=Paper=
{{Paper
|id=Vol-3816/paper77
|storemode=property
|title=Optimising Hierarchical Demand Forecasting with Explainable AI: Insights into Key Drivers
|pdfUrl=https://ceur-ws.org/Vol-3816/paper77.pdf
|volume=Vol-3816
|authors=Mátyás Kuti-Kreszács
|dblpUrl=https://dblp.org/rec/conf/rulemlrr/Kuti-Kreszacs24
}}
==Optimising Hierarchical Demand Forecasting with Explainable AI: Insights into Key Drivers==
https://ceur-ws.org/Vol-3816/paper77.pdf
Optimising Hierarchical Demand Forecasting with
Explainable AI: Insights into Key Drivers
Mátyás Kuti-Kreszács
Babeş-Bolyai University, Cluj-Napoca
Abstract
Demand forecasting is a prediction problem that aims to estimate future needs based on historical data. It serves
as the basis for optimal decision making in multiple areas of value chains such as manufacturing, logistics, and
retail. It is particularly important in demand forecasting models where demand drivers like price, promotions, and
resource planning can help companies optimise pricing, promotional activities, resource planning, and inventory
planning. Our goal is to identify applicable feature importance techniques to hierarchical forecasting problems by
providing insights into feature importance and the underlying decision-making process and helping to understand
the model’s reasoning. We propose applying SHAP values to a forecasting model while using part of a real-world
dataset. The results will provide insight into the key drivers of the forecast and help to understand the impact of
the features on the decisions made by the model.
1. Introduction
Demand forecasting became really important for businesses and serves as the basis for optimal decision
making in multiple areas in value chains such as manufacturing, logistics, and retail. By having multiple
products, manufacturing locations, sales channels, and geographical regions, demand forecasting can
be complex and hierarchical in nature.
However, the problem can be formulated as a regression problem, with the aim of predicting the
future demand based on historical data. This regression problem can be solved using machine learning
models such as random forests, gradient boosting, and neural networks. Unfortunately, these models
are considered black boxes and their predictions are hard to interpret. This is where explainable AI
(XAI) techniques come into play, providing insights into the model’s decision-making process and
helping to understand the underlying rules and reasoning behind the predictions.
One of the most fundamental methods for understanding a model’s reasoning is feature importance
or attribution, which allows identifying key contributor factors to the model’s predictions. This is
especially important in demand forecasting models, where demand drivers, such as price, promotions,
weather, holidays, and economic indicators, can influence demand. Understanding these drivers can
help companies optimise pricing, promotional activities, resource planning, and inventory management.
Our goal is to identify applicable feature importance techniques to demand forecasting models, aiming
to discover key features contributing to the decisions and explain the model’s reasoning at different
levels. The significance of our research is to improve the reasoning and transparency of multiseries
and hierarchical demand forecasting models by providing insights into feature importance and the
underlying rules at various levels. The methods employed are expected to be used not only in demand
forecasting, but also in other grouped and hierarchical forecasting problems in different domains.
1.1. Research Questions
The gap identified in the literature is the lack of studies that apply feature importance techniques to
multiseries models for hierarchical demand forecasting problems and analyse the underlying decision
RuleML+RR’24: Companion Proceedings of the 8th International Joint Conference on Rules and Reasoning, September 16–22, 2024,
Bucharest, Romania
Envelope-Open matyas.kuti@ubbcluj.ro (M. Kuti-Kreszács)
GLOBE https://www.linkedin.com/in/kkmatyas/ (M. Kuti-Kreszács)
Orcid 0009-0004-4997-2000 (M. Kuti-Kreszács)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Workshop ISSN 1613-0073
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drivers at different levels. Other studies focused on the representation of the explanation for sales
forecasting models, but not on the explanation methods themselves[1]. Our research questions are:
• RQ1: Can existing feature importance techniques be applied to multi-series and hierarchical
models to identify key features and explain the underlying decision factors?
• RQ2: How feature importance can be translated to different hierarchical levels?
• RQ3: How do these methods perform when applied to real-world datasets?
• RQ4: How can the results be visualised and interpreted?
• RQ5: What methods are most effective in this context?
In our current work, we partially address RQ1 and RQ2 by proposing a method to apply SHAP values
to a LightGBM model used for forecasting hierarchical time-series data. Furthermore, we make progress
on RQ3 using part of a real-world dataset; however, evaluation is still pending. Last but not least, we
address RQ4 by visualising the results in a way that can be interpreted by the user. RQ5 is still open
and will be addressed in future work.
2. Literature review
2.1. Demand Forecasting with machine learning
Demand forecasting is a prediction problem that aims to estimate future needs based on historical data.
Statistical forecasting methods such as ARIMA[2, 3] and exponential smoothing [3] have been widely
used in demand forecasting. However, they have limitations in intermittent multi-series and hierarchical
forecasting, where machine learning models have shown better performance[4]. An important aspect
also is that there may be multiple exogenous variables so-called demand drivers[5] that can influence
the demand. Internal factors such as price, promotions, and external factors like weather, holidays, and
economic indicators can be considered as demand drivers. These can be used as features in machine
learning models to improve forecast accuracy.
Machine learning models such as tree ensembles and neural networks have been successfully applied
to demand forecasting tasks[4]. Ensemble models in general can be homogeneous with individual models
of the same type or heterogeneous with models of different types. We considered only homogeneous
ensemble tree models because of the applicability of some model-specific explanation methods. To
build tree ensembles, bagging methods such as random forest[6] can be used, which trains multiple
decision trees on different subsets of the data, and the final prediction is the average of the predictions
of the individual models. In addition, boosting methods such as Gradient Boosting Machines (GBM)
[7], XGBoost [8], and LightGBM [9], which train models sequentially on the residuals of the previous
model, in this case using the sum of individual predictions. In a notable forecasting competition [10], a
LightGBM model was the winner and secured four of the top five positions.
2.2. Forecasting techniques
Forecasting techniques can be divided into single-series or multi-series forecasting from the perspective
of the model’s input. Single-series forecasting refers to the prediction of a single time series, while
multiseries forecasting involves the prediction of multiple time series, with the same global model[11].
These series can be related to each other, such as sales of different products, or they can be independent,
such as sales in different regions; therefore, it is important to consider the hierarchical structure of the
data.
Hierarchical forecasting refers to the prediction of multiple time series that are related to each other in
a hierarchical structure[12]. It can be tackled with different single-level approaches, such as bottom-up,
top-down, or middle-out[12]. The top-down approach would involve a single series model for the total
demand and then disaggregating it to the lower levels. The middle-out and bottom-up approach would
involve a multiseries model. Grouped time-series forecasting is a special case of hierarchical forecasting,
where the series are aggregated based on attributes such as product type, region, or sales channel.
[5] suggests three major hierarchies in demand forecasting: product hierarchy, geographical hierarchy,
and time hierarchy. The product hierarchy refers to the categorisation of products according to their
attributes, such as product type, brand, or category. The geographic hierarchy involves the division of
sales regions based on geographic attributes, such as country, state, or city down to the point of sale.
Time hierarchy refers to the temporal structure of the data, such as year, month, week, day, and hour.
2.3. Feature importance
Feature importance (FI) or feature attribution is considered an interpretation method resulting in a
summary statistic that assigns a score to each input feature [13]. Depending on their scope, the FI
methods can be global or local [14, 13]. The global feature importance (GFI) or model feature attribution
methods explain the contribution of features to overall predictions, while the local FI quantifies feature
contributions to specific predictions [13]. Although related, GFI methods differ from feature selection,
which identifies irrelevant features before training. GFI methods can be model-specific, which are
limited to specific model types, while model-agnostic ones are applicable independent of the model
type[13]. Another categorisation of FI methods is given by how it is calculated, in which case the
importance can be based on the model’s structure, while the other approach relies on a dataset.
Among the model-agnostic methods, one of the most common is permutation feature importance
(PFI) which was proposed to measure FI in random forests[15]. It is a model-agnostic, data-dependent
method that measures the decrease in the model’s performance when the features are permuted. The
PFI can be calculated using different metrics such as the mean squared error (MSE), the mean absolute
error (MAE), or the coefficient of determination (𝑅2 ). PFI also has limitations, as it is sensitive to over-
and underfitting[16], in which case the FI differs on training and test data, so the use of both datasets
can be beneficial. In addition, another flaw of the PFI method is that it can generate cases in which the
model does not have training data[17, 18], but other methods were proposed to overcome this[19, 20].
SHAP(SHapley Additive exPlanation)[21] values contribute local explanation for individual predic-
tions, but aggregates of it are useful to assess the importance of global features. For example, the mean
absolute SHAP values quantify the importance of the feature regardless of the direction of the impact
on the prediction. There are different algorithms for approximation from which Kernel SHAP[21] is
one that is model-agnostic. TShap [22] is a method for estimating SHAP values for time series data,
but it uses a surrogate model, so it gives the FI of the surrogate. Another related method is SAGE
(Shapley additive global importance) [19], which estimates the contribution of each feature to the model’s
performance.
Tree specific GFI methods are gain-based importance values which were already introduced with
decision trees [23] It measures of the reduction in mean average error(MAE) made by the decisions
based on the respective feature. Another measure is the split-based importance[8] refers to the number
of decisions made by the model based on a feature. The previously presented SHAP also has a tree
model-specific solution for approximation, called TreeSHAP [24]
2.4. Explainability in forecasting
The number of publications on forecasting explainability is limited. [1] tackled the presentation of
explanations for sales forecasting models, but not the explanation methods themselves. [25] used SHAP
values to explain the prediction of a time series model but on local level and not global level. Skforecast
[11] library extracts model specific global feature importance from tree ensemble models. The work
is focused on either global feature importance or local feature contribution without considering the
multi-series and hierarchical structure of the data.
2.5. Feature importance as a basis for model reasoning
Feature importance methods can provide insight into the model’s decision-making process and help to
understand the underlying rules and reasoning behind the predictions. By including demand drivers
as features in the model, the feature importance methods can help to identify the key drivers of
Data Collection
Tool Evaluation
Model Implementation
Feature Importance Analysis
Model Reasoning
Figure 1: Research design
demand. For external factors such as weather, holidays, and economic indicators, the importance of
the characteristics can help to understand their impact on demand. Through internal factors like price,
promotions, the feature importance can help to understand post-promotion effects and the impact of
price changes on the demand[5]. Knowing the influence of internal factors can help to optimize pricing
strategies and promotional activities. However, causation and correlation are different concepts, and
the feature importance methods can only provide correlation; therefore, the identified key features
should be further analyzed to understand the causation[15]
3. Methodology
Our preliminary research focusses on the methodological aspects of applying feature importance
techniques to hierarchical forecasting models. This includes adaptation of existing methods, but also
tool development to support the analysis of hierarchical forecasting models. Later we plan to conduct
an empirical study to evaluate the methods on real-world datasets.
Our initial research design1 includes the following steps:
• Data collection: identify datasets with hierarchical time series data describing sales/demand for
multiple product categories and regions with exogenous variables.
• Tool evaluation: assess the applicability of existing libraries for hierarchical forecasting and XAI
techniques.
• Model implementation: we build global models that consider multiple series and exogenous
variables.
• Feature importance analysis: We apply model attribution methods and aggregation and decom-
position techniques to identify key features and analyse their impact on the forecast.
• Model reasoning: analyse the feature contributions to forecast and identify underlying rules on
different levels of the hierarchy.
Figure 2: Total weekly sales for the chosen products
Total sales
State TX State WI
Store1 Store2 Store3 Store1 Store2
P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3
Figure 3: Hierarchical structure of product sales data (P1-3 = Product1-3)
3.1. Data collection and preprocessing
To model the hierarchical impact of features on forecasting, we must use datasets with multiple series
and exogenous variables that represent demand drivers. There are multiple open sales data sets available;
however, there are just a few, such as the M5 competition[10] and the Kaggle datasets[26]. For our initial
exploration, we sampled M5 competition[10] dataset, which includes sales data for multiple product
categories and regions. The dataset contains daily sales information for 3049 products in 10 stores over
5 years. For our analysis, we identified three products that have similar sales patterns and are sold
in two states and five stores. As products are from the same category and department, the hierarchy
at the product level was not considered. The reason for this filtering is to reduce the complexity of
the model and to focus on the feature importance analysis. The selected products are FOODS_3_586,
FOODS_3_080, and FOODS_3_555 and are sold in three states of Texas (TX), Wisconsin (WI). The total
sales data for these products are shown in Figure 2. Our hierarchical structure is shown in Figure 3. It
should be mentioned that the hierarchical structure can be inverted, meaning that the products can be
at the top level and the stores at the bottom level, so technically our data set is grouped time series data.
Data preprocessing two main parts: preparing the sales data and the exogenous variables. Sales
data were aggregated at the weekly level. The weeks at the beginning and end of the data set were
removed to have a consistent time period. As features, lagged sales data was included to capture the
temporal dependencies. The exogenous variables were related to pricing and calendar events. The
selling price was already aggregated at the weekly level for each store and product. Calendar events
included whether a day was a holiday, had special events, and if it was a SNAP (Supplemental Nutrition
Assistance Programme) day in a respective state. To include these variables in some way, they were
Feature name Type Description
week_date date Starting date of week used for aggregation
week int Week number of the year
series_id string Unique identifier for the series made of state+store+product combination, represent-
ing the hierarchy
sales float Total sales for the week
lag_n float Sales from the previous n weeks
Table 1
Sales data structure
Feature name Type Description
week_date date Starting date of week used for aggregation
week_of_year int Week number of the year
series_id string Unique identifier for the series made of state+store+product combination, represent-
ing the hierarchy
sell_price float Selling price for the week for
num_of_events int The number of special events and holidays in the week
snap_days int Number of SNAP days in the week
Table 2
Exogenous variables
Parameter Search space Description
n_estimators 50-1000 Number of boosting iterations
max_depth 5-50 Maximum depth of the tree
min_samples_leaf 1-10 Minimum number of samples required to be at a leaf node
num_lagged_sales 4-52 Number of lagged sales records used as features
Table 3
Model hyperparameters search space
counted for each week and state. In addition, the week of the year was included as a feature to capture
seasonality. The data were split into training and test sets, and the last complete year(2015) was used to
test the model. The structure of the data sales data is shown in Table 1 and the exogenous variables in
Table 2.
3.2. Model implementation
The modelling approach is to build a single global on all series and exogenous variables for bottom-up
aggregation. For creating forecast models, the skforecast[27] library was used. The base model for
hierarchical forecasting was LightGBM[9] which we chose because of its efficiency and also because of
its widespread usage in the M5 competition in this data set[10]. Other ensemble models such as Random
Forest or Gradient Boosting Machines could be used as well. Other reasons for choosing LightGBM are
that it can handle categorical variables without the need for one-hot encoding, and that it supports
model-specific split and gain-based global feature importance methods.
Hyperparameter tuning was performed using the Optuna library[28], by Bayesian optimisation.
The search space3 was defined for the parameters of the LightGBM model, including the number of
predictors, the minimum number of samples in the leaf, and the maximum depth of the tree. In addition,
the number of lagged sales records used as features was included in the search space. For the search, the
data was split into training and validation sets, the last year being the validation set used for backtesting.
The performance of the model was evaluated as a mean square error (MSE) in the validation set for
each configuration. The best configuration found was with 239 estimators and a maximum depth of 26
with a backtesting MSE 4263.01 The lagged sales records used as features were 1, 4, 5, 13, and 52 weeks.
The feature input for the final model is a table with the following columns:
• week_of_year represented as numerical values (1-52)
• sell_price for the week for the product in the store
• num_of_events for the week
• snap_days for the week in the state
• lag_n for n in [1, 4, 5, 13, 52] representing the sales from the previous weeks
• series_id noted as (_level_skforecast) encoded as a numerical value representing the series
hierarchy
Series_id could have been encoded as a one-hot encoded vector or as a categorical variable given it is
supported by LightGBM. One-hot encoded vector would have increased the number of features and the
complexity of the model, while with the categorical variable
3.3. Feature importance analysis and model reasoning
Two initial ideas were considered to analyse the importance of characteristics. The first involves
using the mean SHAP values for cohorts representing different levels of the hierarchy, providing
information on the contribution of features throughout the structure. The second approach is based on
conditional permutation importance, which evaluates the importance of features while accounting for
the hierarchical structure on the idea of subgroup-based permutation importance[29]. The first method
was prioritised for implementation due to the availability of support in the SHAP library[30]. Given an
instance 𝑥 for prediction, the SHAP value of the feature 𝑖 is 𝜙𝑖 (𝑥) Each 𝑥 is part of a cohort 𝐶𝑘 based on
the series hierarchy 𝑘. The contribution value or importance of feature 𝑖 for a 𝐶𝑘 cohort is calculated as
1
𝜙𝑖 (𝐶𝑘 ) = ∑ |𝜙 (𝑥)| (1)
|𝐶𝑘 | 𝑥∈𝐶 𝑖
𝑘
where |𝐶𝑘 | is the cardinality of 𝐶𝑘 and |𝜙𝑖 (𝑥)| is the absolute SHAP value of feature, 𝑖 for instance 𝑥.
Steps for the feature importance analysis:
• For each prediction instance 𝑥 and feature 𝑖 calculate SHAP value 𝜙𝑖 (𝑥).
• Split the instances into cohorts according to the hierarchy levels.
• Calculate the mean SHAP values for each cohort 𝐶
• Visualize the mean SHAP values for 𝐶 and summary plots
The reasoning of the model is based on the analysis of the contributions of the features to the forecast.
The aim is to identify the underlying rules and patterns that the model uses to make predictions. The
SHAP values provide a way to understand the impact of the features on the forecast. The analysis can
be done at different levels of the hierarchy, from the global model to the state and store levels.
4. Preliminary Results
The preliminary results focus mainly on the practical application of SHAP values in hierarchical
forecasting models rather than on the theoretical aspects of feature importance. As preliminary results,
we present mean average SHAP values at different aggregation levels. These provide an overview of
the main contributors to the forecast at different levels of the hierarchy.
Furthermore, we visualise the distribution of SHAP values at different aggregation levels using violin
plots, which provide a representation of variability and density of SHAP values for each feature across
the hierarchy. This double representation allows for a more detailed analysis of the contribution of
features to the forecast, for example, if a feature contributes positively or negatively to the forecast. In
the following, we present several cases of different aggregation levels, starting from the global level to
the store and product level, but it is not meant to be exhaustive.
At the global level, the SHAP values4a show that the most important features are the lagged sales
values, especially the sales value of lag 1, which has the highest SHAP value. This is expected as prior
sales are the most important factor in predicting future sales. The violin plot in Figure 4b shows the
distribution of SHAP values for each feature. It shows that the actual impact of the lag value is most of
the time negative, as after a week with higher sales, demand the following week can drop.
(a) Global SHAP values (b) Global SHAP violin plot
Figure 4: Global summary
In case of state-level grouping, the number of samples differs for the two groups, one of them having
only two stores included. This can be observed in the wider distribution on the violin plot of the
Texas(TX) state.
On the lowest level of the hierarchy presented in Figure 7, deviations can be revealed in the order of
importance of the features. For example, in Figure 7d the week-of-year feature has a greater impact on
the forecast than some of the lag values that occurred in other cases in Figure 7c. This can be due to
the fact that the store TX_2 has a different seasonality pattern or might have recurring special events,
since the number of events is also a feature with higher impact on this store. What is problematic in
this case is that, due to the large number of series, representation of the mean absolute SHAP value is
hardly comprehensible in the previous form of the bar plot. As a workaround, the grouping of feature
contribution of each group is presented in Figure 7b. What can be misleading in this case is the lack of
order by impact and a different scale of the 𝑥 axis for each feature.
5. Discussion
In this work, we managed to calculate the Shapley values for the predictions of a hierarchical forecasting
model with some limitations, while we also aggregated these values to different levels of the hierarchy.
By this we addressed the first two research questions. We used a sample of a real-world dataset to
evaluate the proposed method working towards the third research question. We visualised the SHAP
values at different levels of the hierarchy and provided some interpretation of the results. To respond to
the fourth research question, we plan to expand the literature review to include a wider range of XAI
techniques.
This research is expected to contribute in several key areas. First, it will provide an evaluation
of feature importance methods in the context of hierarchical forecasting models. This will help to
identify what methods are most effective and how they can be applied to improve model interpretability.
Second, it aims to provide guidelines, best practices, and limitations of effectively explaining these
models. Finally, the research will support the development of tools that improve the understanding of
hierarchical forecasting models and their underlying rules and reasoning.
The limitations of our study include the handling of categorical variables in the SHAP library. The
effect of ordinal encoding that induces an order on the categorical variables may not be appropriate for all
cases. Low feature importance for the categorical variables may be due to the encoding method. Recent
research [20] proposes a method to handle categorical variables for conditional feature importance. With
regard to data limitations, the dataset used is simplified in multiple dimensions. First, with aggregation
Global SHAP values
State TX SHAP values State WI SHAP values
(a) State level SHAP values
(c) TX state product SHAP summary
(b) State level SHAP values
(d) WI state product SHAP summary
of sales data at the weekly level, multiple exogenous variables such as special events could not be
included. Second, the dataset is limited to a single product category, which may not be representative
of all hierarchical forecast scenarios. Lastly, the input data was limited to the sales lag of the product
without considering other products in the same category. The independence of products in the same
category may not be a realistic assumption.
During the implementation of our initial approach, we encountered several challenges. One of
the main issues was the lack of appropriate tools. For example, the SHAP library does not support
categorical features in the current version. In addition,we faced difficulties in visualising the results;
although the SHAP library offers integrated graphing functions, these have not been effectively used to
deal with a large number of cohors, leading to errors and incomplete plots. Although these issues are
not straightforward to solve, they are a sign of unexplored areas in the field of XAI and hierarchical
forecasting.
There are also potential risks that could impact research in addition to challenges. One of the main
risks is the availability of data, especially real-world datasets that include exogenous variables or demand
drivers. Synthetic datasets can be used as an alternative, but they may not capture the complexity
of real-world scenarios. The evaluation of methods is another potential risk, as it may be difficult to
assess the performance of the explanation methods. In case of application grounded evaluation, it may
be difficult to find experts in the field who can provide meaningful feedback given that each product
category may require different domain knowledge[31].
5.1. Future work
To address the challenges and limitations of the current research, several next steps are proposed for
each part of the research.
• First, the literature review will be extended to include a broader range of XAI techniques. Given
Global SHAP values
State TX SHAP values State WI SHAP values
P1 P2 P3 P1 P2 P3
(a) Aggregation levels
(c) TX state FOODS_3_586 product SHAP summary
(d) WI state FOODS_3_080 product SHAP summary
(b) State and product level SHAP values
Figure 6: State and product level summary
the current context, we focus on feature importance-based evaluation, partial dependence plots,
feature interaction, and other XAI techniques that should be included in the review.
• Data collection will be expanded to include synthetic datasets and additional real-world datasets.
In addition, including more data from the actual dataset and forecasting on the day level can be a
future direction.
• Evaluation and implementation of the tool for other methods will be needed. Conditional
permutation importance can be also evaluated after implementing the method.
• The model implementation could be extended to include additional ML models for hierarchical
forecasting. An additional enhancement to that would be dependent multiseries forecasting, as
usually product sales are not independent of each other, especially in the same product category.
• Rule extraction based on feature importance and interaction can be a future direction.
• After covering the methodological aspects, an empirical study with evaluation in terms of accuracy
and computational efficiency is planned.
5.2. Conclusion
Nowadays every organisation thrives in the direction of becoming data driven. In this context, data-
driven decision making is crucial for optimising business processes to remain competitive. This effort
is supported by the use of data mining, machine learning, and AI techniques. To avoid blindly trusting
Global SHAP values
Store TX_1 Store TX_2 Store TX_3 Store WI_1 Store WI_2
P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3
(a) Aggregation levels
(b) Store and product level SHAP values for one feature
(c) Store TX_3 FOODS_3_586 product SHAP summary (d) Store TX_2 FOODS_3_080 product SHAP summary
Figure 7: Store and product level summary
ML models,it is crucial to understand the reasoning behind their decisions. Our goal is to demystify
hierarchical forecasting models by applying XAI techniques.
This study explores the usage of SHAP values to explain the importance of features in hierarchical
forecasting models. Our preliminary results focused on the practical aspects of aggregating SHAP values
at different levels of hierarchy. This approach provides insights into the model’s reasoning. We plan
to extend this work by evaluating other XAI techniques to enhance the explainability of hierarchical
forecasting models.
Acknowledgement
This work was done in collaboration with my Ph.D. supervisor, Laura Diosan, from Babes-Bolyai
University. I am grateful for her continued support and encouragement throughout this research.
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