), Ramon et al. (2022a)
NWTC |  United States
(39.21, -105.23) |  10 |  2005-2010 (1083) |  87 |  8.37 (16.55) |  Ramon et al. (2020), Ramon et al. (2022b)
Table 1: Static data of the four different sites with hub-height measurements.
Our chosen datasets cover a large range of observation heights as well as
different mean wind speeds and variances.

### Comparing wind speed distributions

To determine whether wind speed distributions from data of different temporal
resolution are statistically different, we compute pairwise Kolmogorov-Smirnov
test statistics of cumulative density distributions

$F_{W}(w)=P(W\leq w).$ (2)

The Kolmogorov-Smirnov statistic $D$ is given by:

$D=\sup\limits_{w}|T(w)-S(w)|$ (3)

where $w$ are the wind speed values, $T$ and $S$ are the wind speed
distributions to be compared, and the supremum, $\sup$, is the largest value
of the set of values $|T(w)-S(w)|$ across all $w$. The Kolmogorov-Smirnov test
only takes the largest absolute difference between the two distributions
across all $w$ values into account and we identify a statistically significant
difference if the $p$-value of an individual test is $p\leq 0.05$.

### Modeling wind speeds using Weibull distributions

The Kolmogorov-Smirnov test can tell us whether wind distributions of
different temporal resolution differ; in order to quantify the differences
found, we model the wind speeds, $w$, as Weibull distributions. This is done
by fitting the parameters of a three-parameter Weibull distribution,

$f(w;\beta,\lambda,\theta)=\frac{\beta}{\lambda}(\frac{w-\theta}{\lambda})^{\beta-1}e^{-(\frac{w-\theta}{\lambda})^{\beta}}$
(4)

to the different temporal resolution datasets. The three-parameter Weibull
distribution described in Equation 4 is defined by $w\geq\theta$ and
$f(w;\beta,\lambda,\theta)=0$ for $w<0$ where $\beta>0$ is the shape
parameter, $\lambda>0$ is the scale parameter and $\theta$ is the location
parameter of the distribution which equals the lowest possible value of the
distribution. For $\beta\approx 3$, the Weibull distribution approximates a
Gaussian distribution, while $3>\beta\geq 1$ corresponds to a right-skewed
distribution, and $\beta>3$ corresponds to a left-skewed distribution, for
$\beta<1$ the density values are steadily decreasing with increasing $w$.
$\lambda$ represents the variability, i.e. smaller values of $\lambda$ are
associated with less variability (Rinne, 2008). We fit the parameters using
Maximum Likelihood Estimation (MLE). This approach requires maximizing the
likelihood function

$L=\prod_{i=1}^{n}f(w_{i};\beta,\lambda,\theta).$ (5)

We can then evaluate the change of the parameters when wind speeds are
averaged or discarded to produce datasets with different temporal resolution.
While Weibull distributions are commonly used to model wind speed
distributions (e.g. Mert and Karakuş, 2015), we additionally use a generalized
Gamma distribution to test the sensitivity of our results to the choice of
underlying distribution. The conclusions are unchanged (figures shown in the
appendix) and we therefore consider our results to be insensitive to the
distribution choice.

### Validating the Weibull parametrization

Using a kernel density estimation we confirm that Weibull distributions are a
reasonable representation of our data, with the exception of monthly averages.
The kernel density estimator $\hat{f}$ of an unknown density $f$ at a point
$x$ is defined by

$\hat{f}_{h}(x)=\frac{1}{nh}\sum_{i=1}^{n}K(\frac{x-x_{i}}{h}),$ (6)

where we choose $K$ to be the Gaussian kernel

$K(x,h)=\exp(-\frac{x^{2}}{2h^{2}}).$ (7)

The band-width $h$ is selected using Scott’s rule (Scott, 2015).

Figures A.2 and A.4 show kernel density estimations for averaged and
instantaneous wind speeds respectively; it is clear that monthly values can
not be described by a Weibull distribution and thus we do not include this
temporal resolution in subsequent analysis. To validate the fit of the Weibull
distributions to the original wind speed distributions we generate quantile-
quantile plots of the observations of length $\frac{l}{t}$ against
$\frac{l}{t}$ randomly drawn samples from the corresponding Weibull
distribution; these plots are shown in Figure A.5 and Figure A.6. The Weibull
distributions generally provide a good fit to the data, although in some
locations the fit is less good at higher wind speeds.

### Power generation and transferability of results

As a last step, we relate the insights from the wind speed distributions to
multi-decadal wind power forecasts. Wind power generation is often forecasted
using hub-height wind speed forecasts with a turbine-specific wind power curve
(Wang et al., 2019) that describes the relationship between wind (speed) and
potential wind power generation and is highly non-linear. We apply the Enercon
E92/2350 wind power curve,visualized in Figure 2. Given the relatively short
duration of the main observational datasets we use in this study (around 3-5
years of non-missing data) and our interest in multi-decadal forecasts, we
study potential power generation using four $18$ to $34$ year long wind speed
datasets.

Additionally, to determine whether the results we find for observational
datasets are also applicable to climate model data, we repeat our analysis
using data generated by the historical run of the MPI-ESM1-2-LR general
circulation model (Eyring et al., 2016) which was the only model with three-
hourly data available on the ESGF node (WCRP CMIP6, 2023) at the time of
research.

Figure 2: Wind power curve of Enercon E92/2350 turbine. The relationship
between wind speed and wind power can be roughly divided into four different
regions: No power is generated if the wind speed is below the cut-in wind
speed or if the wind speed is above the cut-out wind speed where the turbine
is shut down to protect it from damage  . In between, wind power generation
first increases rapidly with increasing wind speed  . Once the maximum wind
speed that the wind turbine can convert to power is reached, the power output
is usually constant until the wind speed exceeds the cut-out wind speed  .

##  3 Results

We analyze the distributions of wind speed averages and instantaneous wind
speed time series, and find consistent results across all sites investigated:
averaging introduces shifts to the wind speed distributions, while three-
hourly and six-hourly instantaneous data are usually close to the original.
This is clearly demonstrated in Figure 3, with differences between 10min wind
speeds and averaged data (left hand column) larger than the differences
between 10min wind speeds and instantaneous wind speeds of lower resolution
(right hand column). Similar patterns can be observed in the four longer
datasets, see Figure A.7.

The impact of averaging can also be seen in the variance of the data: while
averaging does not affect the mean it reduces the variance of the averaged
time series (Figure 4). In contrast, three-hourly and six-hourly instantaneous
values (dashed lines in Figure 4) preserve the variance, with daily
instantaneous values preserving substantially more of the variance than daily
averages, and at some sites, more than six-hourly averages.

Figure 3: Difference of cumulative densities from the 10min data to the other
temporal resolution datasets for average wind speeds (left) and instantaneous
wind speeds (right). It can be seen that the averaged wind speeds are visually
distinguishable, which is less the case for instantaneous wind speeds,
particularly for data with a temporal resolution of six-hourly or higher.
Figure 4: Variances of averaged (solid lines) and instantaneous (dashed lines)
values. The variances steadily decrease when wind speeds are averaged and stay
close to the 10min variances for the instantaneous three-hourly and six-hourly
distributions.

### Kolmogorov-Smirnov tests

To quantify whether the difference between the 10min wind speed distribution
and the averaged and instantaneous distributions of lower resolution are
statistically significant we compute Kolmogorov-Smirnov test statistics of
their cumulative density distributions pairwise. The results for the
Penmanshiel site are presented in Table 2 (for averaged data) and Table 3
(instantaneous data), where values in bold indicate no statistically
significant difference in the distributions, i.e. that using the dataset of
lower temporal resolution may retain all information about the wind speed
distribution. We observe that all averaged distributions, as well as daily
instantaneous distributions, differ significantly from the 10min wind speeds
(right-most column), whilst three- and six-hourly instantaneous values are not
distinguishable from the 10 min data. The results for all other locations are
very similar (see Table A.2 to Table A.15): in general, averages do not
preserve the wind speed distribution, and three- and six-hourly instantaneous
values are almost always not statistically distinguishable from the 10 min
data.

| daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | $\mathbf{1}$ | $1.12\cdot 10^{-3}$ | $2.94\cdot 10^{-5}$ | $1.40\cdot 10^{7}$
six-hourly |  | $\mathbf{1}$ | $\mathbf{4.41\cdot 10^{-1}}$ | $5.83\cdot 10^{-3}$
three-hourly |  |  | $\mathbf{1}$ | $5.83\cdot 10^{-3}$
Table 2: Test statistics of Kolmogorov-Smirnov test for Penmanshiel averages.
We reject the hypothesis that the wind speed distributions are equal if $p\leq
5\cdot 10^{-2}$. Therefore, only the three-hourly averages are not
significantly different from six-hourly averages.

‘  daily six-hourly three-hourly 10min daily $\mathbf{1}$ $3.03\cdot 10^{-2}$
$2.34\cdot 10^{-2}$ $6.62\cdot 10^{-3}$ six-hourly $\mathbf{1}$
$\mathbf{1.00}$ $\mathbf{9.58\cdot 10^{-1}}$ three-hourly $\mathbf{1}$
$\mathbf{8.07\cdot 10^{-1}}$

Table 3: Test statistics of Kolmogorov-Smirnov test for Penmanshiel
instantaneous data. We reject the hypothesis that the wind speed distributions
are equal if $p\leq 5\cdot 10^{-2}$ which reveals that only daily
instantaneous values are significantly different from all other distributions.

### Changes in Weibull parameters

The cumulative density plots and Kolmogorov-Smirnov tests indicate that wind
speed distributions are likely to change when wind speeds are averaged and
likely to stay similar when measurements are discarded, at least until around
six-hourly resolution. The aim of the next steps is to quantify these changes
by parameterizing the distributions as Weibull distributions.

Figure 5 shows the values of the three Weibull parameters for the different
aggregation levels for averaging (top row) and instantaneous (bottom row). The
shape parameter $\beta$ stays approximately constant across all aggregation
levels and types, for both averaged and instantaneous data. For averaging, the
location parameter $\theta$ increases with higher aggregation levels. This is
consistent with the lowest values of the dataset increasing as they are
averaged. Conversely, with lower resolution instantaneous data, the lowest
values remain similar, leading to similar $\theta$ for all resolutions
studied. The scale parameter $\lambda$ decreases with increased averaging
length, and remains relatively constant for instantaneous data, consistent
with the changes in variance shown in Figure 4. We observe very similar
results for $\sim 30$ years of data measured at $10m$ height (see Figure A.8
and Figure A.9). In addition, two of our observational sites, NWTC and Owez,
have wind speed data at multiple heights, ranging from $10m$ to $130m$; the
changes in Weibull parameters seen in Figure 5 are found at all different
heights studied (see Figure A.10).

To test for robustness of our results we also use an MLE to fit a generalized
Gamma distribution (see Appendix for details); both Weibull and Gamma
distributions are regarded as suitable statistical models for wind speed data
(e.g. Mert and Karakuş, 2015). We find very similar results (see Figure A.11
to Figure A.14), with large changes in parameters when averaging data, and
relatively small changes for three- and six-hourly instantaneous values. This
suggests our conclusions are not sensitive to our choice of parameterization.

Figure 5: Parameters when data are averaged (top row) or discarded (bottom
row) for four different datasets a) Kelmarsh b) Penmanshiel c) NWTC d) Owez.
The Weibull parameters change when wind speeds are averaged and stay similar
when values are discarded. The corresponding non-parameterized wind speed
distributions are visualized in Figure A.2 and Figure A.4.

### Implications for multi-decadal wind power forecasting

As introduced in Section 1, wind speeds and wind speed variability are subject
to interannual variability and climate change. Hence, for multi-decadal wind
power forecasts climate models can provide useful information (e.g. Tobin et
al., 2015). To understand how our results apply on multi-decadal timescales,
we repeat our analysis using data from four sites where multiple decades of
$10m$ wind speed observations are available. We also repeat our analysis on
climate model output data to determine whether our conclusions are applicable
to model data. We extract $10m$ wind speeds from the historical run of the
CMIP6 model MPI-ESM1-2-LR for grid points closest to these four longer-term
observational sites (locations shown in Figure A.1). We only use direct output
from the model, and thus at daily temporal resolution we do not have
instantaneous values, only averages.

For these multi-decadal datasets the Kolmogorov-Smirnov tests and Weibull
parameterization analysis produce results comparable to those for the hub-
height observations shown in the previous sections. The parameters of fitted
Weibull distributions behave similarly in response to changing temporal
resolution, with a decrease of $\lambda$ as temporal resolution decreases
indicating a decrease in variability (see Figure 6). The climate model data do
not show an increase in $\theta$ with decreasing temporal resolution, although
for three-hourly and six-hourly averages $\theta$ fitted to the climate model
data is very close to the parameters fitted to the observational data.

So far we have only looked at wind speeds and their distributions. However,
these are just a proxy for wind power – our variable of interest – and wind
power depends non-linearly on wind speed. To estimate the power a wind turbine
could generate over its lifetime, we apply the Enercon E92/2350 power curve
(see Figure 2) to the four multi-decadal observational wind speed datasets as
well as to the wind speeds from the corresponding closest grid-points in the
CMIP6 dataset. We then compare the expected cumulative power generation of the
highest available resolution (10min in observations and three-hourly
instantaneous values in CMIP6 model) to lower resolutions (three-hourly, six-
hourly, and daily averages and instantaneous values). Although the highest
available resolution in the CMIP6 model is only three-hourly, our previous
analysis shows that these values are closely aligned with 10min data.

Figure 7 shows the expected cumulative power generation using wind speed
observations of different resolutions. We show this as a fraction of the total
power generation achieved when applying the wind power curve to the 10min
observational wind speed data and integrating over the whole time period. The
dotted gray line shows 100% and thus if the expected cumulative power
generation reaches this threshold without overshooting we consider the change
introduced by the temporal resolution to be small. The top row of Figure 7
shows that averaging values, particularly to daily, but even to three- or six-
hourly at some locations, can lead to relatively large errors in estimated
power generation, with errors of up to -34.48% (daily average), -15.45% (six-
hourly average), and -10.06% (three-hourly average). Conversely, three-hourly
and six-hourly instantaneous values, shown in the bottom row, reveal very
similar results to the 10min data, with errors less than 2%.

Figure 8 demonstrates that very similar results are found using the output
from the MPI-ESM1-2-LR global climate model, with results given relative to
the total amount of power generated using three-hourly instantaneous values.
In all cases six-hourly instantaneous values are closest to three-hourly
instantaneous values, followed by three-hourly averages and six-hourly
averages; daily averages differ substantially.

Reducing temporal resolution leads to an underestimation of expected power
generation at all sites studied except for daily instantaneous values at
Zugspitze (Figure 7 bottom row, site (f)), a site with relatively high mean
wind speeds and high wind speed variance, situated at a high elevation above
sea level ($2956m$). This is very likely a function of the particular wind
turbine power curve we have chosen. It is important to note that in all cases,
both with observational data and climate projections, the difference in power
generation compared to higher-resolution data increases with an increasing
forecast horizon and does not average out - the shift in the wind distribution
leads to a systematic error in wind power estimation. In general, averaging
tends to result in an underestimation of expected power generation, while
discarding values has only minor impacts.

Figure 6: Parameters of Weibull distribution fitted to observational data
averages (dashed lines) of the four multiple-decadal sites and to the closest
grid points in the CMIP6 MPI-ESM1-2-LR dataset (solid lines). The parameter
$\lambda$ decreases in all cases when comparing three-hourly to daily
averages, indicating a decrease in variability. The location parameter $\beta$
does not show any consistent trends. $\theta$ increases with daily averages in
observational data, but not in the CMIP6 data.  Figure 7: Cumulative power
generation of daily, six-hourly and three-hourly values relative to the total
10min wind power generation computed by feeding the wind speeds into a power
curve. Top row: Average values underestimate wind power generation – the
longer the forecast the higher the error. Bottom row: Instantaneous values
every three or six hours only introduce minor errors. Daily data is in no case
a good proxy. The power generation gap at the Fichtelberg site between $\sim
2005$ and $2010$ stems from various missing observations. Absolute errors are
reported in Table A.16. Figure 8: Cumulative power generation of daily, six-
hourly and three-hourly values relative to three-hourly instantaneous (3h
inst.) wind power generation at the four CMIP6 locations closest to e) Aachen,
f) Zugspitze, g) Boltenhagen, h) Fichtelberg computed by feeding the wind
speeds into a power curve. Top row: Average values underestimate wind power
generation. Bottom row: Instantaneous values (note that daily instantaneous
values were not available as direct output, and so are not included). Using
six-hourly instantaneous values introduces only minor errors relative to
three-hourly instantaneous values.

##  4 Discussion

Global climate models simulate climate dynamics physically using partial
differential equations. Their long forecast horizons make them computationally
expensive with large storage space needs and their output is usually provided
with a temporal resolution ranging from three hours to one month, either as
instantaneous or averaged values. Using hub-height wind speed observations,
this study investigates how the temporal resolution of wind speed data affects
the wind speed distributions. Using multi-decadal observational datasets at
10$m$ height we study the corresponding potential power generation and which
time resolution is actually necessary.

Using hub-height data we find that in all cases investigated, three-hourly and
six-hourly instantaneous observations well-preserve the underlying 10min wind
speed distribution, whilst three- and six-hourly averaging leads to
distributional shifts. However, whether a significant wind speed distribution
shift results in a significant change in wind power generation depends on the
turbine and its corresponding power curve. Our results, using an example
turbine power curve, indicate that the differences in wind distribution
highlighted in this study can lead to accumulating errors when power
generation is forecasted for years to decades (compare Figure 7 and Figure 8).

Our results are consistent across different observational sites and a GCM. We
can therefore give two suggestions when working with wind speed projections
for wind power modelling. First, instantaneous wind speed projections should
be preferred over wind speed averages, as sub-sampling wind speed data
introduces relatively minor errors in contrast to averaging wind speeds, where
we observe a characteristic distributional shift. This shift associated with
averaging data indicates that we might consistently over- or underestimate
wind power generation. Second, instantaneous wind speed projections of six or
three hours suffice, whilst daily data may be too low resolution, even with
instantaneous values. For our sites temporal downscaling or either three- or
six- hourly data is unlikely to provide substantial improvement in accuracy.
For example, instantaneous observational wind speeds with a three-hourly
temporal resolution lead to errors of less than $0.29\%$, with six-hourly data
leading to errors of less than $1.57\%$. The experiments conducted using
climate model data support these results. This knowledge can be useful to
reduce storage space almost loss-free and decrease computational complexity in
further applications using the data.

The primary shortcomings of our investigations include a potential lack of
generalizability across turbines, sensitivity to our underlying highest
temporal data resolution, and uncertain transferability to other climate
models. More specifically, as wind speed data measured at hub-height is often
confidential, we are limited to relatively few datasets. Furthermore, for site
assessment, wind speeds are usually transformed to hub-height which is non-
trivial (e.g. Bañuelos-Ruedas et al., 2010). However, the close agreement of
results from stations across a range of different locations, including
different local geography (off-shore, on-shore, different altitudes and local
topography), suggests that our results likely hold for the majority of
locations. For 7 out of the 8 locations studied, using daily instantaneous or
average values underestimates the power generation (see Figure 7), while six-
hourly values are good approximations. For one station, however, the daily
instantaneous data is an overestimate; understanding the conditions under
which daily data over- or under-estimates the power generation would be useful
for studies in which only daily data is available. In addition, while the
Enercon power curve we use in this study has the characteristic shape of any
modern horizontal wind turbine power curve, it is not necessarily
representative of the turbines at a particular site.

Regarding the sensitivity to the underlying data resolution, we use data in
10min resolution, hence variability on shorter time scales – often associated
with turbulence (e.g. Stull, 1988) – is not preserved. However, higher
temporal resolution data is rarely available or used in wind power forecasting
(Tawn and Browell, 2022), (Effenberger and Ludwig, 2022), and thus we assume
that this is a minor issue. This claim is also supported by Lopez-Villalobos
et al. (2021) who investigate wind power spectra (Van der Hoven, 1957) and
find only small differences between power production given wind speeds of
different resolution between 1min and 6h.

Lastly, climate projections are characterized by different pathways that
describe anthropogenic climate change (Eyring et al., 2016). This makes
handling the data cautiously even more important, especially in wind power
forecasting, where non-linearly dependent wind speed projections are often
used as a proxies for power generation. Our results using one historical CMIP6
run indicate that changes in observational wind speed distributions for
different temporal resolution data can be seen in data from climate models as
well. However, future research has to investigate the sensitivity of these
results to different climate models.

##  5 Conclusion

Wind power generation depends non-linearly on wind speeds. For multi-decadal
wind power forecasts in the order of years to decades, small changes in
modelling the wind speed distributions can result in large systematic errors
in power estimation, with absolute errors increasing with longer forecast
horizon. Using hub-height wind speed observations and climate model output
data, this study investigates how the temporal resolution of wind speed data
affects wind speed distributions and corresponding potential power generation.

We show that instantaneous wind speeds of lower resolution more accurately
represent the underlying distribution of higher resolution data when compared
to averaged wind speeds. Three- and six-hourly instantaneous values preserve
the wind speed distribution of 10min wind speed averages well. Small changes
in the wind speed distribution, through averaging or using daily data, has
significant impacts on the estimated wind power generation of a turbine over
its lifetime. These results hold true across several observational sites and a
global climate model. Based on our results, we argue that modelling wind speed
distributions correctly is what we should aim for in multi-decadal wind power
forecasting.

## Acknowledgements

This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC number 2064/1 –
Project number 390727645 and the Athene Grant of the University of Tübingen.
The authors thank the International Max Planck Research School for Intelligent
Systems (IMPRS-IS) for supporting Nina Effenberger, and the Natural Sciences
and Engineering Research Council of Canada (NSERC) [RGPIN-2020-05783] for
supporting Rachel H. White. We acknowledge support by Open Access Publishing
Fund of University of Tübingen. Open Access funding enabled and organized by
Projekt DEAL. WOA Institution: Eberhard Karls Universitat Tubingen Consortia
Name : Projekt DEAL. Nina wants to thank Roland Stull and the Weather Forecast
Research Team at UBC for (intellectual) resources!

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##  Appendix A Supplementary Material

In the following we present results from additional datasets. Unless specified
otherwise, we assign the following abbreviations to the different datasets: a)
Kelmarsh, b) Penmanshiel, c) NWTC, d) Owez,e) Aachen, f) Zugspitze, g)
Boltenhagen, h) Fichtelberg. They can be found in the top-left corner of all
plots.

Name |  Lat, Lon |  Temporal resolution in $min$ |  Observation period |  Elevation in $m$ |  Mean wind speed (variance) in $\frac{m}{s}$ |  Data source
---|---|---|---|---|---|---
Aachen |  50.78, 6.09 |  10 |  29.04.1993-31.03.2011 |  202.00 |  3.06 (4.19) |  Wetterdienst
Boltenhagen |  54.00, 11.19 |  10 |  11.07.1991-31.12.2022 |  15.00 |  5.54 (8.09) |  Wetterdienst
Fichtelberg |  49.98, 11.84 |  10 |  16.12.1993-31.12.2022 |  654.50 |  2.58 (2.91) |  Wetterdienst
Zugspitze |  47.42, 10.98 |  10 |  31.07.1994-31.12.2022 |  2956.00 |  7.00 (15.86) |  Wetterdienst
Table A.1: Static data of the four longer-term 10m meteorological measurement
sites in Germany. None of the stations was moved during the time period
evaluated.  Figure A.1: Locations of German weather stations that serve as
longer datasets in blue  . The corresponding CMIP6 locations we investigate
for Figure 6 and Figure 8 are marked in orange  . Figure A.2: Kernel density
estimations of averaged wind speeds. The distributions of 10min, three-hourly,
six-hourly and daily data look similar to a Weibull distribution; monthly data
are not Weibull distributed. Figure A.3: Kernel density estimations of
averaged wind speeds of different resolutions excluding monthly data. Figure
A.4: Kernel density estimations of instantaneous wind speeds of different
resolutions. Data at most sites look approximately Weibull distributed. Figure
A.5: QQ-plots of averaged wind speeds and fitted Weibull distributions. From
left to right the data resolution is 10min, three-hourly averages, six-hourly
averages and daily averages. Figure A.6: QQ-plots of instantaneous wind speeds
and fitted Weibull distributions. From left to right the data resolution is
10min, three-hourly averages, six-hourly averages and daily averages. Figure
A.7: Figure 3 for longer datasets. Difference of cumulative densities from the
10min data to the other temporal resolution datasets for average wind speeds
(left) and instantaneous wind speeds (right). It can be seen that the averaged
wind speeds are visually distinguishable, which is less the case for
instantaneous wind speeds, particularly for three- or six-hourly data.

(a) Absolute trends.

(b) Relative trends.

Figure A.8: Weibull parameter trends when fitted to averaged wind speed
distributions.

(a) Absolute trends.

(b) Relative trends.

Figure A.9: Weibull parameter trends when fitted to instantaneous wind speed
distributions.

(a) Parameter trends fitted to averaged wind speed distributions.

(b) Parameter trends fitted to instantaneous wind speed distributions.

Figure A.10: Weibull parameter trends of different observations heights for
the tall towers NWTC and Owez.

To test for robustness of our results we also use MLE to fit a generalized
Gamma distribution

$f(w;a,d,p)=\frac{\frac{p}{a^{d}}x^{d-1}\exp(-(\frac{x}{a})^{p})}{\Gamma(\frac{d}{p})}$
(8)

with

$\Gamma(z)=\int_{0}^{\infty}t^{z-1}\exp(-t)dt$ (9)

instead of fitting a Weibull distribution. Both distributions were regarded to
be suitable statistical models in earlier studies [e.g. Mert and Karakuş,
2015].

Figure A.11: QQ-plots of averaged wind speeds and fitted Gamma distributions.
From left to right the data resolution is 10min, three-hourly averages, six-
hourly averages and daily averages. Figure A.12: QQ-plots of instantaneous
wind speeds and fitted Gamma distributions. From left to right the data
resolution is 10min, three-hourly averages, six-hourly averages and daily
averages.

(a) Absolute trends.

(b) Relative trends.

Figure A.13: Gamma parameter trends when fitted to averaged wind speed
distributions.

(a) Absolute trends.

(b) Relative trends.

Figure A.14: Gamma parameter trends when fitted to instantaneous wind speed
distributions.

| daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $2.24\cdot 10^{-3}$ | $4.30\cdot 10^{-5}$ | $4.98\cdot 10^{-9}$
six-hourly |  | 1 | $\mathbf{4.54\cdot 10^{-1}}$ | $6.88\cdot 10^{-6}$
three-hourly |  |  | 1 | $1.85\cdot 10^{-4}$
Table A.2: $p$-values of Kolmogorov-Smirnov test for averages Kelmarsh. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $1.54\cdot 10^{-2}$ | $5.58\cdot 10^{-3}$ | $2.15\cdot 10^{-3}$
six-hourly |  | 1 | $\mathbf{1.00}$ | $\mathbf{9.81\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{1.00}$
Table A.3: $p$-values of Kolmogorov-Smirnov test for instantaneous Kelmarsh. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $8.15\cdot 10^{-11}$ | $9.13\cdot 10^{-18}$ | $1.35\cdot 10^{-33}$
six-hourly |  | 1 | $5.53\cdot 10^{-4}$ | $2.51\cdot 10^{-35}$
three-hourly |  |  | 1 | $2.45\cdot 10^{-27}$
Table A.4: $p$-values of Kolmogorov-Smirnov test for averaged NWTC wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $\mathbf{1.64\cdot 10^{-1}}$ | $3.84\cdot 10^{-2}$ | $3.61\cdot 10^{-2}$
six-hourly |  | 1 | $\mathbf{7.21\cdot 10^{-1}}$ | $\mathbf{6.20\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{9.01\cdot 10^{-1}}$
Table A.5: $p$-values of Kolmogorov-Smirnov test for instantaneous NWTC wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $3.00\cdot 10^{-3}$ | $2.25\cdot 10^{-4}$ | $1.40\cdot 10^{-5}$
six-hourly |  | 1 | $\mathbf{7.15\cdot 10^{-1}}$ | $\mathbf{9.20\cdot 10^{-2}}$
three-hourly |  |  | 1 | $\mathbf{4.95\cdot 10^{-1}}$
Table A.6: $p$-values of Kolmogorov-Smirnov test for averaged Owez wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $\mathbf{2.14\cdot 10^{-1}}$ | $\mathbf{2.18\cdot 10^{-1}}$ | $\mathbf{1.29\cdot 10^{-1}}$
six-hourly |  | 1 | $\mathbf{1.00}$ | $\mathbf{9.78\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{9.77\cdot 10^{-1}}$
Table A.7: $p$-values of Kolmogorov-Smirnov test for instantaneous Owez wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $3.83\cdot 10^{-14}$ | $1.43\cdot 10^{-21}$ | $3.24\cdot 10^{-52}$
six-hourly |  | 1 | $6.11\cdot 10^{-4}$ | $1.76\cdot 10^{-45}$
three-hourly |  |  | 1 | $1.73\cdot 10^{-44}$
Table A.8: $p$-values of Kolmogorov-Smirnov test for averaged Aachen wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $2.89\cdot 10^{-22}$ | $1.54\cdot 10^{-25}$ | $2.14\cdot 10^{-28}$
six-hourly |  | 1 | $\mathbf{9.35\cdot 10^{-1}}$ | $\mathbf{7.85\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{1.00}$
Table A.9: $p$-values of Kolmogorov-Smirnov test for instantaneous Aachen wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $4.47\cdot 10^{-23}$ | $9.69\cdot 10^{-39}$ | $2.71\cdot 10^{-72}$
six-hourly |  | 1 | $1.16\cdot 10^{-5}$ | $3.01\cdot 10^{-45}$
three-hourly |  |  | 1 | $4.90\cdot 10^{-33}$
Table A.10: $p$-values of Kolmogorov-Smirnov test for averaged Zugspitze wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $5.68\cdot 10^{-22}$ | $3.59\cdot 10^{-24}$ | $3.84\cdot 10^{-27}$
six-hourly |  | 1 | $\mathbf{7.13\cdot 10^{-1}}$ | $\mathbf{3.89\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{1.00}$
Table A.11: $p$-values of Kolmogorov-Smirnov test for instantaneous Zugspitze wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $9.77\cdot 10^{-34}$ | $1.04\cdot 10^{-64}$ | $1.39\cdot 10^{-85}$
six-hourly |  | 1 | $1.04\cdot 10^{-5}$ | $1.69\cdot 10^{-44}$
three-hourly |  |  | 1 | $4.23\cdot 10^{-35}$
Table A.12: $p$-values of Kolmogorov-Smirnov test for averaged Boltenhagen wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $7.14\cdot 10^{-31}$ | $1.72\cdot 10^{-34}$ | $1.09\cdot 10^{-38}$
six-hourly |  | 1 | $\mathbf{7.84\cdot 10^{-1}}$ | $\mathbf{3.59\cdot 10^{-1}}$
three-hourly |  |  | 1 | $\mathbf{1.00}$
Table A.13: $p$-values of Kolmogorov-Smirnov test for instantaneous Boltenhagen wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $4.20\cdot 10^{-108}$ | $2.04\cdot 10^{-163}$ | $3.60\cdot 10^{-265}$
six-hourly |  | 1 | $1.47\cdot 10^{-17}$ | $1.64\cdot 10^{-187}$
three-hourly |  |  | 1 | $6.61\cdot 10^{-184}$
Table A.14: $p$-values of Kolmogorov-Smirnov test for averaged Fichtelberg wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. | daily | six-hourly | three-hourly | 10min
---|---|---|---|---
daily | 1 | $4.34\cdot 10^{-83}$ | $2.06\cdot 10^{-117}$ | $1.27\cdot 10^{-135}$
six-hourly |  | 1 | $9.02\cdot 10^{-6}$ | $7.67\cdot 10^{-12}$
three-hourly |  |  | 1 | $\mathbf{4.24\cdot 10^{-1}}$
Table A.15: $p$-values of Kolmogorov-Smirnov test for instantaneous Fichtelberg wind speed distributions. All values where $p\leq 0.05$ are marked bold and the corresponding distributions are not considered to be significantly different. Location | 10min | 3h avrg. | 3h inst. | 6h inst.
---|---|---|---|---
Aachen | 0 | -40.28 | -3.47 | 1.99
Zugspitze | 0 | -39.16 | 4.76 | 6.06
Boltenhagen | 0 | -111.23 | 9.86 | 3.69
Fichtelberg | 0 | -62.71 | 1.83 | -9.80
Table A.16: Errors (in $MW$) of wind power generation prediction when using 30
years of data of lower resolution (three-hourly average and three-hourly and
six-hourly instantaneous) wind speed observations compared to 10min wind speed
observations.