2022

[1]Kris Sankaran

1]Department of Statistics, University of Wisconsin - Madison, 1300 University
Ave, Madison, 53706, WI

# Source data selection for out-of-domain generalization

Xinran Miao<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

Models that perform out-of-domain generalization borrow knowledge from
heterogeneous source data and apply it to a related but distinct target task.
Transfer learning has proven effective for accomplishing this generalization
in many applications. However, poor selection of a source dataset can lead to
poor performance on the target, a phenomenon called negative transfer. In
order to take full advantage of available source data, this work studies
source data selection with respect to a target task. We propose two source
selection methods that are based on the multi-bandit theory and random search,
respectively. We conduct a thorough empirical evaluation on both simulated and
real data. Our proposals can be also viewed as diagnostics for the existence
of a reweighted source subsamples that perform better than the random
selection of available samples.

###### keywords:

Out-of-domain generalization, Domain shift, Source selection, Transfer
Learning, Multi-armed bandit

##  1 Introduction

In many modern prediction modeling applications, the operating conditions of
an algorithm may not match the setting in which it was trained. For example,
this may result from a change in spatial or temporal context. In either case,
ephemeral predictors — those present in the training data, but not in the
target setting — can result in performance drops (Tsymbal, 2004; Rosenstein .,
2005; Zhang ., 2020). This problem is especially acute when labeled data
scarce in the target regime. As a way of borrowing extra knowledge from out-
of-distribution source data, transfer learning shows great performance in
practice, but its mysterious mechanism makes it difficult to choose a suitable
dataset to pretrain a model on, especially under a computational budget (Wang
., 2019). This is referred to as the out-of-domain generalization problem, and
it has attracted substantial interest from the community over the last two
decades (Moreno-Torres ., 2012; D. Li ., 2018; Yue ., 2019; Recht ., 2019).
Important advances include the use of transfer learning methods to improve
performance on small target datasets (Long ., 2013; Ghifary ., 2014; Guo .,
2018) and the introduction of regularization strategies that encourage models
to learn structure that generalizes across contexts (Kirkpatrick ., 2017;
Chen, Wang, Long Wang, 2019; Chen, Wang, Fu ., 2019).

However, existing approaches are difficult to interpret. When a model performs
much worse outside of the context that it was trained, it can be difficult to
isolate characteristics of the training data or resulting model that are
responsible for the deterioration, which is called negative transfer
(Rosenstein ., 2005; Wang ., 2019). Conversely, performance gains may be
obtained when pretraining models on different source datasets, but the
mechanisms are often unclear. For purely automated systems, prediction
performance is still the key property. However, in an increasing number of
scientific or social applications, models must also be used by domain experts
to better understand their systems of interest, and in this context,
attribution is critical.

Given target and source tasks, one can always supplement data from the target
task with data from the source. The question is: Does the choice make a
difference, and if so, can we understand and take advantage of it? To address
this, we empirically explore several strategies for training models across
non-identically distributed subsets of a source dataset and measuring how
model performance varies across reweighted sources. We consider both linear
regression models trained on tabular and geographic data and deep learning
models trained on imagery of pathology slides. Specifically, we investigate
ensemble and bandit-based approaches for highlighting subsets of the source
data that strongly affect performance on the target. Differential performance
across ensemble members or a strong preference for certain bandit arms are
used as indicators of distributional heterogeneity. Further, we propose
accompanying visualization strategies that clarify the relationship between
source and target data within these algorithms, identifying the optimal source
compositions of the ensemble method and the trajectories of the bandit
selection method. Finally, we study in detail the factors that influence these
source selection methods, including the initial data representation, the
criteria for splitting the sources, and the relative abundance of data similar
to the target.

In section 2 we highlight commonly used approaches to out-of-domain
generalization and then review concepts that directly inform our approach. In
section 3, we detail our source selection strategies. In section 4, we apply
these methods to a simulation reflecting the temporal generalization problem.
Section 5 presents two case studies, one of home prices across California and
another on tumor identification across hospitals with different patient
populations and imaging equipment. We note that all data and code described
below are available: https://github.com/XinranMiao/source_selection.

##  2 Background

###  2.1 Transfer and negative transfer

Consider a prediction task with target data of interest
$\mathcal{D}^{t}=\\{x_{i}^{t},y_{i}^{t}\\}_{i=1}^{n^{t}}$, where $x_{i}^{t}$’s
and $y_{i}^{t}$’s are the observations of variables and responses and $n^{t}$
is the target data size. When $n^{t}$ is small and a related larger
supplementary dataset is available, a common rule-of-thumb is to pretrain the
model on this larger dataset before training it on the target. This practice
is called is transfer learning (Torrey Shavlik, 2010). Denote this related
source dataset by $\mathcal{D}^{s}=\\{x_{i}^{s},y_{i}^{s}\\}_{i=1}^{n^{s}}$
where $n^{s}\gg n^{t}$. We will often have metadata about each observation in
the source. This can be information aside from the predictors or certain data
representations that we are unavailable in the target. Let $p^{t}$ and $p^{s}$
be the unknown distributions of target and source, and $\theta\in\Theta$ be
the parameter of the prediction model. Then transfer learning learns a model
$f_{\theta}$ based on $\mathcal{D}^{s}$, which can be fine-tuned on $D^{t}$
and ideally yields $f_{\theta}(x_{\ast}^{t})\approx y_{\ast}^{t}$ for new
samples $\left(x_{\ast}^{t},y_{\ast}^{t}\right)\sim p^{t}$.

Despite the popularity of transfer learning, negative transfer appears –
training models on source together with target data can be worse than training
on target data alone (Rosenstein ., 2005; Wang ., 2019; Zhang ., 2020). This
can be caused by distributional gaps (Koh ., 2020), where $p^{t}$ and $p^{s}$
are very dissimilar. For example, satellite images from Africa and the Arctic
Pole can be extremely different for landcover segmentation due to dissimilar
feature spaces and prediction rules. Note that both the covariate distribution
and the conditional distribution of response given covariates may change.
(Zhang ., 2020) summarizes methods to overcome negative transfer, including
selecting or reweighting source data (Yao Doretto, 2010; J. Li ., 2019),
improving target data labeling (Gholami ., 2020), addressing domain divergence
(Shen ., 2018), and integrated approaches (Kuzborskij Orabona, 2013).

Instead of working on fixed source and target sets, there also have been
attempts to investigate the transferability of source datasets across an
entire application area. In these studies, the goal is to identify a small
number of source datasets that can be used for transfer learning across a
variety of problems. For example, (Neumann ., 2020) explores qualitatively
important factors in the remote sensing context, identifying source datasets
whose learned representations lead to high performance on various downstream
tasks.

###  2.2 Subset selection

With the purpose of saving computation and avoiding negative transfer,
researchers have developed subset selection strategies to weight the source
dataset effectively with respect to the target task. Suppose we partition the
source into $K$ subsets
$\mathcal{D}^{s}_{1},\mathcal{D}^{s}_{2},\cdots,\mathcal{D}^{s}_{K}$, with
empirical distributions $\hat{p}^{s}_{1},\cdots,\hat{p}^{s}_{K}$. Let
$\\*w=[w_{1},\cdots,w_{K}]$ be the weights of $K$ subsets, then with a
specified loss function $L:\mathcal{D}^{t}\times\Theta\to\mathcal{R}^{+}$, we
wish to minimize the generalization risk

$\displaystyle\min_{\\*w\in\Delta^{K},}\mathbb{E}_{p^{t}}[L(\mathcal{D}^{t},\hat{\theta}\left(\\*w\right))],$
(1)

where the model parameters are fit using

$\displaystyle\hat{\theta}\left(\\*w\right)=\arg\min_{\theta\in\Theta}\mathbb{E}_{\hat{p}(\\*w)}\left[L((x^{s},y^{s}),\theta)\right]$
(2)

and $\hat{p}(\\*w)=\sum_{k=1}^{K}w_{k}\hat{p}^{s}_{k}$ is the weighted
empirical subsampling distribution. The second minimization is the empirical
risk estimator on the $\\*w$-reweighted collection of source datsets. The
first minimization searches across reweightings to minimize the loss on a
dataset $\mathcal{D}^{t}$ drawn from the target $p^{t}$.

Sequential subset selection methods have been proposed under this framework.
Bouneffouf . (2014) treat training examples individually as partitions. Let
$\delta_{x_{i}^{s}}$ be the point mass at $x_{i}^{s}$. Then the weighted
empirical subsampling distribution is
$\frac{1}{n^{s}}\sum_{i=1}^{n^{S}}\delta_{x_{i}^{s}}$, which is a special case
of $\hat{p}\left(\\*w\right)$ with $K=n^{s}$,
$\hat{p}_{i}^{s}=\delta_{x_{i}^{s}}$ and $w_{i}=1/n^{s}$. They model the
effect of adding one example onto the current training set by influence
functions and then add the example with largest effect to the training set at
each iteration. Gutiérrez . (2017) formulates this problem in a more general
source partitioning setting and optimize it using Beta-Bernoulli Thompson
Sampling, which sequentially adds source samples from different partitions
with the choices of partitions updated by target performance. In this case,
the weights $\\*w$ are updated sequentially as a heuristic to optimizing (1).
They demonstrate efficiency in training data selection method on medical image
data. Our approach 3.2 translates this method into the context where source
clusters are learned using alternative representations. Instead of combining
samples from different partitions to train a final model, (Yao Doretto, 2010)
trains weak classifiers sequentially into a stronger one using Boosting. Each
weak classifier is trained by a weighted combination of one source partition
and the target, with choice of partition decided by performance and weights
updated iteratively. Here the weights consist of two numbers indicating one
source partition and the target, which is different from $\\*w$ introduced
before. Despite progress on subset selection algorithms, their dynamics can be
difficult to characterize and diagnostic evaluation is not readily available.
We build on these studies to further investigate performance attribution onto
source subsets for out-of-domain generalization problems.

##  3 Methods

###  3.1 Ensemble Method

The optimizer $(\hat{\\*w},\hat{\theta})$ of (1) cannot be obtained
analytically. One approach to obtaining an approximate empirical solution as
described in Algorithm 1, which we call the ensemble method. For a given
source dataset with $K$ subsets, we randomly sample subsets of size
$n_{\text{training}}$ repeatedly for $J$ times, each time re-weighting the $K$
subsets differently. Specifically, in repetition $j$, we sample weights
$\\*w^{j}$ from a Dirichlet distribution with parameter $\mathbf{1}_{K}$, and
then sample data $d^{j}$ from $\mathcal{D}^{s}$ that are weighted accordingly.
After constructing the reweighted data, we train a model and evaluate it on
the target task, yielding loss $l_{j}$. After $J$ repetitions, we find the $j$
with minimum $l_{j}$. The empirical optimizer would be the $\\*w$ and
$\hat{\theta}$ found in the $j^{th}$ interation, i.e.,
$(\hat{\\*w},\hat{\theta})=(\\*w^{\arg\min_{j}l_{j}},\hat{\theta}^{\arg\min_{j}l_{j}})$.

Algorithm 1 Sample weighting: Ensemble method

Number of source partitions $K$, number of repetitions $J$, and number of
examples in each subsample $n_{\text{training}}$

for $j$ in $1:J$ do

Sample weights
$\\*w^{j}=[w^{j}_{1},w^{j}_{2},\cdots,w^{j}_{K}]\sim\text{Dirichlet}(\mathbf{1}_{K})$
for $K$ subsets.

For each $k$, sample $n_{\text{training}}\times w_{k}$ points from
$\mathcal{D}^{s}_{k}$ to form $d_{k}^{j}$.

Fit the model
$\hat{\theta}\left(w^{j}\right)=\min_{\theta\in\Theta}\sum_{k=1}^{K}\sum_{\left(x_{i},y_{i}\right)\in
d_{k}^{j}}L\left(\left(x_{i}^{t},y_{i}^{t}\right),\theta\right)$

Evaluate the target loss
$l_{j}=\sum_{i=1}^{n^{t}}L\left(\left(x_{i}^{t},y_{i}^{t}\right),\hat{\theta}\left(w^{j}\right)\right)$

end for

The optimizer is
$(\hat{\\*w},\hat{\theta})=\left(\\*w^{j^{\ast}},\hat{\theta}\left(\\*w^{j^{\ast}}\right)\right)$
where $j^{\ast}=\arg\min_{j}l_{j}$

###  3.2 Thompson Sampling with the Beta-Bernoulli Bandit

The multi-armed bandit problem deals with the situation in which an operator
iteratively chooses one of a set of unknown distributions and observes a
reward, with the goal of maximizing the cumulative gain (Lattimore Szepesvári,
2020). An example comes from a gambler at a row of slot machines, who has to
decide which machine to play at each iteration. This setting is analogous to
the sequential subset selection problem if we view different source subsets as
arms in the bandit context.

Consider models trained on each source subset, and denote current parameter
estimates of the reward distributions for each subset by
$\lambda_{1},\cdots,\lambda_{K}$. Starting from a randomly initialized model,
at each iteration $h$, we add a batch of samples from one subset
$\mathcal{D}_{k_{h}}$ and observe the reward $r_{h}$, with the purpose of
maximizing the cumulative expected reward after $H$ rounds:

$\displaystyle
R_{H}=H\max_{h}\mathbb{E}[r_{h}]-\sum\limits_{h=1}^{H}\mathbb{E}[r_{h}].$ (3)

Let reward $r_{h}$ be the indicator of whether adding samples from subset
$k_{h}$ increases the target performance:

$\displaystyle r_{h}=\begin{cases}0,&\text{if adding data from source
}k_{h}\text{increases loss}\\\ 1,&\text{otherwise},\end{cases}$

where $k_{h}\in\\{1,2,\cdots,K\\}$ indexes the source subset sampled at
iteration $h$. We model the probability of this 0-1 reward at the $k^{th}$
source subset using $r_{h}\sim\mathrm{Ber}\left(\lambda_{k_{h}}\right)$.
Assume priors $\lambda_{k}\sim\mathrm{Beta}\left(\alpha_{k},\beta_{k}\right)$.
Then the posterior for $\lambda_{k_{h}}$ at iteration $h$ can be updated to a
Beta distribution with parameters $\alpha_{k_{h-1}}+\mathbb{I}[r_{h}=1]$ and
$\beta_{k_{h-1}}+\mathbb{I}[r_{h}=0]$. At each iteration $h$, the choice of
source subset $D_{k_{h}}$ is obtained by sampling $\hat{\lambda}_{k}$’s from
current posteriors and choosing the arm with the highest probability of
reward, i.e., $k_{h}=\arg\max_{k=1}^{K}\hat{\lambda}_{k}$, where
$\hat{\lambda}_{k}\sim\mathrm{Beta}(\alpha_{k},\beta_{k}),\ k=1,2,\cdots,K$.
Algorithm 2 provides the pseudocode of this Thompson Sampling with Beta-
Bernoulli Bandit method. This approach addresses the trade-off between
exploiting what is known and exploring new potentially useful source data.

Algorithm 2 Source selection: Thompson Sampling with Beta-Bernoulli Bandit

initial hyperparameters $\alpha_{k}=\alpha$, $\beta_{k}=\beta$, source subsets
$\left\\{\mathcal{D}_{k}\right\\}_{k=1}^{K}$, convergence error
$\varepsilon>0$

Randomly initialize model parameter $\hat{\theta}$

Evaluate the model on the target set and obtain accuracy $a_{0}$

while $\lvert a_{h}-a_{h-1}\rvert>\varepsilon$ do

$k_{h}=\arg\max_{k=1}^{K}\hat{\lambda}_{k}$, where
$\hat{\lambda}_{k}\sim\text{Beta}(\alpha_{k},\beta_{k})$

Randomly select a batch of samples from $\mathcal{D}_{k_{h}}$, add them to the
indices of training examples $I_{\text{training}}$

Update
$\hat{\theta}=\arg\min_{\theta\in\Theta}\sum_{\left(x_{i},y_{i}\right)\in
I_{\text{training}}}L((x_{i}^{t},y_{i}^{t}),\theta)$

Compute the prediction accuracy $a_{h}$ of the updated model on target

if $a_{h}>a_{h-1}$ then

$\alpha_{k_{h}}=\alpha_{k_{h}}+1$

else

$\beta_{k_{h}}=\beta_{k_{h}}+1$

end if

end while

###  3.3 Summary statistic

The necessity of subset selection depends on whether reweighting source
subsets has any effect. We summarise the extent to which can this dataset be
reduced to an efficient subset by computing the difference of optimal weights
away from the uniform case

$\displaystyle
D=\frac{1}{K}\sum_{k=1}^{K}\lvert\hat{w}_{k}-\frac{1}{K}\rvert.$ (4)

###  3.4 Protocols

Generalization failures are typically caused by ephemeral predictors or
distributional shifts (Wang ., 2019). We mimic these situations by splitting
the dataset into source and target sets either using metadata information
(section 4 and 5.1) or data representations (5.2). After defining the source
and target, we further split the source into $K$ subsets by either pre-
defining some latent variable (section 4) or clustering features (section 5.1)
and/or their representations (section 5.2). Of the two splitting protocols,
the former fits scenarios where data are collected at different times and
locations with inherent differences, such as satellite images across
continents or stock prices over a long period. Our goal is to determine
whether, and possibly why, certain subsets make better (or worse)
generalizations. The latter can be adopted in a more general situation without
clear information on how distribution varies. In that case, the goal is more
exploratory.

Details of splitting the data are introduced in section 5 and vary among tasks
and datasets. Given the target $\mathcal{D}^{t}$ and $K$ source subsets
$\mathcal{D}^{s}_{1},\mathcal{D}^{s}_{2},\cdots,\mathcal{D}^{s}_{K}$, we
explore the generalization problem using both the ensemble (section 3.1) and
the bandit-selection approaches (section 3.2). We analyze experimental results
both quantitatively (section 3.3) and qualitatively.

Figure 1: The workflow of experiment protocols.

##  4 Simulations

In temporal problems, predictors can be ephemeral, meaning that there is
hidden context that induces changes in the underlying distribution over time
(Tsymbal, 2004). It is therefore important to keep the model updated. Properly
weighting data according to time can help avoid problems brought by
epheremerality while still using of as much historical information as
possible.

As an illustration, we simulate a dataset of size $1000$ where both the
response $y\in\mathcal{R}$ and predictor $x\in\mathcal{R}^{4}$ are collected
over time. We consider time as the meta information $z$. Our target task is to
make predictions on the most recent data (observations with large $z$ values)
given the past. The relationship between $y$ and $x$ depends on $z\in(0,10]$:

$\displaystyle y=b_{z}^{T}x+\varepsilon,$ (5)

where $b_{z}$ is a time-varying coefficient defined by formula (6) and
$\varepsilon\sim\mathcal{N}\left(0,0.1\right)$.

We take $x\sim\mathcal{N}\left(0,I_{p}\right)$. The coordinates of $b_{z}$ are
generated by

$\displaystyle
b_{z,j}=\begin{cases}\beta_{j}\alpha_{j}z+\varepsilon_{j},&0<z\leq 3\\\
\beta_{j}(\alpha_{j}+1)z^{2}+\varepsilon_{j},&3<z\leq 5\\\
\beta_{j}(\alpha_{j}-1)z+\varepsilon_{j},&5<z\leq 10,\end{cases}$ (6)

where
$\beta=[\beta_{1},\beta_{2},\beta_{3},\beta_{4}]^{T}=[.9,.2,-.3,.3]^{T}$,
$\varepsilon_{j}$’s are gaussian with mean zero and standard deviation $.01$,
$.1$, $.04$, $.1$, and $\alpha_{j}$ follows a uniform distribution in
$[-1,1]$. The interpretation that the coefficients in equation (5) are
evolving linearly for the first three timepoints, quadratically for the next
two, and linearly again for the remaining timepoints (but with a different
slope).

We wish to investigate whether there exists a training subset weighted by
time-based clusters that predicts the target better than a balanced subset.
The training model is linear regression. We split source into $K=3$ subsets
according to time $z$, and apply the bandit-selection and ensemble methods.
The experiment follows from the protocol in section 3.4 and values of
parameters are listed in Table 1.

Table 1: Model parameters of Simulation Sampling Procedure | Notation | Description | Value
---|---|---|---
Both methods | $K$ | Number of clusters in source | $3$
Ensemble method | $n_{\mathrm{training}}$ | Subsample size | $1000$
| $n_{\mathrm{simulation}}$ | Number of subsamples to generate | $1000$
Bandit-selection | $H$ | Number of iterations | $30$
| $b$ | Number of samples to add at each iteration | $10$

Fig. 2 shows that bandit selection yields a consistently lower loss compared
with a random selection, and the performance gap increases over iterations.
The final training set of bandit selection mostly comprises data from the
latest source subset, which agrees with the intuition that the most recent
data are most relevant to the present task. The left panel of Fig. 3 shows
that the ensemble method prefers data from the latest subset as well.
Specifically, higher weights on subsets with large values of $z$ improve
performance, reinforcing the conclusions of the bandit method.

Figure 2: Bandit selection results for the simulation in section 4 when $K=3$
and time $z$ has an effect on the relationship between $y$ and $x$ according
to formula 5. The left panel shows the generalization accuracy over iterations
by bandit selection (red) or sequentially adding samples at random (green).
The right panel shows the final occurrence of different source subsets
correspondingly.

Figure 3: Generalization performance across different weights of the ensemble
method for simulation in section 4. The left panel refers to the case when
time has an effect on the relationship between $y$ and $x$ while the right
panel refers to the other. In each ternary plot, vertices and edges represent
three source subsets and the corresponding axes of weights $w^{j}$ in
subsamples, respectively. Each point inside the triangle indicates a weighted
subsample, whose coordinates and color indicate the weights on three source
subsets and the prediction loss, respectively.

We repeat these two methods on a dataset where coefficients are constant over
time, $y=\beta x+\varepsilon$. Fig. 4 and the right panel of Fig. 3 suggest
that neither method improves over a random selection. Under such setting where
data are not heterogeneously distributed across contexts (time, in this
scenario), a source selection will not be useful. In this sense, our methods
can serve as diagnostics on whether an improved source subsample exists.

Figure 4: Bandit selection for simulation in section 4 when $K=3$ and time $z$
doesn’t have an effect on the relationship between $y$ and $x$. The plots
follow the same manners as in Fig. 2

##  5 Experiments

###  5.1 California Housing Prices Data

The California Housing Prices Dataset contains median house prices and ten
explanatory variables derived from the 1990 census (Pace Barry, 1997). The
data have a spatial component, with each home associated with its latitude and
longitude. We use this data to explore the scenario where source and target
samples share the same variables but are geographically dissimilar, and where
a careful selection of source data may support improvement on a target task.
To mimic the situation where the goal is to perform well in a specific region,
we geographically split data into source and target. We first fix a range of
longitude and latitude to be the target region, and then split the remaining
(source) samples via a $K$-means clustering over all variables. The training
model is linear regression. Detailed parameters are listed in Table 2 and
results are given through Figs. 5 \- 10. The high-level takeaways are

  * •

The bandit selection method results in a source subsample that consistently
improves performance relative to a random selection.

  * •

The final bandit selection prefers certain source subsets over others, and the
summary statistic has a high value.

  * •

The ensemble and bandit selection methods agree with each other on the
selected source subsets.

Fig. 5 suggests that bandit selection yields consistently better and more
robust performance than the random selection for $K\in\\{2,\dots,5\\}$. The
two error curves have increasing gaps after overlapping in the first few
iterations. After 100 iterations, the bandit selection error curve
consistently has lower target set error than random selection. From Fig. 6,
the increasing summary statistic in bandit selection reflects the increasing
nonuniformity of weights across source subsets. The selection makes certain
subsets more visible, with individual subsets contributing more than 50% of
the total weight. In comparison, the random selection shows decreasing summary
statistics and finally results in almost uniform weights over the source
subsets. Further, the bandit selection’s preference on source subsets also
agrees with the ensemble search; e.g. from Fig. 7 and 10 both methods favor
the first source subset when $K=3$. Feature histograms for each source subset
and the target (Fig. 9) suggests that this preferred subset
$\mathcal{D}_{1}^{s}$ is more similar with the target $\mathcal{D}^{t}$ with
respect to the explanatory variable income and the response variable housing
price. Those two variables may be responsible for the selection of specific
source clusters for this given target.

Table 2: Model parameters of the California housing dataset Sampling Procedure | Notation | Description | Value
---|---|---|---
Both methods | $K$ | Number of clusters in source | $2,3,4,5$
Ensemble method | $n_{\mathrm{training}}$ | Subsample size | $200$
| $n_{\mathrm{simulation}}$ | Number of subsamples to generate | $200$
Bandit-selection | $H$ | Number of iterations | $200$
| $b$ | Number of samples to add at each iteration | $20$
Figure 5: Bandit selection error over iterations with (columns) $K$ from $2$
to $5$ for (red) cluster-based and (green) random source splits in the
California Housing Prices experiment (section 5.1). The learned mixture
outperforms its alternative consistently with less fluctuations. Figure 6:
Subset weights and the summary statistic over iterations in the bandit
selection of California Housing Prices experiment (section 5.1). Columns refer
to different choices of $K$ and rows represent whether the selection is based
on (top) multi-armed bandit or (bottom) by random. For each subplot, the
x-axis represents iterations, while the y-axis includes two parts: a line plot
(top) indicating the summary statistic and stacked bins (bottom) indicating
subset weights. Figure 7: The occurrence of three source subsets over 200
iterations for bandit (red) and random (blue) strategy when $K=3$ in the
California Housing Prices experiment (5.2.1). Compared with the random
selection which selects almost uniformly across subsets, bandit-based
selection selects the first subset $\mathcal{D}_{1}^{s}$. Figure 8: The
source-target split on California Housing Prices dataset for $K=3$ with color
indicating source subsets or the target dataset (section 5.2.1). Figure 9:
Variable histograms of California Housing Prices dataset (section 5.2.1). Rows
correspond to source subsets and target, while columns represent different
variables. The most favorable source subset, $\mathcal{D}_{1}^{s}$, has
similar histograms on explanatory variable income and the response variable
housing price, whiles its histograms of latitude and longitude show less in
common with the target $\mathcal{D}^{t}$ compared with other source subsets
$\mathcal{D}_{2}^{s}$ and $\mathcal{D}_{3}^{s}$. Figure 10: Generalization
performance across different weights of ensemble method when $K=3$ in the
California housing experiment (section 5.1). The ternary plot follows the same
format with Fig. 3. Since points near the $\mathcal{D}_{1}^{s}$ vertex have
lowest loss values, we can draw the same conclusion as drawn from Fig. 7 that
$\mathcal{D}_{1}^{s}$ is more informative.

###  5.2 Camelyon17-wilds dataset

In breast cancer diagnosis, pathologists detect tumors manually by screening
slides of lymph nodes under microscope. While automated algorithms generally
yield good results, they may generalize poorly across facilities or equipment
due to differences in data collection and processing (Tellez ., 2019). We
study the tumor classification task on Camelyon17-wilds dataset, where inputs
are $96\times 96$ patches of whole-slide images of sentinel lymph nodes, and
outputs are their binary tumor indicator labels for breast cancer metastases
detection (Koh ., 2020). The data consist of $455,955$ patches collected from
five separate hospitals, across which generalization performances differ
substantially.

The Camelyon17-wilds dataset distinguishes itself from the simulation (section
4) and California Housing Prices experiment (section 5.1) in two aspects.
First, it consists of images whose original features, pixels, are only weakly
predictive; this makes extracting higher level representations necessary.
Since deep learning approaches are usually used for imagery data, the model
parameters are not directly interpretable with respect to input features.
High-dimensionality of feature representations also requires dimensionality
reduction methods be employed before reaching conclusions. Second, data are
collected from five hospitals, which can be treated as inherent group splits
with potential distributional shifts. As discussed by Koh . (2020),
generalization from one hospital to another may have bad performance. This
makes source data selection realistic and sensible.

The experimental protocol follows from section 3.4. The classification model
we use is the Residual Network (ResNet) proposed by He . (2016) with
implementation details described in Appendix A.1. For the source / target
split, we set one hospital as the target. For the source, we use either
samples from all but the target hospital or samples from all hospitals
(including the target hospital). Notice that for the second arrangement, the
source and target still don’t contain overlapping data points, although some
observations in the source come from the same hospital as the target. The
source data are further split by either (i) hospitals themselves or (ii)
clustered deep neural network representations of the imagery. By changing the
target hospital, we can also evaluate whether samples of certain hospitals are
“uniformly easier” to predict, and whether the good generalization is
commutative between hospitals.

We provide details of the source splitting approaches. Approach (i) uses the
original hospital IDs as source subset splits ($K=5$) to mimic the situation
when we select source data from facilities available. We also assign subsets
randomly as a comparison. In contrast, approach (ii) is based on extracting
features from a ResNet. Specifically, we pass a pretrained ResNet through
source images and obtain activations after convolution layers. Each
convolution layer corresponds to an array of activations with first dimension
equal to the number of data points. After flattening the activation array and
performing Principal Component Analysis with $50$ components, we have a
$455,955\times 50$ matrix of feature representations from each convolution
layer. We use activations from either shallow or deep convolutional layers to
study the influence of this choice. In principle, representations from deeper
layers in the network capture higher level structures. However, it may reveal
too much about the class label and thus cannot be a good criteria to split the
source images. For example, if the deepest activations are discriminative
enough for classification, then splitting the source accordingly can result in
highly imbalanced subsets with respect to labels. With either deep or shallow
ResNet feature representations, we cluster the source into $K$ subsets via
$K$-means.

####  5.2.1 Experimental results

We start with the case where the target is hospital 5 and the source contains
data from all hospitals. First, we investigate the influence of source
splitting criteria on target performance. When we use hospitals as source
subsets ($K=5$), we would expect to select most samples from hospital 5, since
variation among facilities may affect generalization. Fig. 11 confirms this
conjecture under the bandit selection, since the final source composition is
dominated by hospital 5. Deciding source subsets by clustering ResNet features
yields a comparable accuracy over bandit iterations (Fig. 12) – accuracy on
the target rises in the first 10 iterations and stabilizes afterwards. The
choice of $K$ does not noticeably affect accuracy, but does affect the
distribution of source subsets used. When $K=5$, certain clusters are
preferred and almost dominate the selection (Fig. 13). The summary statistic
reaches a similar value as the hospital split case. A larger choice of $K$
results in a more uniform spread of the ultimate subset weights as well as a
lower summary statistic. This is expected since finer clustering makes
clusters less distinguishable from one other.

Next, we change the source by excluding data points from hospital 5. Compared
with previous results, the accuracy of bandit selection fluctuates much more
after consistently increasing in the first 10 iterations, and it converges to
a lower overall accuracy (Fig. 14). This further confirms that including data
from hospital 5 in the source supports generalization. Another observation is
that in both source settings, splitting the source randomly yields a
comparable bandit selection. The source split by clustering deep neural
network features does not ensure a better source subsample. We discuss this
phenomenon further in section 5.2.2.

We have focused on results where the target consists of hospital 5 samples.
Other choices of target hospitals yield similar results. The only noticeable
difference is that, when we use hospital 1 as the target and all hospitals as
source and split the source by hospitals, the bandit selection does not prefer
data from hospital 1 over other hospitals. Complete experimental results can
be found in Appendix A.2. If we treat our method as a diagnostic, we would
conclude that prediction on hospital 1 does not require source selection. This
agrees with (Koh ., 2020), which frames hospital 5 as the out-of-distribution
data and other hospitals as in-distribution data. In the following subsection,
we explore to what extent having hospital 5 data in the source benefits
generalization.

Figure 11: Hospital-based subset weights as a function of bandit selection
iteration. Target data consists of hospital 5 only, while source consists of
observations from all hospitals. We split the source into $K=5$ subsets
according to hospitals. The plot follows the same format as each subplot of
Fig. 6. Figure 12: Model accuracy over bandit selection. Target data consists
of hospital 5 observations only, while source consists of observations from
all hospitals. The source is split into $K$ subsets either at random, by
hospital ID (only when $K=5$), by clustering the deep ResNet representations,
or by clustering the shallow ResNet representations. Column represents number
of source subsets $K$ and in each subplot, $x$ and $y$-axes represent
iterations and accuracy, respectively, with color indicating the ways we split
the source. In each bandit selection, the model converges within the first
fifteen iterations maintains accuracy between $0.8$ and $0.9$ with small
fluctuations. The result doesn’t have an obvious dependence on $K$ or how we
split the source. Figure 13: Hospital-based subset weights as a function of
bandit selection iteration in the Camelyon17-wilds experiment (section 5.2.1).
Target data consists of hospital 5 only, while source consists of observations
from all hospitals. We cluster the source into $K$ subsets ($K$ indicated by
column) according to deep (top) or shallow (bottom) ResNet representations.
Each subplot follows the same format as each subplot of Fig. 6. Figure 14:
Model accuracy over bandit selection iterations in the Camelyon17-wilds
experiment for generalization to hospital 5(section 5.2.1). The experimental
settings are the same as Fig. 12 except that data from hospital 5 are excluded
from the source. In each bandit selection, model accuracy stabilizes between
$0.6$ and $0.75$. Performance has a positive relationship with $K$, but does
not have a clear dependence on split criteria.

####  5.2.2 Transitional study

Having source samples from hospital 5 benefits generalization performance. A
follow-up question is the extent to which improvement depends on the
proportion of source data points available from hospital 5. For example, are a
few samples from hospital 5 sufficient to guarantee strong generalization?
This relates to the situation where we must determine the number of samples
from the hospital of interest that will need to be labeled, assuming plentiful
data from other hospitals. Such knowledge is especially important when
labeling is scarce and costly.

In the following experiment, we perform bandit selection with the source
containing $1500$ examples from hospitals 1 through 4 each and varying numbers
of examples from hospital 5. We call the size of hospital 5 data in the source
compared with others the “proportion” parameter. When the proportion equals to
0, then we do not have data from hospital 5 in the source; when the proportion
equals to 1, then all 1500 source examples are from hospital 5. We initialize
the model by training a ResNet classifier with data from hospital 1 to 4 in
the source until its training accuracy reaches $0.7$. That mimics the scenario
where a model must be adapted to a target facility, after having been
initialized using others. At each bandit iteration, we update the model with
one selected source subset for 5 epochs. For source split criteria, we
continue to use hospitals, clusters, and random assignments.

Fig. 15 suggests that adding hospital 5 data points benefits cluster-based
bandit selection and random selection almost equally. In contrast, the
hospital-based selection starts with a lower performance at first, increases
faster and outperforms them within ten iterations. There is a distinct gap
between the performance of hospital-based selection and the rest. However,
hospital-based selection seems to benefit little from adding further samples
from hospital 5, and its advantage over others becomes less obvious when the
source includes as many data points from hospital 5 as data from any other
hospital.

Figure 15: Transitional study: generalization accuracy over iterations in
bandit selection with “proportion” parameter denoted by columns (section
5.2.2). The format of each subplot is the same as subplots in Fig. 12.

Fig. 16 explains the behavior of hospital-based selection. Good overall
performance results from the fact that hospital 5 is eventually selected with
high probability. Unstable initial performance comes from the randomness of
initialization in bandit selection – by chance, the bandit may initially place
low weight on hospital 5. In contrast, in the other two scenarios, the
training set always contains data from hospital 5, regardless of the choices
from the bandit selection. This ensures a relatively high starting
performance. Since the bandit selection allows training examples being
repeatedly chosen, increasing the number of data points from hospital 5 in the
source does not necessarily benefit performance. In fact, performance even
drops slightly when the proportion is close to $1$. An intuition behind this
is, when we have few samples from hospital 5, the algorithm tends to
repeatedly select and train on the same data, while it tends to train on a
variety of data points from hospital 5 otherwise. In this sense, we don’t
necessarily need as many samples from the source.

Figure 16: Hospital-based subset weights as iteration goes by in bandit
selection of transitional study (section 5.2.2). Target data consists of
hospital 5 only, while source consists of observations from all hospitals.
Columns refer to the choices of “proportion” parameter and subplots follow the
same manner as Fig. 11.

We make two secondary comments on Fig. 15. First, similar behavior of two
repetitions suggests the robustness of the algorithm. Second, clustering
according to either deep or shallow ResNet feature yields similar results.
This contradicts the intuition that deep features may capture dissimilarity
across facilities. In this problem, in order to have good generalization on
hospital 5, we need to include (a few) data points from hospital 5, and allow
weight to concentrate on that source.

##  6 Conclusion

We have studied two source selection methods in out-of-distribution
generalization problems. We demonstrate how our methods work in one simulated
and two real-life datasets, discussing reasons why selection may or may not
help across specific contexts. Both methods can also serve as diagnostics and
provide quantitative and qualitative illustrations of dataset heterogeneity.
One direction for further study is further investigation of criteria for
source splitting. For example, the choice of data representations and number
of source subsets would benefit from further analysis.

##  7 Acknowledgement

This research was performed using the compute resources and assistance of the
UW-Madison Center For High Throughput Computing (CHTC) in the Department of
Computer Sciences. The CHTC is supported by UW-Madison, the Advanced Computing
Initiative, the Wisconsin Alumni Research Foundation, the Wisconsin Institutes
for Discovery, and the National Science Foundation, and is an active member of
the OSG Consortium, which is supported by the National Science Foundation and
the U.S. Department of Energy’s Office of Science.

##  Appendix A Supplementary materials of experiments on Camelyon 17-wilds
datset

###  A.1 Model

Our training model is the ResNet model (He ., 2016) with 18 layers pretrained
by PyTorch (resnet18). In the experiments. we randomly select $5000$ data
points evenly distributed over hospitals and labels. The batch size is $16$.
In the bandit selection of section 5.2.1, we add $b=30$ data points from the
chosen source subset to the training data and train the whole model for $30$
epochs at each of the $H=100$ iterations. In the bandit selection of section
5.2.2, we initiate with the model trained on data points from hospitals 1 to 4
with training accuracy $0.70$ (it hasn’t reached its capacity). At each of the
$H=100$ iterations, we add $b=100$ samples from one source subset into the
training set and train the model for $3$ epochs.

In the case that we cluster the source via features, we extract the features
of the first and last convolutional layers by passing the source data (image
arrays) through the ResNet model we mentioned earlier. After that, we reduce
the dimension of either case into $50$ using PCA. For simplicity, we denote
the resulting arrays of the first layer as shallow features and that of the
last layer as called deep features.

###  A.2 Experimental results

When we use hospital 1 as target and data from all hospitals as source and
perform a bandit selection, there isn’t a distinguishable difference between
learned mixtures and a random sample with respect to generalization
performance, regardless of whether we are using deep or shallow features, or
how many subsets we split the source into (Fig. 17). The behavior of summary
statistic between these two scenarios over iterations (Fig. 18 and 19) is also
similar: it goes down after a high initial value due to randomness, and then
decreases after the model converges since none of the subsets are selected. We
also notice that when we use hospitals to split the source, the bandit
selection isn’t dominantly selecting data from hospital 1 (Fig. 20). This
suggests that making predictions on hospital 1 doesn’t require such a source
data selection under this setting.

Figure 17: Model accuracy over bandit selection. Target data consists of
hospital 1 observations only, while source consist of observations from all
hospitals. The experiment and visualization settings are the same as Fig. 12
except that the target hospital is hospital 1. Figure 18: Hospital-based
subset weights as iteration goes by in bandit selection. The experiment and
visualization settings are the same as Fig. 13 except that the target hospital
is hospital 1. Figure 19: Hospital-based subset weights as iteration goes by
in bandit selection. The experiment and visualization settings are the same as
Fig. 18 except that the source is split randomly. Figure 20: Hospital-based
subset weights as iteration goes by in bandit selection. The experiment and
visualization settings are the same as Fig. 11 except that the target hospital
is hospital 1.

Furthermore, we apply a source / target split where the source excludes data
from hospital 1, as what we have done for hospital 5 in section 5.2. There
isn’t an obvious performance drop (Fig. 21). The summary statistic and source
subset weights also behave similar as the previous experiment (Fig. 22 and
Fig. 23). This confirms with the idea that in a diagnostic framework, we will
conclude that source selection isn’t that effective when our target is
hospital 1.

Figure 21: Model accuracy over bandit selection. The experiment and
visualization settings are the same as Fig. 17 except that we exclude data
points of hospital 1 from the source. Figure 22: Hospital-based subset weights
as iteration goes by in bandit selection. The experiment and visualization
settings are the same as Fig. 18 except that we exclude data points of
hospital 1 from the source. Figure 23: Hospital-based subset weights as
iteration goes by in bandit selection. The experiment and visualization
settings are the same as Fig. 22 except that we split the source randomly.

###  A.3 Transitional study

Figure 24: Transitional study for cluster-based source split. The experiment
and visualization settings are the same as Fig. 16 except that the source is
split by clustering (top) deep or (shallow) ResNet representations. Figure 25:
Transitional study for randomly-split source. The experiment and visualization
settings are the same as Fig. 16 except that the source is split randomly.

As supplementary figures for the transitional study of Camelyon17-wilds
dataset in section 5.2.2, Fig. 24 and Fig. 25 show the source subset weights
over the bandit selection when source is split by clustering ResNet features
and randomly, respectively.

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# The Mellin Transform and Non-local Derivatives of Fractal Calculus

Alireza Khalili Golmankhaneh 111Corresponding author
Department of Physics, Urmia Branch, Islamic Azad University, Urmia 63896,
Iran
Kerri Welch
Faculty at California Institute of Integral Studies, San Francisco, CA, U.S.A.
Cristina Serpa
Instituto Superior de Engenharia de Lisboa (ISEL),
Instituto Politécnico de Lisboa, Lisbon, Portugal
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional
(CMAFcIO),
Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal
Palle E. T. Jørgensen
Department of Mathematics,The University of Iowa, Iowa City, IA 52242-1419,
U.S.A

###### Abstract

In this paper, the fractal calculus of fractal sets and fractal curves are
compared. The analogues of the Riemann-Liouville and the Caputo integrals and
derivatives are defined for the fractal curves which are non-local
derivatives. The analogous for the fractional Laplace concepts are defined to
solve fractal non-local differential equations on fractal curves. The fractal
local Mellin and fractal non-local transforms are defined to solve fractal
differential equations. We present tables and examples to illustrate the
results.

Keywords: Fractal local Mellin, fractal non-local transforms, fractal non-
local derivatives
2010 Mathematics Subject Classification: 28A80,26A33

##  1 Introduction

Fractal geometry was introduced by Benoit Mandelbrot [1, 2] which involves
complex geometric shapes where their fractal dimensions exceed their
topological dimensions [3]. The fractals are self-similar and often have non-
integer dimensions [4]. Since fractals have different measures, such as the
Hausdorff measure, length, surface, and volume, the measures of Euclidean
geometric shapes, cannot be used in fractal analysis [5, 6].
The fractal analysis was formulated by many researchers using harmonic
analysis [7, 8], measure theory [9], probabilistic method [10], fractional
space [11], fractional calculus [12], and non-standard methods [13]. The
development of fractal calculus benefits also from new techniques to obtain
fractal functions from real data, e.g., the fractal regression gives applied
scientists the fractal shape of data [14]. The conjugation of theoretical
studies with fitting approximative functions is a strong means to better
understand some concrete problems of applied science that do not fit the
classical geometric setting.
Fractal calculus is formulated by a generalization of ordinary calculus which
involves differential equations. Their solutions are functions with fractal
support such as fractal sets and curves [15, 16, 17, 18]. Fractal calculus is
algorithmic and simpler compared with other methods [18]. The fractal calculus
approach with the fractional spaces method is compared in applications such as
Einstein’s field equations [19]. Fractal stochastic differential equations are
defined and the fractional Brownian motion and diffusion in the medium with
fractal structures are categorized [20, 21, 22, 23, 24, 25, 26, 27]. The
fractal differential equations were solved by different methods and their
stability conditions are given [28, 29, 30, 31]. The fractal calculus was
generalized to involve Cantor cubes and Cantor tartan [32] and defined the
Laplace equation [33]. The derivative and integral of the Weierstrass function
are given [34].
The layout of the manuscript is as follows:
In Section 2, we compare and review the fractal calculus of fractal sets and
curves. We define non-local fractal derivatives and integrals such as
analogues of the Riemann-Liouville derivatives and Caputo fractional
derivatives in Section 3. The analogues Laplace transform of Riemann-Liouville
derivatives and Caputo derivatives are defined to solve non-local differential
equations on fractals in Section 4. In Section 5, the analogue of the Mellin
transform is defined and used to solve a fractal differential equation. We
devoted Section 6 to a conclusion.

##  2 Basic tools

In this section, we summarize the fractal calculus on fractal curves and
fractal sets [15, 16, 17, 18].

###  2.1 Fractal calculus on fractal curves

######  Definition 1.

Consider a fractal curve $\mathfrak{F}$ and a subdivision
$\mathfrak{P}_{[c_{1},c_{2}]},[c_{1},c_{2}]\in[a_{0},b_{0}]\subset\mathbb{R}$.
Then the mass function is defined by

$\zeta^{\alpha}(\mathfrak{F},c_{1},c_{2})=\lim_{\delta\rightarrow
0}\inf_{|P|\leq\delta}\sum_{i=0}^{n-1}\frac{|\mathbf{v}(z_{i+1})-\mathbf{v}(z_{i})|^{\alpha}}{\Gamma(\alpha+1)},$
(1)

where $|.|$ is the Euclidean norm of $\mathbb{R}^{n}$, $1\leq\alpha\leq n$,
$P_{[c_{1},c_{2}]}=\\{c_{1}=z_{0},...,z_{n}=c_{2}\\}$, $|P|=\max_{0\leq i\leq
n-1}(t_{i+1}-t_{i})$ for a subdivision $\mathfrak{P}$ and $\Gamma(*)$ is the
gamma function.

######  Definition 2.

The dimension of $\mathfrak{F}$ is defined by

$\displaystyle\dim_{\zeta}(\mathfrak{F})$
$\displaystyle=\inf\\{\alpha:\zeta^{\alpha}(\mathfrak{F},c_{1},c_{2})=0\\}$
$\displaystyle=\sup\\{\alpha:\zeta^{\alpha}(\mathfrak{F},c_{1},c_{2})=\infty\\}.$
(2)

######  Definition 3.

The rise function of $\mathfrak{F}$ is defined by

$S_{\mathfrak{F}}^{\alpha}(z)=\left\\{\begin{array}[]{ll}\zeta^{\alpha}(\mathfrak{F},q_{0},z),&z\geq
q_{0};\\\ -\zeta^{\alpha}(\mathfrak{F},z,q_{0}),&z<q_{0},\end{array}\right.$
(3)

where $z\in[a_{0},b_{0}]$, and $S_{\mathfrak{F}}^{\alpha}(z)$ gives the mass
of $\mathfrak{F}$ upto point $z$.

######  Definition 4.

The $\mathfrak{F}^{\alpha}$-derivative is defined by

$D_{\mathfrak{F}}^{\alpha}f(\theta)=\underset{\theta^{\prime}\rightarrow\theta}{\mathfrak{F}_{-}lim}\leavevmode\nobreak\
\frac{f(\theta^{\prime})-f(\theta)}{J(\theta^{\prime})-J(\theta)},$ (4)

where $\mathfrak{F}_{-}lim$ indicates the fractal limit (see in [16]), and
$\mathbf{v}(z)=\theta$ and $S_{\mathfrak{F}}^{\alpha}(z)=J(\theta)$.

######  Remark 1.

We note that the Euclidean distance from the origin upto a point
$\theta=\mathbf{v}(z)$ is given by
$\mathfrak{L}(\theta)=\mathfrak{L}(\mathbf{v}(z))=|\mathbf{v}(z)|.$

######  Definition 5.

The $\mathfrak{F}^{\alpha}$-integral is defined by

$\displaystyle\int_{D(c_{1},c_{2})}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta$
$\displaystyle=\sup_{\mathfrak{P}[c_{1},c_{2}]}\sum_{i=0}^{n-1}\inf_{\theta\in
C(z_{i},z_{i+1})}f(\theta)(J(\theta_{i+1})-J(\theta_{i}))$
$\displaystyle=\inf_{\mathfrak{P}[c_{1},c_{2}]}\sum_{i=0}^{n-1}\sup_{\theta\in
C(z_{i},z_{i+1})}f(\theta)(J(\theta_{i+1})-J(\theta_{i})),$ (5)

where $z_{i}=\mathbf{v}^{-1}(\theta_{i})$, and $D(c_{1},c_{2})$ is the section
of between points $\mathbf{v}(c_{1})$ and $\mathbf{v}(c_{2})$ on the fractal
curve $\mathfrak{F}$ [16].

###  2.2 Fractal calculus for fractal sets

Here, we give a summary of fractal calculus of sets
$\mathbf{F}\subset[c_{1},c_{2}]\subset\mathbb{R}$ [15].

######  Definition 6.

The flag function of $\mathbf{F}$ is defined by

$\rho(\mathbf{F},I)=\left\\{\begin{array}[]{ll}1,&\textmd{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbf{F}\cap I\neq\emptyset;\\\
0,&\textmd{otherwise},\end{array}\right.$ (6)

where $I=[c_{1},c_{2}]\subset\mathbb{R}$

######  Definition 7.

The coarse-grained mass of $\mathbf{F}\cap[c_{1},c_{2}]$ is defined by

$\xi_{\delta}^{\alpha}(\mathbf{F},c_{1},c_{2})=\inf_{|\mathfrak{P}|\leq\delta}\sum_{i=0}^{n-1}\Gamma(\alpha+1)(z_{i+1}-z_{i})^{\alpha}\rho(\mathbf{F},[z_{i},z_{i+1}]),$
(7)

where

$|\mathfrak{P}|=\max_{0\leq i\leq n-1}(z_{i+1}-z_{i}),$ (8)

and $0<\alpha\leq 1$.

######  Definition 8.

The mass function of $\mathbf{F}$ is defined by

$\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=\lim_{\delta\rightarrow
0}\xi_{\delta}^{\alpha}(\mathbf{F},c_{1},c_{2}).$ (9)

######  Definition 9.

The fractal dimension of $\mathbf{F}\cap[c_{1},c_{2}]$, is defined by

$\displaystyle\dim_{\xi}(\mathbf{F}\cap[c_{1},c_{2}])$
$\displaystyle=\inf\\{\alpha:\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=0\\}$
$\displaystyle=\sup\\{\alpha:\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=\infty\\}.$
(10)

######  Definition 10.

The integral staircase function of $\mathbf{F}$ is defined by

$S_{\mathbf{F}}^{\alpha}(z)=\left\\{\begin{array}[]{ll}\xi^{\alpha}(\mathbf{F},c_{0},z),&if\leavevmode\nobreak\
z\geq c_{0};\\\
-\xi^{\alpha}(\mathbf{F},z,c_{0}),&otherwise,\end{array}\right.$ (11)

where $c_{0}\in\mathbb{R}$ is a fixed number.

######  Definition 11.

For a function $g$ on an $\alpha$-perfect fractal set, $F^{\alpha}$-derivative
of $g$ at $x$ in is defined by

$D_{\mathbf{F}}^{\alpha}g(z)=\left\\{\begin{array}[]{ll}\underset{y\rightarrow
z}{\mathbf{F}_{-}lim}\leavevmode\nobreak\
\frac{g(y)-g(z)}{S_{\mathbf{F}}^{\alpha}(y)-S_{\mathbf{F}}^{\alpha}(z)},&if\leavevmode\nobreak\
z\in\mathbf{F};\\\ 0,&otherwise,\end{array}\right.$ (12)

if the fractal limit $\mathbf{F}_{-}lim$ exists [15].

######  Definition 12.

The $\mathbf{F}^{\alpha}$-integral of a bounded function $g(z),g\in B(F)$
(i.e., $g$ is a bounded function of $F$), is defined by

$\displaystyle\int_{a}^{b}g(z)d_{\mathbf{F}}^{\alpha}z$
$\displaystyle=\sup_{\mathfrak{P}_{[c_{1},c_{2}]}}\sum_{i=0}^{n-1}\inf_{z\in\mathbf{F}\cap
I}g(z)(S_{\mathbf{F}}^{\alpha}(z_{i+1})-S_{F}^{\alpha}(z_{i}))$
$\displaystyle=\inf_{\mathfrak{P}_{[c_{1},c_{2}]}}\sum_{i=0}^{n-1}\sup_{z\in\mathbf{F}\cap
I}g(z)(S_{\mathbf{F}}^{\alpha}(z_{i+1})-S_{\mathbf{F}}^{\alpha}(z_{i})),$ (13)

where $z\in\mathbf{F}$, and infimum or supremum are taking on the all
subdivision $\mathfrak{P}_{[c_{1},c_{2}]}$.

######  Example 1.

Consider the fractal discretional equation

$D_{G}^{\alpha}y=2y-4,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ y(0)=5,$ (14)

where $G$ is fractal set or fractal curve. Utilizing the conjugacy of fractal
calculus with ordinary calculus [15, 17, 18] the solution of Eq.(14) on a
fractal set is;

$y(x)=2+3\exp(2S_{\mathbf{F}}^{\alpha}(x)),$ (15)

and the solution of Eq.(14) on a fractal curve is;

$y(\theta)=2+3\exp(2S_{\mathfrak{F}}^{\alpha}(\theta)),$ (16)

In Figures 1 and 2, we have plotted Eqs.(15) and (16), respectively.

Figure 1: The graph of Eq.(15) Figure 2: The graph of Eq.(16)

##  3 Non-local derivatives on fractals

In the following, the non-local integrals and derivatives on fractal spaces
are defined [35].

###  3.1 Non-local derivatives of fractal sets

######  Definition 13.

The right-sided Riemann-Liouville $F^{\alpha}$-integral of order
$\beta\in\mathbb{R}$ of $g$ (function on a fractal set) is defined by

${}_{z}\mathcal{I}_{b}^{\beta}g(z)=\frac{1}{\Gamma(\beta)}\int_{z}^{b}\frac{g(\tau)}{(S_{\mathbf{F}}^{\alpha}(\tau)-S_{\mathbf{F}}^{\alpha}(z))^{1-\beta}}d_{\mathbf{F}}^{\alpha}\tau,$
(17)

and the left-sided Riemann-Liouville $F^{\alpha}$-integral of $g$ is given by

${}_{a}\mathcal{I}_{z}^{\beta}g(z)=\frac{1}{\Gamma(\beta)}\int_{a}^{z}\frac{g(\tau)}{(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(\tau))^{1-\beta}}d_{\mathbf{F}}^{\alpha}\tau.$
(18)

######  Definition 14.

The right-sided Riemann-Liouville $F^{\alpha}$-derivative of order
$\beta\in\mathbb{R}$ of $g$ is defined by

${}_{z}\mathcal{D}_{b}^{\beta}g(z)=\frac{1}{\Gamma(n-\beta)}(-D_{\mathbf{F}}^{\alpha})^{n}\int_{z}^{b}\frac{g(\tau)}{(S_{\mathbf{F}}^{\alpha}(\tau)-S_{\mathbf{F}}^{\alpha}(z))^{-n+\beta+1}}d_{\mathbf{F}}^{\alpha}\tau,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ g\in C^{n}(\mathbf{F}),$ (19)

and the left-sided Riemann-Liouville $F^{\alpha}$-derivative of $g$ is defined
by

${}_{a}\mathcal{D}_{z}^{\beta}g(z)=\frac{1}{\Gamma(n-\beta)}(D_{\mathbf{F}}^{\alpha})^{n}\int_{a}^{z}\frac{g(\tau)}{(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(\tau))^{-n+\beta+1}}d_{\mathbf{F}}^{\alpha}\tau,$
(20)

where $n\alpha-\alpha\leq\beta<\alpha n$.

######  Definition 15.

The right-sided Caputo $F^{\alpha}$-derivative of order $\beta\in\mathbb{R}$
of $g$ is defined by

${}_{z}^{C}\mathcal{D}_{b}^{\beta}g(z)=\frac{1}{\Gamma(n-\beta)}\int_{z}^{b}(S_{\mathbf{F}}^{\alpha}(\tau)-S_{\mathbf{F}}^{\alpha}(z))^{n-\beta-1}(-D_{\mathbf{F}}^{\alpha})^{n}g(\tau)d_{\mathbf{F}}^{\alpha}\tau,$
(21)

and the right-sided Caputo $F^{\alpha}$-derivative of $g$ is defined by

${}_{a}^{C}\mathcal{D}_{z}^{\beta}g(z)=\frac{1}{\Gamma(n-\beta)}\int_{a}^{z}(S_{\mathbf{F}}^{\alpha}(z)-S_{F}^{\alpha}(\tau))^{n-\beta-1}(D_{\mathbf{F}}^{\alpha})^{n}g(\tau)d_{\mathbf{F}}^{\alpha}\tau,$
(22)

where $n\alpha-\alpha\leq\beta<\alpha n$.

Ordinary calculus | Fractal calculus on sets | Fractal calculus on curves
---|---|---
$\int_{0}^{z}\tau^{m}d\tau=\frac{1}{m+1}z^{m+1}$ |  $\int_{0}^{z}(S_{\mathbf{F}}^{\alpha}(\tau))^{m}d_{\mathbf{F}}^{\alpha}\tau=\frac{1}{m+1}(S_{\mathbf{F}}^{\alpha}(z))^{m+1}$ |  $\int_{0}^{\theta}J_{\mathfrak{F}}^{\alpha}(\tau)^{m}d_{\mathfrak{F}}^{\alpha}\tau=\frac{1}{m+1}(J(\theta))^{m+1}$
$\frac{d}{dz}z^{m}=mz^{m-1}$ |  $D_{\mathbf{F}}^{\alpha}(S_{\mathbf{F}}^{\alpha}(z))^{m}=m(S_{\mathbf{F}}^{\alpha}(z))^{m-1}\chi_{\mathbf{F}}(z)$ |  $D_{\mathfrak{F}}^{\alpha}(J(\theta))^{m}=m(J(\theta))^{m-1}\chi_{\mathfrak{F}}(\theta)$
${}_{0}I_{z}^{\beta}z^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m+\beta+1)}z^{m+\beta}$ |  ${}_{0}\mathcal{I}_{z}^{\beta}(S_{\mathbf{F}}^{\alpha}(z))^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m+\beta+1)}(S_{\mathbf{F}}^{\alpha}(z))^{m+\beta}$ |  ${}_{0}\mathcal{I}_{\theta}^{\beta}J(\theta)^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m+\beta+1)}J(\theta)^{m+\beta}$
${}_{0}D_{z}^{\beta}z^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}z^{m-\beta}$ |  ${}_{0}\mathcal{D}_{z}^{\beta}(S_{\mathbf{F}}^{\alpha}(z))^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}(S_{\mathbf{F}}^{\alpha}(z))^{m-\beta}\chi_{\mathbf{F}}(z)$ |  ${}_{0}\mathcal{D}_{\theta}^{\beta}J(\theta)^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}(J(\theta))^{m-\beta}\chi_{\mathfrak{F}}(\theta)$
${}_{a}I_{z}^{\beta}(z-a)^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(\beta+m+1)}(z-a)^{\beta+m}$ |  ${}_{a}\mathcal{I}_{z}^{\beta}(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(a))^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(\beta+m+1)}(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(a))^{\beta+m}$ |  ${}_{a}\mathcal{I}_{\theta}^{\beta}(J(\theta)-J(a))^{m}$ $\displaystyle=\frac{\Gamma(m+1)}{\Gamma(\beta+m+1)}(J(\theta)-J(a))^{\beta+n}$
${}_{a}D^{\beta}_{z}(z-a)^{m}=$ $\displaystyle\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}(z-a)^{m-\beta}$ |  ${}_{a}\mathcal{D}^{\beta}_{z}(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(a))^{m}=$ $\displaystyle\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}(S_{\mathbf{F}}^{\alpha}(z)-S_{\mathbf{F}}^{\alpha}(a))^{m-\beta}$ |  ${}_{a}\mathcal{D}^{\beta}_{\theta}(J(\theta)-J(a))^{m}=$ $\displaystyle\frac{\Gamma(m+1)}{\Gamma(m-\beta+1)}(J(\theta)-J(a))^{n-\beta}$
${}_{-\infty}I_{z}^{\beta}\exp(\lambda z)=\lambda^{\beta}\exp(\lambda z)$ |  ${}_{-\infty}\mathcal{I}_{z}^{\beta}\exp(\lambda S_{\mathbf{F}}^{\alpha}(z))=\lambda^{\beta}\exp(\lambda S_{\mathbf{F}}^{\alpha}(z))$ |  ${}_{-\infty}\mathcal{I}_{\theta}^{\beta}\exp(\lambda J(\theta))=\lambda^{\beta}\exp(\lambda J(\theta))$
${}_{-\infty}D_{z}^{\beta}\cos(z)=\cos\left(z+\frac{\pi}{2}\beta\right)$ |  ${}_{-\infty}\mathcal{D}_{z}^{\beta}\cos(S_{\mathbf{F}}^{\alpha}(z))=\cos\left(S_{\mathbf{F}}^{\alpha}(z)+\frac{\pi}{2}\beta\right)$ |  ${}_{-\infty}\mathcal{D}_{\theta}^{\beta}\cos(J(\theta))=\cos\left(J(\theta)+\frac{\pi}{2}\beta\right)$
${}_{-\infty}D_{z}^{\beta}\exp(\lambda z)=\lambda^{\beta}\exp(\lambda z)$ |  ${}_{-\infty}\mathcal{D}_{z}^{\beta}\exp(\lambda S_{\mathbf{F}}^{\alpha}(z))=\lambda^{\beta}\exp(\lambda S_{\mathbf{F}}^{\alpha}(z))$ |  ${}_{-\infty}\mathcal{D}_{\theta}^{\beta}\exp(\lambda J(\theta))=\lambda^{\beta}\exp(\lambda J(\theta))$
Table 1: Some analogies of fractal calculus and standard calculus.

###  3.2 Non-local derivatives on fractal curves

######  Definition 16.

The right-sided Riemann-Liouville $F^{\alpha}$-integral of order
$\beta\in\mathbb{R}$ of $f$ (on a fractal curve $F$) is defined by

${}_{\theta}\mathcal{I}_{b}^{\beta}f(\theta)=\frac{1}{\Gamma(\beta)}\int_{D(\theta,b)}\frac{f(\tau)}{(J(t)-J(\theta))^{1-\beta}}d_{\mathfrak{F}}^{\alpha}\tau,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \theta\in\mathfrak{F},$ (23)

and the left-sided Riemann-Liouville $F^{\alpha}$-integral of $f$ is defined
by

$_{a}\mathcal{I}_{\theta}^{\beta}f(\theta)=\frac{1}{\Gamma(\beta)}\int_{D(a,\theta)}\frac{f(\tau)}{(J(\theta)-J(t))^{1-\beta}}d_{\mathfrak{F}}^{\alpha}\tau.$
(24)

######  Definition 17.

The right-sided Riemann-Liouville $F^{\alpha}$-derivative of order
$\beta\in\mathbb{R}$ of $f$ is defined by

${}_{\theta}\mathcal{D}_{b}^{\beta}f(\theta)=\frac{1}{\Gamma(n-\beta)}(-D_{\mathfrak{F}}^{\alpha})^{n}\int_{D(\theta,b)}\frac{f(\tau)}{(J(\tau)-J(\theta))^{-n+\beta+1}}d_{\mathfrak{F}}^{\alpha}\tau,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ f\in C^{n}(\mathfrak{F}),$ (25)

and the left-sided Riemann-Liouville $F^{\alpha}$-derivative of $f$ is defined
by

$_{a}\mathcal{D}_{\theta}^{\beta}f(\theta)=\frac{1}{\Gamma(n-\beta)}(D_{\mathfrak{F}}^{\alpha})^{n}\int_{D(a,\theta)}\frac{f(\tau)}{(J(\theta)-J(\tau))^{-n+\beta+1}}d_{\mathfrak{F}}^{\alpha}\tau,$
(26)

where $n\alpha-\alpha\leq\beta<\alpha n$.

######  Definition 18.

The right-sided Caputo $F^{\alpha}$-derivative of order $\beta\in\mathbb{R}$
of $f$ is defined by

${}_{\theta}^{C}\mathcal{D}_{b}^{\beta}f(\theta)=\frac{1}{\Gamma(n-\beta)}\int_{D(\theta,b)}(J(\tau)-J(\theta))^{n-\beta-1}(-D_{\mathfrak{F}}^{\alpha})^{n}f(\tau)d_{\mathfrak{F}}^{\alpha}\tau,$
(27)

and the right-sided Caputo $F^{\alpha}$-derivative of order
$\beta\in\mathbb{R}$ of $f$ is defined by

${}_{a}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)=\frac{1}{\Gamma(n-\beta)}\int_{D(a,\theta)}(J(\theta)-J(\tau))^{n-\beta-1}(D_{F}^{\alpha})^{n}f(\tau)d_{\mathfrak{F}}^{\alpha}\tau,$
(28)

where $n\alpha-\alpha\leq\beta<\alpha n$.

######  Example 2.

Consider the following function on a fractal curve

$f(\theta)=(J(\theta)-J(a))^{\nu}.$ (29)

Then its non-local derivative is

$\displaystyle_{a}\mathcal{I}_{\theta}^{\beta}(J(\theta)-J(a))^{\nu}$
$\displaystyle=\frac{1}{\Gamma(\beta)}\int_{D(a,\theta)}(J(t)-J(a))^{\nu}(J(\theta)-J(t))^{\beta-1}d_{\mathfrak{F}}^{\alpha}t.$
(30)

By the change of variable $J(t)=J(a)+\zeta(J(\theta)-J(a))$, Eq.(30), turns
into

$\displaystyle_{a}\mathcal{I}_{\theta}^{\beta}(J(\theta)-J(a))^{\nu}$
$\displaystyle=\frac{(J(\theta)-J(a))^{\beta+\nu}}{\Gamma(\beta)}\int_{D(0,1)}(1-\zeta)^{\beta-1}\zeta^{\nu}d_{F}^{\alpha}\zeta$
$\displaystyle=\frac{B(\nu+1,\beta)}{\Gamma(\beta)}(J(\theta)-J(a))^{\beta+\nu}$
$\displaystyle=\frac{\Gamma(\nu+1)}{\Gamma(\beta+\nu+1)}(J(\theta)-J(a))^{\beta+\nu},$
(31)

where $B(.,.)$ is the Beta function, which is defined by

$B(\nu+1,\beta)=\int_{D(0,1)}(1-\zeta)^{\beta-1}\zeta^{\nu}d_{F}^{\alpha}\zeta=\frac{\Gamma(\nu+1)\Gamma(\beta)}{\Gamma(\nu+\beta+1)}.$
(32)

##  4  Laplace transform of the fractal Riemann-Lioville differintegral

In this section, we explore the Laplace transform of fractal Riemann-Lioville
differintegral [36].

######  Corollary 1.

Laplace transform of the fractal Riemann-Lioville integral is defined by

$\mathcal{L}[_{0}I_{\theta}^{\beta}f(\theta)]=J(s)^{\beta}F(s),$ (33)

where

$F(s)=\mathcal{L}[f(\theta)]=\int_{D(0,\infty)}\exp(-J(s)J(\theta))f(\theta)d_{\mathfrak{F}}^{\alpha}\theta.$
(34)

######  Corollary 2.

The Laplace transform of the fractal Riemann-Liouville derivative is defined
by

$\mathcal{L}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=J(s)^{\beta}F(s)-\sum_{k=0}^{n-1}J(s)^{k}[_{0}\mathcal{D}_{\theta}^{\beta-k-1}f(\theta)]_{\theta=0},$
(35)

where $n\alpha-\alpha\leq\beta<n\alpha$.

######  Remark 2.

We note that Eq.(35), for the case $n=1$, becomes

$\mathcal{L}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=J(s)^{\beta}F(s)-\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{\beta-1}f(\theta)|_{\theta=0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0\leq\beta<\alpha,$ (36)

and for $n=2$, it follows from Eq.(35) that

$\mathcal{L}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=J(s)^{\beta}F(s)-\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{\beta-1}f(\theta)|_{\theta=0}-J(s)_{0}\mathcal{D}_{\theta}^{\beta-2}f(\theta)|_{\theta=0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \alpha\leq\beta<2\alpha.$ (37)

######  Theorem 1.

The fractal Laplace transform of function $J(\theta)^{\eta
m+\mu-1}E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})$ is

$\mathcal{L}[J(\theta)^{\eta
m+\mu-1}E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})]=\frac{m!J(s)^{\eta-\mu}}{(J(s)^{\eta}-a)^{m+1}},$
(38)

where

$E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})=\sum_{k=0}^{\infty}\frac{(k+m)!}{k!}\frac{a^{k}J(\theta)^{k\eta}}{\Gamma(\eta
k+\eta m+\mu)}.$ (39)

###### Proof.

Using the linearity of the fractal Laplace transform and Eq.(39), we can
rewrite Eq.(38) as

$\mathcal{L}[J(\theta)^{\eta
m+\mu-1}E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})]=\sum_{k=0}^{\infty}\frac{(k+m)!a^{k}}{k!\Gamma(\eta
k+\eta m+\mu)}\mathcal{L}[J(\theta)^{\eta k+\eta m+\mu-1}].$ (40)

Applying the fractal Laplace transform of the power function
$\mathcal{L}[J(\theta)^{\beta-1}]=\Gamma(\beta)J(s)^{-\beta}$, Eq.(40) turns
into the following form

$\displaystyle\mathcal{L}[J(\theta)^{\eta
m+\mu-1}E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})]$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(k+m)!a^{k}}{k!\Gamma(\eta k+\eta
m+\mu)}\frac{\Gamma(\eta k+\eta m+\mu)}{J(s)^{\eta k+\eta m+\mu}}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(k+m)!}{k!}\frac{a^{k}}{J(s)^{\eta
k+\eta m+\mu}}$ $\displaystyle=J(s)^{-\eta
m-\mu}\sum_{k=0}^{\infty}\frac{(k+m)!}{k!}\bigg{(}\frac{a}{J(s)^{\eta}}\bigg{)}^{k}$
$\displaystyle=J(s)^{-\eta
m-\mu}D_{\mathfrak{F}}^{m}\sum_{k=m}^{\infty}\bigg{(}\frac{a}{J(s)^{\eta}}\bigg{)}^{k}$
$\displaystyle=J(s)^{-\eta
m-\mu}D_{\mathfrak{F}}^{m}\frac{1}{(1-\frac{a}{J(s)^{\eta}})}$
$\displaystyle=J(s)^{-\eta m-\mu}\frac{m!}{(1-\frac{a}{J(s)^{\eta}})^{m+1}}$
$\displaystyle=\frac{m!J(s)^{\eta-\mu}}{(J(s)^{\eta}-a)^{m+1}},$ (41)

which completes the proof. ∎

Function in $\theta$-space | Fractal Laplace Transform $s$-space
---|---
$\frac{J(\theta)^{\beta-1}}{\Gamma(\beta)}$ |  $\frac{1}{J(s)^{\beta}}$
$J(\theta)^{\beta-1}E_{\beta,\beta}(aJ(\theta)^{\beta})$ |  $\frac{1}{J(s)^{\beta}-a}$
$E_{\beta}(-aJ(\theta)^{\beta})$ |  $\frac{J(s)^{\beta}}{J(s)(J(s)^{\beta}+a)}$
$1-E_{\beta}(-aJ(\theta)^{\beta})$ |  $\frac{a}{J(s)(J(s)^{\beta}+a)}$
$J(\theta)^{\beta}E_{1,\beta+1}(aJ(\theta))$ |  $\frac{1}{J(s)^{\beta}(J(s)-a)}$
$J(\theta)^{\beta-1}E_{\nu,\beta}(aJ(\theta)^{\nu})$ |  $\frac{J(s)^{\nu-\beta}}{(J(s)^{\nu}-a)}$
$J(\theta)^{\eta m+\mu-1}E^{m}_{\eta,\mu}(aJ(\theta)^{\eta})$ |  $\frac{m!J(s)^{\eta-\mu}}{(J(s)^{\eta}-a)^{m+1}}$
Table 2: Some formulas of the fractal Laplace transforms.

######  Corollary 3.

The Laplace transform of the Caputo derivative is defined by

$\mathcal{L}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]=J(s)^{\beta}F(s)-\sum_{k=0}^{n-1}J(s)^{\beta-k-1}[(D_{\mathfrak{F}}^{\alpha})^{k}f(\theta)]_{\theta=0},$
(42)

where $n\alpha-\alpha<\beta\leq n\alpha$.

######  Example 3.

Consider the non-local fractal differential equation

$_{0}D_{\theta}^{\beta}y(\theta)-\lambda
y(\theta)=h(\theta),\leavevmode\nobreak\ \leavevmode\nobreak\ (\theta>0),$
(43)

with the non-local initial condition

$[_{0}D_{\theta}^{\beta-k}y(\theta)]_{\theta=0}=c_{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (k=1,2,...,n),$ (44)

where $n\alpha-\alpha\leq\beta<n\alpha$ and $\lambda$ is constant.
To solve this problem, taking the fractal non-local Laplace transform of
Eq.(43) and using Eq.(35) we obtain

$J(s)^{\beta}Y(J(s))-\lambda Y(J(s))=H(J(s))+\sum_{k=1}^{n}c_{k}J(s)^{k-1}.$
(45)

Solving for $Y(J(s))$, we have

$Y(J(s))=\frac{H(J(s))}{J(s)^{\beta}-\lambda}+\sum_{k=1}^{n}c_{k}\frac{J(s)^{k-1}}{J(s)^{\beta}-\lambda}.$
(46)

The fractal inverse Laplace transform and Eq.(38) give the solution

$y(\theta)=\sum_{k=1}^{n}c_{k}J(\theta)^{\beta-k}E_{\beta,\beta-k+1}(\lambda
J(\theta)^{\beta})+\int_{D(0,\theta)}(J(\theta)-J(t))^{\beta-1}E_{\beta,\beta}(\lambda(J(\theta)-J(t))^{\beta})h(t)d_{\mathfrak{F}}^{\alpha}t.$
(47)

##  5 The analogue of Mellin integral transform on fractal calculus

In this section, we generalize the Mellin integral transform to include a
function with fractal support.

######  Definition 19.

The Mellin integral transform $M(s)$ of a function $f$ on fractals is defined
by

$M(s)=\int_{D(0,\infty)}f(\theta)J(\theta)^{J(s)-1}d_{\mathfrak{F}}^{\alpha}\theta,$
(48)

where $J(s)$ is complex, such as $\lambda_{1}<Re(J(s))<\lambda_{2}$. The
Mellin transform exists if $f$ is fractal piecewise continuous in every closed
interval $D(a,b)\subset D(0,\infty)$ and

$\int_{D(0,1)}|f(\theta)|J(\theta)^{\lambda_{1}-1}d_{\mathfrak{F}}^{\alpha}\theta<\infty,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{D(1,\infty)}|f(\theta)|J(\theta)^{\lambda_{2}-1}d_{\mathfrak{F}}^{\alpha}\theta<\infty.$
(49)

######  Definition 20.

The fractal inverse Mellin transform formula is defined by

$f(\theta)=\frac{1}{2\pi
i}\int_{\lambda-i\infty}^{\lambda+i\infty}M(z)J(\theta)^{-z}dz,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (0<J(\theta)<\infty),$ (50)

where $\lambda_{1}<\lambda<\lambda_{2}$.

######  Example 4.

The fractal Mellin transform of function $J(\theta)^{\nu}f(\theta)$ is

$\mathcal{M}(J(\theta)^{\nu}f(\theta))=\int_{D(0,\infty)}J(\theta)^{\nu}f(\theta)J(\theta)^{J(s)-1}d_{\mathfrak{F}}^{\alpha}\theta=M(\nu+J(s)).$
(51)

######  Definition 21.

The fractal Mellin convolution is defined by

$f(\theta)*h(\theta)=\int_{D(0,\infty)}f(J(\theta)\tau)h(\tau)d_{\mathfrak{F}}^{\alpha}\tau,$
(52)

and its fractal Mellin transform is

$\mathcal{M}\\{f(J(\theta)\tau)g(\tau)d_{\mathfrak{F}}^{\alpha}\tau\\}=M_{1}(J(s))M_{2}(1-J(s)),$
(53)

where $M_{1}(.)$ and $M_{2}(.)$ are the fractal Mellin transform of
$f(\theta)$ and $h(\theta)$, respectively.

######  Corollary 4.

The fractal Mellin transform holds

$\mathcal{M}\bigg{[}J(\theta)^{\kappa}\int_{D(0,\infty)}J(t)^{\tau}f(J(\theta)t)g(t)d_{\mathfrak{F}}^{\alpha}t\bigg{]}=M_{1}(J(s)+\kappa)M_{2}(1-J(s)-\kappa+\tau).$
(54)

######  Theorem 2.

The fractal Mellin transform of $n$-th $F^{\alpha}$-derivative is

$\mathcal{M}[(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)]=\frac{\Gamma(1-J(s)+n)}{\Gamma((1-J(s))}M(J(s)-n).$
(55)

###### Proof.

To prove of this theorem, we use the fractal integral by part as follows

$\displaystyle\mathcal{M}[(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)]$
$\displaystyle=\int_{D(0,\infty)}(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)J(\theta)^{J(s)-1}d_{\mathfrak{F}}^{\alpha}\theta$
$\displaystyle=[(D_{\mathfrak{F}}^{\alpha})^{n-1}f(\theta)J(\theta)^{J(s)-1}\bigg{|}_{0}^{\infty}-(J(s)-1)\int_{D(0,\infty)}D_{\mathfrak{F}}^{\alpha})^{n-1}f(\theta)J(\theta)^{J(s)-2}d_{\mathfrak{F}}^{\alpha}\theta$
$\displaystyle=[(D_{\mathfrak{F}}^{\alpha})^{n-1}f(\theta)J(\theta)^{J(s)-1}\bigg{|}_{0}^{\infty}-(J(s)-1)\mathcal{M}[(D_{\mathfrak{F}}^{\alpha})^{n-1}f(\theta)]$
$\displaystyle=...$
$\displaystyle=\sum_{k=0}^{n-1}(-1)^{k}\frac{\Gamma(J(s))}{\Gamma(J(s)-k)}[D_{\mathfrak{F}}^{\alpha})^{n-k-1}f(\theta)J(\theta)^{J(s)-k-1}\bigg{|}_{0}^{\infty}+(-1)^{n}\frac{\Gamma(J(s))}{\Gamma(J(s)-k)}M(J(s)-n)$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(1-J(s)+k)}{\Gamma(1-J(s))}[D_{\mathfrak{F}}^{\alpha})^{n-k-1}f(\theta)J(\theta)^{J(s)-k-1}\bigg{|}_{0}^{\infty}+\frac{\Gamma(1-J(s)+n)}{\Gamma(1-J(s))}M(J(s)-n).$
(56)

The first term of the right side of Eq.(5) as the limits $t=0$ and $t=\infty$
is zero. Then Eq.(5) takes the form

$\mathcal{M}[(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)]=\frac{\Gamma(1-J(s)+n)}{\Gamma(1-J(s))}M(J(s)-n),$
(57)

which completes the proof. ∎

######  Theorem 3.

The Mellin transform of the fractal Riemann-Liouville is

$\mathcal{M}[_{0}\mathcal{I}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)+\beta),$
(58)

where $M(J(s))$ is the Mellin transform of the function $f(J(\theta))$.

###### Proof.

To prove this theorem, recalling the Definition 24 and setting $a=0$ we have

$\displaystyle_{0}\mathcal{I}_{\theta}^{\beta}f(\theta)$
$\displaystyle=\frac{1}{\Gamma(\beta)}\int_{D(0,\theta)}(J(\theta)-J(t))^{\beta-1}f(t)d_{\mathfrak{F}}^{\alpha}t,$
by setting $J(t)=J(\theta)\zeta$ we have
${}_{0}\mathcal{I}_{\theta}^{\beta}f(\theta)$
$\displaystyle=\frac{J(\theta)^{\beta}}{\Gamma(\beta)}\int_{D(0,1)}(1-\zeta)^{\beta-1}f(J(\theta)\zeta)d_{\mathfrak{F}}^{\alpha}\zeta\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle=\frac{J(\theta)^{\beta}}{\Gamma(\beta)}\int_{D(0,\infty)}f(J(\theta)\zeta)h(\zeta)d_{\mathfrak{F}}^{\alpha}\zeta,$
(59)

where

$h(\theta)=\left\\{\begin{array}[]{ll}(1-\theta)^{\beta-1},&(0\leq\theta<1);\\\
0,&(\theta\geq 1).\end{array}\right.$ (60)

The fractal Mellin transform of $h(\theta)$ is

$\mathcal{M}[h(\theta)]=B(\beta,J(s))=\frac{\Gamma(\beta)\Gamma(J(s))}{\Gamma(\beta+J(s))},$
(61)

where $B(.,.)$ is the Beta function. Using Corollary 4, Eq.(5) and Eq.(61), we
have

$\mathcal{M}[_{0}\mathcal{I}_{\theta}^{\beta}f(\theta)]=\frac{1}{\Gamma(\beta)}M(J(s)+\beta)B(\beta,1-J(s)-\beta),$
(62)

or

$\mathcal{M}[_{0}\mathcal{I}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)+\beta),$
(63)

which completes the proof. ∎

######  Theorem 4.

The fractal Mellin transform of the Riemann-Liouville derivative is

$\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(64)

###### Proof.

By view of Definitions 26 and 19, we have

$\displaystyle\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\mathcal{M}[(D_{F}^{\alpha})^{n}\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{\beta-n}f(\theta)],$ (65)

and setting $h(\theta)=\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{-(n-\beta)}f(\theta)$ we have

$\displaystyle\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\mathcal{M}[(D_{F}^{\alpha})^{n}h(\theta)].$
(66)

Recalling Theorem 2, we can write

$\displaystyle\mathcal{M}[(D_{\mathfrak{F}}^{\alpha})^{n}h(\theta)]$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(1-J(s)+k)}{\Gamma(1-J(s))}[D_{\mathfrak{F}}^{\alpha})^{n-k-1}h(\theta)J(\theta)^{J(s)-k-1}\bigg{|}_{0}^{\infty}+\frac{\Gamma(1-J(s)+n)}{\Gamma(1-J(s))}H(J(s)-n),$
(67)

where $H(J(s))$ is the fractal Mellin transform of $h(J(\theta))$. By
replacing $h(\theta)$, we rewrite Eq.(67) as

$\displaystyle\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(1-J(s)+k)}{\Gamma(1-J(s))}[D_{\mathfrak{F}}^{\alpha})^{n-k-1}\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{-(n-\beta)}f(\theta)J(\theta)^{J(s)-k-1}\bigg{|}_{0}^{\infty}$
$\displaystyle+\frac{\Gamma(1-J(s)+n)}{\Gamma(1-J(s))}\frac{\Gamma(1-(J(s)-n)-(n-\beta))}{\Gamma(1-(J(s)-n))}M((J(s)-n)+(n-\beta)),$
(68)

or

$\displaystyle\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(1-J(s)+k)}{\Gamma(1-J(s))}[_{0}\mathcal{D}_{\theta}^{\beta-k-1}f(\theta)J(\theta)^{J(s)-k-1}\bigg{]}_{0}^{\infty}+\frac{\Gamma(1-J(s)+\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(69)

In the case of $1<\beta<\alpha$, Eq.(69) turns into

$\displaystyle\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\leavevmode\nobreak\
_{0}\mathcal{D}_{\theta}^{\beta-1}f(\theta)J(\theta)^{J(s)-1}\bigg{]}_{0}^{\infty}+\frac{\Gamma(1-J(s)+\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(70)

By substituting the limits $\theta=0$ and $\theta=\infty$, the first term of
right side of Eq.(70) is zero, and Eq.(70) on the simplest becomes

$\mathcal{M}[_{0}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)+\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(71)

∎

######  Theorem 5.

The fractal Mellin transform of the Caputo derivative is

$\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)+\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(72)

###### Proof.

For proving this theorem, by virtue of Definition 19 we have

$\displaystyle\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\mathcal{M}[_{0}\mathcal{D}_{\theta}^{-(n-\beta)}(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)].$
(73)

By supposing $h(\theta)=(D_{\mathfrak{F}}^{\alpha})^{n}f(\theta)$ we get

$\displaystyle\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\mathcal{M}[_{0}\mathcal{D}_{\theta}^{-(n-\beta)}h(\theta)]$
$\displaystyle=\frac{1-J(s)-(n-\beta)}{\Gamma(1-J(s))}H(J(s)+(n+\beta))$
$\displaystyle=\frac{\Gamma(1-J(s)-n+\beta))}{\Gamma(1-J(s))}\bigg{\\{}\sum_{k=0}^{n-1}\frac{\Gamma(1-(J(s)+n-\beta)+k)}{\Gamma(1-(J(s)+n-\beta))}\bigg{[}(D_{\mathfrak{F}}^{\alpha})^{n-k-1}f(\theta)J(\theta)^{(J(s)+n-\beta)-k-1}\bigg{]}_{0}^{\infty}$
$\displaystyle+\frac{\Gamma(1-(J(s)+n-\beta)+n)}{\Gamma(J(s)+n-\beta-n)}M((J(s)+n-\beta)-n)\bigg{\\}}$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(1-J(s)-n+\beta+k)}{\Gamma(1-J(s))}\bigg{[}(D_{\mathfrak{F}}^{\alpha})^{n-k-1}f(\theta)J(\theta)^{J(s)+n-\beta-k-1}\bigg{]}_{0}^{\infty}$
$\displaystyle+\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)-\beta),$ (74)

or

$\displaystyle\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]$
$\displaystyle=\sum_{k=0}^{n-1}\frac{\Gamma(\beta-
n-J(s))}{\Gamma(1-J(s))}\bigg{[}(D_{\mathfrak{F}}^{\alpha})^{k}f(\theta)J(\theta)^{J(s)-\beta+k}\bigg{]}_{0}^{\infty}$
$\displaystyle+\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$ (75)

For the case of $1<\beta<\alpha$, Eq.(5) turn into

$\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(\beta-J(s))}{\Gamma(1-J(s))}\bigg{[}f(\theta)J(\theta)^{J(s)-\beta}\bigg{]}_{0}^{\infty}+\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(76)

By taking into account of the limits $\theta=0$ and $\theta=\infty$, we arrive
at

$\mathcal{M}[_{0}^{C}\mathcal{D}_{\theta}^{\beta}f(\theta)]=\frac{\Gamma(1-J(s)-\beta)}{\Gamma(1-J(s))}M(J(s)-\beta).$
(77)

Then the proof is complete. ∎

######  Example 5.

Consider the fractal differential equation

$D_{\mathfrak{F}}^{\alpha}f(\theta)+f(\theta)=0,$ (78)

By taking the fractal Mellin transform we have

$-(J(s)-1)M(J(s)-1)+M(J(s))=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{or}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ M(J(s)-1)=J(s)M(J(s)).$ (79)

The solution of Eq.(79) is $M(J(s))=\Gamma(J(s))$. Then the fractal inverse
Mellin transform gives the solution of Eq.(78) by

$\displaystyle f(\theta)$ $\displaystyle=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}J(\theta)^{-z}\Gamma(z)d_{\mathfrak{F}}^{\alpha}z$
$\displaystyle=\sum_{n=0}^{\infty}Res_{z=-n}J(s)^{-z}\Gamma(z)$
$\displaystyle=\sum_{n=0}^{\infty}J(s)^{n}\frac{(-1)^{n}}{n!}=\exp(-J(\theta)).$
(80)

##  6 Conclusion

In this paper, the Riemann-Liouville integrals and derivatives and Caputo
derivatives of fractal sets and curves have been defined. The processes on
fractal spaces with memory effect can be modeled by them. Fractal differential
equations have been solved by using the Mellin transform. The solution of the
fractal differential equation is not differentiable in the standard calculus.
The results become standard cases if we choose $\alpha=1$.

##  7 Appendix

In this section, we give some formulas used in the paper [37, 38].
The Mittag-Leffler one-parameter function is defined by

$E_{\alpha}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(\alpha
k+1)},\leavevmode\nobreak\ \leavevmode\nobreak\ (\alpha>0,\leavevmode\nobreak\
\leavevmode\nobreak\ \beta>0).$ (81)

The Mittag-Leffler two-parameter function is defined by

$E_{\alpha,\beta}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(\alpha
k+\beta)},\leavevmode\nobreak\ \leavevmode\nobreak\
(\alpha>0,\leavevmode\nobreak\ \leavevmode\nobreak\ \beta>0),$ (82)

where $E_{\alpha,1}(x)=E_{\alpha}(x)$.
The gamma function is defined by

$\Gamma(x)=\int_{0}^{\infty}e^{-\tau}\tau^{x-1}d\tau.$ (83)

The beta function is defined by

$\mathbf{B}(x,\nu)=\int_{0}^{1}\tau^{x-1}(1-\tau)^{\nu-1}d\tau,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (Re(x)>0,\leavevmode\nobreak\
\leavevmode\nobreak\ Re(\nu)>0).$ (84)

The beta function relates to the gamma function by the following equation

$\mathbf{B}(x,\nu)=\frac{\Gamma(x)\Gamma(\nu)}{\Gamma(x+\nu)}.$ (85)

Acknowledgements: Cristina Serpa acknowledges partial funding by national
funds through FCT-Foundation for Science and Technology, project reference:
UIDB/04561/2020.

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